http://www.map.mpim-bonn.mpg.de/api.php?action=feedcontributions&user=Daniel+M%C3%BCllner&feedformat=atomManifold Atlas - User contributions [en]2022-10-06T14:01:55ZUser contributionsMediaWiki 1.18.4http://www.map.mpim-bonn.mpg.de/2-manifolds2-manifolds2022-02-27T15:10:35Z<p>Daniel Müllner: /* Classification */</p>
<hr />
<div>{{Authors|Daniel Müllner}}<br />
== Introduction ==<br />
<wikitex>;<br />
A surface is a synonym for a $2$-dimensional manifold. Complex $2$-dimensional (real $4$-dimensional) complex manifolds are also called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections [[#Surfaces with boundary|2]] and [[#Classification|3]]), surfaces without boundary are considered.<br />
</wikitex><br />
== First construction: connected sum ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
The two simplest closed [[Wikipedia:Orientable|orientable]] $2$-manifolds are:<br />
* the $2$-sphere: $S^2 := \{ (x, y) \in \Rr^2 | x^2 + y^2 = 1 \}$,<br />
* the $2$-torus: $T^2 := S^1 \times S^1$, the [[Wikipedia:Cartesian_product|Cartesian product]] of two [[1-manifolds#Exmaples|circles]].<br />
<br />
All orientable surfaces are homeomorphic to the [[Connected sum|connected sum]] of $g$ [[Torus|tori]] $T^2$ ($g\geq 0$) and so we define<br />
* $F_g := \sharp_g T^2 = T^2 \sharp \dots \sharp T^2$, the $g$-fold connected sum of the $2$-torus.<br />
The case $g=0$ refers to the 2-[[sphere]] $S^2$. The number $g$ is called the [[Wikipedia:Genus_(mathematics)#Orientable_surface|''genus'']] of the surface: for more on the concept of genus see, e.g. \cite{Hirzebruch&Kreck2009}.<br />
[[Image:Surfaces.png|none|frame|A 2-sphere (genus 0), a torus (genus 1) and an orientable surface of higher genus]]<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
The simplest non-orientable surface is the [[Wikipedia:Projective_plane|real projective plane]] $\RP^2$: for the history of the discovery of this interesting manifold see the page [[Projective plane: a history]]. <br />
<br />
All non-orientable surfaces are homeomorphic to the connected sum of $h$ real projective planes and and so for all $h \geq 1$ we define<br />
* $R_h := \sharp_h \RP^2 = \RP^2 \sharp \dots \sharp \RP^2$, to be the $h$-fold connected sum of $\RP^2$. <br />
For example, the ''Klein bottle'' is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2$. The number $h$ is called the ''genus'' or ''cross-cap number'' of the surface.<br />
[[Image:Boy surface.jpg|right|thumb|310px|The Boy surface, an immersion of $\RP^2$ in $\Rr^3$. This steel sculpture stands in front of the [http://www.mfo.de Oberwolfach Institute].]]<br />
[[Image:Klein_bottle.png|left|frame|A Klein bottle (non-orientable, genus 2) immersed in $\Rr^3$]]{{-|left}}<br />
</wikitex><br />
<br />
== Surfaces with boundary ==<br />
The boundary of a surface is a disjoint union (possibly empty) of circles. Surfaces with boundary can be constructed by removing open discs from surfaces without boundary.<br />
<br />
== Classification ==<br />
<wikitex>;<br />
Compact, connected surfaces are classified by orientability (yes/no), the number of boundary components (a nonnegative integer) and the genus after filling the bounday circles by disks (an integer $\geq 0$ in the orientable case, $\geq 1$ in the non-orientable case). Instead of the genus, also e. g. the Euler characteristic can be used in the classification. The classifications up to homotopy equivalence, homeomorphism, PL-equivalence and diffeomorphism coincide.<br />
<br />
References: {{cite|Ahlfors&Sario1960|Thm. 1.42A, 1§8}}, {{cite|Hirsch1994|Thm. 9.3.11}}, {{cite|Moise1977|Thm. 8.3, Thm. 8.5, Thm. 22.9}}{{-}}<br />
<br />
The connected sum of $T^2$ with $\RP^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$. Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect to connected sum, and is generated by $T^2$ and<br />
$\RP^2$, with the sole relation $T^2\mathbin{\sharp}\RP^2=\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$.<br />
<br />
Closed 2-manifolds are homeomorphic if and only if they have isomorphic [[Intersection_form|intersection forms]]. Likewise, compact 2-manifolds with non-empty boundary are homeomorphic if and only if they have isomorphic intersection forms. Cf. [[4-manifolds:_1-connected#Topological_classification|the topological classification of simply-connected 4-manifolds]].<br />
</wikitex><br />
<br />
== More constructions ==<br />
<br />
=== By polygons ===<br />
<wikitex>;<br />
Each orientable surface of genus $g>0$ can be constructed by identifying pairs of edges in a regular polygon with $4g$ sides. Label the edges by the sequence $a_1,b_1,\bar a_1,\bar b_1,\ldots,a_g,b_g,\bar a_g,\bar b_g$. Also orient the edges such that those labeled without an overbar are oriented in one direction (e. g. clockwise) and those with an overbar are oppositely oriented. Now identify corresponding edges, respecting the orientation.<br />
<br />
The 2-sphere can be obtained from a 2-gon with edges labeled $a,\bar a$.<br />
<br />
Each non-orientable surface of genus $h$ can be obtained from a $2h$-gon with edges labeled $a_1,a_1,\ldots,a_h,a_h$.<br />
<br />
{|<br />
|<br />
[[Image:Polygon_construction.png|frame|The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.]]<br />
|<br />
[[Image:Polygon_sphere.png|frame|The 2-sphere can be obtained by identifying the edges of a 2-gon.]]<br />
|<br />
[[Image:Polygon_RP2.png|frame|Construction of $\RP^2$.]]<br />
|<br />
[[Image:Polygon_Klein_bottle.png|frame|Construction of the Klein bottle.]]<br />
|}<br />
Reference: {{cite|Massey1977|Section 1.5}}<br />
</wikitex><br />
<br />
=== By gluing handles and crosscaps ===<br />
<wikitex>;<br />
An orientable surface of genus $g$ can be obtained by successively gluing $g$ 1-handles to the 2-sphere such that the embeddings of $S^0\times D^2$ in $S_{g-1}$ are in each case either orientation-preserving or orientation-reversing on both components $\{1\}\times D^2\cong D^2$ and $\{-1\}\times D^2\cong -D^2$.<br />
<br />
A non-orientable surface of genus $h$ can be obtained by gluing $h$ ''crosscaps'' to $S^2$. For this, embed $D^2$ in $S^2$ (or $R_{h-1}$ from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary $S^1$. The result of attaching a non-orientable handle to $S^2$ or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.<br />
<br />
Reference: {{cite|Hirsch1994|Section 9.1}}<br />
</wikitex><br />
<br />
=== By branched coverings ===<br />
<wikitex>;<br />
Every orientable surface of genus $g$ can be obtained as the branched double covering of the 2-sphere with $2g+2$ branching points.<br />
<br />
{{beginproof}}<br />
The branched double covering of the 2-sphere, with $2g+2$ branch points, has Euler characteristic $4-(2g+2)=2(1-g)$. Thus every orientable closed surface is such a branched covering.<br />
{{endproof}}<br />
</wikitex><br />
<br />
=== As complex curves ===<br />
<wikitex>;<br />
A smooth, irreducible, plane, complex projective-algebraic curve of order $d$ (i. e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree $d$ in $\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus<br />
$$<br />
g = \frac{(d-1)(d-2)}2.<br />
$$<br />
<br />
Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.<br />
<br />
Reference: {{cite|Brieskorn&Knörrer1986|Lemma 9.2.1}}<br />
</wikitex><br />
<br />
== Properties ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
Let $S_g$ denote an oriented surface of genus $g$.<br />
<br />
* By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, $2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps.<br />
<br />
* Therefore, the homology groups with any coefficients $G$ are given by $H_0\cong G$, $H_1\cong G^{2g}$, $H_2\cong G$.<br />
<br />
* The integral cohomology ring is completely determined by the intersection form on $H^1$, which is necessarily isomorphic to the hyperbolic form. A basis for $H^1(S_g)$, for which the intersection form is the standard hyperbolic form, is given by the loops $(a_1,b_1,\ldots,a_g,b_g)$ in the polygon construction above.<br />
<br />
* Homotopy groups<br />
** For the homotopy group of $S^2$, see the article about [[sphere|spheres]].<br />
** All surfaces of higher genus $g\geq1$ are aspherical with fundamental group $\pi_1(S_g,*)\cong \langle a_1,b_1,\ldots, a_g,b_g \mid a_1b_1\cdots a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2$.<br />
<br />
* All orientable surfaces can be embedded in $\Rr^3$.<br />
<br />
* Every surface can be given a complex structure. Together with the complex structure, it is a [[wikipedia:Riemann surface|Riemann surface]]<br />
<br />
* Characteristic classes<br />
** All Stiefel-Whitney classes vanish.<br />
** All Pontrjagin classes vanish.<br />
** The Euler characteristic is $2-2g$.<br />
** Given a complex structure on the surface, the first Chern class is equal to the Euler class.<br />
<br />
* All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in $\Rr^3$ has constant positive curvature, the torus can be given a flat metric, and all surfaces of higher genus admit metrics of constant negative curvature.<br />
<br />
* All surfaces are smoothly [[amphicheiral]].<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
Let $R_h$ denote an non-orientable surface of genus $h$.<br />
<br />
* Again, each non-orientable surface has a cell decomposition with one 0-cell, $h$ 1-cells and one 2-cell. The differential $C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$ with respect to the basis given by the loops the loops $(a_1,\ldots,a_h)$ in the polygon construction is the matrix $(2,\ldots, 2)$.<br />
<br />
* Therefore, the integral homology groups are isomorphic to $H_0\cong\Zz$, $H_1\cong \Zz^{h-1}\oplus \Zz/2$, $H_2\cong 0$.<br />
<br />
* The mod-2 homology groups are $H_0=\Zz/2$, $H_1\cong(\Zz/2)^h$, $H_2=\Zz/2$ since the chain complex above is acyclic mod 2. The intersection form on $H_1(R_h;\Zz/2)$ with respect to this cellular basis is given by the identity matrix.<br />
<br />
* Homotopy groups<br />
** The fundamental group of $R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$.<br />
** The orientation double covering of $R_h$ is $S_{h-1}$. Therefore, all higher homotopy groups of $\RP^2$ equal those of $S^2$, and the non-orientable surfaces of higher genus are aspherical.<br />
<br />
* The orientation double covering also determines the curvature properties: $\RP^2$ can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.<br />
<br />
* Non-orientable surfaces cannot be embedded in $\Rr^3$. However, they do embed in $\Rr^4$.<br />
<br />
* Characteristic classes<br />
** The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism $\pi_1(R_h,*)\to\Zz/2$ which maps each generator $a_1,\ldots,a_h$ to the generator of $\Zz/2$. The second Stiefel-Whitney class is zero if $h$ is even and is the non-zero element of $H^2(R_h;\Zz)=H^2(R_h;\Zz/2)=\Zz/2$ if $h$ is odd. In all cases $w_2=w_1^2$.<br />
** The Euler characteristic is $2-h$.<br />
</wikitex><br />
<br />
=== General ===<br />
There are purely point-set topology characterizations of surfaces; see {{cite|Wilder1949}}.<br />
<br />
== References ==<br />
{{#RefList:}}<br />
== External links ==<br />
* The Wikipedia page about [[Wikipedia:Surface|surfaces]]<br />
<br />
[[Category:Manifolds]]<br />
<!--[[Category:Dimension 2]] --><br />
[[Category:Aspherical manifolds]]<br />
<!-- [[Category:Constant curvature]] --></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/2-manifolds2-manifolds2022-02-27T15:07:56Z<p>Daniel Müllner: /* Orientable surfaces */</p>
<hr />
<div>{{Authors|Daniel Müllner}}<br />
== Introduction ==<br />
<wikitex>;<br />
A surface is a synonym for a $2$-dimensional manifold. Complex $2$-dimensional (real $4$-dimensional) complex manifolds are also called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections [[#Surfaces with boundary|2]] and [[#Classification|3]]), surfaces without boundary are considered.<br />
</wikitex><br />
== First construction: connected sum ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
The two simplest closed [[Wikipedia:Orientable|orientable]] $2$-manifolds are:<br />
* the $2$-sphere: $S^2 := \{ (x, y) \in \Rr^2 | x^2 + y^2 = 1 \}$,<br />
* the $2$-torus: $T^2 := S^1 \times S^1$, the [[Wikipedia:Cartesian_product|Cartesian product]] of two [[1-manifolds#Exmaples|circles]].<br />
<br />
All orientable surfaces are homeomorphic to the [[Connected sum|connected sum]] of $g$ [[Torus|tori]] $T^2$ ($g\geq 0$) and so we define<br />
* $F_g := \sharp_g T^2 = T^2 \sharp \dots \sharp T^2$, the $g$-fold connected sum of the $2$-torus.<br />
The case $g=0$ refers to the 2-[[sphere]] $S^2$. The number $g$ is called the [[Wikipedia:Genus_(mathematics)#Orientable_surface|''genus'']] of the surface: for more on the concept of genus see, e.g. \cite{Hirzebruch&Kreck2009}.<br />
[[Image:Surfaces.png|none|frame|A 2-sphere (genus 0), a torus (genus 1) and an orientable surface of higher genus]]<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
The simplest non-orientable surface is the [[Wikipedia:Projective_plane|real projective plane]] $\RP^2$: for the history of the discovery of this interesting manifold see the page [[Projective plane: a history]]. <br />
<br />
All non-orientable surfaces are homeomorphic to the connected sum of $h$ real projective planes and and so for all $h \geq 1$ we define<br />
* $R_h := \sharp_h \RP^2 = \RP^2 \sharp \dots \sharp \RP^2$, to be the $h$-fold connected sum of $\RP^2$. <br />
For example, the ''Klein bottle'' is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2$. The number $h$ is called the ''genus'' or ''cross-cap number'' of the surface.<br />
[[Image:Boy surface.jpg|right|thumb|310px|The Boy surface, an immersion of $\RP^2$ in $\Rr^3$. This steel sculpture stands in front of the [http://www.mfo.de Oberwolfach Institute].]]<br />
[[Image:Klein_bottle.png|left|frame|A Klein bottle (non-orientable, genus 2) immersed in $\Rr^3$]]{{-|left}}<br />
</wikitex><br />
<br />
== Surfaces with boundary ==<br />
The boundary of a surface is a disjoint union (possibly empty) of circles. Surfaces with boundary can be constructed by removing open discs from surfaces without boundary.<br />
<br />
== Classification ==<br />
<wikitex>;<br />
Compact, connected surfaces are classified by orientability (yes/no), the number of boundary components (a nonnegative integer) and the genus after filling the bounday circles by disks (an integer $\geq 0$ in the orientable case, $\geq 1$ in the non-orientable case). Instead of the genus, also e. g. the Euler characteristic can be used in the classification. The classifications up to homotopy equivalence, homeomorphism, PL-equivalence and diffeomorphism coincide.<br />
<br />
References: {{cite|Ahlfors&Sario1960|Thm. 1.42A, 1§8}}, {{cite|Hirsch1994|Thm. 9.3.11}}, {{cite|Moise1977|Thm. 8.3, Thm. 8.5, Thm. 22.9}}{{-}}<br />
<br />
The connected sum of $T^2$ with $\RP^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$. Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect to connected sum, and is generated by $T^2$ and<br />
$\RP^2$, with the sole relation $T^2\mathbin{\sharp}\RP^2=\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$.<br />
<br />
Compact 2-manifolds (possibly with boundary) are homeomorphic if and only if they have isomorphic [[Intersection_form|intersection forms]]. Cf. [[4-manifolds:_1-connected#Topological_classification|the topological classification of simply-connected 4-manifolds]].<br />
</wikitex><br />
<br />
== More constructions ==<br />
<br />
=== By polygons ===<br />
<wikitex>;<br />
Each orientable surface of genus $g>0$ can be constructed by identifying pairs of edges in a regular polygon with $4g$ sides. Label the edges by the sequence $a_1,b_1,\bar a_1,\bar b_1,\ldots,a_g,b_g,\bar a_g,\bar b_g$. Also orient the edges such that those labeled without an overbar are oriented in one direction (e. g. clockwise) and those with an overbar are oppositely oriented. Now identify corresponding edges, respecting the orientation.<br />
<br />
The 2-sphere can be obtained from a 2-gon with edges labeled $a,\bar a$.<br />
<br />
Each non-orientable surface of genus $h$ can be obtained from a $2h$-gon with edges labeled $a_1,a_1,\ldots,a_h,a_h$.<br />
<br />
{|<br />
|<br />
[[Image:Polygon_construction.png|frame|The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.]]<br />
|<br />
[[Image:Polygon_sphere.png|frame|The 2-sphere can be obtained by identifying the edges of a 2-gon.]]<br />
|<br />
[[Image:Polygon_RP2.png|frame|Construction of $\RP^2$.]]<br />
|<br />
[[Image:Polygon_Klein_bottle.png|frame|Construction of the Klein bottle.]]<br />
