1-manifolds
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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 neighbourhood homeomorphic either to the real line or to the half-line .
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 . To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.
For other expositions about -manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Sections 3.1.1.16-19].
2 Examples
- The real line:
- The half-line:
- The circle:
- The closed interval:
3 Topological classification
3.1 Reduction to classification of connected manifolds
The following elementary facts hold for -manifolds of any dimension .
Any manifold is homeomorphic to the disjoint sum of its connected components.
A connected component of an -manifold is a -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
- half-line
- circle
- closed interval .
No two of these manifolds are homeomorphic to each other.
3.3 Characterizing the topological type of a connected 1-manifold
Theorem 3.2.
- Any connected non-compact 1-manifold without boundary is homeomorphic to .
- Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to .
- Any connected closed 1-manifold is homeomorphic to .
- Any connected compact 1-manifold with non-empty boundary is homeomorphic to .
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.
Theorems 3.1 and 3.2 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.
3.4 About proofs of the classification theorems
The proofs of Theorems 3.1 and 3.2 above are elementary. They can be found, e.g., in [Fuks&Rokhlin1984, Sections 3.1.1.16-19]. The core of them are the following simple lemmas:
Lemma 3.4. If a topological space can be represented as the union of a nondecreasing sequence of open subsets, all homeomorphic to , then is homeomorphic to .
3.5 Corollary: homotopy classification
Theorem 3.5. Each connected 1-manifold is either contractible, or homotopy equivalent to circle.
It follows immediately from Theorem 3.1.
3.6 Corollary: cobordisms of 0-manifolds
Theorem 3.6. A compact 0-manifold bounds a compact 1-manifold iff the number of points in is even.
Corollary 3.7. Two compact 0-manifolds are cordant iff their numbers of points are congruent modulo 2.
4 Orders and orientations
4.1 Interval topology
Most of properties specific for 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.
Open intervals and open rays and form the base of the (standard) topology on . This way of introducing a topological structure can be applied in any (linearly) ordered set . On and , the (standard) topology is induced from the standard topology on , and can be described in terms of the induced order.
Theorem 4.1. Every connected non-closed 1-manifold admits exactly two linear orders defining its topology.
Proof. By Theorem 3.2, a coonected non-closed 1-manifold is homeomorphic either to , or , or . On each of these 1-manifolds there are two linear orders, and , defining the topology. We have to prove that there is no other linear order defining the same topology.
A linear order on a set is encoded in the system of sets for . A system of susbsets with comes from a linear order on iff either , or , or for any and only if . In the interval topology on , each and is open.
For each interior point of a connected non-closed 1-manifold , its complement is a disjoint union of two open connected sets and . Any linear order on defining the topology of would give rise to a splitting of each of these two sets into its intesections with disjoint sets and . If defines the topology on , then and are open. Due to connectedness of , either , or . Hence the order coincides with one of the two standard orders.
4.2 Orientations
An orientation of a 1-manifold can be interpreted via linear orders defining the topology on its open subsets homeomorphic to or . An orientation of any of these two 1-manifolds is nothing but such linear order. For a general 1-manifold one need to globalize this idea. It can be done in the same way as in higher dimensions, but there are specifically 1-dimensional approaches to the globalizing.
First, due to the topological classification, one can restrict to just four model 1-manifolds , , and . For the first three of them, an orientation still can be defined as a linear order determining the topology of the manifold. For this approach does not work, but can be adjusted: instead of linear orders one can rely on cyclic orders that define the topology. It is a bit cumbersome, because cyclic orders are more cumbersome than usual linear orders.
There is a more conceptual way, which immitate the classical definition of orientations of differentiable manifolds, but rely, instead of coordinate charts, on local linear orders.
Let be a 1-manifold. A local orientation of at a point is a pair consisting of a neighborhood of homeomorphic to or and a linear order on defining the topology on . Denote by the set of all local orientations of .
An orientation on is a map such that for any and any connected component of the restrictions of and to coincide iff .
Theorem 4.2. On any connected 1-manifold there exists exactly two orientations.
Proof. If is non-closed connected 1-manifold, then by Theorem \ref{thm:characterization} it can be identified by a homeomorphism to either , or , or . An orientation is defined by its value on . Indeed, for any open set homeomorphic to or , the set is connected, on it there are exactly two linear orders defining the topology: one of them is induced by , the other by . The value of on the corresponding elements of are defined by . On the other hand, there exists an orientation taking each of the two possible values on , because the construction used for the proof of existence above, gives a map satisfying the definition of orientation.
If is a closed connected 1-manifold, then by Theorem \ref{thm:characterization} it can be identified by a homeomorphism with .
As a cyclic order is determined by a linear order on the complement of any point, orientations of a connected closed 1-manifold are defined by orientations of the complement of any point. By Theorem 4.1, the complement has exactly two orientations, so the circle cannot have more.
Corollary 4.3. Any 1-manifold admits an orientation. If the 1-manifold consists of connected components, then it admits orientations.
4.3 Self-homeomorphisms
Theorem 4.4. A map is a homeomorphism iff is a monotone bijection.
Proof. Let be a homeomorphism. First, observe that maps every ray to a ray. Indeed, for any , the map induces a homeomorphism . The rays and are connected components of . Therefore their images are connected components and of .
Observe that rays have the same direction iff one of them is contained in the other one. Therefore two rays of the same dirction are mapped by to rays with the same direction. Thus rays are mapped either all to rays or all to . Thus is monotone.
Let be a monotone bijection. Then the image and preimage under of any open interval are open intervals. Therefore, both and are continuous, and hence is a homeomorphism.
The following theorem can be proved similarly or can be deduced from Theorem 4.4
Theorem 4.5.
- A map is a homeomorphism iff is a monotone bijection.
- A map is a homeomorphism iff is a monotone increasing bijection.
- A map is a homeomorphism iff is a bijection that either preserves or reverses the cyclic order of points on .
A self-homeomorphism of a connected 1-manifold increases with respect to one order or cyclic order iff it increases with respect to the opposite order. In other words, it preserves an orientation iff it preserves the opposite orientation. Since there are only two orientations, this is a property of homeomorphism which does not depend on orientation. Any self-homeomorphism of a connected 1-manifold either preserves orientation, or reverses it.
The half-line does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to admits an orientation reversing map.
Thus, there are 5 topological types of oriented connected 1-manifolds: the topological type of the non-oriented half-line splits into the oriented topological types of and with the orientations induced by the standard order.
4.4 Orientations via coordinate charts
Definition of orientations of 1-manifolds given above is specific for dimension 1, it has no a direct high-dimensional generalization. Here is a more conventional approach to orientations of 1-manifolds, still specific for dimension 1.
Let be a 1-manifold, an open connected set. A homeomorphism , where is or is called a chart or local coordinate system.
4.5 Characterizations of connected compact 1-manifolds in terms of separating points
A subset of a topological space is said to separate if can be presented as a union of two disjoint open sets.
Theorem 4.6. (See [Moore1920].) Let be a connected compact Hausdorff second countable topological space.
- If every two points separate , then is homeomorphic to the circle.
- If each point, with two exceptions, separates , then is homeomorphic to .
Any point splits to two disjoint open rays and .
Theorem 4.7. (See [Ward1936].) Let be a connected locally compact Hausdorff second countable topological space.
- If the complement of each point in consists of two connected components, then is homeomorphic to .
- If contains a point such that is connected and consists of two connected components for each , , then is homeomorphic to .
5 Invariants
5.1 Basic invariants
As follows from the Theorems 3.1 and 3.2 above, the following invariants
- the number of connected components,
- the compactness of each connected component,
- and the number of boundary points of each connected component
determine the topological type of a 1-manifold.
5.2 Homotopy invariants
The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions are trivial. The fundamental group is infinite cyclic group, if the connected component of containing is homeomorphic to circle, and trivial otherwise.
5.3 Homology invariants
Let be a 1-manifold with a finite number of connected components. By Theorem 3.1, it is homeomorphic to a disjoint union
of some numbers of copies of , , and :Then is a free abelian group of rank equal to the number of all connected components of and is a free abelian group of rank equal to the number of closed (compact without boundary) components of .
