# 2-manifolds

## 1 Introduction

A surface is a synonym for a $2$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}2$-dimensional manifold. Complex $2$$2$-dimensional (real $4$$4$-dimensional) complex manifolds are also called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections 2 and 3), surfaces without boundary are considered.

## 2 First construction: connected sum

### 2.1 Orientable surfaces

The two simplest closed orientable $2$$2$-manifolds are:

• the $2$$2$-sphere: $S^2 := \{ (x, y) \in \Rr^2 | x^2 + y^2 = 1 \}$$S^2 := \{ (x, y) \in \Rr^2 | x^2 + y^2 = 1 \}$,
• the $2$$2$-torus: $T^2 := S^1 \times S^1$$T^2 := S^1 \times S^1$, the Cartesian product of two circles.

All orientable surfaces are homeomorphic to the connected sum of $g$$g$ tori $T^2$$T^2$ ($g\geq 0$$g\geq 0$) and so we define

• $F_g := \sharp_g T^2 = T^2 \sharp \dots \sharp T^2$$F_g := \sharp_g T^2 = T^2 \sharp \dots \sharp T^2$, the $g$$g$-fold connected sum of the $2$$2$-torus.

The case $g=0$$g=0$ refers to the 2-sphere $S^2$$S^2$. The number $g$$g$ is called the genus of the surface: for more on the concept of genus see, e.g. [Hirzebruch&Kreck2009].

A 2-sphere (genus 0), a torus (genus 1) and an orientable surface of higher genus

### 2.2 Non-orientable surfaces

The simplest non-orientable surface is the real projective plane $\RP^2$$\RP^2$: for the history of the discovery of this interesting manifold see the page Projective plane: a history.

All non-orientable surfaces are homeomorphic to the connected sum of $h$$h$ real projective planes and and so for all $h \geq 1$$h \geq 1$ we define

• $R_h := \sharp_h \RP^2 = \RP^2 \sharp \dots \sharp \RP^2$$R_h := \sharp_h \RP^2 = \RP^2 \sharp \dots \sharp \RP^2$, to be the $h$$h$-fold connected sum of $\RP^2$$\RP^2$.

For example, the Klein bottle is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2$$\RP^2\mathbin{\sharp}\RP^2$. The number $h$$h$ is called the genus or cross-cap number of the surface.

The Boy surface, an immersion of $\RP^2$$\RP^2$ in $\Rr^3$$\Rr^3$. This steel sculpture stands in front of the Oberwolfach Institute.
A Klein bottle (non-orientable, genus 2) immersed in $\Rr^3$$\Rr^3$

## 3 Surfaces with boundary

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.

## 4 Classification

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$$\geq 0$ in the orientable case, $\geq 1$$\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.

References: [Ahlfors&Sario1960, Thm. 1.42A, 1§8], [Hirsch1994, Thm. 9.3.11], [Moise1977, Thm. 8.3, Thm. 8.5, Thm. 22.9]

The connected sum of $T^2$$T^2$ with $\RP^2$$\RP^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$$\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$. Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect to connected sum, and is generated by $T^2$$T^2$ and $\RP^2$$\RP^2$, with the sole relation $T^2\mathbin{\sharp}\RP^2=\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$$T^2\mathbin{\sharp}\RP^2=\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$.

Closed 2-manifolds are homeomorphic if and only if they have isomorphic intersection forms. Likewise, compact 2-manifolds with non-empty boundary are homeomorphic if and only if they have isomorphic intersection forms. Cf. the topological classification of simply-connected 4-manifolds.

## 5 More constructions

### 5.1 By polygons

Each orientable surface of genus $g>0$$g>0$ can be constructed by identifying pairs of edges in a regular polygon with $4g$$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$$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.

The 2-sphere can be obtained from a 2-gon with edges labeled $a,\bar a$$a,\bar a$.

Each non-orientable surface of genus $h$$h$ can be obtained from a $2h$$2h$-gon with edges labeled $a_1,a_1,\ldots,a_h,a_h$$a_1,a_1,\ldots,a_h,a_h$.

 The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon. The 2-sphere can be obtained by identifying the edges of a 2-gon. Construction of $\RP^2$$\RP^2$. Construction of the Klein bottle.

Reference: [Massey1977, Section 1.5]

### 5.2 By gluing handles and crosscaps

An orientable surface of genus $g$$g$ can be obtained by successively gluing $g$$g$ 1-handles to the 2-sphere such that the embeddings of $S^0\times D^2$$S^0\times D^2$ in $S_{g-1}$$S_{g-1}$ are in each case either orientation-preserving or orientation-reversing on both components $\{1\}\times D^2\cong D^2$$\{1\}\times D^2\cong D^2$ and $\{-1\}\times D^2\cong -D^2$$\{-1\}\times D^2\cong -D^2$.

A non-orientable surface of genus $h$$h$ can be obtained by gluing $h$$h$ crosscaps to $S^2$$S^2$. For this, embed $D^2$$D^2$ in $S^2$$S^2$ (or $R_{h-1}$$R_{h-1}$ from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary $S^1$$S^1$. The result of attaching a non-orientable handle to $S^2$$S^2$ or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.