|}<br />
Reference: {{cite|Massey1977|Section 1.5}}<br />
</wikitex><br />
<br />
=== By gluing handles and crosscaps ===<br />
<wikitex>;<br />
An orientable surface of genus $g$ can be obtained by successively gluing $g$ 1-handles to the 2-sphere such that the embeddings of $S^0\times D^2$ in $S_{g-1}$ are in each case either orientation-preserving or orientation-reversing on both components $\{1\}\times D^2\cong D^2$ and $\{-1\}\times D^2\cong -D^2$.<br />
<br />
A non-orientable surface of genus $h$ can be obtained by gluing $h$ ''crosscaps'' to $S^2$. For this, embed $D^2$ in $S^2$ (or $R_{h-1}$ from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary $S^1$. The result of attaching a non-orientable handle to $S^2$ or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.<br />
<br />
Reference: {{cite|Hirsch1994|Section 9.1}}<br />
</wikitex><br />
<br />
=== By branched coverings ===<br />
<wikitex>;<br />
Every orientable surface of genus $g$ can be obtained as the branched double covering of the 2-sphere with $2g+2$ branching points.<br />
<br />
{{beginproof}}<br />
The branched double covering of the 2-sphere, with $2g+2$ branch points, has Euler characteristic $4-(2g+2)=2(1-g)$. Thus every orientable closed surface is such a branched covering.<br />
{{endproof}}<br />
</wikitex><br />
<br />
=== As complex curves ===<br />
<wikitex>;<br />
A smooth, irreducible, plane, complex projective-algebraic curve of order $d$ (i. e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree $d$ in $\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus<br />
$$<br />
g = \frac{(d-1)(d-2)}2.<br />
$$<br />
<br />
Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.<br />
<br />
Reference: {{cite|Brieskorn&Knörrer1986|Lemma 9.2.1}}<br />
</wikitex><br />
<br />
== Properties ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
Let $S_g$ denote an oriented surface of genus $g$.<br />
<br />
* By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, $2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps.<br />
<br />
* Therefore, the homology groups with any coefficients $G$ are given by $H_0\cong G$, $H_1\cong G^{2g}$, $H_2\cong G$.<br />
<br />
* The integral cohomology ring is completely determined by the intersection form on $H^1$, which is necessarily isomorphic to the hyperbolic form. A basis for $H^1(S_g)$, for which the intersection form is the standard hyperbolic form, is given by the loops $(a_1,b_1,\ldots,a_g,b_g)$ in the polygon construction above.<br />
<br />
* Homotopy groups<br />
** For the homotopy group of $S^2$, see the article about [[sphere|spheres]].<br />
** All surfaces of higher genus $g\geq1$ are aspherical with fundamental group $\pi_1(S_g,*)\cong \langle a_1,b_1,\ldots, a_g,b_g \mid a_1b_1\cdots a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2$.<br />
<br />
* All orientable surfaces can be embedded in $\Rr^3$.<br />
<br />
* Every surface can be given a complex structure. Together with the complex structure, it is a [[wikipedia:Riemann surface|Riemann surface]]<br />
<br />
* Characteristic classes<br />
** All Stiefel-Whitney classes vanish.<br />
** All Pontrjagin classes vanish.<br />
** The Euler characteristic is $2-2g$.<br />
** Given a complex structure on the surface, the first Chern class is equal to the Euler class.<br />
<br />
* All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in $\Rr^3$ has constant positive curvature, the torus can be given a flat metric, and all surfaces of higher genus admit metrics of constant negative curvature.<br />
<br />
* All surfaces are smoothly [[amphicheiral]].<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
Let $R_h$ denote an non-orientable surface of genus $h$.<br />
<br />
* Again, each non-orientable surface has a cell decomposition with one 0-cell, $h$ 1-cells and one 2-cell. The differential $C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$ with respect to the basis given by the loops the loops $(a_1,\ldots,a_h)$ in the polygon construction is the matrix $(2,\ldots, 2)$.<br />
<br />
* Therefore, the integral homology groups are isomorphic to $H_0\cong\Zz$, $H_1\cong \Zz^{h-1}\oplus \Zz/2$, $H_2\cong 0$.<br />
<br />
* The mod-2 homology groups are $H_0=\Zz/2$, $H_1\cong(\Zz/2)^h$, $H_2=\Zz/2$ since the chain complex above is acyclic mod 2. The intersection form on $H_1(R_h;\Zz/2)$ with respect to this cellular basis is given by the identity matrix.<br />
<br />
* Homotopy groups<br />
** The fundamental group of $R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$.<br />
** The orientation double covering of $R_h$ is $S_{h-1}$. Therefore, all higher homotopy groups of $\RP^2$ equal those of $S^2$, and the non-orientable surfaces of higher genus are aspherical.<br />
<br />
* The orientation double covering also determines the curvature properties: $\RP^2$ can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.<br />
<br />
* Non-orientable surfaces cannot be embedded in $\Rr^3$. However, they do embed in $\Rr^4$.<br />
<br />
* Characteristic classes<br />
** The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism $\pi_1(R_h,*)\to\Zz/2$ which maps each generator $a_1,\ldots,a_h$ to the generator of $\Zz/2$. The second Stiefel-Whitney class is zero if $h$ is even and is the non-zero element of $H^2(R_h;\Zz)=H^2(R_h;\Zz/2)=\Zz/2$ if $h$ is odd. In all cases $w_2=w_1^2$.<br />
** The Euler characteristic is $2-h$.<br />
</wikitex><br />
<br />
=== General ===<br />
There are purely point-set topology characterizations of surfaces; see {{cite|Wilder1949}}.<br />
<br />
== References ==<br />
{{#RefList:}}<br />
== External links ==<br />
* The Wikipedia page about [[Wikipedia:Surface|surfaces]]<br />
<br />
[[Category:Manifolds]]<br />
<!--[[Category:Dimension 2]] --><br />
[[Category:Aspherical manifolds]]<br />
<!-- [[Category:Constant curvature]] --></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/2-manifolds2-manifolds2019-03-30T19:37:56Z<p>Daniel Müllner: /* Classification */</p>
<hr />
<div>{{Authors|Daniel Müllner}}{{MediaWiki:Being refereed}}<br />
== Introduction ==<br />
<wikitex>;<br />
A surface is a synonym for a $2$-dimensional manifold. Complex $2$-dimensional (real $4$-dimensional) complex manifolds are also called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections [[#Surfaces with boundary|2]] and [[#Classification|3]]), surfaces without boundary are considered.<br />
</wikitex><br />
== First construction: connected sum ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
The two simplest closed [[Wikipedia:Orientable|orientable]] $2$-manifolds are:<br />
* the $2$-sphere: $S^2 := \{ (x, y) \in \Rr^2 | x^2 + y^2 = 1 \}$,<br />
* the $2$-torus: $T^2 := S^1 \times S^1$, the [[Wikipedia:Cartesian_product|Cartesian product]] of two [[1-manifolds#Exmaples|circles]].<br />
<br />
All orientable surfaces are homeomorphic to the [[Parametric connected sum#Connected sum|connected sum]] of $g$ [[Torus|tori]] $T^2$ ($g\geq 0$) and so we define<br />
* $F_g := \sharp_g T^2 = T^2 \sharp \dots \sharp T^2$, the $g$-fold connected sum of the $2$-torus.<br />
The case $g=0$ refers to the 2-[[sphere]] $S^2$. The number $g$ is called the [[Wikipedia:Genus_(mathematics)#Orientable_surface|''genus'']] of the surface: for more on the concept of genus see, e.g. \cite{Hirzebruch&Kreck2009}.<br />
[[Image:Surfaces.png|none|frame|A 2-sphere (genus 0), a torus (genus 1) and an orientable surface of higher genus]]<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
The simplest non-orientable surface is the [[Wikipedia:Projective_plane|real projective plane]] $\RP^2$: for the history of the discovery of this interesting manifold see the page [[Projective plane: a history]]. <br />
<br />
All non-orientable surfaces are homeomorphic to the connected sum of $h$ real projective planes and and so for all $h \geq 1$ we define<br />
* $R_h := \sharp_h \RP^2 = \RP^2 \sharp \dots \sharp \RP^2$, to be the $h$-fold connected sum of $\RP^2$. <br />
For example, the ''Klein bottle'' is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2$. The number $h$ is called the ''genus'' or ''cross-cap number'' of the surface.<br />
[[Image:Boy surface.jpg|right|thumb|310px|The Boy surface, an immersion of $\RP^2$ in $\Rr^3$. This steel sculpture stands in front of the [http://www.mfo.de Oberwolfach Institute].]]<br />
[[Image:Klein_bottle.png|left|frame|A Klein bottle (non-orientable, genus 2) immersed in $\Rr^3$]]{{-|left}}<br />
</wikitex><br />
<br />
== Surfaces with boundary ==<br />
The boundary of a surface is a disjoint union (possibly empty) of circles. Surfaces with boundary can be constructed by removing open discs from surfaces without boundary.<br />
<br />
== Classification ==<br />
<wikitex>;<br />
Compact, connected surfaces are classified by orientability (yes/no), the number of boundary components (a nonnegative integer) and the genus after filling the bounday circles by disks (an integer $\geq 0$ in the orientable case, $\geq 1$ in the non-orientable case). Instead of the genus, also e. g. the Euler characteristic can be used in the classification. The classifications up to homotopy equivalence, homeomorphism, PL-equivalence and diffeomorphism coincide.<br />
<br />
References: {{cite|Ahlfors&Sario1960|Thm. 1.42A, 1§8}}, {{cite|Hirsch1994|Thm. 9.3.11}}, {{cite|Moise1977|Thm. 8.3, Thm. 8.5, Thm. 22.9}}{{-}}<br />
<br />
The connected sum of $T^2$ with $\RP^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$. Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect to connected sum, and is generated by $T^2$ and<br />
$\RP^2$, with the sole relation $T^2\mathbin{\sharp}\RP^2=\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$.<br />
<br />
Compact 2-manifolds (possibly with boundary) are homeomorphic if and only if they have isomorphic [[Intersection_form|intersection forms]]. Cf. [[4-manifolds:_1-connected#Topological_classification|the topological classification of simply-connected 4-manifolds]].<br />
</wikitex><br />
<br />
== More constructions ==<br />
<br />
=== By polygons ===<br />
<wikitex>;<br />
Each orientable surface of genus $g>0$ can be constructed by identifying pairs of edges in a regular polygon with $4g$ sides. Label the edges by the sequence $a_1,b_1,\bar a_1,\bar b_1,\ldots,a_g,b_g,\bar a_g,\bar b_g$. Also orient the edges such that those labeled without an overbar are oriented in one direction (e. g. clockwise) and those with an overbar are oppositely oriented. Now identify corresponding edges, respecting the orientation.<br />
<br />
The 2-sphere can be obtained from a 2-gon with edges labeled $a,\bar a$.<br />
<br />
Each non-orientable surface of genus $h$ can be obtained from a $2h$-gon with edges labeled $a_1,a_1,\ldots,a_h,a_h$.<br />
<br />
{|<br />
|<br />
[[Image:Polygon_construction.png|frame|The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.]]<br />
|<br />
[[Image:Polygon_sphere.png|frame|The 2-sphere can be obtained by identifying the edges of a 2-gon.]]<br />
|<br />
[[Image:Polygon_RP2.png|frame|Construction of $\RP^2$.]]<br />
|<br />
[[Image:Polygon_Klein_bottle.png|frame|Construction of the Klein bottle.]]<br />
|}<br />
Reference: {{cite|Massey1977|Section 1.5}}<br />
</wikitex><br />
<br />
=== By gluing handles and crosscaps ===<br />
<wikitex>;<br />
An orientable surface of genus $g$ can be obtained by successively gluing $g$ 1-handles to the 2-sphere such that the embeddings of $S^0\times D^2$ in $S_{g-1}$ are in each case either orientation-preserving or orientation-reversing on both components $\{1\}\times D^2\cong D^2$ and $\{-1\}\times D^2\cong -D^2$.<br />
<br />
A non-orientable surface of genus $h$ can be obtained by gluing $h$ ''crosscaps'' to $S^2$. For this, embed $D^2$ in $S^2$ (or $R_{h-1}$ from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary $S^1$. The result of attaching a non-orientable handle to $S^2$ or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.<br />
<br />
Reference: {{cite|Hirsch1994|Section 9.1}}<br />
</wikitex><br />
<br />
=== By branched coverings ===<br />
<wikitex>;<br />
Every orientable surface of genus $g$ can be obtained as the branched double covering of the 2-sphere with $2g+2$ branching points.<br />
<br />
{{beginproof}}<br />
The branched double covering of the 2-sphere, with $2g+2$ branch points, has Euler characteristic $4-(2g+2)=2(1-g)$. Thus every orientable closed surface is such a branched covering.<br />
{{endproof}}<br />
</wikitex><br />
<br />
=== As complex curves ===<br />
<wikitex>;<br />
A smooth, irreducible, plane, complex projective-algebraic curve of order $d$ (i. e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree $d$ in $\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus<br />
$$<br />
g = \frac{(d-1)(d-2)}2.<br />
$$<br />
<br />
Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.<br />
<br />
Reference: {{cite|Brieskorn&Knörrer1986|Lemma 9.2.1}}<br />
</wikitex><br />
<br />
== Properties ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
Let $S_g$ denote an oriented surface of genus $g$.<br />
<br />
* By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, $2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps.<br />
<br />
* Therefore, the homology groups with any coefficients $G$ are given by $H_0\cong G$, $H_1\cong G^{2g}$, $H_2\cong G$.<br />
<br />
* The integral cohomology ring is completely determined by the intersection form on $H^1$, which is necessarily isomorphic to the hyperbolic form. A basis for $H^1(S_g)$, for which the intersection form is the standard hyperbolic form, is given by the loops $(a_1,b_1,\ldots,a_g,b_g)$ in the polygon construction above.<br />
<br />
* Homotopy groups<br />
** For the homotopy group of $S^2$, see the article about [[sphere|spheres]].<br />
** All surfaces of higher genus $g\geq1$ are aspherical with fundamental group $\pi_1(S_g,*)\cong \langle a_1,b_1,\ldots, a_g,b_g \mid a_1b_1\cdots a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2$.<br />
<br />
* All orientable surfaces can be embedded in $\Rr^3$.<br />
<br />
* Every surface can be given a complex structure. Together with the complex structure, it is a [[wikipedia:Riemann surface|Riemann surface]]<br />
<br />
* Characteristic classes<br />
** All Stiefel-Whitney classes vanish.<br />
** All Pontrjagin classes vanish.<br />
** The Euler characteristic is $2-2g$.<br />
** Given a complex structure on the surface, the first Chern class is equal to the Euler class.<br />
<br />
* All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in $\Rr^3$ has constant positive curvature, the torus can be given a flat metric, and all surfaces of higher genus admit metrics of constant negative curvature.<br />
<br />
* All surfaces are smoothly [[amphicheiral]].<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
Let $R_h$ denote an non-orientable surface of genus $h$.<br />
<br />
* Again, each non-orientable surface has a cell decomposition with one 0-cell, $h$ 1-cells and one 2-cell. The differential $C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$ with respect to the basis given by the loops the loops $(a_1,\ldots,a_h)$ in the polygon construction is the matrix $(2,\ldots, 2)$.<br />
<br />
* Therefore, the integral homology groups are isomorphic to $H_0\cong\Zz$, $H_1\cong \Zz^{h-1}\oplus \Zz/2$, $H_2\cong 0$.<br />
<br />
* The mod-2 homology groups are $H_0=\Zz/2$, $H_1\cong(\Zz/2)^h$, $H_2=\Zz/2$ since the chain complex above is acyclic mod 2. The intersection form on $H_1(R_h;\Zz/2)$ with respect to this cellular basis is given by the identity matrix.<br />
<br />
* Homotopy groups<br />
** The fundamental group of $R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$.<br />
** The orientation double covering of $R_h$ is $S_{h-1}$. Therefore, all higher homotopy groups of $\RP^2$ equal those of $S^2$, and the non-orientable surfaces of higher genus are aspherical.<br />
<br />
* The orientation double covering also determines the curvature properties: $\RP^2$ can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.<br />
<br />
* Non-orientable surfaces cannot be embedded in $\Rr^3$. However, they do embed in $\Rr^4$.<br />
<br />
* Characteristic classes<br />
** The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism $\pi_1(R_h,*)\to\Zz/2$ which maps each generator $a_1,\ldots,a_h$ to the generator of $\Zz/2$. The second Stiefel-Whitney class is zero if $h$ is even and is the non-zero element of $H^2(R_h;\Zz)=H^2(R_h;\Zz/2)=\Zz/2$ if $h$ is odd. In all cases $w_2=w_1^2$.<br />
** The Euler characteristic is $2-h$.<br />
</wikitex><br />
<br />