Relative homology groups: is a free abelian group of rank equal to the number of connected components of without boundary; is a free abelian group of rank equal to the number of compact components of . So,
Numbers , , , and and the topological type of can be recovered from the ranks of these groups.
Above by homology we mean homology with compact support. The homology with closed support (Borel-Moore homology):
The Poincare duality is an isomorphism between usual cohomogy (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So
A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over has non-trivial monodromy, then all the homology groups are trivial.
5.4 Tangent bundle invariants
The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial.
6 Additional structures
6.1 Triangulations
Any 1-manifold admits 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 . Similarly, the topological type of a triangulation of is defined by the number of 1-simplices, which can take any positive integral value.
6.2 Smooth structures
Any 1-manifold admits a smooth structure.
If smooth 1-manifolds and are homeomorphic, then they are also diffeomorphic. Moreover,
Theorem 6.1. Any homeomorphism between two smooth 1-manifolds can be approximated in the -topology by a diffeomorphism.
Proof. By Theorems 4.4 and 4.5, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points.
6.3 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 in a metric space with metric is .
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 with metric is .
For each value of the diameter there is a standard model for the inner metric space. For the four homeomorphism types of connected 1-manifolds these standard models are as follows.
- For with diameter this is .
- For with diameter this is .
- For a circle with inner metric of diameter this is the circle of radius on the plane with the inner metric.
- For with diameter this is .
An inner metric on a connected 1-manifold 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.
7 Constructions
7.1 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 changes 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.
7.2 Connected sums
The notion of connected sum is defined for 1-manifolds, but the connectivity of the outcome is different in dimension 1 compared to other dimensions. Indeed term connected sum can be misleading in dimension 1 since a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of is a disjoint sum of two copies of .
Note that connected sum is only a well defined operation on oriented manifolds and one has to be careful with the orientations. For example
8 Groups of self-homeomorphisms
8.1 Mapping class groups
Recall that the mapping class group of a manifold is the quotient group of the group of all homeomorphisms by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of is .
An orientation reversing homeomorphism cannot be isotopic to an orientation preserving homeomorphism. For auto-homeomorphisms of a connected 1-manifold this is the only obstruction to being isotopic:
Theorem 8.1. Any two auto-homeomorphisms of a connected 1-manifold that are either both orientation preserving, or both orientation reversing are isotopic.
This is a corollary of the following two obvious lemmas.
Lemma 8.2. On rectilinear isotopy. Let be one of the following 1-manifolds: , , or . Let be two monotone bijections that are either both increasing or both decreasing. Then the family with consists of monotone bijections (and hence is an isotopy between and ).
Lemma 8.3. Let be two bijections that either both preserve or both reverse the standard cyclic order of points on . Let and coincide at . Then and are isotopic via the canonical isotopy which is stationary at and is provided on the complement of by stereographic projections and the rectilinear isotopy from Lemma 8.2 of the corresponding self-homeomorphisms of .
Corollary 8.4. and
Remark. All the statements in this section remains true, if everywhere the word homeomorphism is replaced by the word diffeomorphism and is replaced by .
8.2 Homotopy types of groups of self-homeomorphisms
The group contains as a subgroup, which is its deformation retract. It follows from Lemma 8.3. More precisely, for each point , Lemma 8.3 provides a deformation retraction .
Similarly, the group of self-homeomorphisms of isotopic to identity contains as a subgroup, which is its deformation retract.
The groups of self-homeomorphisms of , and which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 8.2 applied to and an arbitrary .
Thus for each connected 1-manifold the group of homeomorphisms isotopic to identity is homotopy equivalent to .
9 Finite group actions
Consider an action of a finite group on a 1-manifold .
9.1 Free actions
For any point , its orbit is a finite set and has an invariant neighborhood whose connected components are disjoint open sets, each of them contains exactly one point of , and either all the components are homeomorphic to , or all homeomorphic to .
If the action is free, then the orbit space is a 1-manifold and the natural projection is a covering.
Therefore the theory of coverings gives a simple classification of free finite group actions on 1-manifolds.
A contractible 1-manifold has no non-trivial covering. Thus, if a free finite group action on 1-manifold has a contractible orbit space , then is a disjoint union of copies of and permutes these copies. In particular, there is no non-trivial free group action on a connected 1-manifold having contractible orbit space.
Coverings with connected are in one-to-one correspondence with subgroups of finite indices of . For each index there is one subgroup, and hence one covering. The total space is homeomorphic to , and the covering is equivalent to . In the corresponding action, the group is cyclic of order , it acts on by rotations.
In this classification of free finite group actions on connected 1-manifolds, the orbit space plays the main role. However, it is easy to reformulate it with emphasis the 1-manifold on which the group acts. This is done in the next two theorems.
Theorem 9.1. There is no non-trivial free finite group action on a contractible 1-manifold.
Theorem 9.2. If a finite group acts freely on the circle than is cyclic. Any finite cyclic group has a linear free action on . Any free action of a finite cyclic group on is conjugate to a linear action.
9.2 Asymmetry of a half-line
Theorem 9.3. There is no non-trivial action of a finite group in $\Rr_+$.
Proof. We will prove that the only homeomorphism $h:\Rr_+\to\Rr_+$ of finite order is the identity. Obsreve, first that any homeomorphism $\Rr_+\to\Rr_+$ preserves the only boundary point $0\in\Rr_+$. Assume that $h$ is a homeomorphism $\Rr_+\to\Rr_+$ of finite order $m$, and there exists $a\in\Rr_+$ such that $h(a)=b\ne a$. Then $h([0,a])=[0,b]$. Without loss of generality, we may assume that $b<a$ (otherwise just replace $h$ by $h^{-1}$).
Then $h([0,a]=[0,b]\subset[0,a]$ and $a\not\in h([0,a])$. By the assumption, $h^m(a)=a$. On the other hand, $h^m(a)\in h^m([0,a])\subset h^{m-1}([0,a])\subset\dots \subset h([0,a])\not\ni a$.
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9.3 Actions on line and segment
Theorem 9.4. The only orientation preserving homeomorphism of finite order is the identity.
Proof. Fix a homeomorphism (say, define it by formula ). Consider a homeomorphism . It preserves orientation (since preserves orientation). So, it is a monotone increasing bijection of finite order. It can be extended to by letting . The extended homeomorphism has the same finite order. But by Theorem 9.3 any such homeomorphism is the identity.
Theorem 9.5. Any orientation reversing homeomorphism of finite order is of order two. It is conjugate to the symmetry against a point.
Proof. An orientation reversing homeomorphism is a monotone decreasing bijection. Consider the function . It is also monotone decreasing bijection and hence there exists a unique such that , that is .
The homeomorphism maps each connected component of to a connected component of . The connected components are open rays and . If each of them is mapped to itself, then defines a homeomorphism of a finite order of the closed rays and . Then by Theorem 9.3, is identity, which contradicts to our assumption. Thus, and . Then preserves the rays, and, by Theorem 9.3, is identity. Thus has order two.
Choose a homeomorphism . Define function by formula . It's a homeomorphism. Together, and form a homeomorphism . As easy to check, .
Theorem 9.6. A non-trivial finite group acting effectively on is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
Proof. As follows from Corollaries 9.4 and 9.5, any non-trivial element of the group is an orientation reversing involution. We have to prove that the group contains at most one such element. Assume that there are two orientation reversing homeomorphisms, and of the line . Their composition preserves orientation. Since it belongs to a finite group, it has finite order. By Theorem 9.4, it is identity. So, and hence . But . Therefore and .
Corollary 9.7. A non-trivial finite group acting effectively on is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
Proof. Any auto-homeomorphism of preserves the boundary and the interior of . Hence an effective finite group action on induces an action of finite group on the interior of . An auto-homeomorphism of is recovered from its restriction to the interior. Moreover, any auto-homeomorphism of has a unique extension to auto-homeomorphism of . The interior is homeomorphic to .
9.4 Actions on circle
{{beginthm|Theorem} Any periodic orientation reversing homeomorphism is an involution (i.e., has period 2). It is conjugate to a symmetry of against its diameter. </div>
Proof. Observe first that any orientation reversing auto-homeomorphism of the circle has a fixed point. One can prove this by elementary arguments, but we just refer to the Lefschetz Fixed Point Theorem: the Lefschetz number of such homeomorphism is 2.
