1-manifolds

An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 08:32, 18 July 2013 and the changes since publication.

1 Introduction

According to the general definition of manifold, a manifold of dimension 1 is a topological space which is second countable (i.e., its topological structure has a countable base), satisfies the Hausdorff axiom (any two different points have disjoint neighborhoods) and each point of which has a neighborhood homeomorphic either to line $\Rr$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; According to the general definition of manifold, a manifold of dimension 1 is a topological space which is second countable (i.e., its topological structure has a countable base), satisfies the Hausdorff axiom (any two different points have disjoint neighborhoods) and each point of which has a neighborhood homeomorphic either to line $\Rr $ or half-line $\Rr_+=\{x\in\Rr\mid x\ge0\}$. Manifolds of dimension 1 are called ''curves'', but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term ''curve'' may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euclidean space $\Rr^n$. To be on the safe side, we use an unambiguous term ''manifold of dimension 1'' or ''1-manifold''. Specific properties of 1-manifolds can be related to the fact that the topological structure in a 1-manifold is defined by linear or cyclic ordering of points. </wikitex> == Examples == <wikitex>; * Real line $\mathbb R$ * Half-line $\mathbb R_+$ * Circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$ * Closed interval $I=[0,1]$ </wikitex> == Classification == ==== Reduction to classification of connected manifolds ==== <!-- COMMMENT O.V. This subsection has to be moved to some general theory of manifolds, because it contains almost nothing specific about dimension 1 (at the only occasion 1-manifolds have to be replaced by $n$-manifolds). Only a reference or, rather, a link to this must be left here. END OF COMMENT--> <wikitex>; Any manifold is homeomorphic to the disjoint sum of its connected components. <!-- Should we have here a reference to a definition of disjoint sum of topological spaces? --> A connected component of a 1-manifold is a 1-manifold. Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic. </wikitex> ==== Topological classification of connected 1-manifolds ==== <wikitex>; {{beginthm|Theorem}} Any connected 1-manifold is homeomorphic to one of the following 4 manifolds: * real line $\mathbb R$ * half-line $\mathbb R_+$ * circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$ * closed interval $I=[0,1]$. No two of these manifolds are homeomorphic to each other. {{endthm}} </wikitex> ==== Characterizing the topological type of a connected 1-manifold ==== <wikitex>; * Any connected closed 1-manifold is homeomorphic to $S^1$. * Any connected compact 1-manifold with non-empty boundary is homeomorphic to $I$. * Any connected non-compact 1-manifold without boundary is homeomorphic to $\Rr$. * Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to $\Rr_+$. Thus for connected 1-manifolds two invariants, compactness and presence of boundary form a complete system of topological invariants. Each of the invariants takes two values. </wikitex> ==== Remarks ==== <wikitex>; Proofs of the results above are elementary. The core of them is the following simple {{beginthm|Lemma}} Any connected 1-manifold covered by two open subsets homeomorphic to $\Rr$ is homeomorphic either to $\Rr$ or $S^1$.{{endthm}} The theorems above solve the topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many Topology textbooks manage not to mention this fundamental result. If we enlarge the collection of spaces by getting rid of the Hausdorff property, then the number of topological types of connected spaces becomes uncountable. Indeed, one can take two copies of line $\Rr$ and identify an open set in one of them with its copy in the other one by the identity map. The result satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. </wikitex> === Corollaries === ==== Homotopy classification of 1-manifolds ==== <wikitex>; Each connected 1-manifold is either contractible, or homotopy equivalent to circle. </wikitex> ==== 0-manifolds cobordant to zero ==== <wikitex>; A compact 0-manifold $X$ bounds a compact 1-manifold iff the number of points in $X$ is even. </wikitex> ==== Smooth structures ==== <wikitex>; ''Any 1-manifold admits a smooth structure.'' If smooth 1-manifolds $X$ and $Y$ are homeomorphic, then they are also diffeomorphic. Moreover, <br> ''any homeomorphism $X\to Y$ can be approximated in the $C^0$-topology by a diffeomorphism.'' Technically this can be considered as a corollary of the following simple theorem:<br> ''a map $\Rr\to\Rr$ is a homeomorphism iff it is a continuous monotone bijection.'' </wikitex> == Invariants == <!