=== General ===<br />
There are purely point-set topology characterizations of surfaces; see {{cite|Wilder1949}}.<br />
<br />
== References ==<br />
{{#RefList:}}<br />
== External links ==<br />
* The Wikipedia page about [[Wikipedia:Surface|surfaces]]<br />
<br />
[[Category:Manifolds]]<br />
<!--[[Category:Dimension 2]] --><br />
[[Category:Aspherical manifolds]]<br />
<!-- [[Category:Constant curvature]] --></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/2-manifolds2-manifolds2011-01-14T15:53:12Z<p>Daniel Müllner: </p>
<hr />
<div>{{Authors|Daniel Müllner}}{{MediaWiki:Being refereed}}<br />
== Introduction ==<br />
<wikitex>;<br />
A surface is a synonym for a $2$-dimensional manifold. Complex $2$-dimensional (real $4$-dimensional) complex manifolds are also called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections [[#Surfaces with boundary|2]] and [[#Classification|3]]), surfaces without boundary are considered.<br />
</wikitex><br />
== First construction: connected sum ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
The two simplest closed [[Wikipedia:Orientable|orientable]] $2$-manifolds are:<br />
* the $2$-sphere: $S^2 := \{ (x, y) \in \Rr^2 | x^2 + y^2 = 1 \}$,<br />
* the $2$-torus: $T^2 := S^1 \times S^1$, the [[Wikipedia:Cartesian_product|Cartesian product]] of two [[1-manifolds#Exmaples|circles]].<br />
<br />
All orientable surfaces are homeomorphic to the [[Parametric connected sum#Connected sum|connected sum]] of $g$ [[Torus|tori]] $T^2$ ($g\geq 0$) and so we define<br />
* $F_g := \sharp_g T^2 = T^2 \sharp \dots \sharp T^2$, the $g$-fold connected sum of the $2$-torus.<br />
The case $g=0$ refers to the 2-[[sphere]] $S^2$. The number $g$ is called the [[Wikipedia:Genus_(mathematics)#Orientable_surface|''genus'']] of the surface: for more on genus see, e.g. \cite{Hirzebruch&Kreck2009}.<br />
[[Image:Surfaces.png|none|frame|A 2-sphere (genus 0), a torus (genus 1) and an orientable surface of higher genus]]<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex><br />
The simplest non-orientable surface is the [[Wikipedia:Projective_plane|real projective plane]] $\RP^2$: for the history of the discovery of this wonderful manifold see the page [[Projective plane: a history]]. <br />
<br />
All non-orientable surfaces are homeomorphic to the connected sum of $h$ real projective planes and and so for all $h \geq 1$ we define<br />
* $R_h := \sharp_h \RP^2 = \RP^2 \sharp \dots \sharp \RP^2$, to be the $h$-fold connected sum of $\RP^2$. <br />
For example, the ''Klein bottle'' is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2$. The number $h$ is called the ''genus'' or ''cross-cap number'' of the surface.<br />
[[Image:Boy surface.jpg|right|thumb|310px|The Boy surface, an immersion of $\RP^2$ in $\Rr^3$. This steel sculpture stands in front of the [http://www.mfo.de Oberwolfach Institute].]]<br />
[[Image:Klein_bottle.png|left|frame|A Klein bottle (non-orientable, genus 2) immersed in $\Rr^3$]]{{-|left}}<br />
</wikitex><br />
<br />
== Surfaces with boundary ==<br />
The boundary of a surface is a disjoint union (possibly empty) of circles. Surfaces with boundary can be constructed by removing open discs from surfaces without boundary.<br />
<br />
== Classification ==<br />
<wikitex>;<br />
Compact, connected surfaces are classified by orientability (yes/no), the number of boundary components (a nonnegative integer) and the genus after filling the bounday circles by disks (an integer $\geq 0$ in the orientable case, $\geq 1$ in the non-orientable case). Instead of the genus, also e. g. the Euler characteristic can be used in the classification. The classifications up to homotopy equivalence, homeomorphism, PL-equivalence and diffeomorphism coincide.<br />
<br />
References: {{cite|Ahlfors&Sario1960|Thm. 1.42A, 1§8}}, {{cite|Hirsch1994|Thm. 9.3.11}}, {{cite|Moise1977|Thm. 8.3, Thm. 8.5, Thm. 22.9}}{{-}}<br />
<br />
The connected sum of $T^2$ with $\RP^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$. Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect to connected sum, and is generated by $T^2$ and<br />
$\RP^2$, with the sole relation $T^2\mathbin{\sharp}\RP^2=\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$.<br />
</wikitex><br />
<br />
== More constructions ==<br />
<br />
=== By polygons ===<br />
<wikitex>;<br />
Each orientable surface of genus $g>0$ can be constructed by identifying pairs of edges in a regular polygon with $4g$ sides. Label the edges by the sequence $a_1,b_1,\bar a_1,\bar b_1,\ldots,a_g,b_g,\bar a_g,\bar b_g$. Also orient the edges such that those labeled without an overbar are oriented in one direction (e. g. clockwise) and those with an overbar are oppositely oriented. Now identify corresponding edges, respecting the orientation.<br />
<br />
The 2-sphere can be obtained from a 2-gon with edges labeled $a,\bar a$.<br />
<br />
Each non-orientable surface of genus $h$ can be obtained from a $2h$-gon with edges labeled $a_1,a_1,\ldots,a_h,a_h$.<br />
<br />
{|<br />
|<br />
[[Image:Polygon_construction.png|frame|The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.]]<br />
|<br />
[[Image:Polygon_sphere.png|frame|The 2-sphere can be obtained by identifying the edges of a 2-gon.]]<br />
|<br />
[[Image:Polygon_RP2.png|frame|Construction of $\RP^2$.]]<br />
|<br />
[[Image:Polygon_Klein_bottle.png|frame|Construction of the Klein bottle.]]<br />
|}<br />
Reference: {{cite|Massey1977|Section 1.5}}<br />
</wikitex><br />
<br />
=== By gluing handles and crosscaps ===<br />
<wikitex>;<br />
An orientable surface of genus $g$ can be obtained by successively gluing $g$ 1-handles to the 2-sphere such that the embeddings of $S^0\times D^2$ in $S_{g-1}$ are in each case either orientation-preserving or orientation-reversing on both components $\{1\}\times D^2\cong D^2$ and $\{-1\}\times D^2\cong -D^2$.<br />
<br />
A non-orientable surface of genus $h$ can be obtained by gluing $h$ ''crosscaps'' to $S^2$. For this, embed $D^2$ in $S^2$ (or $R_{h-1}$ from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary $S^1$. The result of attaching a non-orientable handle to $S^2$ or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.<br />
<br />
Reference: {{cite|Hirsch1994|Section 9.1}}<br />
</wikitex><br />
<br />
=== By branched coverings ===<br />
<wikitex>;<br />
Every orientable surface of genus $g$ can be obtained as the branched double covering of the 2-sphere with $2g+2$ branching points.<br />
<br />
{{beginproof}}<br />
The branched double covering of the 2-sphere, with $2g+2$ branch points, has Euler characteristic $4-(2g+2)=2(1-g)$. Thus every orientable closed surface is such a branched covering.<br />
{{endproof}}<br />
</wikitex><br />
<br />
=== As complex curves ===<br />
<wikitex>;<br />
A smooth, irreducible, plane, complex projective-algebraic curve of order $d$ (i. e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree $d$ in $\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus<br />
$$<br />
g = \frac{(d-1)(d-2)}2.<br />
$$<br />
<br />
Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.<br />
<br />
Reference: {{cite|Brieskorn&Knörrer1986|Lemma 9.2.1}}<br />
</wikitex><br />
<br />
== Properties ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
Let $S_g$ denote an oriented surface of genus $g$.<br />
<br />
* By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, $2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps.<br />
<br />
* Therefore, the homology groups with any coefficients $G$ are given by $H_0\cong G$, $H_1\cong G^{2g}$, $H_2\cong G$.<br />
<br />
* The integral cohomology ring is completely determined by the intersection form on $H^1$, which is necessarily isomorphic to the hyperbolic form. A basis for $H^1(S_g)$, for which the intersection form is the standard hyperbolic form, is given by the loops $(a_1,b_1,\ldots,a_g,b_g)$ in the polygon construction above.<br />
<br />
* Homotopy groups<br />
** For the homotopy group of $S^2$, see the article about [[sphere|spheres]].<br />
** All surfaces of higher genus $g\geq1$ are aspherical with fundamental group $\pi_1(S_g,*)\cong \langle a_1,b_1,\ldots, a_g,b_g \mid a_1b_1\cdots a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2$.<br />
<br />
* All orientable surfaces can be embedded in $\Rr^3$.<br />
<br />
* Every surface can be given a complex structure. Together with the complex structure, it is a [[wikipedia:Riemann surface|Riemann surface]]<br />
<br />
* Characteristic classes<br />
** All Stiefel-Whitney classes vanish.<br />
** All Pontrjagin classes vanish.<br />
** The Euler characteristic is $2-2g$.<br />
** Given a complex structure on the surface, the first Chern class is equal to the Euler class.<br />
<br />
* All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in $\Rr^3$ has constant positive curvature, the torus can be given a flat metric, and all surfaces of higher genus admit metrics of constant negative curvature.<br />
<br />
* All surfaces are smoothly [[amphicheiral]].<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
Let $R_h$ denote an non-orientable surface of genus $h$.<br />
<br />
* Again, each non-orientable surface has a cell decomposition with one 0-cell, $h$ 1-cells and one 2-cell. The differential $C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$ with respect to the basis given by the loops the loops $(a_1,\ldots,a_h)$ in the polygon construction is the matrix $(2,\ldots, 2)$.<br />
<br />
* Therefore, the integral homology groups are isomorphic to $H_0\cong\Zz$, $H_1\cong \Zz^{h-1}\oplus \Zz/2$, $H_2\cong 0$.<br />
<br />
* The mod-2 homology groups are $H_0=\Zz/2$, $H_1\cong(\Zz/2)^h$, $H_2=\Zz/2$ since the chain complex above is acyclic mod 2. The intersection form on $H_1(R_h;\Zz/2)$ with respect to this cellular basis is given by the identity matrix.<br />
<br />
* Homotopy groups<br />
** The fundamental group of $R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$.<br />
** The orientation double covering of $R_h$ is $S_{h-1}$. Therefore, all higher homotopy groups of $\RP^2$ equal those of $S^2$, and the non-orientable surfaces of higher genus are aspherical.<br />
<br />
* The orientation double covering also determines the curvature properties: $\RP^2$ can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.<br />
<br />
* Non-orientable surfaces cannot be embedded in $\Rr^3$. However, they do embed in $\Rr^4$.<br />
<br />
* Characteristic classes<br />
** The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism $\pi_1(R_h,*)\to\Zz/2$ which maps each generator $a_1,\ldots,a_h$ to the generator of $\Zz/2$. The second Stiefel-Whitney class is zero if $h$ is even and is the non-zero element of $H^2(R_h;\Zz)=H^2(R_h;\Zz/2)=\Zz/2$ if $h$ is odd. In all cases $w_2=w_1^2$.<br />
** The Euler characteristic is $2-h$.<br />
</wikitex><br />
<br />
=== General ===<br />
There are purely point-set topology characterizations of surfaces; see {{cite|Wilder1949}}.<br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Manifolds]]<br />
<!--[[Category:Dimension 2]] --><br />
[[Category:Aspherical manifolds]]<br />
<!-- [[Category:Constant curvature]] --></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/2-manifolds2-manifolds2010-12-16T23:35:14Z<p>Daniel Müllner: </p>
<hr />
<div>{{Authors|Daniel Müllner}}{{MediaWiki:Being refereed}}<br />
A surface is a synonym for a 2-dimensional manifold. Complex 2-dimensional (real 4-dimensional) complex manifolds are also called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections [[#Surfaces with boundary|2]] and [[#Classification|3]]), surfaces without boundary are considered.<br />
<br />
== First construction: connected sum ==<br />
<wikitex>;<br />
All orientable surfaces are homeomorphic to the connected sum of $g$ [[Torus|tori]] $T^2$ ($g\geq 0$). The case $g=0$ refers to the 2-[[sphere]] $S^2$. All non-orientable surfaces are homeomorphic to the connected sum of $h$ [[real projective spaces]] $\RP^2$ ($h\geq 1$). For example, the ''Klein bottle'' is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2$. The numbers $g$ and $h$ are called the ''genus'', if the surface is orientable, and genus or ''cross-cap number'', if the surface is non-orientable.<br />
<br />
[[Image:Surfaces.png|left|frame|A 2-sphere (genus 0), a torus (genus 1) and an orientable surface of higher genus]]<br />
[[Image:Boy surface.jpg|right|thumb|310px|The Boy surface, an immersion of $\RP^2$ in $\Rr^3$. This steel sculpture stands in front of the [http://www.mfo.de Oberwolfach Institute].]]<br />
[[Image:Klein_bottle.png|left|frame|A Klein bottle (non-orientable, genus 2) immersed in $\Rr^3$]]{{-|left}}<br />
<br />
The connected sum of $T^2$ with $\RP^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$. Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect to connected sum, and is generated by $T^2$ and<br />
$\RP^2$, with the sole relation $T^2\mathbin{\sharp}\RP^2=\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$.<br />
</wikitex><br />
<br />
== Surfaces with boundary ==<br />
The boundary of a surface is a disjoint union (possibly empty) of circles. Surfaces with boundary can be constructed by removing open discs from surfaces without boundary.<br />
<br />
== Classification ==<br />
<wikitex>;<br />
Compact, connected surfaces are classified by orientability (yes/no), the number of boundary components (a nonnegative integer) and the genus after filling the bounday circles by disks (an integer $\geq 0$ in the orientable case, $\geq 1$ in the non-orientable case). Instead of the genus, also e. g. the Euler characteristic can be used in the classification. The classifications up to homotopy equivalence, homeomorphism, PL-equivalence and diffeomorphism coincide.<br />
<br />
References: {{cite|Ahlfors&Sario1960|Thm. 1.42A, 1§8}}, {{cite|Hirsch1994|Thm. 9.3.11}}, {{cite|Moise1977|Thm. 8.3, Thm. 8.5, Thm. 22.9}}{{-}}<br />
</wikitex><br />
<br />
== More constructions ==<br />
<br />
=== By polygons ===<br />
<wikitex>;<br />
Each orientable surface of genus $g>0$ can be constructed by identifying pairs of edges in a regular polygon with $4g$ sides. Label the edges by the sequence $a_1,b_1,\bar a_1,\bar b_1,\ldots,a_g,b_g,\bar a_g,\bar b_g$. Also orient the edges such that those labeled without an overbar are oriented in one direction (e. g. clockwise) and those with an overbar are oppositely oriented. Now identify corresponding edges, respecting the orientation.<br />
<br />
The 2-sphere can be obtained from a 2-gon with edges labeled $a,\bar a$.<br />
<br />
Each non-orientable surface of genus $h$ can be obtained from a $2h$-gon with edges labeled $a_1,a_1,\ldots,a_h,a_h$.<br />
<br />
{|<br />
|<br />
[[Image:Polygon_construction.png|frame|The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.]]<br />
|<br />
[[Image:Polygon_sphere.png|frame|The 2-sphere can be obtained by identifying the edges of a 2-gon.]]<br />
|<br />
[[Image:Polygon_RP2.png|frame|Construction of $\RP^2$.]]<br />
|<br />
[[Image:Polygon_Klein_bottle.png|frame|Construction of the Klein bottle.]]<br />
|}<br />
Reference: {{cite|Massey1977|Section 1.5}}<br />
</wikitex><br />
<br />
=== By gluing handles and crosscaps ===<br />
<wikitex>;<br />
An orientable surface of genus $g$ can be obtained by successively gluing $g$ 1-handles to the 2-sphere such that the embeddings of $S^0\times D^2$ in $S_{g-1}$ are in each case either orientation-preserving or orientation-reversing on both components $\{1\}\times D^2\cong D^2$ and $\{-1\}\times D^2\cong -D^2$.<br />
<br />
A non-orientable surface of genus $h$ can be obtained by gluing $h$ ''crosscaps'' to $S^2$. For this, embed $D^2$ in $S^2$ (or $R_{h-1}$ from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary $S^1$. The result of attaching a non-orientable handle to $S^2$ or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.<br />
<br />
Reference: {{cite|Hirsch1994|Section 9.1}}<br />
</wikitex><br />
<br />
=== By branched coverings ===<br />
<wikitex>;<br />
Every orientable surface of genus $g$ can be obtained as the branched double covering of the 2-sphere with $2g+2$ branching points.<br />
<br />
{{beginproof}}<br />
The branched double covering of the 2-sphere, with $2g+2$ branch points, has Euler characteristic $4-(2g+2)=2(1-g)$. Thus every orientable closed surface is such a branched covering.<br />
{{endproof}}<br />
</wikitex><br />
<br />
=== As complex curves ===<br />
<wikitex>;<br />
A smooth, irreducible, plane, complex projective-algebraic curve of order $d$ (i. e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree $d$ in $\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus<br />