Consider the complement of a fixed point. The homeomorphism resticted to it satisfies the conditions of Corollary 9.5, which gives the required result.
Observe that by theorems 9.5, 9.7, and 9.7 any non-identity periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.
Theorem 9.8. A periodic non-identity orientation preserving homeomorphism has no fixed point. It is conjugate to a rotation.
Proof. If if had a fixed point, then we could consider its restriction to the complement of this point, and by Theorem 9.4 would conclude that it is identity and hence the whole homeomorphism is identity.
For the same reasons, the non-identity powers of our periodic non-identity orientation preserving homeomorphism have no fixed points. Therefore, these powers form a cyclic group freely acting on . See Theorem 9.2.
Theorem 9.9. A finite group acting effectively on 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.
Proof. If all the homeomorphisms in the action preserve orientation, then by Theorem 9.8 the action is free, and the result follows from Theorem 9.2.
Assume that the action contains an orientation reversing homeomorphism. The orientation preserving homeomorphisms from the action form a cyclic subgroup as above. It is of index 2. Its complement consists of orientation reversing involutions. If the subgroup of orientation preserving homeomorphisms is trivial, then the whole group is of order 2 and the only non-trivial element is an orientation resersing involution. When the group contains two orientation preserving homeomorphisms, the whole group is the cartesian product of two cyclic groups of order 2. It is called Klein's Vierergruppe or dihedral group . If the number of orientation preserving homeomorphisms is , then the whole group is called the dihedral group . It is the symmetry group of an -sided regular polygon.
10 Relatives of 1-manifolds
10.1 Non-Hausdorff 1-manifolds
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 the disjoint union of two copies of the line and identify an open set in one of them with its copy in the other one by the identity map. The quotient space is connected and satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. In this way one can construct uncountably many pairwise non-homeomorphic spaces. To prove that they are not homeomorphic, one can use, for example, the topological type of the subset formed by those points that do not separate the space.
11 References
- [Fuks&Rokhlin1984] D. B. Fuks and V. A. Rokhlin, Beginner's course in topology. Geometric chapters. Translated from the Russian by A. Iacob. Universitext. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1984. MR759162 (86a:57001) Zbl 0562.54003
- [Gale1987] D. Gale, The Teaching of Mathematics: The Classification of 1-Manifolds: A Take-Home Exam, Amer. Math. Monthly 94 (1987), no.2, 170–175. MR1541035 Zbl 0621.57001
- [Ghys2001] E. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no.3-4, 329–407. MR1876932 (2003a:37032) Zbl 1044.37033
- [Moore1920] R. L. Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), no.3, 333–347. MR1501148 () Zbl 47.0519.08
- [Ward1936] A. J. Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198. MR1577110 Zbl 62.0693.02
12 External links
- The Encylopedia of Mathematics article on one-dimensional manifolds.
- The Encylopedia of Mathematics article on lines.
- The Wikipedia page about curves.
{{beginthm|Theorem}} Any homeomorphism between two smooth 1-manifolds can be approximated in the $C^0$-topology by a diffeomorphism. {{endthm}} {{beginproof}} By Theorems \ref{thm:homeomorphisms-of-line} and \ref{thm:homeomorphisms-of-others}, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points. {{endproof}}
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 . To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.
For other expositions about -manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Sections 3.1.1.16-19].
2 Examples
- The real line:
- The half-line:
- The circle:
- The closed interval:
3 Topological classification
3.1 Reduction to classification of connected manifolds
The following elementary facts hold for -manifolds of any dimension .
Any manifold is homeomorphic to the disjoint sum of its connected components.
A connected component of an -manifold is a -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
- half-line
- circle
- closed interval .
No two of these manifolds are homeomorphic to each other.
3.3 Characterizing the topological type of a connected 1-manifold
Theorem 3.2.
- Any connected non-compact 1-manifold without boundary is homeomorphic to .
- Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to .
- Any connected closed 1-manifold is homeomorphic to .
- Any connected compact 1-manifold with non-empty boundary is homeomorphic to .
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.
Theorems 3.1 and 3.2 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.
3.4 About proofs of the classification theorems
The proofs of Theorems 3.1 and 3.2 above are elementary. They can be found, e.g., in [Fuks&Rokhlin1984, Sections 3.1.1.16-19]. The core of them are the following simple lemmas:
Lemma 3.4. If a topological space can be represented as the union of a nondecreasing sequence of open subsets, all homeomorphic to , then is homeomorphic to .
3.5 Corollary: homotopy classification
Theorem 3.5. Each connected 1-manifold is either contractible, or homotopy equivalent to circle.
It follows immediately from Theorem 3.1.
3.6 Corollary: cobordisms of 0-manifolds
Theorem 3.6. A compact 0-manifold bounds a compact 1-manifold iff the number of points in is even.
Corollary 3.7. Two compact 0-manifolds are cordant iff their numbers of points are congruent modulo 2.
4 Orders and orientations
4.1 Interval topology
Most of properties specific for 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.
Open intervals and open rays and form the base of the (standard) topology on . This way of introducing a topological structure can be applied in any (linearly) ordered set . On and , the (standard) topology is induced from the standard topology on , and can be described in terms of the induced order.
Theorem 4.1. Every connected non-closed 1-manifold admits exactly two linear orders defining its topology.
Proof. By Theorem 3.2, a coonected non-closed 1-manifold is homeomorphic either to , or , or . On each of these 1-manifolds there are two linear orders, and , defining the topology. We have to prove that there is no other linear order defining the same topology.
A linear order on a set is encoded in the system of sets for . A system of susbsets with comes from a linear order on iff either , or , or for any and only if . In the interval topology on , each and is open.
For each interior point of a connected non-closed 1-manifold , its complement is a disjoint union of two open connected sets and . Any linear order on defining the topology of would give rise to a splitting of each of these two sets into its intesections with disjoint sets and . If defines the topology on , then and are open. Due to connectedness of , either , or . Hence the order coincides with one of the two standard orders.
4.2 Orientations
An orientation of a 1-manifold can be interpreted via linear orders defining the topology on its open subsets homeomorphic to or . An orientation of any of these two 1-manifolds is nothing but such linear order. For a general 1-manifold one need to globalize this idea. It can be done in the same way as in higher dimensions, but there are specifically 1-dimensional approaches to the globalizing.
First, due to the topological classification, one can restrict to just four model 1-manifolds , , and . For the first three of them, an orientation still can be defined as a linear order determining the topology of the manifold. For this approach does not work, but can be adjusted: instead of linear orders one can rely on cyclic orders that define the topology. It is a bit cumbersome, because cyclic orders are more cumbersome than usual linear orders.
There is a more conceptual way, which immitate the classical definition of orientations of differentiable manifolds, but rely, instead of coordinate charts, on local linear orders.
Let be a 1-manifold. A local orientation of at a point is a pair consisting of a neighborhood of homeomorphic to or and a linear order on defining the topology on . Denote by the set of all local orientations of .
An orientation on is a map such that for any and any connected component of the restrictions of and to coincide iff .
Theorem 4.2. On any connected 1-manifold there exists exactly two orientations.
Proof. If is non-closed connected 1-manifold, then by Theorem \ref{thm:characterization} it can be identified by a homeomorphism to either , or , or . An orientation is defined by its value on . Indeed, for any open set homeomorphic to or , the set is connected, on it there are exactly two linear orders defining the topology: one of them is induced by , the other by . The value of on the corresponding elements of are defined by . On the other hand, there exists an orientation taking each of the two possible values on , because the construction used for the proof of existence above, gives a map satisfying the definition of orientation.
If is a closed connected 1-manifold, then by Theorem \ref{thm:characterization} it can be identified by a homeomorphism with .
As a cyclic order is determined by a linear order on the complement of any point, orientations of a connected closed 1-manifold are defined by orientations of the complement of any point. By Theorem 4.1, the complement has exactly two orientations, so the circle cannot have more.
Corollary 4.3. Any 1-manifold admits an orientation. If the 1-manifold consists of connected components, then it admits orientations.
4.3 Self-homeomorphisms
Theorem 4.4. A map is a homeomorphism iff is a monotone bijection.
Proof. Let be a homeomorphism. First, observe that maps every ray to a ray. Indeed, for any , the map induces a homeomorphism . The rays and are connected components of . Therefore their images are connected components and of .