-- COMMENT O.V.: About usage of Euler characteristic. A non-compact space has many Euler characteristics. The notion of Euler characteristic depends on the properties that one wants to preserve. If one wants to have invariance under homotopy equivalences, then the additivity would be lost. If one wants to keep additivity, then it is not the alternating sum of Betti numbers. If one wants to keep topological invariance and additivity, then $\chi(\Rr)=\chi(I)-2\chi(pt)=1-2=-1$, similarly $\chi(\Rr_+)=0$. In all the cases under consideration the latter Euler characteristic is the alternating sum of the ranks of homology groups with closed support (Borel-Moore homology). Pretty legitimate! I would not recommend to use the Euler characteristic in a non-compact situation without a detailed specification of the definition. The problem of recognizing of the topological type is solved by the two invariants that I mentioned below: compactness and presence of boundary. Other invariants can be used and deserve consideration. However, I suggest to place the section Invariants after the section Classification. END OF COMMENT --> <wikitex>; As follows from the classification theorems, * the ''number of connected components'', * ''compactness of a connected component'', * and the number of boundary points of a connected component<br> play fundamental role in topology of 1-manifolds. Homotopy invariants are extremely simple. All homology and homotopy groups of dimensions $>1$ are trivial. Tangent bundles of 1-manifolds are trivial. </wikitex> == Further discussion == ==== Orientations ==== <wikitex>; Orientation of a 1-manifold can be interpreted via linear or cyclic orderings of their points. An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold. An orientation of a connected closed 1-manifold is a cyclic order on the set of its points such that the topology of this cyclic order coincides with the topology of the 1-manifold. An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation). Any 1-manifold admits an orientation. Half-line $\Rr_+$ does not admit a homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$ admits an orientation reversing map. Thus, there are 5 topological types of oriented connected 1-manifolds. </wikitex> ==== Triangulations ==== <wikitex>; Any 1-manifold admit a triangulation. A triangulation of a non-compact connected 1-manifold is unique up to homeomorphism. A compact 1-manifold has non-homeomorphic triangulations, but they are easy to classify up to homeomorphism. On circle the topological type of a triangulation is defined by the number of 1-simplices. This number can take any integral value $\ge3$. Similarly, the topological type of a triangulation of $I$ is defined by the number of 1-simplices, which can take any positive integral value. </wikitex> ==== Inner metrics ==== <wikitex>; Recall that a metric on a path-connected space is said to be ''inner'' if the distance between any two points is equal to the infimum of lengths of paths connecting the points, and that the length of a path $s:I\to X$ in a metric space $X$ with metric $d:X\times X\to \Rr_+$ is $\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$. Any connected 1-manifold admits an inner metric. A connected 1-manifold with an inner metric is defined up to isometry by the diameter of the space. Recall that the diameter of a metric space $X$ with metric $d:X\times X\to\Rr$ is $\sup\{ d(x,y)\mid x,y\in X\}$. For each value of diameter there is a standard model for the inner metric space. For a circle with inner metric of diameter $D\in (0,\infty)$ this is the circle $\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$ of radius $D/\pi$ on the plane with the inner metric. For $I$ with diameter $D\in(0,\infty)$ this is $[0,D]$. For $\Rr_+$ with diameter $D\in(0,\infty]$ this is $[0,D)$. For $\Rr$ with diameter $D\in (0,\infty]$ this is $(-D/2,D/2)$. An inner metric on a connected 1-manifolds defines a unique smooth structure on the manifold, namely, the smooth structure induced by the isometry to the corresponding standard model from the list above. </wikitex> ==== Mapping