Reference: [Hirsch1994, Section 9.1]

### 5.3 By branched coverings

Every orientable surface of genus $g$$g$ can be obtained as the branched double covering of the 2-sphere with $2g+2$$2g+2$ branching points.

Proof. The branched double covering of the 2-sphere, with $2g+2$$2g+2$ branch points, has Euler characteristic $4-(2g+2)=2(1-g)$$4-(2g+2)=2(1-g)$. Thus every orientable closed surface is such a branched covering.

$\square$$\square$

### 5.4 As complex curves

A smooth, irreducible, plane, complex projective-algebraic curve of order $d$$d$ (i. e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree $d$$d$ in $\CP^2$$\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus

$\displaystyle g = \frac{(d-1)(d-2)}2.$

Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.

Reference: [Brieskorn&Knörrer1986, Lemma 9.2.1]

## 6 Properties

### 6.1 Orientable surfaces

Let $S_g$$S_g$ denote an oriented surface of genus $g$$g$.

• By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, $2g$$2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps.
• Therefore, the homology groups with any coefficients $G$$G$ are given by $H_0\cong G$$H_0\cong G$, $H_1\cong G^{2g}$$H_1\cong G^{2g}$, $H_2\cong G$$H_2\cong G$.
• The integral cohomology ring is completely determined by the intersection form on $H^1$$H^1$, which is necessarily isomorphic to the hyperbolic form. A basis for $H^1(S_g)$$H^1(S_g)$, for which the intersection form is the standard hyperbolic form, is given by the loops $(a_1,b_1,\ldots,a_g,b_g)$$(a_1,b_1,\ldots,a_g,b_g)$ in the polygon construction above.
• Homotopy groups
• For the homotopy group of $S^2$$S^2$, see the article about spheres.
• All surfaces of higher genus $g\geq1$$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$$\pi_1(S_g,*)\cong \langle a_1,b_1,\ldots, a_g,b_g \mid a_1b_1\cdots a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_ObSc51$$\Zz^2$.
• All orientable surfaces can be embedded in $\Rr^3$$\Rr^3$.
• Every surface can be given a complex structure. Together with the complex structure, it is a Riemann surface
• Characteristic classes
• All Stiefel-Whitney classes vanish.
• All Pontrjagin classes vanish.
• The Euler characteristic is $2-2g$$2-2g$.
• Given a complex structure on the surface, the first Chern class is equal to the Euler class.
• All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in $\Rr^3$$\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.

### 6.2 Non-orientable surfaces

Let $R_h$$R_h$ denote an non-orientable surface of genus $h$$h$.

• Again, each non-orientable surface has a cell decomposition with one 0-cell, $h$$h$ 1-cells and one 2-cell. The differential $C_1\to C_0$$C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$$C_2\to C_1$ with respect to the basis given by the loops the loops $(a_1,\ldots,a_h)$$(a_1,\ldots,a_h)$ in the polygon construction is the matrix $(2,\ldots, 2)$$(2,\ldots, 2)$.
• Therefore, the integral homology groups are isomorphic to $H_0\cong\Zz$$H_0\cong\Zz$, $H_1\cong \Zz^{h-1}\oplus \Zz/2$$H_1\cong \Zz^{h-1}\oplus \Zz/2$, $H_2\cong 0$$H_2\cong 0$.
• The mod-2 homology groups are $H_0=\Zz/2$$H_0=\Zz/2$, $H_1\cong(\Zz/2)^h$$H_1\cong(\Zz/2)^h$, $H_2=\Zz/2$$H_2=\Zz/2$ since the chain complex above is acyclic mod 2. The intersection form on $H_1(R_h;\Zz/2)$$H_1(R_h;\Zz/2)$ with respect to this cellular basis is given by the identity matrix.
• Homotopy groups
• The fundamental group of $R_h$$R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$$\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdots a_h^2\rangle$.
• The orientation double covering of $R_h$$R_h$ is $S_{h-1}$$S_{h-1}$. Therefore, all higher homotopy groups of $\RP^2$$\RP^2$ equal those of $S^2$$S^2$, and the non-orientable surfaces of higher genus are aspherical.
• The orientation double covering also determines the curvature properties: $\RP^2$$\RP^2$ can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.
• Non-orientable surfaces cannot be embedded in $\Rr^3$$\Rr^3$. However, they do embed in $\Rr^4$$\Rr^4$.
• Characteristic classes
• The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism $\pi_1(R_h,*)\to\Zz/2$$\pi_1(R_h,*)\to\Zz/2$ which maps each generator $a_1,\ldots,a_h$$a_1,\ldots,a_h$ to the generator of $\Zz/2$$\Zz/2$. The second Stiefel-Whitney class is zero if $h$$h$ is even and is the non-zero element of $H^2(R_h;\Zz)=H^2(R_h;\Zz/2)=\Zz/2$$H^2(R_h;\Zz)=H^2(R_h;\Zz/2)=\Zz/2$ if $h$$h$ is odd. In all cases $w_2=w_1^2$$w_2=w_1^2$.
• The Euler characteristic is $2-h$$2-h$.

### 6.3 General

There are purely point-set topology characterizations of surfaces; see [Wilder1949].