Observe that rays have the same direction iff one of them is contained in the other one. Therefore two rays of the same dirction are mapped by to rays with the same direction. Thus rays are mapped either all to rays or all to . Thus is monotone.
Let be a monotone bijection. Then the image and preimage under of any open interval are open intervals. Therefore, both and are continuous, and hence is a homeomorphism.
The following theorem can be proved similarly or can be deduced from Theorem 4.4
Theorem 4.5.
- A map is a homeomorphism iff is a monotone bijection.
- A map is a homeomorphism iff is a monotone increasing bijection.
- A map is a homeomorphism iff is a bijection that either preserves or reverses the cyclic order of points on .
A self-homeomorphism of a connected 1-manifold increases with respect to one order or cyclic order iff it increases with respect to the opposite order. In other words, it preserves an orientation iff it preserves the opposite orientation. Since there are only two orientations, this is a property of homeomorphism which does not depend on orientation. Any self-homeomorphism of a connected 1-manifold either preserves orientation, or reverses it.
The half-line does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to admits an orientation reversing map.
Thus, there are 5 topological types of oriented connected 1-manifolds: the topological type of the non-oriented half-line splits into the oriented topological types of and with the orientations induced by the standard order.
4.4 Orientations via coordinate charts
Definition of orientations of 1-manifolds given above is specific for dimension 1, it has no a direct high-dimensional generalization. Here is a more conventional approach to orientations of 1-manifolds, still specific for dimension 1.
Let be a 1-manifold, an open connected set. A homeomorphism , where is or is called a chart or local coordinate system.
4.5 Characterizations of connected compact 1-manifolds in terms of separating points
A subset of a topological space is said to separate if can be presented as a union of two disjoint open sets.
Theorem 4.6. (See [Moore1920].) Let be a connected compact Hausdorff second countable topological space.
- If every two points separate , then is homeomorphic to the circle.
- If each point, with two exceptions, separates , then is homeomorphic to .
Any point splits to two disjoint open rays and .
Theorem 4.7. (See [Ward1936].) Let be a connected locally compact Hausdorff second countable topological space.
- If the complement of each point in consists of two connected components, then is homeomorphic to .
- If contains a point such that is connected and consists of two connected components for each , , then is homeomorphic to .
5 Invariants
5.1 Basic invariants
As follows from the Theorems 3.1 and 3.2 above, the following invariants
- the number of connected components,
- the compactness of each connected component,
- and the number of boundary points of each connected component
determine the topological type of a 1-manifold.
5.2 Homotopy invariants
The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions are trivial. The fundamental group is infinite cyclic group, if the connected component of containing is homeomorphic to circle, and trivial otherwise.
5.3 Homology invariants
Let be a 1-manifold with a finite number of connected components. By Theorem 3.1, it is homeomorphic to a disjoint union
of some numbers of copies of , , and :Then is a free abelian group of rank equal to the number of all connected components of and is a free abelian group of rank equal to the number of closed (compact without boundary) components of .
Relative homology groups: is a free abelian group of rank equal to the number of connected components of without boundary; is a free abelian group of rank equal to the number of compact components of . So,
Numbers , , , and and the topological type of can be recovered from the ranks of these groups.
Above by homology we mean homology with compact support. The homology with closed support (Borel-Moore homology):
The Poincare duality is an isomorphism between usual cohomogy (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So
A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over has non-trivial monodromy, then all the homology groups are trivial.
5.4 Tangent bundle invariants
The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial.
6 Additional structures
6.1 Triangulations
Any 1-manifold admits 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 . Similarly, the topological type of a triangulation of is defined by the number of 1-simplices, which can take any positive integral value.
6.2 Smooth structures
Any 1-manifold admits a smooth structure.
If smooth 1-manifolds and are homeomorphic, then they are also diffeomorphic. Moreover,
Theorem 6.1. Any homeomorphism between two smooth 1-manifolds can be approximated in the -topology by a diffeomorphism.
Proof. By Theorems 4.4 and 4.5, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points.
6.3 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 in a metric space with metric is .
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 with metric is .
For each value of the diameter there is a standard model for the inner metric space. For the four homeomorphism types of connected 1-manifolds these standard models are as follows.
- For with diameter this is .
- For with diameter this is .
- For a circle with inner metric of diameter this is the circle of radius on the plane with the inner metric.
- For with diameter this is .
An inner metric on a connected 1-manifold 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.
7 Constructions
7.1 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 changes 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.
7.2 Connected sums
The notion of connected sum is defined for 1-manifolds, but the connectivity of the outcome is different in dimension 1 compared to other dimensions. Indeed term connected sum can be misleading in dimension 1 since a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of is a disjoint sum of two copies of .
Note that connected sum is only a well defined operation on oriented manifolds and one has to be careful with the orientations. For example
8 Groups of self-homeomorphisms
8.1 Mapping class groups
Recall that the mapping class group of a manifold is the quotient group of the group of all homeomorphisms by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of is .
An orientation reversing homeomorphism cannot be isotopic to an orientation preserving homeomorphism. For auto-homeomorphisms of a connected 1-manifold this is the only obstruction to being isotopic:
Theorem 8.1. Any two auto-homeomorphisms of a connected 1-manifold that are either both orientation preserving, or both orientation reversing are isotopic.
This is a corollary of the following two obvious lemmas.
Lemma 8.2. On rectilinear isotopy. Let be one of the following 1-manifolds: , , or . Let be two monotone bijections that are either both increasing or both decreasing. Then the family with consists of monotone bijections (and hence is an isotopy between and ).
Lemma 8.3. Let be two bijections that either both preserve or both reverse the standard cyclic order of points on . Let and coincide at . Then and are isotopic via the canonical isotopy which is stationary at and is provided on the complement of by stereographic projections and the rectilinear isotopy from Lemma 8.2 of the corresponding self-homeomorphisms of .
Corollary 8.4. and
Remark. All the statements in this section remains true, if everywhere the word homeomorphism is replaced by the word diffeomorphism and is replaced by .
8.2 Homotopy types of groups of self-homeomorphisms
The group contains as a subgroup, which is its deformation retract. It follows from Lemma 8.3. More precisely, for each point , Lemma 8.3 provides a deformation retraction .
Similarly, the group of self-homeomorphisms of isotopic to identity contains as a subgroup, which is its deformation retract.
The groups of self-homeomorphisms of , and which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 8.2 applied to and an arbitrary .
Thus for each connected 1-manifold the group of homeomorphisms isotopic to identity is homotopy equivalent to .
9 Finite group actions
Consider an action of a finite group on a 1-manifold .
9.1 Free actions
For any point , its orbit is a finite set and has an invariant neighborhood whose connected components are disjoint open sets, each of them contains exactly one point of , and either all the components are homeomorphic to , or all homeomorphic to .
If the action is free, then the orbit space is a 1-manifold and the natural projection is a covering.
Therefore the theory of coverings gives a simple classification of free finite group actions on 1-manifolds.
A contractible 1-manifold has no non-trivial covering. Thus, if a free finite group action on 1-manifold has a contractible orbit space , then is a disjoint union of copies of and permutes these copies. In particular, there is no non-trivial free group action on a connected 1-manifold having contractible orbit space.
Coverings with connected are in one-to-one correspondence with subgroups of finite indices of . For each index there is one subgroup, and hence one covering. The total space is homeomorphic to , and the covering is equivalent to . In the corresponding action, the group is cyclic of order , it acts on by rotations.
In this classification of free finite group actions on connected 1-manifolds, the orbit space plays the main role. However, it is easy to reformulate it with emphasis the 1-manifold on which the group acts. This is done in the next two theorems.
Theorem 9.1. There is no non-trivial free finite group action on a contractible 1-manifold.
Theorem 9.2. If a finite group acts freely on the circle than is cyclic. Any finite cyclic group has a linear free action on . Any free action of a finite cyclic group on is conjugate to a linear action.
9.2 Asymmetry of a half-line
Theorem 9.3. There is no non-trivial action of a finite group in $\Rr_+$.