class groups ==== <wikitex>; Recall that the ''mapping class group'' of a manifold $X$ is the quotient group of the group $\operatorname{Homeo} (X)$ of all homeomorphisms $X\to X$ by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of $X$ is $\pi_0(\operatorname{Homeo}(X))$. A reversing orientation homeomorphism cannot be isotopic to the identity. For an auto-homeomorphism of a connected 1-manifold this is the only obstruction to being isotopic to the identity. Therefore $\pi_0(\operatorname{Homeo}(S^1))=\pi_0(\operatorname{Homeo}(\Rr))=\pi_0(\operatorname{Homeo}(I))=\mathbb{Z}/_2$, while $\pi_0(\operatorname{Homeo} \Rr_+)=0$. </wikitex> ==== Homotopy types of groups of auto-homeomorphisms ==== <wikitex>; The group $\operatorname{Homeo}(S^1)$ contains $O(2)$ as a subgroup, which is its deformation retract. Similarly, the group of auto-homeomorphisms of $S^1$ isotopic to identity contains $SO(2)=S^1$ as a subgroup, which is its deformation retract. Groups of auto-homeomorphisms of $\Rr$, $\Rr_+$ and $I$ isotopic to identity are contractible. Thus for each connected 1-manifold $X$ the group of homeomorphisms $X\to X$ isotopic to identity is homotopy equivalent to $X$. </wikitex> ==== Finite group actions ==== <wikitex>; There is no non-trivial free finite group actions on contractible 1-manifolds. On circle if a finite group $G$ has a free action than $G$ is cyclic. Any cyclic group has a linear free action on $S^1$. Any free action of a cyclic group on $S^1$ is conjugate to a linear action. Any periodic orientation reversing homeomorphism $S^1\to S^1$ is an involution (i.e., has period 2). Any non-trivial periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation. A finite group acting effectively on $S^1$ is either cyclic or dihedral, and the action is conjugate to a linear one. The standard actions of cyclic and dihedral groups on circle are provided by the symmetry groups of regular polygons. </wikitex> ==== Surgery ==== <wikitex>; Any 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary. If given two 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen preserving the orientation (this means that the corresponding cobordism is an oriented 2-manifold and its orientation induces on the boundary the given orientation on one of the 1-manifolds and the orientation opposite to the given one on the other 1-manifold). An index 1 surgery preserving orientation on closed 1-manifold chnages the number of connected components by 1. An index 1 surgery on a closed 1-manifold, which does not preserve any orientation, preserves the number of connected components. </wikitex> ==== Connected sums ==== <wikitex>; The notion of connected sum is defined for 1-manifolds, but it does not work in the same way as for manifolds of higher dimensions. Recall that a manifold $X$ of dimension $n$ is a ''connected sum'' of manifolds $A$ and $B$ if there are $n$-disks $D_A\subset A$ and $D_B\subset B$ and a homeomorphism $h:D_A\to D_B$ such that $X$ is homeomorphic to $(A\cup_h B)\smallsetminus\operatorname{Int}D_A$ (that is to the result of attaching of $A$ to $B$ by $h$ with the image of interior of the identified disks $D_A$ and $D_B$ removed). One writes $X=A\sharp B$, however the topology of $X$ may depend not only on $A$ and $B$, but also on $D_A$, $D_B$ and $h$. The dependence disappears if all connected components of $A$ are homeomorphic to each other, all connected components of $B$ are homeomorphic to each other, and a homeomorphism $D_A\to D_A$ reversing orientation can be extended to a homeomorphism $A\to A$, or a homeomorphism $D_B\to D_B$ reversing orientation can be extended to a homeomorphism $B\to B$. For a collection of manifolds satisfying these conditions, a connected sum can be considered as operation on topological types. <!-- COMMENT BY O.V. The paragraph above should be moved to a general theory of manifolds, but a reference to it must be left END OF COMMENT --> The very term ''connected sum'' is compromised in dimension 1. Indeed, a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of $\Rr$ is a disjoint sum of two copies of $\Rr$. Moreover, connected sum, as an operation on topological types of 1-manifolds, is not well-defined. Indeed, both disjoint sums $I\amalg\Rr$ and $\Rr_+\amalg\Rr_+$ can be presented as a connected sum of two copies of $\Rr_+$. </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\Rr$ or half-line $\Rr_+=\{x\in\Rr\mid x\ge0\}$$\Rr_+=\{x\in\Rr\mid x\ge0\}$.