$$<br />
g = \frac{(d-1)(d-2)}2.<br />
$$<br />
<br />
Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.<br />
<br />
Reference: {{cite|Brieskorn&Knörrer1986|Lemma 9.2.1}}<br />
</wikitex><br />
<br />
== Properties ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
Let $S_g$ denote an oriented surface of genus $g$.<br />
<br />
* By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, $2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps.<br />
<br />
* Therefore, the homology groups with any coefficients $G$ are given by $H_0\cong G$, $H_1\cong G^{2g}$, $H_2\cong G$.<br />
<br />
* The integral cohomology ring is completely determined by the intersection form on $H^1$, which is necessarily isomorphic to the hyperbolic form. A basis for $H^1(S_g)$, for which the intersection form is the standard hyperbolic form, is given by the loops $(a_1,b_1,\ldots,a_g,b_g)$ in the polygon construction above.<br />
<br />
* Homotopy groups<br />
** For the homotopy group of $S^2$, see the article about [[sphere|spheres]].<br />
** All surfaces of higher genus $g\geq1$ are aspherical with fundamental group $\pi_1(S_g,*)\cong \langle a_1,b_1,\ldots, a_g,b_g \mid a_1b_1\cdots a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2$.<br />
<br />
* All orientable surfaces can be embedded in $\Rr^3$.<br />
<br />
* Every surface can be given a complex structure. Together with the complex structure, it is a [[wikipedia:Riemann surface|Riemann surface]]<br />
<br />
* Characteristic classes<br />
** All Stiefel-Whitney classes vanish.<br />
** All Pontrjagin classes vanish.<br />
** The Euler characteristic is $2-2g$.<br />
** Given a complex structure on the surface, the first Chern class is equal to the Euler class.<br />
<br />
* All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in $\Rr^3$ has constant positive curvature, the torus can be given a flat metric, and all surfaces of higher genus admit metrics of constant negative curvature.<br />
<br />
* All surfaces are smoothly [[amphicheiral]].<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
Let $R_h$ denote an non-orientable surface of genus $h$.<br />
<br />
* Again, each non-orientable surface has a cell decomposition with one 0-cell, $h$ 1-cells and one 2-cell. The differential $C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$ with respect to the basis given by the loops the loops $(a_1,\ldots,a_h)$ in the polygon construction is the matrix $(2,\ldots, 2)$.<br />
<br />
* Therefore, the integral homology groups are isomorphic to $H_0\cong\Zz$, $H_1\cong \Zz^{h-1}\oplus \Zz/2$, $H_2\cong 0$.<br />
<br />
* The mod-2 homology groups are $H_0=\Zz/2$, $H_1\cong(\Zz/2)^h$, $H_2=\Zz/2$ since the chain complex above is acyclic mod 2. The intersection form on $H_1(R_h;\Zz/2)$ with respect to this cellular basis is given by the identity matrix.<br />
<br />
* Homotopy groups<br />
** The fundamental group of $R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$.<br />
** The orientation double covering of $R_h$ is $S_{h-1}$. Therefore, all higher homotopy groups of $\RP^2$ equal those of $S^2$, and the non-orientable surfaces of higher genus are aspherical.<br />
<br />
* The orientation double covering also determines the curvature properties: $\RP^2$ can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.<br />
<br />
* Non-orientable surfaces cannot be embedded in $\Rr^3$. However, they do embed in $\Rr^4$.<br />
<br />
* Characteristic classes<br />
** The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism $\pi_1(R_h,*)\to\Zz/2$ which maps each generator $a_1,\ldots,a_h$ to the generator of $\Zz/2$. The second Stiefel-Whitney class is zero if $h$ is even and is the non-zero element of $H^2(R_h;\Zz)=H^2(R_h;\Zz/2)=\Zz/2$ if $h$ is odd. In all cases $w_2=w_1^2$.<br />
** The Euler characteristic is $2-h$.<br />
</wikitex><br />
<br />
=== General ===<br />
<br />
There are purely point-set topology characterizations of surfaces; see {{cite|Wilder1949}}.<br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Manifolds]]<br />
<!--[[Category:Dimension 2]] --><br />
[[Category:Aspherical manifolds]]<br />
<!-- [[Category:Constant curvature]] --></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/2-manifolds2-manifolds2010-12-16T23:09:24Z<p>Daniel Müllner: Changes from the referee report.</p>
<hr />
<div>{{Authors|Daniel Müllner}}{{MediaWiki:Being refereed}}<br />
A surface is a synonym for a 2-dimensional manifold. Complex 2-dimensional (real 4-dimensional) complex manifolds are also called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections [[#Surfaces with boundary|2]] and [[#Classification|3]]), surfaces without boundary are considered.<br />
<br />
== First construction: connected sum ==<br />
<wikitex>;<br />
All orientable surfaces are homeomorphic to the connected sum of $g$ [[Torus|tori]] $T^2$ ($g\geq 0$). The case $g=0$ refers to the 2-[[sphere]] $S^2$. All non-orientable surfaces are homeomorphic to the connected sum of $h$ [[real projective spaces]] $\RP^2$ ($h\geq 1$). For example, the ''Klein bottle'' is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2$. The numbers $g$ and $h$ are called the ''genus'', if the surface is orientable, and genus or ''cross-cap number'', if the surface is non-orientable.<br />
<br />
[[Image:Surfaces.png|left|frame|A 2-sphere (genus 0), a torus (genus 1) and an orientable surface of higher genus]]<br />
[[Image:Boy surface.jpg|right|thumb|310px|The Boy surface, an immersion of $\RP^2$ in $\Rr^3$. This steel sculpture stands in front of the [http://www.mfo.de Oberwolfach Institute].]]<br />
[[Image:Klein_bottle.png|left|frame|A Klein bottle (non-orientable, genus 2) immersed in $\Rr^3$]]{{-|left}}<br />
<br />
The connected sum of $T^2$ with $\RP^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$. Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect to connected sum, and is generated by $T^2$ and<br />
$\RP^2$, with the sole relation $T^2\mathbin{\sharp}\RP^2=\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$.<br />
</wikitex><br />
<br />
== Surfaces with boundary ==<br />
The boundary of a surface is a disjoint union (possibly empty) of circles. Surfaces with boundary can be constructed by removing open discs from surfaces without boundary.<br />
<br />
== Classification ==<br />
<wikitex>;<br />
Compact, connected surfaces are classified by orientability (yes/no), the number of boundary components (a nonnegative integer) and the genus after filling the bounday circles by disks (an integer $\geq 0$ in the orientable case, $\geq 1$ in the non-orientable case). Instead of the genus, also e. g. the Euler characteristic can be used in the classification. The classifications up to homotopy equivalence, homeomorphism, PL-equivalence and diffeomorphism coincide.<br />
<br />
References: {{cite|Ahlfors&Sario1960|Thm. 1.42A, 1§8}}, {{cite|Hirsch1994|Thm. 9.3.11}}, {{cite|Moise1977|Thm. 8.5, Thm. 22.9}}{{-}}<br />
</wikitex><br />
<br />
== More constructions ==<br />
<br />
=== By polygons ===<br />
<wikitex>;<br />
Each orientable surface of genus $g>0$ can be constructed by identifying pairs of edges in a regular polygon with $4g$ sides. Label the edges by the sequence $a_1,b_1,\bar a_1,\bar b_1,\ldots,a_g,b_g,\bar a_g,\bar b_g$. Also orient the edges such that those labeled without an overbar are oriented in one direction (e. g. clockwise) and those with an overbar are oppositely oriented. Now identify corresponding edges, respecting the orientation.<br />
<br />
The 2-sphere can be obtained from a 2-gon with edges labeled $a,\bar a$.<br />
<br />
Each non-orientable surface of genus $h$ can be obtained from a $2h$-gon with edges labeled $a_1,a_1,\ldots,a_h,a_h$.<br />
<br />
{|<br />
|<br />
[[Image:Polygon_construction.png|frame|The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.]]<br />
|<br />
[[Image:Polygon_sphere.png|frame|The 2-sphere can be obtained by identifying the edges of a 2-gon.]]<br />
|<br />
[[Image:Polygon_RP2.png|frame|Construction of $\RP^2$.]]<br />
|<br />
[[Image:Polygon_Klein_bottle.png|frame|Construction of the Klein bottle.]]<br />
|}<br />
Reference: {{cite|Massey1977|Section 1.5}}<br />
</wikitex><br />
<br />
=== By gluing handles and crosscaps ===<br />
<wikitex>;<br />
An orientable surface of genus $g$ can be obtained by successively gluing $g$ 1-handles to the 2-sphere such that the embeddings of $S^0\times D^2$ in $S_{g-1}$ are in each case either orientation-preserving or orientation-reversing on both components $\{1\}\times D^2\cong D^2$ and $\{-1\}\times D^2\cong -D^2$.<br />
<br />
A non-orientable surface of genus $h$ can be obtained by gluing $h$ ''crosscaps'' to $S^2$. For this, embed $D^2$ in $S^2$ (or $R_{h-1}$ from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary $S^1$. The result of attaching a non-orientable handle to $S^2$ or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.<br />
<br />
Reference: {{cite|Hirsch1994|Section 9.1}}<br />
</wikitex><br />
<br />
=== By branched coverings ===<br />
<wikitex>;<br />
Every orientable surface of genus $g$ can be obtained as the branched double covering of the 2-sphere with $2g+2$ branching points.<br />
<br />
{{beginproof}}<br />
The branched double covering of the 2-sphere, with $2g+2$ branch points, has Euler characteristic $4-(2g+2)=2(1-g)$. Thus every orientable closed surface is such a branched covering.<br />
{{endproof}}<br />
</wikitex><br />
<br />
=== As complex curves ===<br />
<wikitex>;<br />
A smooth, irreducible, plane, complex projective-algebraic curve of order $d$ (i. e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree $d$ in $\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus<br />
$$<br />
g = \frac{(d-1)(d-2)}2.<br />
$$<br />
<br />
Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.<br />
<br />
Reference: {{cite|Brieskorn&Knörrer1986|Lemma 9.2.1}}<br />
</wikitex><br />
<br />
== Properties ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
Let $S_g$ denote an oriented surface of genus $g$.<br />
<br />
* By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, $2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps.<br />
<br />
* Therefore, the homology groups with any coefficients $G$ are given by $H_0\cong G$, $H_1\cong G^{2g}$, $H_2\cong G$.<br />
<br />
* The integral cohomology ring is completely determined by the intersection form on $H^1$, which is necessarily isomorphic to the hyperbolic form. A basis for $H^1(S_g)$, for which the intersection form is the standard hyperbolic form, is given by the loops $(a_1,b_1,\ldots,a_g,b_g)$ in the polygon construction above.<br />
<br />
* Homotopy groups<br />
** For the homotopy group of $S^2$, see the article about [[sphere|spheres]].<br />
** All surfaces of higher genus $g\geq1$ are aspherical with fundamental group $\pi_1(S_g,*)\cong \langle a_1,b_1,\ldots, a_g,b_g \mid a_1b_1\cdots a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2$.<br />
<br />
* All orientable surfaces can be embedded in $\Rr^3$.<br />
<br />
* Every surface can be given a complex structure. Together with the complex structure, it is a [[wikipedia:Riemann surface|Riemann surface]]<br />
<br />
* Characteristic classes<br />
** All Stiefel-Whitney classes vanish.<br />
** All Pontrjagin classes vanish.<br />
** The Euler characteristic is $2-2g$.<br />
** Given a complex structure on the surface, the first Chern class is equal to the Euler class.<br />
<br />
* All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in $\Rr^3$ has constant positive curvature, the torus can be given a flat metric, and all surfaces of higher genus admit metrics of constant negative curvature.<br />
<br />
* All surfaces are smoothly [[amphicheiral]].<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
Let $R_h$ denote an non-orientable surface of genus $h$.<br />
<br />
* Again, each non-orientable surface has a cell decomposition with one 0-cell, $h$ 1-cells and one 2-cell. The differential $C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$ with respect to the basis given by the loops the loops $(a_1,\ldots,a_h)$ in the polygon construction is the matrix $(2,\ldots, 2)$.<br />
<br />
* Therefore, the integral homology groups are isomorphic to $H_0\cong\Zz$, $H_1\cong \Zz^{h-1}\oplus \Zz/2$, $H_2\cong 0$.<br />
<br />
* The mod-2 homology groups are $H_0=\Zz/2$, $H_1\cong(\Zz/2)^h$, $H_2=\Zz/2$ since the chain complex above is acyclic mod 2. The intersection form on $H_1(R_h;\Zz/2)$ with respect to this cellular basis is given by the identity matrix.<br />
<br />
* Homotopy groups<br />
** The fundamental group of $R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$.<br />
** The orientation double covering of $R_h$ is $S_{h-1}$. Therefore, all higher homotopy groups of $\RP^2$ equal those of $S^2$, and the non-orientable surfaces of higher genus are aspherical.<br />
<br />
* The orientation double covering also determines the curvature properties: $\RP^2$ can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.<br />
<br />
* Non-orientable surfaces cannot be embedded in $\Rr^3$. However, they do embed in $\Rr^4$.<br />
<br />
* Characteristic classes<br />
** The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism $\pi_1(R_h,*)\to\Zz/2$ which maps each generator $a_1,\ldots,a_h$ to the generator of $\Zz/2$. The second Stiefel-Whitney class is zero if $h$ is even and is the non-zero element of $H^2(R_h;\Zz)=H^2(R_h;\Zz/2)=\Zz/2$ if $h$ is odd. In all cases $w_2=w_1^2$.<br />
** The Euler characteristic is $2-h$.<br />
</wikitex><br />
<br />
=== General ===<br />
<br />
There are purely point-set topology characterizations of surfaces; see {{cite|Wilder1949}}.<br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Manifolds]]<br />
<!--[[Category:Dimension 2]] --><br />
[[Category:Aspherical manifolds]]<br />
<!-- [[Category:Constant curvature]] --></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/2-manifolds2-manifolds2010-09-16T00:36:09Z<p>Daniel Müllner: /* By gluing handles and crosscaps */</p>
<hr />
<div>A surface is a synonym for a 2-dimensional manifold. Also complex 2-dimensional (real 4-dimensional) complex manifolds are called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections [[#Surfaces with boundary|2]] and [[#Classification|3]]), surfaces without boundary are considered.<br />
<br />
== First construction: connected sum ==<br />
<wikitex>;<br />
All orientable surfaces are homeomorphic to the connected sum of $g$ [[Torus|tori]] $T^2$ ($g\geq 0$). The case $g=0$ refers to the 2-[[sphere]] $S^2$. All non-orientable surfaces are homeomorphic to the connected sum of $h$ [[real projective spaces]] $\RP^2$ ($h\geq 1$). For example, the ''Klein bottle'' is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2$. The numbers $g$ and $h$ are called the ''genus'' of the orientable/non-orientable surface.<br />
<br />
[[Image:Surfaces.png|left|frame|A 2-sphere (genus 0), a torus (genus 1) and an orientable surface of higher genus]]<br />
[[Image:Boy surface.jpg|right|thumb|310px|The Boy surface, an immersion of $\RP^2$ in $\Rr^3$. This steel sculpture stands in front of the [http://www.mfo.de Oberwolfach Institute].]]<br />
[[Image:Klein_bottle.png|left|frame|A Klein bottle (non-orientable, genus 2) immersed in $\Rr^3$]]{{-|left}}<br />
<br />
The connected sum of $\RP^2$ with $T^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$.<br />
</wikitex><br />
<br />
== Surfaces with boundary ==<br />
The boundary of a surface is a disjoint union (possibly empty) of circles. Surfaces with boundary can be constructed as boundaryless surfaces with open 2-disks removed.<br />
<br />
== Classification ==<br />
<wikitex>;<br />
Compact, connected surfaces are classified by orientability (yes/no), the number of boundary components (a nonnegative integer) and the genus after filling the bounday circles by disks (an integer $\geq 0$ in the orientable case, $\geq 1$ in the non-orientable case). Instead of the genus, also e.&thinsp;g. the Euler characteristic can be used in the classification. The classifications up to homotopy equivalence, homeomorphism, PL-equivalence and diffeomorphism coincide.<br />
<br />
References: {{cite|Ahlfors&Sario1960|Thm. 1.42A, 1&sect;8}}, {{cite|Hirsch1994|Thm. 9.3.11}}, {{cite|Moise1977|Thm. 8.5, Thm. 22.9}}{{-}}<br />
</wikitex><br />
<br />
== More constructions ==<br />
<br />
=== By polygons ===<br />
<wikitex>;<br />