Proof. We will prove that the only homeomorphism $h:\Rr_+\to\Rr_+$ of finite order is the identity. Obsreve, first that any homeomorphism $\Rr_+\to\Rr_+$ preserves the only boundary point $0\in\Rr_+$. Assume that $h$ is a homeomorphism $\Rr_+\to\Rr_+$ of finite order $m$, and there exists $a\in\Rr_+$ such that $h(a)=b\ne a$. Then $h([0,a])=[0,b]$. Without loss of generality, we may assume that $b<a$ (otherwise just replace $h$ by $h^{-1}$).
Then $h([0,a]=[0,b]\subset[0,a]$ and $a\not\in h([0,a])$. By the assumption, $h^m(a)=a$. On the other hand, $h^m(a)\in h^m([0,a])\subset h^{m-1}([0,a])\subset\dots \subset h([0,a])\not\ni a$.
</wikitex>
9.3 Actions on line and segment
Theorem 9.4. The only orientation preserving homeomorphism of finite order is the identity.
Proof. Fix a homeomorphism (say, define it by formula ). Consider a homeomorphism . It preserves orientation (since preserves orientation). So, it is a monotone increasing bijection of finite order. It can be extended to by letting . The extended homeomorphism has the same finite order. But by Theorem 9.3 any such homeomorphism is the identity.
Theorem 9.5. Any orientation reversing homeomorphism of finite order is of order two. It is conjugate to the symmetry against a point.
Proof. An orientation reversing homeomorphism is a monotone decreasing bijection. Consider the function . It is also monotone decreasing bijection and hence there exists a unique such that , that is .
The homeomorphism maps each connected component of to a connected component of . The connected components are open rays and . If each of them is mapped to itself, then defines a homeomorphism of a finite order of the closed rays and . Then by Theorem 9.3, is identity, which contradicts to our assumption. Thus, and . Then preserves the rays, and, by Theorem 9.3, is identity. Thus has order two.
Choose a homeomorphism . Define function by formula . It's a homeomorphism. Together, and form a homeomorphism . As easy to check, .
Theorem 9.6. A non-trivial finite group acting effectively on is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
Proof. As follows from Corollaries 9.4 and 9.5, any non-trivial element of the group is an orientation reversing involution. We have to prove that the group contains at most one such element. Assume that there are two orientation reversing homeomorphisms, and of the line . Their composition preserves orientation. Since it belongs to a finite group, it has finite order. By Theorem 9.4, it is identity. So, and hence . But . Therefore and .
Corollary 9.7. A non-trivial finite group acting effectively on is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
Proof. Any auto-homeomorphism of preserves the boundary and the interior of . Hence an effective finite group action on induces an action of finite group on the interior of . An auto-homeomorphism of is recovered from its restriction to the interior. Moreover, any auto-homeomorphism of has a unique extension to auto-homeomorphism of . The interior is homeomorphic to .
9.4 Actions on circle
{{beginthm|Theorem} Any periodic orientation reversing homeomorphism is an involution (i.e., has period 2). It is conjugate to a symmetry of against its diameter. </div>
Proof. Observe first that any orientation reversing auto-homeomorphism of the circle has a fixed point. One can prove this by elementary arguments, but we just refer to the Lefschetz Fixed Point Theorem: the Lefschetz number of such homeomorphism is 2.
Consider the complement of a fixed point. The homeomorphism resticted to it satisfies the conditions of Corollary 9.5, which gives the required result.
Observe that by theorems 9.5, 9.7, and 9.7 any non-identity periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.
Theorem 9.8. A periodic non-identity orientation preserving homeomorphism has no fixed point. It is conjugate to a rotation.
Proof. If if had a fixed point, then we could consider its restriction to the complement of this point, and by Theorem 9.4 would conclude that it is identity and hence the whole homeomorphism is identity.
For the same reasons, the non-identity powers of our periodic non-identity orientation preserving homeomorphism have no fixed points. Therefore, these powers form a cyclic group freely acting on . See Theorem 9.2.
Theorem 9.9. A finite group acting effectively on 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.
Proof. If all the homeomorphisms in the action preserve orientation, then by Theorem 9.8 the action is free, and the result follows from Theorem 9.2.
Assume that the action contains an orientation reversing homeomorphism. The orientation preserving homeomorphisms from the action form a cyclic subgroup as above. It is of index 2. Its complement consists of orientation reversing involutions. If the subgroup of orientation preserving homeomorphisms is trivial, then the whole group is of order 2 and the only non-trivial element is an orientation resersing involution. When the group contains two orientation preserving homeomorphisms, the whole group is the cartesian product of two cyclic groups of order 2. It is called Klein's Vierergruppe or dihedral group . If the number of orientation preserving homeomorphisms is , then the whole group is called the dihedral group . It is the symmetry group of an -sided regular polygon.
10 Relatives of 1-manifolds
10.1 Non-Hausdorff 1-manifolds
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 the disjoint union of two copies of the line and identify an open set in one of them with its copy in the other one by the identity map. The quotient space is connected and satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. In this way one can construct uncountably many pairwise non-homeomorphic spaces. To prove that they are not homeomorphic, one can use, for example, the topological type of the subset formed by those points that do not separate the space.
11 References
- [Fuks&Rokhlin1984] D. B. Fuks and V. A. Rokhlin, Beginner's course in topology. Geometric chapters. Translated from the Russian by A. Iacob. Universitext. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1984. MR759162 (86a:57001) Zbl 0562.54003
- [Gale1987] D. Gale, The Teaching of Mathematics: The Classification of 1-Manifolds: A Take-Home Exam, Amer. Math. Monthly 94 (1987), no.2, 170–175. MR1541035 Zbl 0621.57001
- [Ghys2001] E. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no.3-4, 329–407. MR1876932 (2003a:37032) Zbl 1044.37033
- [Moore1920] R. L. Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), no.3, 333–347. MR1501148 () Zbl 47.0519.08
- [Ward1936] A. J. Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198. MR1577110 Zbl 62.0693.02
12 External links
- The Encylopedia of Mathematics article on one-dimensional manifolds.
- The Encylopedia of Mathematics article on lines.
- The Wikipedia page about curves.
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 . To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.
For other expositions about -manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Sections 3.1.1.16-19].
2 Examples
- The real line:
- The half-line:
- The circle:
- The closed interval:
3 Topological classification
3.1 Reduction to classification of connected manifolds
The following elementary facts hold for -manifolds of any dimension .
Any manifold is homeomorphic to the disjoint sum of its connected components.
A connected component of an -manifold is a -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
- half-line
- circle
- closed interval .
No two of these manifolds are homeomorphic to each other.
3.3 Characterizing the topological type of a connected 1-manifold
Theorem 3.2.
- Any connected non-compact 1-manifold without boundary is homeomorphic to .
- Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to .
- Any connected closed 1-manifold is homeomorphic to .
- Any connected compact 1-manifold with non-empty boundary is homeomorphic to .
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.
Theorems 3.1 and 3.2 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.
3.4 About proofs of the classification theorems
The proofs of Theorems 3.1 and 3.2 above are elementary. They can be found, e.g., in [Fuks&Rokhlin1984, Sections 3.1.1.16-19]. The core of them are the following simple lemmas:
Lemma 3.4. If a topological space can be represented as the union of a nondecreasing sequence of open subsets, all homeomorphic to , then is homeomorphic to .
3.5 Corollary: homotopy classification
Theorem 3.5. Each connected 1-manifold is either contractible, or homotopy equivalent to circle.
It follows immediately from Theorem 3.1.
3.6 Corollary: cobordisms of 0-manifolds
Theorem 3.6. A compact 0-manifold bounds a compact 1-manifold iff the number of points in is even.
Corollary 3.7. Two compact 0-manifolds are cordant iff their numbers of points are congruent modulo 2.
4 Orders and orientations
4.1 Interval topology
Most of properties specific for 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.
Open intervals and open rays and form the base of the (standard) topology on . This way of introducing a topological structure can be applied in any (linearly) ordered set . On and , the (standard) topology is induced from the standard topology on , and can be described in terms of the induced order.
Theorem 4.1. Every connected non-closed 1-manifold admits exactly two linear orders defining its topology.
Proof. By Theorem 3.2, a coonected non-closed 1-manifold is homeomorphic either to , or , or . On each of these 1-manifolds there are two linear orders, and , defining the topology. We have to prove that there is no other linear order defining the same topology.