Manifolds of dimension 1 are called curves, but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term curve may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euclidean space $\Rr^n$$\Rr^n$. To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.

Specific properties of 1-manifolds can be related to the fact that the topological structure in a 1-manifold is defined by linear or cyclic ordering of points.

2 Examples

• Real line $\mathbb R$$\mathbb R$

• Half-line $\mathbb R_+$$\mathbb R_+$

• Circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$

• Closed interval $I=[0,1]$$I=[0,1]$

3 Classification

3.1 Reduction to classification of connected manifolds

Any manifold is homeomorphic to the disjoint sum of its connected components.

A connected component of a 1-manifold is a 1-manifold.

Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic.

3.2 Topological classification of connected 1-manifolds

Theorem 3.1. Any connected 1-manifold is homeomorphic to one of the following 4 manifolds:

• real line $\mathbb R$$\mathbb R$
• half-line $\mathbb R_+$$\mathbb R_+$
• circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$
• closed interval $I=[0,1]$$I=[0,1]$.

No two of these manifolds are homeomorphic to each other.

3.3 Characterizing the topological type of a connected 1-manifold

• Any connected closed 1-manifold is homeomorphic to $S^1$$S^1$.

• Any connected compact 1-manifold with non-empty boundary is homeomorphic to $I$$I$.

• Any connected non-compact 1-manifold without boundary is homeomorphic to $\Rr$$\Rr$.

• Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to $\Rr_+$$\Rr_+$.

Thus for connected 1-manifolds two invariants, compactness and presence of boundary form a complete system of topological invariants. Each of the invariants takes two values.

3.4 Remarks

Proofs of the results above are elementary. The core of them is the following simple

Lemma 3.2. Any connected 1-manifold covered by two open subsets homeomorphic to $\Rr$$\Rr$ is homeomorphic either to $\Rr$$\Rr$ or $S^1$$S^1$.

The theorems above solve the topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many Topology textbooks manage not to mention this fundamental result.

If we enlarge the collection of spaces by getting rid of the Hausdorff property, then the number of topological types of connected spaces becomes uncountable. Indeed, one can take two copies of line $\Rr$$\Rr$ and identify an open set in one of them with its copy in the other one by the identity map. The result satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom.

3.5 Corollaries

3.5.1 Homotopy classification of 1-manifolds

Each connected 1-manifold is either contractible, or homotopy equivalent to circle.

3.5.2 0-manifolds cobordant to zero

A compact 0-manifold $X$$X$ bounds a compact 1-manifold iff the number of points in $X$$X$ is even.

3.5.3 Smooth structures

Any 1-manifold admits a smooth structure.

If smooth 1-manifolds $X$$X$ and $Y$$Y$ are homeomorphic, then they are also diffeomorphic. Moreover,
any homeomorphism $X\to Y$$X\to Y$ can be approximated in the $C^0$$C^0$-topology by a diffeomorphism.

Technically this can be considered as a corollary of the following simple theorem:
a map $\Rr\to\Rr$$\Rr\to\Rr$ is a homeomorphism iff it is a continuous monotone bijection.

4 Invariants

As follows from the classification theorems,

• the number of connected components,
• compactness of a connected component,
• and the number of boundary points of a connected component
  

play fundamental role in topology of 1-manifolds.

Homotopy invariants are extremely simple. All homology and homotopy groups of dimensions $>1$$>1$ are trivial.

Tangent bundles of 1-manifolds are trivial.