Each orientable surface of genus $g>0$ can be constructed by identifying pairs of edges in a regular polygon with $4g$ sides. Label the edges by the sequence $a_1,b_1,\bar a_1,\bar b_1,\ldots,a_g,b_g,\bar a_g,\bar b_g$. Also orient the edges such that those labeled without an overbar are oriented in one direction (e.&thinsp;g. clockwise) and those with an overbar are oppositely oriented. Now identify corresponding edges, respecting the orientation.<br />
<br />
The 2-sphere can be obtained from a 2-gon with edges labeled $a,\bar a$.<br />
<br />
Each non-orientable surface of genus $h$ can be obtained from a $2h$-gon with edges labeled $a_1,a_1,\ldots,a_h,a_h$.<br />
<br />
{|<br />
|<br />
[[Image:Polygon_construction.png|frame|The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.]]<br />
|<br />
[[Image:Polygon_sphere.png|frame|The 2-sphere can be obtained by identifying the edges of a 2-gon.]]<br />
|<br />
[[Image:Polygon_RP2.png|frame|Construction of $\RP^2$.]]<br />
|<br />
[[Image:Polygon_Klein_bottle.png|frame|Construction of the Klein bottle.]]<br />
|}<br />
Reference: {{cite|Massey1977|Section 1.5}}<br />
</wikitex><br />
<br />
=== By gluing handles and crosscaps ===<br />
<wikitex>;<br />
An orientable surface of genus $g$ can be obtained by successively gluing $g$ 1-handles to the 2-sphere such that the embeddings of $S^0\times D^2$ in $S_{g-1}$ are in each case either orientation-preserving or orientation-reversing on both components $\{1\}\times D^2\cong D^2$ and $\{-1\}\times D^2\cong -D^2$.<br />
<br />
A non-orientable surface of genus $h$ can be obtained by gluing $h$ ''crosscaps'' to $S^2$. For this, embed $D^2$ in $S^2$ (or $R_{h-1}$ from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary $S^1$. The result of attaching a non-orientable handle to $S^2$ or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.<br />
<br />
Reference: {{cite|Hirsch1994|Section 9.1}}<br />
</wikitex><br />
<br />
=== By branched coverings ===<br />
<wikitex>;<br />
Every orientable surface of genus $g$ can be obtained as the branched double covering of the 2-sphere with $2g+2$ branching points.<br />
</wikitex><br />
<br />
=== As complex curves ===<br />
<wikitex>;<br />
A smooth, irreducible, plane, complex projective-algebraic curve of order $d$ (i.&thinsp;e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree $d$ in $\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus<br />
$$<br />
g = \frac{(d-1)(d-2)}2.<br />
$$<br />
<br />
Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.<br />
<br />
Reference: {{cite|Brieskorn&Knörrer1986|Lemma 9.2.1}}<br />
</wikitex><br />
<br />
== Properties ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
Let $S_g$ denote an oriented surface of genus $g$.<br />
<br />
* By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, $2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps.<br />
<br />
* Therefore, the homology groups with any coefficients $G$ are given by $H_0\cong G$, $H_1\cong G^{2g}$, $H_2\cong G$.<br />
<br />
* The integral cohomology ring is completely determined by the intersection form on $H^1$, which is necessarily isomorphic to the hyperbolic form. A basis for $H^1(S_g)$, for which the intersection form is the standard hyperbolic form, is given by the loops $(a_1,b_1,\ldots,a_g,b_g)$ in the polygon construction above.<br />
<br />
* Homotopy groups<br />
** For the homotopy group of $S^2$, see the article about [[sphere|spheres]].<br />
** All surfaces of higher genus $g\geq1$ are aspherical with fundamental group $\pi_1(S_g,*)\cong \langle a_1,b_1,\ldots, a_g,b_g \mid a_1b_1\cdots a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2$.<br />
<br />
* All orientable surfaces can be embedded in $\Rr^3$.<br />
<br />
* Every surface can be given a complex structure. Together with the complex structure, it is a [[wikipedia:Riemann surface|Riemann surface]]<br />
<br />
* Characteristic classes<br />
** All Stiefel-Whitney classes vanish.<br />
** All Pontrjagin classes vanish.<br />
** The Euler characteristic is $2-2g$.<br />
** Given a complex structure on the surface, the first Chern class is equal to the Euler class.<br />
<br />
* All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in $\Rr^3$ has constant positive curvature, the torus can be given a flat metric, and all surfaces of higher genus admit metrics of constant negative curvature.<br />
<br />
* All surfaces are smoothly [[amphicheiral]].<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
Let $R_h$ denote an non-orientable surface of genus $h$.<br />
<br />
* Again, each non-orientable surface has a cell decomposition with one 0-cell, $h$ 1-cells and one 2-cell. The differential $C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$ with respect to the basis given by the loops the loops $(a_1,\ldots,a_h)$ in the polygon construction is the matrix $(2,\ldots, 2)$.<br />
<br />
* Therefore, the integral homology groups are isomorphic to $H_0\cong\Zz$, $H_1\cong \Zz^{h-1}\oplus \Zz/2$, $H_2\cong 0$.<br />
<br />
* The mod-2 homology groups are $H_0=\Zz/2$, $H_1\cong(\Zz/2)^h$, $H_2=\Zz/2$ since the chain complex above is acyclic mod 2. The intersection form on $H_1(R_h;\Zz/2)$ with respect to this cellular basis is given by the identity matrix.<br />
<br />
* Homotopy groups<br />
** The fundamental group of $R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$.<br />
** The orientation double covering of $R_h$ is $S_{h-1}$. Therefore, all higher homotopy groups of $\RP^2$ equal those of $S^2$, and the non-orientable surfaces of higher genus are aspherical.<br />
<br />
* The orientation double covering also determines the curvature properties: $\RP^2$ can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.<br />
<br />
* Non-orientable surfaces cannot be embedded in $\Rr^3$.<br />
<br />
* Characteristic classes<br />
** The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism $\pi_1(R_h,*)\to\Zz/2$ which maps each generator $a_1,\ldots,a_h$ to the generator of $\Zz/2$. The second Stiefel-Whitney class is zero if $h$ is even and the non-zero element of $H^2(R_h;\Zz)=\Zz/2$ if $h$ is odd.<br />
** The Euler characteristic is $2-h$.<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Manifolds]]<br />
<!--[[Category:Dimension 2]] --><br />
[[Category:Aspherical manifolds]]<br />
[[Category:Constant curvature]]</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/2-manifolds2-manifolds2010-09-16T00:33:00Z<p>Daniel Müllner: </p>
<hr />
<div>A surface is a synonym for a 2-dimensional manifold. Also complex 2-dimensional (real 4-dimensional) complex manifolds are called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections [[#Surfaces with boundary|2]] and [[#Classification|3]]), surfaces without boundary are considered.<br />
<br />
== First construction: connected sum ==<br />
<wikitex>;<br />
All orientable surfaces are homeomorphic to the connected sum of $g$ [[Torus|tori]] $T^2$ ($g\geq 0$). The case $g=0$ refers to the 2-[[sphere]] $S^2$. All non-orientable surfaces are homeomorphic to the connected sum of $h$ [[real projective spaces]] $\RP^2$ ($h\geq 1$). For example, the ''Klein bottle'' is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2$. The numbers $g$ and $h$ are called the ''genus'' of the orientable/non-orientable surface.<br />
<br />
[[Image:Surfaces.png|left|frame|A 2-sphere (genus 0), a torus (genus 1) and an orientable surface of higher genus]]<br />
[[Image:Boy surface.jpg|right|thumb|310px|The Boy surface, an immersion of $\RP^2$ in $\Rr^3$. This steel sculpture stands in front of the [http://www.mfo.de Oberwolfach Institute].]]<br />
[[Image:Klein_bottle.png|left|frame|A Klein bottle (non-orientable, genus 2) immersed in $\Rr^3$]]{{-|left}}<br />
<br />
The connected sum of $\RP^2$ with $T^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$.<br />
</wikitex><br />
<br />
== Surfaces with boundary ==<br />
The boundary of a surface is a disjoint union (possibly empty) of circles. Surfaces with boundary can be constructed as boundaryless surfaces with open 2-disks removed.<br />
<br />
== Classification ==<br />
<wikitex>;<br />
Compact, connected surfaces are classified by orientability (yes/no), the number of boundary components (a nonnegative integer) and the genus after filling the bounday circles by disks (an integer $\geq 0$ in the orientable case, $\geq 1$ in the non-orientable case). Instead of the genus, also e.&thinsp;g. the Euler characteristic can be used in the classification. The classifications up to homotopy equivalence, homeomorphism, PL-equivalence and diffeomorphism coincide.<br />
<br />
References: {{cite|Ahlfors&Sario1960|Thm. 1.42A, 1&sect;8}}, {{cite|Hirsch1994|Thm. 9.3.11}}, {{cite|Moise1977|Thm. 8.5, Thm. 22.9}}{{-}}<br />
</wikitex><br />
<br />
== More constructions ==<br />
<br />
=== By polygons ===<br />
<wikitex>;<br />
Each orientable surface of genus $g>0$ can be constructed by identifying pairs of edges in a regular polygon with $4g$ sides. Label the edges by the sequence $a_1,b_1,\bar a_1,\bar b_1,\ldots,a_g,b_g,\bar a_g,\bar b_g$. Also orient the edges such that those labeled without an overbar are oriented in one direction (e.&thinsp;g. clockwise) and those with an overbar are oppositely oriented. Now identify corresponding edges, respecting the orientation.<br />
<br />
The 2-sphere can be obtained from a 2-gon with edges labeled $a,\bar a$.<br />
<br />
Each non-orientable surface of genus $h$ can be obtained from a $2h$-gon with edges labeled $a_1,a_1,\ldots,a_h,a_h$.<br />
<br />
{|<br />
|<br />
[[Image:Polygon_construction.png|frame|The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.]]<br />
|<br />
[[Image:Polygon_sphere.png|frame|The 2-sphere can be obtained by identifying the edges of a 2-gon.]]<br />
|<br />
[[Image:Polygon_RP2.png|frame|Construction of $\RP^2$.]]<br />
|<br />
[[Image:Polygon_Klein_bottle.png|frame|Construction of the Klein bottle.]]<br />
|}<br />
Reference: {{cite|Massey1977|Section 1.5}}<br />
</wikitex><br />
<br />
=== By gluing handles and crosscaps ===<br />
<wikitex>;<br />
An orientable surface of genus $g$ can be obtained by successively gluing $g$ 1-handles to the 2-sphere such that the embeddings of $S^0\times D^2$ in $S_{g-1}$ are in each case either orientation-preserving or orientation-reversing on both components $\{1\}\times D^2\cong D^2$ and $\{-1\}\times D^2\cong -D^2$.<br />
<br />
An non-orientable surface of genus $h$ can be obtained by gluing $h$ ''crosscaps'' to $S^2$. For this, embed $D^2$ in $S^2$ (or $R_{h-1}$ from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary $S^1$. The result of attaching a non-orientable handle to $S^2$ or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.<br />
<br />
Reference: {{cite|Hirsch1994|Section 9.1}}<br />
</wikitex><br />
<br />
=== By branched coverings ===<br />
<wikitex>;<br />
Every orientable surface of genus $g$ can be obtained as the branched double covering of the 2-sphere with $2g+2$ branching points.<br />
</wikitex><br />
<br />
=== As complex curves ===<br />
<wikitex>;<br />
A smooth, irreducible, plane, complex projective-algebraic curve of order $d$ (i.&thinsp;e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree $d$ in $\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus<br />
$$<br />
g = \frac{(d-1)(d-2)}2.<br />
$$<br />
<br />
Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.<br />
<br />
Reference: {{cite|Brieskorn&Knörrer1986|Lemma 9.2.1}}<br />
</wikitex><br />
<br />
== Properties ==<br />
=== Orientable surfaces ===<br />
<wikitex>;<br />
Let $S_g$ denote an oriented surface of genus $g$.<br />
<br />
* By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, $2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps.<br />
<br />
* Therefore, the homology groups with any coefficients $G$ are given by $H_0\cong G$, $H_1\cong G^{2g}$, $H_2\cong G$.<br />
<br />
* The integral cohomology ring is completely determined by the intersection form on $H^1$, which is necessarily isomorphic to the hyperbolic form. A basis for $H^1(S_g)$, for which the intersection form is the standard hyperbolic form, is given by the loops $(a_1,b_1,\ldots,a_g,b_g)$ in the polygon construction above.<br />
<br />
* Homotopy groups<br />
** For the homotopy group of $S^2$, see the article about [[sphere|spheres]].<br />
** All surfaces of higher genus $g\geq1$ are aspherical with fundamental group $\pi_1(S_g,*)\cong \langle a_1,b_1,\ldots, a_g,b_g \mid a_1b_1\cdots a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2$.<br />
<br />
* All orientable surfaces can be embedded in $\Rr^3$.<br />
<br />
* Every surface can be given a complex structure. Together with the complex structure, it is a [[wikipedia:Riemann surface|Riemann surface]]<br />
<br />
* Characteristic classes<br />
** All Stiefel-Whitney classes vanish.<br />
** All Pontrjagin classes vanish.<br />
** The Euler characteristic is $2-2g$.<br />
** Given a complex structure on the surface, the first Chern class is equal to the Euler class.<br />
<br />
* All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in $\Rr^3$ has constant positive curvature, the torus can be given a flat metric, and all surfaces of higher genus admit metrics of constant negative curvature.<br />
<br />
* All surfaces are smoothly [[amphicheiral]].<br />
</wikitex><br />
<br />
=== Non-orientable surfaces ===<br />
<wikitex>;<br />
Let $R_h$ denote an non-orientable surface of genus $h$.<br />
<br />
* Again, each non-orientable surface has a cell decomposition with one 0-cell, $h$ 1-cells and one 2-cell. The differential $C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$ with respect to the basis given by the loops the loops $(a_1,\ldots,a_h)$ in the polygon construction is the matrix $(2,\ldots, 2)$.<br />
<br />
* Therefore, the integral homology groups are isomorphic to $H_0\cong\Zz$, $H_1\cong \Zz^{h-1}\oplus \Zz/2$, $H_2\cong 0$.<br />
<br />
* The mod-2 homology groups are $H_0=\Zz/2$, $H_1\cong(\Zz/2)^h$, $H_2=\Zz/2$ since the chain complex above is acyclic mod 2. The intersection form on $H_1(R_h;\Zz/2)$ with respect to this cellular basis is given by the identity matrix.<br />
<br />
* Homotopy groups<br />
** The fundamental group of $R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$.<br />
** The orientation double covering of $R_h$ is $S_{h-1}$. Therefore, all higher homotopy groups of $\RP^2$ equal those of $S^2$, and the non-orientable surfaces of higher genus are aspherical.<br />
<br />
* The orientation double covering also determines the curvature properties: $\RP^2$ can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.<br />
<br />
* Non-orientable surfaces cannot be embedded in $\Rr^3$.<br />
<br />
* Characteristic classes<br />
** The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism $\pi_1(R_h,*)\to\Zz/2$ which maps each generator $a_1,\ldots,a_h$ to the generator of $\Zz/2$. The second Stiefel-Whitney class is zero if $h$ is even and the non-zero element of $H^2(R_h;\Zz)=\Zz/2$ if $h$ is odd.<br />
** The Euler characteristic is $2-h$.<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Manifolds]]<br />
<!--[[Category:Dimension 2]] --><br />
[[Category:Aspherical manifolds]]<br />
[[Category:Constant curvature]]</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/AmphicheiralAmphicheiral2009-11-21T18:48:38Z<p>Daniel Müllner: Redirected page to Chirality</p>
<hr />
<div>#REDIRECT [[Chirality]]</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Template:M%C3%BCllner2009Template:Müllner20092009-11-21T18:47:26Z<p>Daniel Müllner: </p>
<hr />
<div>Daniel Müllner, ''[http://dx.doi.org/10.2140/agt.2009.9.2361 Orientation reversal of manifolds''], Algebr. Geom. Topol. '''9''' (2009), no. 4, 2361–2390.<br />
<noinclude>[[Category:Bibliography]]</noinclude></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Chiral_manifoldChiral manifold2009-11-21T18:45:50Z<p>Daniel Müllner: Created page with 'A closed, connected, orientable manifold in one of the categories TOP, PL or DIFF is called ''chiral'' if it does not admit an orientation-reversing automorphism in the respectiv…'</p>
<hr />
<div>A closed, connected, orientable manifold in one of the categories TOP, PL or DIFF is called ''chiral'' if it does not admit an orientation-reversing automorphism in the respective category and ''amphicheiral'' if it does. For the sake of clarity, the category should be indicated by adverbs: e.&thinsp;g. a ''topologically chiral'' manifold does not admit an orientation-reversing self-homeomorphism, whereas a ''smoothly amphicheiral'' manifold is a differentiable manifold which admits an orientation-reversing self-diffeomorphism. <br />
<br />