A linear order on a set is encoded in the system of sets for . A system of susbsets with comes from a linear order on iff either , or , or for any and only if . In the interval topology on , each and is open.
For each interior point of a connected non-closed 1-manifold , its complement is a disjoint union of two open connected sets and . Any linear order on defining the topology of would give rise to a splitting of each of these two sets into its intesections with disjoint sets and . If defines the topology on , then and are open. Due to connectedness of , either , or . Hence the order coincides with one of the two standard orders.
4.2 Orientations
An orientation of a 1-manifold can be interpreted via linear orders defining the topology on its open subsets homeomorphic to or . An orientation of any of these two 1-manifolds is nothing but such linear order. For a general 1-manifold one need to globalize this idea. It can be done in the same way as in higher dimensions, but there are specifically 1-dimensional approaches to the globalizing.
First, due to the topological classification, one can restrict to just four model 1-manifolds , , and . For the first three of them, an orientation still can be defined as a linear order determining the topology of the manifold. For this approach does not work, but can be adjusted: instead of linear orders one can rely on cyclic orders that define the topology. It is a bit cumbersome, because cyclic orders are more cumbersome than usual linear orders.
There is a more conceptual way, which immitate the classical definition of orientations of differentiable manifolds, but rely, instead of coordinate charts, on local linear orders.
Let be a 1-manifold. A local orientation of at a point is a pair consisting of a neighborhood of homeomorphic to or and a linear order on defining the topology on . Denote by the set of all local orientations of .
An orientation on is a map such that for any and any connected component of the restrictions of and to coincide iff .
Theorem 4.2. On any connected 1-manifold there exists exactly two orientations.
Proof. If is non-closed connected 1-manifold, then by Theorem \ref{thm:characterization} it can be identified by a homeomorphism to either , or , or . An orientation is defined by its value on . Indeed, for any open set homeomorphic to or , the set is connected, on it there are exactly two linear orders defining the topology: one of them is induced by , the other by . The value of on the corresponding elements of are defined by . On the other hand, there exists an orientation taking each of the two possible values on , because the construction used for the proof of existence above, gives a map satisfying the definition of orientation.
If is a closed connected 1-manifold, then by Theorem \ref{thm:characterization} it can be identified by a homeomorphism with .
As a cyclic order is determined by a linear order on the complement of any point, orientations of a connected closed 1-manifold are defined by orientations of the complement of any point. By Theorem 4.1, the complement has exactly two orientations, so the circle cannot have more.
Corollary 4.3. Any 1-manifold admits an orientation. If the 1-manifold consists of connected components, then it admits orientations.
4.3 Self-homeomorphisms
Theorem 4.4. A map is a homeomorphism iff is a monotone bijection.
Proof. Let be a homeomorphism. First, observe that maps every ray to a ray. Indeed, for any , the map induces a homeomorphism . The rays and are connected components of . Therefore their images are connected components and of .
Observe that rays have the same direction iff one of them is contained in the other one. Therefore two rays of the same dirction are mapped by to rays with the same direction. Thus rays are mapped either all to rays or all to . Thus is monotone.
Let be a monotone bijection. Then the image and preimage under of any open interval are open intervals. Therefore, both and are continuous, and hence is a homeomorphism.
The following theorem can be proved similarly or can be deduced from Theorem 4.4
Theorem 4.5.
- A map is a homeomorphism iff is a monotone bijection.
- A map is a homeomorphism iff is a monotone increasing bijection.
- A map is a homeomorphism iff is a bijection that either preserves or reverses the cyclic order of points on .
A self-homeomorphism of a connected 1-manifold increases with respect to one order or cyclic order iff it increases with respect to the opposite order. In other words, it preserves an orientation iff it preserves the opposite orientation. Since there are only two orientations, this is a property of homeomorphism which does not depend on orientation. Any self-homeomorphism of a connected 1-manifold either preserves orientation, or reverses it.
The half-line does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to admits an orientation reversing map.
Thus, there are 5 topological types of oriented connected 1-manifolds: the topological type of the non-oriented half-line splits into the oriented topological types of and with the orientations induced by the standard order.
4.4 Orientations via coordinate charts
Definition of orientations of 1-manifolds given above is specific for dimension 1, it has no a direct high-dimensional generalization. Here is a more conventional approach to orientations of 1-manifolds, still specific for dimension 1.
Let be a 1-manifold, an open connected set. A homeomorphism , where is or is called a chart or local coordinate system.
4.5 Characterizations of connected compact 1-manifolds in terms of separating points
A subset of a topological space is said to separate if can be presented as a union of two disjoint open sets.
Theorem 4.6. (See [Moore1920].) Let be a connected compact Hausdorff second countable topological space.
- If every two points separate , then is homeomorphic to the circle.
- If each point, with two exceptions, separates , then is homeomorphic to .
Any point splits to two disjoint open rays and .
Theorem 4.7. (See [Ward1936].) Let be a connected locally compact Hausdorff second countable topological space.
- If the complement of each point in consists of two connected components, then is homeomorphic to .
- If contains a point such that is connected and consists of two connected components for each , , then is homeomorphic to .
5 Invariants
5.1 Basic invariants
As follows from the Theorems 3.1 and 3.2 above, the following invariants
- the number of connected components,
- the compactness of each connected component,
- and the number of boundary points of each connected component
determine the topological type of a 1-manifold.
5.2 Homotopy invariants
The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions are trivial. The fundamental group is infinite cyclic group, if the connected component of containing is homeomorphic to circle, and trivial otherwise.
5.3 Homology invariants
Let be a 1-manifold with a finite number of connected components. By Theorem 3.1, it is homeomorphic to a disjoint union
of some numbers of copies of , , and :Then is a free abelian group of rank equal to the number of all connected components of and is a free abelian group of rank equal to the number of closed (compact without boundary) components of .
Relative homology groups: is a free abelian group of rank equal to the number of connected components of without boundary; is a free abelian group of rank equal to the number of compact components of . So,
Numbers , , , and and the topological type of can be recovered from the ranks of these groups.
Above by homology we mean homology with compact support. The homology with closed support (Borel-Moore homology):
The Poincare duality is an isomorphism between usual cohomogy (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So
A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over has non-trivial monodromy, then all the homology groups are trivial.
5.4 Tangent bundle invariants
The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial.
6 Additional structures
6.1 Triangulations
Any 1-manifold admits 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 . Similarly, the topological type of a triangulation of is defined by the number of 1-simplices, which can take any positive integral value.
6.2 Smooth structures
Any 1-manifold admits a smooth structure.
If smooth 1-manifolds and are homeomorphic, then they are also diffeomorphic. Moreover,
Theorem 6.1. Any homeomorphism between two smooth 1-manifolds can be approximated in the -topology by a diffeomorphism.
Proof. By Theorems 4.4 and 4.5, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points.
6.3 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 in a metric space with metric is .
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 with metric is .
For each value of the diameter there is a standard model for the inner metric space. For the four homeomorphism types of connected 1-manifolds these standard models are as follows.
- For with diameter this is .
- For with diameter this is .
- For a circle with inner metric of diameter this is the circle of radius on the plane with the inner metric.
- For with diameter this is .
An inner metric on a connected 1-manifold 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.
7 Constructions
7.1 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 changes 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.
7.2 Connected sums
The notion of connected sum is defined for 1-manifolds, but the connectivity of the outcome is different in dimension 1 compared to other dimensions. Indeed term connected sum can be misleading in dimension 1 since a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of is a disjoint sum of two copies of .
Note that connected sum is only a well defined operation on oriented manifolds and one has to be careful with the orientations. For example
8 Groups of self-homeomorphisms
8.1 Mapping class groups
Recall that the mapping class group of a manifold is the quotient group of the group of all homeomorphisms by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of is .
An orientation reversing homeomorphism cannot be isotopic to an orientation preserving homeomorphism. For auto-homeomorphisms of a connected 1-manifold this is the only obstruction to being isotopic:
Theorem 8.1. Any two auto-homeomorphisms of a connected 1-manifold that are either both orientation preserving, or both orientation reversing are isotopic.
This is a corollary of the following two obvious lemmas.