5 Further discussion

5.1 Orientations

Orientation of a 1-manifold can be interpreted via linear or cyclic orderings of their points.

An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold.

An orientation of a connected closed 1-manifold is a cyclic order on the set of its points such that the topology of this cyclic order coincides with the topology of the 1-manifold.

An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation).

Any 1-manifold admits an orientation.

Half-line $\Rr_+$$\Rr_+$ does not admit a homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$$\Rr_+$ admits an orientation reversing map. Thus, there are 5 topological types of oriented connected 1-manifolds.

5.2 Triangulations

Any 1-manifold admit a triangulation. A triangulation of a non-compact connected 1-manifold is unique up to homeomorphism.

A compact 1-manifold has non-homeomorphic triangulations, but they are easy to classify up to homeomorphism. On circle the topological type of a triangulation is defined by the number of 1-simplices. This number can take any integral value $\ge3$$\ge3$. Similarly, the topological type of a triangulation of $I$$I$ is defined by the number of 1-simplices, which can take any positive integral value.

5.3 Inner metrics

Recall that a metric on a path-connected space is said to be inner if the distance between any two points is equal to the infimum of lengths of paths connecting the points, and that the length of a path $s:I\to X$$s:I\to X$ in a metric space $X$$X$ with metric $d:X\times X\to \Rr_+$$d:X\times X\to \Rr_+$ is $\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$$\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$.

Any connected 1-manifold admits an inner metric. A connected 1-manifold with an inner metric is defined up to isometry by the diameter of the space. Recall that the diameter of a metric space $X$$X$ with metric $d:X\times X\to\Rr$$d:X\times X\to\Rr$ is $\sup\{ d(x,y)\mid x,y\in X\}$$\sup\{ d(x,y)\mid x,y\in X\}$.

For each value of diameter there is a standard model for the inner metric space. For a circle with inner metric of diameter $D\in (0,\infty)$$D\in (0,\infty)$ this is the circle $\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$$\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$ of radius $D/\pi$$D/\pi$ on the plane with the inner metric. For $I$$I$ with diameter $D\in(0,\infty)$$D\in(0,\infty)$ this is $[0,D]$$[0,D]$. For $\Rr_+$$\Rr_+$ with diameter $D\in(0,\infty]$$D\in(0,\infty]$ this is $[0,D)$$[0,D)$. For $\Rr$$\Rr$ with diameter $D\in (0,\infty]$$D\in (0,\infty]$ this is $(-D/2,D/2)$$(-D/2,D/2)$.

An inner metric on a connected 1-manifolds defines a unique smooth structure on the manifold, namely, the smooth structure induced by the isometry to the corresponding standard model from the list above.

5.4 Mapping class groups

Recall that the mapping class group of a manifold $X$$X$ is the quotient group of the group $\operatorname{Homeo} (X)$$\operatorname{Homeo} (X)$ of all homeomorphisms $X\to X$$X\to X$ by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of $X$$X$ is $\pi_0(\operatorname{Homeo}(X))$$\pi_0(\operatorname{Homeo}(X))$.

A reversing orientation homeomorphism cannot be isotopic to the identity. For an auto-homeomorphism of a connected 1-manifold this is the only obstruction to being isotopic to the identity. Therefore $\pi_0(\operatorname{Homeo}(S^1))=\pi_0(\operatorname{Homeo}(\Rr))=\pi_0(\operatorname{Homeo}(I))=\mathbb{Z}/_2$$\pi_0(\operatorname{Homeo}(S^1))=\pi_0(\operatorname{Homeo}(\Rr))=\pi_0(\operatorname{Homeo}(I))=\mathbb{Z}/_2$, while $\pi_0(\operatorname{Homeo} \Rr_+)=0$$\pi_0(\operatorname{Homeo} \Rr_+)=0$.