This definition can be extended by the notion of ''homotopical chirality/amphicheirality'' when homotopy self-equivalences are considered. Chiral manifolds in the strongest sense do not admit self-maps of degree &minus;1; they are called ''strongly chiral'' and ''weakly amphicheiral'' in the opposite case.<br />
<br />
(The words ''amphicheiral'', ''amphichiral'' and ''achiral'' are synonyms. ''Amphicheiral'' is most frequently used in [http://www.ams.org/mathscinet/ MathSciNet].)<br />
<br />
== References ==<br />
<br />
* {{Müllner2009}}<br />
<br />
[[Category:Theory]]</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Main_PageMain Page2009-09-22T13:28:59Z<p>Daniel Müllner: </p>
<hr />
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== Welcome to the Manifold Atlas Project<br>sponsored by the [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics] ==<br />
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== Register for the Manifold Atlas ==<br />
Please '''[[Manifold Atlas:Registration|register]]''' with a verified email address in order to write and discuss in the Manifold Atlas.</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Main_PageMain Page2009-09-22T13:26:10Z<p>Daniel Müllner: </p>
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== Welcome to the Manifold Atlas Project<br>{{MediaWiki:Sponsormessage}} ==<br />
The [[Manifold Atlas:About|Manifold Atlas]] is an '''evolving''' and '''scientifically citable online resource''' for researchers and students interested in [[Manifold Atlas:Definition of “manifold”|manifolds]]. It contains '''four chapters'''.<br />
* '''[[:Category:Manifolds|Manifolds]]:''' pages about '''constructions''' of manifolds and the '''invariants''' and '''properties''' of these manifolds.<br />
* '''[[:Category:Problems|Problems]]:''' pages listing '''open questions''' about manifolds, their status and history.<br />
* '''[[:Category:Theory|Theory]]:''' pages summarising useful '''general facts''' and '''theories''' about manifolds. <br />
* '''[[:Category:History|History]]:''' pages about the '''history''' of the study of manifolds.<br />
You will find the articles listed alphabetically by clicking each chapter heading.<br />
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A '''[[Special:Search|full text search]]''' is available: a quick version lies in the box to the left of screen.<br />
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[[Manifold Atlas:About#Scientific goals and structure|'''Citable pages''']] in the Atlas have been [[Manifold Atlas:Editorial Process|'''refereed''']] and approved by the [[Manifold Atlas:Editorial Board|'''editorial board''']].<br />
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* You will find the articles listed alphabetically by clicking each chapter. <br />
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== General information ==<br />
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* As of September and October 2009 the Atlas [[Manifold Atlas:Diary#Forming the Editorial Board|'''forming its first editorial board''']].<br />
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You can either '''[[Manifold Atlas:Create_a_new page|create a new page]]''' and '''discuss''' or '''modify''' all articles by clicking on <b>'discussion'</b> or <b>'edit'</b> at the top of each page. Don't forget to [[Special:UserLogin|log in]]! <br />
<br />
The [[Manifold Atlas:Community Portal|Community Portal]] is a general discussion forum.<br />
<br />
Read the [[Manifold Atlas:Author information|author information]] for important information about writing articles in the Manifold Atlas.--><br />
<br />
== Register for the Manifold Atlas ==<br />
Please '''[[Manifold Atlas:Registration|register]]''' with a verified email address in order to write and discuss in the Manifold Atlas.</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/SandboxSandbox2009-09-22T13:17:39Z<p>Daniel Müllner: </p>
<hr />
<div>Write here...<br />
<br />
{{Special:ListUsers}}</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/SandboxSandbox2009-09-22T13:17:14Z<p>Daniel Müllner: </p>
<hr />
<div>Write here...<br />
<br />
{{Special:UserList}}</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Main_PageMain Page2009-09-22T12:56:57Z<p>Daniel Müllner: /* Welcome to the Manifold Atlas Projecttext */</p>
<hr />
<div>__NOTOC__<br />
{| style="float:right; margin-left:20px; max-width:20em; border:solid thin black; padding:.5ex;"<br />
|-<br />
|<!-- The 10 --> Recent changes:<br />
<DPL><br />
namespace=Main<br />
nottitlematch=Main_Page|Sandbox|Links<br />
ordermethod=lastedit<br />
order=descending<br />
count=10<br />
</DPL><br />
|-<br />
|}<br />
== Welcome to the Manifold Atlas Project<br>sponsored by the [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics] ==<br />
The [[Manifold Atlas:About|Manifold Atlas]] is an '''evolving''' and '''scientifically citable online resource''' for researchers and students interested in [[Manifold Atlas:Definition of “manifold”|manifolds]]. It contains '''four chapters'''.<br />
* '''[[:Category:Manifolds|Manifolds]]:''' pages about '''constructions''' of manifolds and the '''invariants''' and '''properties''' of these manifolds.<br />
* '''[[:Category:Problems|Problems]]:''' pages listing '''open questions''' about manifolds, their status and history.<br />
* '''[[:Category:Theory|Theory]]:''' pages summarising useful '''general facts''' and '''theories''' about manifolds. <br />
* '''[[:Category:History|History]]:''' pages about the '''history''' of the study of manifolds.<br />
You will find the articles listed alphabetically by clicking each chapter heading.<br />
<br />
A '''[[Special:Search|full text search]]''' is available: a quick version lies in the box to the left of screen.<br />
<br />
[[Manifold Atlas:About#Scientific goals and structure|'''Citable pages''']] in the Atlas have been [[Manifold Atlas:Editorial Process|'''refereed''']] and approved by the [[Manifold Atlas:Editorial Board|'''editorial board''']].<br />
<!--== Search the Manifold Atlas ==<br />
* You will find the articles listed alphabetically by clicking each chapter. <br />
* A '''[[Special:Search|full text search]]''' is available: a quick version lies in the box to the left of screen. --><br />
<br />
== General information ==<br />
* The Manifold Atlas is hosted by the [http://www.hausdorff-research-institute.uni-bonn.de/index Hausdorff Institute for Mathematics].<br />
* As of September and October 2009 the Atlas [[Manifold Atlas:Diary#Forming the Editorial Board|'''forming its first editorial board''']].<br />
* For more information please read '''[[Manifold Atlas:About|about the Atlas]]''' or visit the [[Manifold Atlas:Diary|'''Atlas diary''']] or the [[Manifold Atlas:Community_Portal|'''Community portal''']].<br />
<br />
== Write in the Manifold Atlas ==<br />
* '''[[Manifold Atlas:Create_a_new open-edit page|Create a new open-edit page]]'''.<br />
* '''[[Manifold Atlas:Create_a_new author-based page|Create a new author-based page]]'''.<br />
* '''[[Manifold Atlas:Orientation for authors#Scientific goals and style|Add to or edit ]]''' existing articles.<br />
* '''[[Manifold Atlas:Orientation_for_authors#Writing_on_discussion_pages|Join the discussion]]''' about a page.<br />
<br />
'''[[Manifold Atlas:Author orientation|Author orientation]]''' gives information about writing in the atlas. '''[[Manifold Atlas:User orientation|User orientation]]''' gives information for registered users. <br />
<!--<br />
The '''[[Manifold Atlas:Community Portal|community portal]]''' is a general discussion forum.<br />
<br />
You can either '''[[Manifold Atlas:Create_a_new page|create a new page]]''' and '''discuss''' or '''modify''' all articles by clicking on <b>'discussion'</b> or <b>'edit'</b> at the top of each page. Don't forget to [[Special:UserLogin|log in]]! <br />
<br />
The [[Manifold Atlas:Community Portal|Community Portal]] is a general discussion forum.<br />
<br />
Read the [[Manifold Atlas:Author information|author information]] for important information about writing articles in the Manifold Atlas.--><br />
<br />
== Register for the Manifold Atlas ==<br />
Please '''[[Manifold Atlas:Registration|register]]''' with a verified email address in order to write and discuss in the Manifold Atlas.</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Main_PageMain Page2009-09-22T12:55:59Z<p>Daniel Müllner: </p>
<hr />
<div>__NOTOC__<br />
{| style="float:right; margin-left:20px; max-width:20em; border:solid thin black; padding:.5ex;"<br />
|-<br />
|<!-- The 10 --> Recent changes:<br />
<DPL><br />
namespace=Main<br />
nottitlematch=Main_Page|Sandbox|Links<br />
ordermethod=lastedit<br />
order=descending<br />
count=10<br />
</DPL><br />
|-<br />
|}<br />
== Welcome to the Manifold Atlas Project<br>text ==<br />
The [[Manifold Atlas:About|Manifold Atlas]] is an '''evolving''' and '''scientifically citable online resource''' for researchers and students interested in [[Manifold Atlas:Definition of “manifold”|manifolds]]. It contains '''four chapters'''.<br />
* '''[[:Category:Manifolds|Manifolds]]:''' pages about '''constructions''' of manifolds and the '''invariants''' and '''properties''' of these manifolds.<br />
* '''[[:Category:Problems|Problems]]:''' pages listing '''open questions''' about manifolds, their status and history.<br />
* '''[[:Category:Theory|Theory]]:''' pages summarising useful '''general facts''' and '''theories''' about manifolds. <br />
* '''[[:Category:History|History]]:''' pages about the '''history''' of the study of manifolds.<br />
You will find the articles listed alphabetically by clicking each chapter heading.<br />
<br />
A '''[[Special:Search|full text search]]''' is available: a quick version lies in the box to the left of screen.<br />
<br />
[[Manifold Atlas:About#Scientific goals and structure|'''Citable pages''']] in the Atlas have been [[Manifold Atlas:Editorial Process|'''refereed''']] and approved by the [[Manifold Atlas:Editorial Board|'''editorial board''']].<br />
<!--== Search the Manifold Atlas ==<br />
* You will find the articles listed alphabetically by clicking each chapter. <br />
* A '''[[Special:Search|full text search]]''' is available: a quick version lies in the box to the left of screen. --><br />
== General information ==<br />
* The Manifold Atlas is hosted by the [http://www.hausdorff-research-institute.uni-bonn.de/index Hausdorff Institute for Mathematics].<br />
* As of September and October 2009 the Atlas [[Manifold Atlas:Diary#Forming the Editorial Board|'''forming its first editorial board''']].<br />
* For more information please read '''[[Manifold Atlas:About|about the Atlas]]''' or visit the [[Manifold Atlas:Diary|'''Atlas diary''']] or the [[Manifold Atlas:Community_Portal|'''Community portal''']].<br />
<br />
== Write in the Manifold Atlas ==<br />
* '''[[Manifold Atlas:Create_a_new open-edit page|Create a new open-edit page]]'''.<br />
* '''[[Manifold Atlas:Create_a_new author-based page|Create a new author-based page]]'''.<br />
* '''[[Manifold Atlas:Orientation for authors#Scientific goals and style|Add to or edit ]]''' existing articles.<br />
* '''[[Manifold Atlas:Orientation_for_authors#Writing_on_discussion_pages|Join the discussion]]''' about a page.<br />
<br />
'''[[Manifold Atlas:Author orientation|Author orientation]]''' gives information about writing in the atlas. '''[[Manifold Atlas:User orientation|User orientation]]''' gives information for registered users. <br />
<!--<br />
The '''[[Manifold Atlas:Community Portal|community portal]]''' is a general discussion forum.<br />
<br />
You can either '''[[Manifold Atlas:Create_a_new page|create a new page]]''' and '''discuss''' or '''modify''' all articles by clicking on <b>'discussion'</b> or <b>'edit'</b> at the top of each page. Don't forget to [[Special:UserLogin|log in]]! <br />
<br />
The [[Manifold Atlas:Community Portal|Community Portal]] is a general discussion forum.<br />
<br />
Read the [[Manifold Atlas:Author information|author information]] for important information about writing articles in the Manifold Atlas.--><br />
<br />
== Register for the Manifold Atlas ==<br />
Please '''[[Manifold Atlas:Registration|register]]''' with a verified email address in order to write and discuss in the Manifold Atlas.</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:AboutManifold Atlas:About2009-09-22T12:38:05Z<p>Daniel Müllner: /* “What is a manifold" */</p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
The mission of the Manifold Atlas is to empower and engage topologists and geometers to collect and develop information about [[Manifold Atlas:Definition of “manifold”|manifolds]], in particular [[:Category:Manifolds|constructions and invariants]] and [[:Category:Problems|problems]] but also [[:Category:Theory|general]] and [[:Category:History|historical]] information.<br />
<br />
== “What is a manifold" ==<br />
We use the term manifold broadly to mean any second countable Hausdorff space which is locally Euclidean of a fixed dimension and which may, or may not, be equipped with extra structures: for a precise definition see the [[Manifold Atlas:Definition of “manifold”|'''definition of “manifold"''']].<br />
<br />
== Scientific goals and structure ==<br />
* The Manifold Atlas aims to serve as a journal standard, citable reference for the study of manifolds. <br />
* There are two sorts of pages in the Atlas: evolving pages and static pages. <br />
* '''Evolving pages''' are continually open to improvement and expansion but '''are not strongly scientifically quotable'''. <br />
* '''[[Manifold Atlas:Static pages|Static pages]]''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']]. <br />
* Static pages are instantly recognisable via the suffix '''<tt>/nth Edition</tt>''':<br />
* Static pages are statically preserved as '''scientifically citable documents'''. <br />
<!-- The Manifold Atlas aims to serve as a journal standard quotable reference for the study of manifolds. It contains evolving pages and quotable pages. '''Evolving pages''' pages are continually open to improvement and expansion but '''are not scientifically quotable'''. '''Static pages''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']] and are preserved as '''scientifically quotable documents'''. --><br />
<br />
== Writing in the Manifold Atlas ==<br />
* The Manifold Atlas supports two styles of articles: open-editing articles and author-based articles.<br />
* [[Manifold Atlas:Writing style#Open-editing pages|'''Open-editing articles''']] can be edited openly by any registered user.<br />
* [[Manifold Atlas:Writing style#Author-based pages|'''Author-based articles''']] are written by a single author or team of authors.<br />
* All content in the Manifold Atlas is '''freely available''' on the world wide web as described on the [[Manifold Atlas:User rights|'''user rights page''']].<br />
<br />
== Staff ==<br />
* The managing editor of the Manifold Atlas is [http://www.him.uni-bonn.de/kreck Matthias Kreck]. <br />
* The scientific administrators of the Manifold Atlas are [http://www.dcrowley.net/ Diarmuid Crowley] and [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].<br />
<br />
== Affiliation ==<br />
* The Manifold Atlas is hosted by the [http://www.him.uni-bonn.de Hausdorff Institute for Mathematics] and financed by the [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics].<br />
<br />
== Platform ==<br />
The platform for the Manifold Atlas is [http://www.mediawiki.org MediaWiki]: special local features were developed by [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:AboutManifold Atlas:About2009-09-22T12:36:30Z<p>Daniel Müllner: /* Platform */</p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
The mission of the Manifold Atlas is to empower and engage topologists and geometers to collect and develop information about [[Manifold Atlas:Definition of “manifold”|manifolds]], in particular [[:Category:Manifolds|constructions and invariants]] and [[:Category:Problems|problems]] but also [[:Category:Theory|general]] and [[:Category:History|historical]] information.<br />
<br />
== “What is a manifold" ==<br />
We use the term manifold broadly to mean any second countable Hausdorff space which is locally Euclidean of a fixed dimension and which may, or may not, be equipped with extra structures: for a precise definition see the [[Manifold Atlas:Definition of “manifold”|'''defintion of “manifold"''']].<br />
<br />
== Scientific goals and structure ==<br />
* The Manifold Atlas aims to serve as a journal standard, citable reference for the study of manifolds. <br />