Lemma 8.2. On rectilinear isotopy. Let be one of the following 1-manifolds: , , or . Let be two monotone bijections that are either both increasing or both decreasing. Then the family with consists of monotone bijections (and hence is an isotopy between and ).
Lemma 8.3. Let be two bijections that either both preserve or both reverse the standard cyclic order of points on . Let and coincide at . Then and are isotopic via the canonical isotopy which is stationary at and is provided on the complement of by stereographic projections and the rectilinear isotopy from Lemma 8.2 of the corresponding self-homeomorphisms of .
Corollary 8.4. and
Remark. All the statements in this section remains true, if everywhere the word homeomorphism is replaced by the word diffeomorphism and is replaced by .
8.2 Homotopy types of groups of self-homeomorphisms
The group contains as a subgroup, which is its deformation retract. It follows from Lemma 8.3. More precisely, for each point , Lemma 8.3 provides a deformation retraction .
Similarly, the group of self-homeomorphisms of isotopic to identity contains as a subgroup, which is its deformation retract.
The groups of self-homeomorphisms of , and which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 8.2 applied to and an arbitrary .
Thus for each connected 1-manifold the group of homeomorphisms isotopic to identity is homotopy equivalent to .
9 Finite group actions
Consider an action of a finite group on a 1-manifold .
9.1 Free actions
For any point , its orbit is a finite set and has an invariant neighborhood whose connected components are disjoint open sets, each of them contains exactly one point of , and either all the components are homeomorphic to , or all homeomorphic to .
If the action is free, then the orbit space is a 1-manifold and the natural projection is a covering.
Therefore the theory of coverings gives a simple classification of free finite group actions on 1-manifolds.
A contractible 1-manifold has no non-trivial covering. Thus, if a free finite group action on 1-manifold has a contractible orbit space , then is a disjoint union of copies of and permutes these copies. In particular, there is no non-trivial free group action on a connected 1-manifold having contractible orbit space.
Coverings with connected are in one-to-one correspondence with subgroups of finite indices of . For each index there is one subgroup, and hence one covering. The total space is homeomorphic to , and the covering is equivalent to . In the corresponding action, the group is cyclic of order , it acts on by rotations.
In this classification of free finite group actions on connected 1-manifolds, the orbit space plays the main role. However, it is easy to reformulate it with emphasis the 1-manifold on which the group acts. This is done in the next two theorems.
Theorem 9.1. There is no non-trivial free finite group action on a contractible 1-manifold.
Theorem 9.2. If a finite group acts freely on the circle than is cyclic. Any finite cyclic group has a linear free action on . Any free action of a finite cyclic group on is conjugate to a linear action.
9.2 Asymmetry of a half-line
Theorem 9.3. There is no non-trivial action of a finite group in $\Rr_+$.
Proof. We will prove that the only homeomorphism $h:\Rr_+\to\Rr_+$ of finite order is the identity. Obsreve, first that any homeomorphism $\Rr_+\to\Rr_+$ preserves the only boundary point $0\in\Rr_+$. Assume that $h$ is a homeomorphism $\Rr_+\to\Rr_+$ of finite order $m$, and there exists $a\in\Rr_+$ such that $h(a)=b\ne a$. Then $h([0,a])=[0,b]$. Without loss of generality, we may assume that $b<a$ (otherwise just replace $h$ by $h^{-1}$).
Then $h([0,a]=[0,b]\subset[0,a]$ and $a\not\in h([0,a])$. By the assumption, $h^m(a)=a$. On the other hand, $h^m(a)\in h^m([0,a])\subset h^{m-1}([0,a])\subset\dots \subset h([0,a])\not\ni a$.
</wikitex>
9.3 Actions on line and segment
Theorem 9.4. The only orientation preserving homeomorphism of finite order is the identity.
Proof. Fix a homeomorphism (say, define it by formula ). Consider a homeomorphism . It preserves orientation (since preserves orientation). So, it is a monotone increasing bijection of finite order. It can be extended to by letting . The extended homeomorphism has the same finite order. But by Theorem 9.3 any such homeomorphism is the identity.
Theorem 9.5. Any orientation reversing homeomorphism of finite order is of order two. It is conjugate to the symmetry against a point.
Proof. An orientation reversing homeomorphism is a monotone decreasing bijection. Consider the function . It is also monotone decreasing bijection and hence there exists a unique such that , that is .
The homeomorphism maps each connected component of to a connected component of . The connected components are open rays and . If each of them is mapped to itself, then defines a homeomorphism of a finite order of the closed rays and . Then by Theorem 9.3, is identity, which contradicts to our assumption. Thus, and . Then preserves the rays, and, by Theorem 9.3, is identity. Thus has order two.
Choose a homeomorphism . Define function by formula . It's a homeomorphism. Together, and form a homeomorphism . As easy to check, .
Theorem 9.6. A non-trivial finite group acting effectively on is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
Proof. As follows from Corollaries 9.4 and 9.5, any non-trivial element of the group is an orientation reversing involution. We have to prove that the group contains at most one such element. Assume that there are two orientation reversing homeomorphisms, and of the line . Their composition preserves orientation. Since it belongs to a finite group, it has finite order. By Theorem 9.4, it is identity. So, and hence . But . Therefore and .
Corollary 9.7. A non-trivial finite group acting effectively on is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
Proof. Any auto-homeomorphism of preserves the boundary and the interior of . Hence an effective finite group action on induces an action of finite group on the interior of . An auto-homeomorphism of is recovered from its restriction to the interior. Moreover, any auto-homeomorphism of has a unique extension to auto-homeomorphism of . The interior is homeomorphic to .
9.4 Actions on circle
{{beginthm|Theorem} Any periodic orientation reversing homeomorphism is an involution (i.e., has period 2). It is conjugate to a symmetry of against its diameter. </div>
Proof. Observe first that any orientation reversing auto-homeomorphism of the circle has a fixed point. One can prove this by elementary arguments, but we just refer to the Lefschetz Fixed Point Theorem: the Lefschetz number of such homeomorphism is 2.
Consider the complement of a fixed point. The homeomorphism resticted to it satisfies the conditions of Corollary 9.5, which gives the required result.
Observe that by theorems 9.5, 9.7, and 9.7 any non-identity periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.
Theorem 9.8. A periodic non-identity orientation preserving homeomorphism has no fixed point. It is conjugate to a rotation.
Proof. If if had a fixed point, then we could consider its restriction to the complement of this point, and by Theorem 9.4 would conclude that it is identity and hence the whole homeomorphism is identity.
For the same reasons, the non-identity powers of our periodic non-identity orientation preserving homeomorphism have no fixed points. Therefore, these powers form a cyclic group freely acting on . See Theorem 9.2.
Theorem 9.9. A finite group acting effectively on 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.
Proof. If all the homeomorphisms in the action preserve orientation, then by Theorem 9.8 the action is free, and the result follows from Theorem 9.2.
Assume that the action contains an orientation reversing homeomorphism. The orientation preserving homeomorphisms from the action form a cyclic subgroup as above. It is of index 2. Its complement consists of orientation reversing involutions. If the subgroup of orientation preserving homeomorphisms is trivial, then the whole group is of order 2 and the only non-trivial element is an orientation resersing involution. When the group contains two orientation preserving homeomorphisms, the whole group is the cartesian product of two cyclic groups of order 2. It is called Klein's Vierergruppe or dihedral group . If the number of orientation preserving homeomorphisms is , then the whole group is called the dihedral group . It is the symmetry group of an -sided regular polygon.
10 Relatives of 1-manifolds
10.1 Non-Hausdorff 1-manifolds
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 the disjoint union of two copies of the line and identify an open set in one of them with its copy in the other one by the identity map. The quotient space is connected and satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. In this way one can construct uncountably many pairwise non-homeomorphic spaces. To prove that they are not homeomorphic, one can use, for example, the topological type of the subset formed by those points that do not separate the space.