5.5 Homotopy types of groups of auto-homeomorphisms

The group $\operatorname{Homeo}(S^1)$$\operatorname{Homeo}(S^1)$ contains $O(2)$$O(2)$ as a subgroup, which is its deformation retract. Similarly, the group of auto-homeomorphisms of $S^1$$S^1$ isotopic to identity contains $SO(2)=S^1$$SO(2)=S^1$ as a subgroup, which is its deformation retract.

Groups of auto-homeomorphisms of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$ and $I$$I$ isotopic to identity are contractible.

Thus for each connected 1-manifold $X$$X$ the group of homeomorphisms $X\to X$$X\to X$ isotopic to identity is homotopy equivalent to $X$$X$.

5.6 Finite group actions

There is no non-trivial free finite group actions on contractible 1-manifolds. On circle if a finite group $G$$G$ has a free action than $G$$G$ is cyclic. Any cyclic group has a linear free action on $S^1$$S^1$. Any free action of a cyclic group on $S^1$$S^1$ is conjugate to a linear action.

Any periodic orientation reversing homeomorphism $S^1\to S^1$$S^1\to S^1$ is an involution (i.e., has period 2). Any non-trivial periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.

A finite group acting effectively on $S^1$$S^1$ is either cyclic or dihedral, and the action is conjugate to a linear one. The standard actions of cyclic and dihedral groups on circle are provided by the symmetry groups of regular polygons.

5.7 Surgery

Any 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary.

If given two 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen preserving the orientation (this means that the corresponding cobordism is an oriented 2-manifold and its orientation induces on the boundary the given orientation on one of the 1-manifolds and the orientation opposite to the given one on the other 1-manifold).

An index 1 surgery preserving orientation on closed 1-manifold chnages the number of connected components by 1. An index 1 surgery on a closed 1-manifold, which does not preserve any orientation, preserves the number of connected components.

5.8 Connected sums

The notion of connected sum is defined for 1-manifolds, but it does not work in the same way as for manifolds of higher dimensions.

Recall that a manifold $X$$X$ of dimension $n$$n$ is a connected sum of manifolds $A$$A$ and $B$$B$ if there are $n$$n$-disks $D_A\subset A$$D_A\subset A$ and $D_B\subset B$$D_B\subset B$ and a homeomorphism $h:D_A\to D_B$$h:D_A\to D_B$ such that $X$$X$ is homeomorphic to $(A\cup_h B)\smallsetminus\operatorname{Int}D_A$$(A\cup_h B)\smallsetminus\operatorname{Int}D_A$ (that is to the result of attaching of $A$$A$ to $B$$B$ by $h$$h$ with the image of interior of the identified disks $D_A$$D_A$ and $D_B$$D_B$ removed). One writes $X=A\sharp B$$X=A\sharp B$, however the topology of $X$$X$ may depend not only on $A$$A$ and $B$$B$, but also on $D_A$$D_A$, $D_B$$D_B$ and $h$$h$. The dependence disappears if all connected components of $A$$A$ are homeomorphic to each other, all connected components of $B$$B$ are homeomorphic to each other, and a homeomorphism $D_A\to D_A$$D_A\to D_A$ reversing orientation can be extended to a homeomorphism $A\to A$$A\to A$, or a homeomorphism $D_B\to D_B$$D_B\to D_B$ reversing orientation can be extended to a homeomorphism $B\to B$$B\to B$. For a collection of manifolds satisfying these conditions, a connected sum can be considered as operation on topological types.

The very term connected sum is compromised in dimension 1. Indeed, a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of $\Rr$$\Rr$ is a disjoint sum of two copies of $\Rr$$\Rr$.

Moreover, connected sum, as an operation on topological types of 1-manifolds, is not well-defined. Indeed, both disjoint sums $I\amalg\Rr$$I\amalg\Rr$ and $\Rr_+\amalg\Rr_+$$\Rr_+\amalg\Rr_+$ can be presented as a connected sum of two copies of $\Rr_+$$\Rr_+$.

6 References

This page has not been refereed. The information given here might be incomplete or provisional.