* There are two sorts of pages in the Atlas: evolving pages and static pages. <br />
* '''Evolving pages''' are continually open to improvement and expansion but '''are not strongly scientifically quotable'''. <br />
* '''[[Manifold Atlas:Static pages|Static pages]]''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']]. <br />
* Static pages are instantly recognisable via the suffix '''<tt>/nth Edition</tt>''':<br />
* Static pages are statically preserved as '''scientifically citable documents'''. <br />
<!-- The Manifold Atlas aims to serve as a journal standard quotable reference for the study of manifolds. It contains evolving pages and quotable pages. '''Evolving pages''' pages are continually open to improvement and expansion but '''are not scientifically quotable'''. '''Static pages''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']] and are preserved as '''scientifically quotable documents'''. --><br />
<br />
== Writing in the Manifold Atlas ==<br />
* The Manifold Atlas supports two styles of articles: open-editing articles and author-based articles.<br />
* [[Manifold Atlas:Writing style#Open-editing pages|'''Open-editing articles''']] can be edited openly by any registered user.<br />
* [[Manifold Atlas:Writing style#Author-based pages|'''Author-based articles''']] are written by a single author or team of authors.<br />
* All content in the Manifold Atlas is '''freely available''' on the world wide web as described on the [[Manifold Atlas:User rights|'''user rights page''']].<br />
<br />
== Staff ==<br />
* The managing editor of the Manifold Atlas is [http://www.him.uni-bonn.de/kreck Matthias Kreck]. <br />
* The scientific administrators of the Manifold Atlas are [http://www.dcrowley.net/ Diarmuid Crowley] and [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].<br />
<br />
== Affiliation ==<br />
* The Manifold Atlas is hosted by the [http://www.him.uni-bonn.de Hausdorff Institute for Mathematics] and financed by the [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics].<br />
<br />
== Platform ==<br />
The platform for the Manifold Atlas is [http://www.mediawiki.org MediaWiki]: special local features were developed by [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:AboutManifold Atlas:About2009-09-22T12:36:16Z<p>Daniel Müllner: /* Platform */</p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
The mission of the Manifold Atlas is to empower and engage topologists and geometers to collect and develop information about [[Manifold Atlas:Definition of “manifold”|manifolds]], in particular [[:Category:Manifolds|constructions and invariants]] and [[:Category:Problems|problems]] but also [[:Category:Theory|general]] and [[:Category:History|historical]] information.<br />
<br />
== “What is a manifold" ==<br />
We use the term manifold broadly to mean any second countable Hausdorff space which is locally Euclidean of a fixed dimension and which may, or may not, be equipped with extra structures: for a precise definition see the [[Manifold Atlas:Definition of “manifold”|'''defintion of “manifold"''']].<br />
<br />
== Scientific goals and structure ==<br />
* The Manifold Atlas aims to serve as a journal standard, citable reference for the study of manifolds. <br />
* There are two sorts of pages in the Atlas: evolving pages and static pages. <br />
* '''Evolving pages''' are continually open to improvement and expansion but '''are not strongly scientifically quotable'''. <br />
* '''[[Manifold Atlas:Static pages|Static pages]]''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']]. <br />
* Static pages are instantly recognisable via the suffix '''<tt>/nth Edition</tt>''':<br />
* Static pages are statically preserved as '''scientifically citable documents'''. <br />
<!-- The Manifold Atlas aims to serve as a journal standard quotable reference for the study of manifolds. It contains evolving pages and quotable pages. '''Evolving pages''' pages are continually open to improvement and expansion but '''are not scientifically quotable'''. '''Static pages''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']] and are preserved as '''scientifically quotable documents'''. --><br />
<br />
== Writing in the Manifold Atlas ==<br />
* The Manifold Atlas supports two styles of articles: open-editing articles and author-based articles.<br />
* [[Manifold Atlas:Writing style#Open-editing pages|'''Open-editing articles''']] can be edited openly by any registered user.<br />
* [[Manifold Atlas:Writing style#Author-based pages|'''Author-based articles''']] are written by a single author or team of authors.<br />
* All content in the Manifold Atlas is '''freely available''' on the world wide web as described on the [[Manifold Atlas:User rights|'''user rights page''']].<br />
<br />
== Staff ==<br />
* The managing editor of the Manifold Atlas is [http://www.him.uni-bonn.de/kreck Matthias Kreck]. <br />
* The scientific administrators of the Manifold Atlas are [http://www.dcrowley.net/ Diarmuid Crowley] and [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].<br />
<br />
== Affiliation ==<br />
* The Manifold Atlas is hosted by the [http://www.him.uni-bonn.de Hausdorff Institute for Mathematics] and financed by the [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics].<br />
<br />
== Platform ==<br />
The platform for the Manifold Atlas is [http:www.mediawiki.org MediaWiki]: special local features were developed by [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:AboutManifold Atlas:About2009-09-22T12:35:25Z<p>Daniel Müllner: /* Affiliation */</p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
The mission of the Manifold Atlas is to empower and engage topologists and geometers to collect and develop information about [[Manifold Atlas:Definition of “manifold”|manifolds]], in particular [[:Category:Manifolds|constructions and invariants]] and [[:Category:Problems|problems]] but also [[:Category:Theory|general]] and [[:Category:History|historical]] information.<br />
<br />
== “What is a manifold" ==<br />
We use the term manifold broadly to mean any second countable Hausdorff space which is locally Euclidean of a fixed dimension and which may, or may not, be equipped with extra structures: for a precise definition see the [[Manifold Atlas:Definition of “manifold”|'''defintion of “manifold"''']].<br />
<br />
== Scientific goals and structure ==<br />
* The Manifold Atlas aims to serve as a journal standard, citable reference for the study of manifolds. <br />
* There are two sorts of pages in the Atlas: evolving pages and static pages. <br />
* '''Evolving pages''' are continually open to improvement and expansion but '''are not strongly scientifically quotable'''. <br />
* '''[[Manifold Atlas:Static pages|Static pages]]''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']]. <br />
* Static pages are instantly recognisable via the suffix '''<tt>/nth Edition</tt>''':<br />
* Static pages are statically preserved as '''scientifically citable documents'''. <br />
<!-- The Manifold Atlas aims to serve as a journal standard quotable reference for the study of manifolds. It contains evolving pages and quotable pages. '''Evolving pages''' pages are continually open to improvement and expansion but '''are not scientifically quotable'''. '''Static pages''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']] and are preserved as '''scientifically quotable documents'''. --><br />
<br />
== Writing in the Manifold Atlas ==<br />
* The Manifold Atlas supports two styles of articles: open-editing articles and author-based articles.<br />
* [[Manifold Atlas:Writing style#Open-editing pages|'''Open-editing articles''']] can be edited openly by any registered user.<br />
* [[Manifold Atlas:Writing style#Author-based pages|'''Author-based articles''']] are written by a single author or team of authors.<br />
* All content in the Manifold Atlas is '''freely available''' on the world wide web as described on the [[Manifold Atlas:User rights|'''user rights page''']].<br />
<br />
== Staff ==<br />
* The managing editor of the Manifold Atlas is [http://www.him.uni-bonn.de/kreck Matthias Kreck]. <br />
* The scientific administrators of the Manifold Atlas are [http://www.dcrowley.net/ Diarmuid Crowley] and [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].<br />
<br />
== Affiliation ==<br />
* The Manifold Atlas is hosted by the [http://www.him.uni-bonn.de Hausdorff Institute for Mathematics] and financed by the [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics].<br />
<br />
== Platform ==<br />
The platform for the Manifold Atlas is [[metawikipedia:Main_Page|MediaWiki]]: special local features were developed by [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:AboutManifold Atlas:About2009-09-22T12:34:14Z<p>Daniel Müllner: /* Affiliation */</p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
The mission of the Manifold Atlas is to empower and engage topologists and geometers to collect and develop information about [[Manifold Atlas:Definition of “manifold”|manifolds]], in particular [[:Category:Manifolds|constructions and invariants]] and [[:Category:Problems|problems]] but also [[:Category:Theory|general]] and [[:Category:History|historical]] information.<br />
<br />
== “What is a manifold" ==<br />
We use the term manifold broadly to mean any second countable Hausdorff space which is locally Euclidean of a fixed dimension and which may, or may not, be equipped with extra structures: for a precise definition see the [[Manifold Atlas:Definition of “manifold”|'''defintion of “manifold"''']].<br />
<br />
== Scientific goals and structure ==<br />
* The Manifold Atlas aims to serve as a journal standard, citable reference for the study of manifolds. <br />
* There are two sorts of pages in the Atlas: evolving pages and static pages. <br />
* '''Evolving pages''' are continually open to improvement and expansion but '''are not strongly scientifically quotable'''. <br />
* '''[[Manifold Atlas:Static pages|Static pages]]''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']]. <br />
* Static pages are instantly recognisable via the suffix '''<tt>/nth Edition</tt>''':<br />
* Static pages are statically preserved as '''scientifically citable documents'''. <br />
<!-- The Manifold Atlas aims to serve as a journal standard quotable reference for the study of manifolds. It contains evolving pages and quotable pages. '''Evolving pages''' pages are continually open to improvement and expansion but '''are not scientifically quotable'''. '''Static pages''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']] and are preserved as '''scientifically quotable documents'''. --><br />
<br />
== Writing in the Manifold Atlas ==<br />
* The Manifold Atlas supports two styles of articles: open-editing articles and author-based articles.<br />
* [[Manifold Atlas:Writing style#Open-editing pages|'''Open-editing articles''']] can be edited openly by any registered user.<br />
* [[Manifold Atlas:Writing style#Author-based pages|'''Author-based articles''']] are written by a single author or team of authors.<br />
* All content in the Manifold Atlas is '''freely available''' on the world wide web as described on the [[Manifold Atlas:User rights|'''user rights page''']].<br />
<br />
== Staff ==<br />
* The managing editor of the Manifold Atlas is [http://www.him.uni-bonn.de/kreck Matthias Kreck]. <br />
* The scientific administrators of the Manifold Atlas are [http://www.dcrowley.net/ Diarmuid Crowley] and [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].<br />
<br />
== Affiliation ==<br />
* The Manifold Atlas is hosted by the [http://www.him.uni-bonn.de/index Hausdorff Institute for Mathematics] and financed by the [http://www.hcm.uni-bonn.de Hausdorff Center for Mathematics].<br />
<br />
== Platform ==<br />
The platform for the Manifold Atlas is [[metawikipedia:Main_Page|MediaWiki]]: special local features were developed by [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:AboutManifold Atlas:About2009-09-22T12:32:30Z<p>Daniel Müllner: /* Affiliation */</p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}}<br />
The mission of the Manifold Atlas is to empower and engage topologists and geometers to collect and develop information about [[Manifold Atlas:Definition of “manifold”|manifolds]], in particular [[:Category:Manifolds|constructions and invariants]] and [[:Category:Problems|problems]] but also [[:Category:Theory|general]] and [[:Category:History|historical]] information.<br />
<br />
== “What is a manifold" ==<br />
We use the term manifold broadly to mean any second countable Hausdorff space which is locally Euclidean of a fixed dimension and which may, or may not, be equipped with extra structures: for a precise definition see the [[Manifold Atlas:Definition of “manifold”|'''defintion of “manifold"''']].<br />
<br />
== Scientific goals and structure ==<br />
* The Manifold Atlas aims to serve as a journal standard, citable reference for the study of manifolds. <br />
* There are two sorts of pages in the Atlas: evolving pages and static pages. <br />
* '''Evolving pages''' are continually open to improvement and expansion but '''are not strongly scientifically quotable'''. <br />
* '''[[Manifold Atlas:Static pages|Static pages]]''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']]. <br />
* Static pages are instantly recognisable via the suffix '''<tt>/nth Edition</tt>''':<br />
* Static pages are statically preserved as '''scientifically citable documents'''. <br />
<!-- The Manifold Atlas aims to serve as a journal standard quotable reference for the study of manifolds. It contains evolving pages and quotable pages. '''Evolving pages''' pages are continually open to improvement and expansion but '''are not scientifically quotable'''. '''Static pages''' have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']] and are preserved as '''scientifically quotable documents'''. --><br />
<br />
== Writing in the Manifold Atlas ==<br />
* The Manifold Atlas supports two styles of articles: open-editing articles and author-based articles.<br />
* [[Manifold Atlas:Writing style#Open-editing pages|'''Open-editing articles''']] can be edited openly by any registered user.<br />
* [[Manifold Atlas:Writing style#Author-based pages|'''Author-based articles''']] are written by a single author or team of authors.<br />
* All content in the Manifold Atlas is '''freely available''' on the world wide web as described on the [[Manifold Atlas:User rights|'''user rights page''']].<br />
<br />
== Staff ==<br />
* The managing editor of the Manifold Atlas is [http://www.him.uni-bonn.de/kreck Matthias Kreck]. <br />
* The scientific administrators of the Manifold Atlas are [http://www.dcrowley.net/ Diarmuid Crowley] and [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].<br />
<br />
== Affiliation ==<br />
* The Manifold Atlas is hosted by the [http://www.hausdorff-research-institute.uni-bonn.de/index Hausdorff Institute for Mathematics].<br />
<br />
== Platform ==<br />
The platform for the Manifold Atlas is [[metawikipedia:Main_Page|MediaWiki]]: special local features were developed by [http://www.math.uni-bonn.de/people/muellner/ Daniel Müllner].</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:Evolving_pages_and_static_pagesManifold Atlas:Evolving pages and static pages2009-09-22T12:31:06Z<p>Daniel Müllner: /* What will be up-dated */</p>
<hr />
<div>This page describes the special features of static pages in the Manifold Atlas.<br />
<br />
An essential function of the Manifold Atlas is to serve as a journal standard, citable reference for the study of manifolds. The static pages of the Atlas realise this function: whereas the [[evolving pages]] are for Wikipedia style dynamic development of knowledge.<br />
<br />
* Static pages have been approved by the '''[[Manifold Atlas:Editorial Board|editorial board]]''' via a rigorous [[Manifold Atlas:Editorial Process|'''editorial process''']]. <br />
* Static pages are instantly recognisable by<br />
** the blue [[MediaWiki:Approved-Static|approval message]] they carry in their header<br />
** the suffix '''<tt>/nth Edition</tt>''' in their title.<br />
* Static pages will be preserved as '''scientifically citable documents''' in the strong sense that their hard-copy text is preserved for precise reference.<br />
<br />
== What is preserved? ==<br />
* As a citable scientific document, a static article should be viewed with “hard-copy vision": that is the content of this article is what you would have if you printed it out: '''the hyperlinks are not part of the text'''.<br />
* Any attached PDF files are part of the the text and will be preserved as accompanying remarks.<br />
<br />
== What will be up-dated ==<br />
* The administrators of the Atlas will consider any appropriate up-dates of static pages which do not effect their “hard-copy form".<br />
* As a web-page, as static-page still contains links to the Atlas and the world wide web. <br />
* The administrators of the Atlas will up-date these links and also add new categories to the page as appropriate.<br />
<br />
== What can be change? ==<br />
* Also, just as journals typeset their articles, minor type-setting adjustments can occur to the “hard-copy view" of static pages.</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:Editorial_PolicyManifold Atlas:Editorial Policy2009-09-22T12:23:31Z<p>Daniel Müllner: /* Editorial guidelines */</p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}} <br />
* This page describes the proposed editorial process and guidelines in the Manifold Atlas. <br />
<br />
* If you have any comments, questions or criticisms, please write them on the discussion page.<br />
<br />
* The refereeing process is organised and overseen by the [[Manifold Atlas: Editorial Board|'''editorial board''']].<br />
<br />
== The editorial process ==<br />
* After a page reaches [[Manifold Atlas:Editorial Process#Editorial guidelines|'''maturity''']], the editorial board will organise for it to be reviewed by at least two members of the board.<br />
<br />
* When an evolving page is approved, a static, citable version of that page bearing a '''[[MediaWiki:Approved-Static|static editorial message]]''' is created which links back to the evolving page.<br />
<br />
* Evolving pages which have been approved by the editorial board bear an '''[[MediaWiki:Approved-Dynamic|editorial message]]''' which links to the corresponding static version of the page.<br />
<br />
== Editorial guidelines ==<br />
* An article is called '''mature''' when it satisfies the criteria of correctness, completeness and clarity.<br />
<br />
* '''Correctness''': the manifold Atlas is a reliable reference for research about manifolds. Therefore an essential criteria for an article is that all information in it is correct.<br />
<br />
* '''Completeness''': since the Manifold Atlas is a reference source, articles on a given subject should not ignore important aspects of that subject. In particular, an mature article will '''thoroughly reference''' the relevant mathematical literature. <br />
<br />
* '''Clarity''': it is important that information is readily and clearly available for readers. This is especially important for the [[Manifold Atlas:Structure of a Manifolds page#Invariants|invariants]] section of articles in the Manifolds chapter but holds for all parts of all pages.</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:SidebarMediaWiki:Sidebar2009-09-22T12:15:46Z<p>Daniel Müllner: </p>
<hr />
<div>* navigation<br />
** mainpage|mainpage-description<br />
** Category:Manifolds|Manifolds<br />
** Category:Problems|Problems<br />
** Category:Theory|Theory<br />
** Category:History|chapter:History<br />
** Manifold Atlas:Bibliography|Bibliography<br />
** Manifold Atlas:Writing tools|Writing tools<br />
** portal-url|portal<br />
<!--** Manifold Atlas:Diary|Diary--><br />
<!--** Links to the web|Links to the web--><br />
<!--** currentevents-url|currentevents--><br />
<!--** recentchanges-url|recentchanges--><br />
<!--** randompage-url|randompage--><br />
** helppage|help<br />
* SEARCH<br />
* TOOLBOX<br />
* LANGUAGES</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:SidebarMediaWiki:Sidebar2009-09-22T12:15:19Z<p>Daniel Müllner: </p>
<hr />
<div>Hosted by: HIM<br />
* something<br />
something<br />
* navigation<br />
** mainpage|mainpage-description<br />
** Category:Manifolds|Manifolds<br />
** Category:Problems|Problems<br />
** Category:Theory|Theory<br />
** Category:History|chapter:History<br />
** Manifold Atlas:Bibliography|Bibliography<br />
** Manifold Atlas:Writing tools|Writing tools<br />
** portal-url|portal<br />
<!--** Manifold Atlas:Diary|Diary--><br />
<!--** Links to the web|Links to the web--><br />
<!--** currentevents-url|currentevents--><br />
<!--** recentchanges-url|recentchanges--><br />
<!--** randompage-url|randompage--><br />
** helppage|help<br />
* SEARCH<br />
* TOOLBOX<br />
* LANGUAGES<br />
abcde</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:SidebarMediaWiki:Sidebar2009-09-22T12:11:50Z<p>Daniel Müllner: </p>
<hr />
<div>Hosted by: HIM<br />
* something<br />
something<br />
* navigation<br />
** mainpage|mainpage-description<br />
** Category:Manifolds|Manifolds<br />
** Category:Problems|Problems<br />
** Category:Theory|Theory<br />
** Category:History|chapter:History<br />
** Manifold Atlas:Bibliography|Bibliography<br />
** Manifold Atlas:Writing tools|Writing tools<br />
** portal-url|portal<br />
<!--** Manifold Atlas:Diary|Diary--><br />
<!--** Links to the web|Links to the web--><br />
<!--** currentevents-url|currentevents--><br />
<!--** recentchanges-url|recentchanges--><br />
<!--** randompage-url|randompage--><br />
** helppage|help<br />
* SEARCH<br />
* TOOLBOX<br />
* LANGUAGES</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:SidebarMediaWiki:Sidebar2009-09-22T12:11:28Z<p>Daniel Müllner: </p>
<hr />
<div>Hosted by: HIM<br />
* something<br />
* navigation<br />
** mainpage|mainpage-description<br />
** Category:Manifolds|Manifolds<br />
** Category:Problems|Problems<br />
** Category:Theory|Theory<br />
** Category:History|chapter:History<br />
** Manifold Atlas:Bibliography|Bibliography<br />
** Manifold Atlas:Writing tools|Writing tools<br />
** portal-url|portal<br />
<!--** Manifold Atlas:Diary|Diary--><br />
<!--** Links to the web|Links to the web--><br />
<!--** currentevents-url|currentevents--><br />
<!--** recentchanges-url|recentchanges--><br />
<!--** randompage-url|randompage--><br />
** helppage|help<br />
* SEARCH<br />
* TOOLBOX<br />
* LANGUAGES</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:SidebarMediaWiki:Sidebar2009-09-22T12:10:28Z<p>Daniel Müllner: </p>
<hr />
<div>Hosted by: HIM<br />
* navigation<br />
** mainpage|mainpage-description<br />
** Category:Manifolds|Manifolds<br />
** Category:Problems|Problems<br />
** Category:Theory|Theory<br />
** Category:History|chapter:History<br />
** Manifold Atlas:Bibliography|Bibliography<br />
** Manifold Atlas:Writing tools|Writing tools<br />
** portal-url|portal<br />
<!--** Manifold Atlas:Diary|Diary--><br />
<!--** Links to the web|Links to the web--><br />
<!--** currentevents-url|currentevents--><br />
<!--** recentchanges-url|recentchanges--><br />
<!--** randompage-url|randompage--><br />
** helppage|help<br />
* SEARCH<br />
* TOOLBOX<br />
* LANGUAGES</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Manifold_Atlas:DiaryManifold Atlas:Diary2009-09-22T10:24:34Z<p>Daniel Müllner: </p>
<hr />
<div>{{DISPLAYTITLE:{{NAMESPACE}}: {{PAGENAME}}}} <br />
This page records interesting events in the history of the Atlas as well as its current phase of development.<br />
<br />
== Forming the Editorial Board - September and October 2009 ==<br />
In September and October 2009 the Manifold Atlas is forming it's editorial board. Prospective editors are invited to [[Manifold Atlas:Registration|register]] in the Atlas, to browse its current pages and to begin using it: in particular by [[Manifold Atlas:Author orientation|writing]] and [[Manifold Atlas:User orientation#Writing on discussion pages|discussing]] the first articles of the Atlas. Prospective editors may also wish to:<br />
* review the [[Manifold Atlas:Editorial Process|proposed editorial process]],<br />
* contribute to the [[Manifold Atlas:Community_Portal|community portal]],<br />
* add to the discussion on [{{fullurl:Special:PrefixIndex}}?namespace=4 other project pages].<br />
<br />
Prospective editors may of course wish to remain as Manifold Atlas users even if they would not like to take on the responsibilities of being an editor.<br />
<br />
== First steps: idea and the platform - March to September 2009 ==<br />
From March 2009 until September 2009, [[User:Matthias Kreck|Matthias Kreck]], [[User:Daniel Müllner|Daniel Müllner]], [[User:Christian Ausoni|Christian Ausoni]] and [[User:Diarmuid Crowley|Diarmuid Crowley]] began the process of creating the Atlas. A conceptual challenge which arose was how to combine the flexibility and openness of the world wide web and a Wikipedia style of knowledge creation with the desire to have scientifically reliable and citable information. The solution was two fold: to have both [[Manifold Atlas:Writing style#Open-editing pages|open-editing]] and [[Manifold Atlas:Writing style#Author-based pages|author-based]] options for page creation and development and to have [[Manifold Atlas:About#Scientific goals and structure|citable]], editorially approved versions of pages which are statically preserved for reference as well as the usual [[Manifold Atlas:About#Scientific goals and structure|evolving pages]] in the Wikipedia style.<br />
<br />
As of writing the hope is that the above concepts will combine the best aspects of flexibility and openness with scientific reliablity.</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Template:BibitemTemplate:Bibitem2009-09-21T12:41:48Z<p>Daniel Müllner: </p>
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<div><span id="{{anchorencode:{{{1}}}}}">[[Template:{{{1}}}|&#91;{{{1}}}&#93;]] {{{{{1}}}}}</span></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Template:BibitemTemplate:Bibitem2009-09-21T12:41:07Z<p>Daniel Müllner: </p>
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<div><span id="{{anchorencode:{{{1}}}}}">[[{{{1}}}|&#91;{{{1}}}&#93;]] {{{{{1}}}}}</span></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Template:BibitemTemplate:Bibitem2009-09-21T12:40:23Z<p>Daniel Müllner: </p>
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<div><span id="{{anchorencode:{{{1}}}}}">[[ &#91;{{{1}}}&#93; ]] {{{{{1}}}}}</span></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Template:BibitemTemplate:Bibitem2009-09-21T12:39:57Z<p>Daniel Müllner: </p>
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<div><span id="{{anchorencode:{{{1}}}}}">[[&#91;{{{1}}}&#93;]] {{{{{1}}}}}</span></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/Template:BibitemTemplate:Bibitem2009-09-21T12:38:24Z<p>Daniel Müllner: </p>
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<div><span id="{{anchorencode:{{{1}}}}}">[[[{{{1}}}]]] {{{{{1}}}}}</span></div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:31:01Z<p>Daniel Müllner: </p>
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/* WikiTex extensions */<br />
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div.texdisplay {<br />
margin: 3px; padding: 3px;<br />
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text-align:center;<br />
}<br />
/* change to float:left if you want formula numbers appear on the left */<br />
div.texdisplay span.dispno {<br />
float: right;<br />
}<br />
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/* You may like it or not... */<br />
/* h2 span.mw-headline { font-weight:bold; } */<br />
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/* Experimental! */<br />
/* a:link { color:#00FF00; } */<br />
/* a:visited { color:#007F00; } */</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:08:25Z<p>Daniel Müllner: </p>
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div.texdisplay {<br />
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background-color: transparent;<br />
text-align:center;<br />
}<br />
/* change to float:left if you want formula numbers appear on the left */<br />
div.texdisplay span.dispno {<br />
float: right;<br />
}<br />
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/* You may like it or not... */<br />
h2 span.mw-headline { font-weight:bold; }<br />
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/* Experimental! */<br />
a:link { color:#00FF00; }<br />
a:visited { color:#007F00; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:07:10Z<p>Daniel Müllner: </p>
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text-align:center;<br />
}<br />
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div.texdisplay span.dispno {<br />
float: right;<br />
}<br />
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/* You may like it or not... */<br />
h2 span.mw-headline { font-weight:bold; }<br />
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/* Experimental! */<br />
a:link { color:#00FF00; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:05:41Z<p>Daniel Müllner: </p>
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text-align:center;<br />
}<br />
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div.texdisplay span.dispno {<br />
float: right;<br />
}<br />
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/* You may like it or not... */<br />
h2 span.mw-headline { font-weight:bold; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:05:06Z<p>Daniel Müllner: </p>
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background-color: transparent;<br />
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float: right;<br />
}<br />
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h2 span.mw-headline { color:red; font-weight:bold; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:04:07Z<p>Daniel Müllner: </p>
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<div>/* CSS placed here will be applied to all skins */<br />
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}<br />
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div.texdisplay span.dispno {<br />
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#mw-headline { color:red; font-weight:bold; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:03:54Z<p>Daniel Müllner: </p>
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margin: 3px; padding: 3px;<br />
background-color: transparent;<br />
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}<br />
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div.texdisplay span.dispno {<br />
float: right;<br />
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#mw-headline h2 span { color:red; font-weight:bold; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:03:38Z<p>Daniel Müllner: </p>
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margin: 3px; padding: 3px;<br />
background-color: transparent;<br />
text-align:center;<br />
}<br />
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div.texdisplay span.dispno {<br />
float: right;<br />
}<br />
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#mw-headline h2 span { font-weight:bold; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:01:41Z<p>Daniel Müllner: </p>
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background-color: transparent;<br />
text-align:center;<br />
}<br />
/* change to float:left if you want formula numbers appear on the left */<br />
div.texdisplay span.dispno {<br />
float: right;<br />
}<br />
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#mw-headline h2 { font-weight:bold; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:00:43Z<p>Daniel Müllner: </p>
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text-align:center;<br />
}<br />
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div.texdisplay span.dispno {<br />
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h2 { font-weight:bold; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T09:00:05Z<p>Daniel Müllner: </p>
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div.texdisplay span.dispno {<br />
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h2 { color:blue; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T08:59:53Z<p>Daniel Müllner: </p>
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}<br />
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div.texdisplay span.dispno {<br />
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h1 { color:blue; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T08:59:37Z<p>Daniel Müllner: </p>
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#firstHeading { color:green; }</div>Daniel Müllnerhttp://www.map.mpim-bonn.mpg.de/MediaWiki:Common.cssMediaWiki:Common.css2009-09-21T08:55:14Z<p>Daniel Müllner: </p>
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/* change to float:left if you want formula numbers appear on the left */<br />
div.texdisplay span.dispno {<br />
float: right;<br />
}<br />
<br />
h1 { color:green; }</div>Daniel Müllner