11 References
- [Fuks&Rokhlin1984] D. B. Fuks and V. A. Rokhlin, Beginner's course in topology. Geometric chapters. Translated from the Russian by A. Iacob. Universitext. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1984. MR759162 (86a:57001) Zbl 0562.54003
- [Gale1987] D. Gale, The Teaching of Mathematics: The Classification of 1-Manifolds: A Take-Home Exam, Amer. Math. Monthly 94 (1987), no.2, 170–175. MR1541035 Zbl 0621.57001
- [Ghys2001] E. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no.3-4, 329–407. MR1876932 (2003a:37032) Zbl 1044.37033
- [Moore1920] R. L. Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), no.3, 333–347. MR1501148 () Zbl 47.0519.08
- [Ward1936] A. J. Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198. MR1577110 Zbl 62.0693.02
12 External links
- The Encylopedia of Mathematics article on one-dimensional manifolds.
- The Encylopedia of Mathematics article on lines.
- The Wikipedia page about curves.
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 . To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.
For other expositions about -manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Sections 3.1.1.16-19].
2 Examples
- The real line:
- The half-line:
- The circle:
- The closed interval:
3 Topological classification
3.1 Reduction to classification of connected manifolds
The following elementary facts hold for -manifolds of any dimension .
Any manifold is homeomorphic to the disjoint sum of its connected components.
A connected component of an -manifold is a -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
- half-line
- circle
- closed interval .
No two of these manifolds are homeomorphic to each other.
3.3 Characterizing the topological type of a connected 1-manifold
Theorem 3.2.
- Any connected non-compact 1-manifold without boundary is homeomorphic to .
- Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to .
- Any connected closed 1-manifold is homeomorphic to .
- Any connected compact 1-manifold with non-empty boundary is homeomorphic to .
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.
Theorems 3.1 and 3.2 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.
3.4 About proofs of the classification theorems
The proofs of Theorems 3.1 and 3.2 above are elementary. They can be found, e.g., in [Fuks&Rokhlin1984, Sections 3.1.1.16-19]. The core of them are the following simple lemmas:
Lemma 3.4. If a topological space can be represented as the union of a nondecreasing sequence of open subsets, all homeomorphic to , then is homeomorphic to .
3.5 Corollary: homotopy classification
Theorem 3.5. Each connected 1-manifold is either contractible, or homotopy equivalent to circle.
It follows immediately from Theorem 3.1.
3.6 Corollary: cobordisms of 0-manifolds
Theorem 3.6. A compact 0-manifold bounds a compact 1-manifold iff the number of points in is even.
Corollary 3.7. Two compact 0-manifolds are cordant iff their numbers of points are congruent modulo 2.
4 Orders and orientations
4.1 Interval topology
Most of properties specific for 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.
Open intervals and open rays and form the base of the (standard) topology on . This way of introducing a topological structure can be applied in any (linearly) ordered set . On and , the (standard) topology is induced from the standard topology on , and can be described in terms of the induced order.
Theorem 4.1. Every connected non-closed 1-manifold admits exactly two linear orders defining its topology.
Proof. By Theorem 3.2, a coonected non-closed 1-manifold is homeomorphic either to , or , or . On each of these 1-manifolds there are two linear orders, and , defining the topology. We have to prove that there is no other linear order defining the same topology.
A linear order on a set is encoded in the system of sets for . A system of susbsets with comes from a linear order on iff either , or , or for any and only if . In the interval topology on , each and is open.
For each interior point of a connected non-closed 1-manifold , its complement is a disjoint union of two open connected sets and . Any linear order on defining the topology of would give rise to a splitting of each of these two sets into its intesections with disjoint sets and . If defines the topology on , then and are open. Due to connectedness of , either , or . Hence the order coincides with one of the two standard orders.
4.2 Orientations
An orientation of a 1-manifold can be interpreted via linear orders defining the topology on its open subsets homeomorphic to or . An orientation of any of these two 1-manifolds is nothing but such linear order. For a general 1-manifold one need to globalize this idea. It can be done in the same way as in higher dimensions, but there are specifically 1-dimensional approaches to the globalizing.
First, due to the topological classification, one can restrict to just four model 1-manifolds , , and . For the first three of them, an orientation still can be defined as a linear order determining the topology of the manifold. For this approach does not work, but can be adjusted: instead of linear orders one can rely on cyclic orders that define the topology. It is a bit cumbersome, because cyclic orders are more cumbersome than usual linear orders.
There is a more conceptual way, which immitate the classical definition of orientations of differentiable manifolds, but rely, instead of coordinate charts, on local linear orders.
Let be a 1-manifold. A local orientation of at a point is a pair consisting of a neighborhood of homeomorphic to or and a linear order on defining the topology on . Denote by the set of all local orientations of .
An orientation on is a map such that for any and any connected component of the restrictions of and to coincide iff .
Theorem 4.2. On any connected 1-manifold there exists exactly two orientations.
Proof. If is non-closed connected 1-manifold, then by Theorem \ref{thm:characterization} it can be identified by a homeomorphism to either , or , or . An orientation is defined by its value on . Indeed, for any open set homeomorphic to or , the set is connected, on it there are exactly two linear orders defining the topology: one of them is induced by , the other by . The value of on the corresponding elements of are defined by . On the other hand, there exists an orientation taking each of the two possible values on , because the construction used for the proof of existence above, gives a map satisfying the definition of orientation.
If is a closed connected 1-manifold, then by Theorem \ref{thm:characterization} it can be identified by a homeomorphism with .
As a cyclic order is determined by a linear order on the complement of any point, orientations of a connected closed 1-manifold are defined by orientations of the complement of any point. By Theorem 4.1, the complement has exactly two orientations, so the circle cannot have more.
Corollary 4.3. Any 1-manifold admits an orientation. If the 1-manifold consists of connected components, then it admits orientations.
4.3 Self-homeomorphisms
Theorem 4.4. A map is a homeomorphism iff is a monotone bijection.
Proof. Let be a homeomorphism. First, observe that maps every ray to a ray. Indeed, for any , the map induces a homeomorphism . The rays and are connected components of . Therefore their images are connected components and of .
Observe that rays have the same direction iff one of them is contained in the other one. Therefore two rays of the same dirction are mapped by to rays with the same direction. Thus rays are mapped either all to rays or all to . Thus is monotone.
Let be a monotone bijection. Then the image and preimage under of any open interval are open intervals. Therefore, both and are continuous, and hence is a homeomorphism.
The following theorem can be proved similarly or can be deduced from Theorem 4.4
Theorem 4.5.
- A map is a homeomorphism iff is a monotone bijection.
- A map is a homeomorphism iff is a monotone increasing bijection.
- A map is a homeomorphism iff is a bijection that either preserves or reverses the cyclic order of points on .
A self-homeomorphism of a connected 1-manifold increases with respect to one order or cyclic order iff it increases with respect to the opposite order. In other words, it preserves an orientation iff it preserves the opposite orientation. Since there are only two orientations, this is a property of homeomorphism which does not depend on orientation. Any self-homeomorphism of a connected 1-manifold either preserves orientation, or reverses it.
The half-line does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to admits an orientation reversing map.
Thus, there are 5 topological types of oriented connected 1-manifolds: the topological type of the non-oriented half-line splits into the oriented topological types of and with the orientations induced by the standard order.
4.4 Orientations via coordinate charts
Definition of orientations of 1-manifolds given above is specific for dimension 1, it has no a direct high-dimensional generalization. Here is a more conventional approach to orientations of 1-manifolds, still specific for dimension 1.
Let be a 1-manifold, an open connected set. A homeomorphism , where is or is called a chart or local coordinate system.
4.5 Characterizations of connected compact 1-manifolds in terms of separating points
A subset of a topological space is said to separate if can be presented as a union of two disjoint open sets.
Theorem 4.6. (See [Moore1920].) Let be a connected compact Hausdorff second countable topological space.
- If every two points separate , then is homeomorphic to the circle.
- If each point, with two exceptions, separates , then is homeomorphic to .
Any point splits to two disjoint open rays and .
Theorem 4.7. (See [Ward1936].) Let be a connected locally compact Hausdorff second countable topological space.
- If the complement of each point in consists of two connected components, then is homeomorphic to .
- If contains a point such that is connected and consists of two connected components for each , , then is homeomorphic to .
5 Invariants
5.1 Basic invariants
As follows from the Theorems 3.1 and 3.2 above, the following invariants
- the number of connected components,
- the compactness of each connected component,
- and the number of boundary points of each connected component
determine the topological type of a 1-manifold.
5.2 Homotopy invariants
The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions are trivial. The fundamental group is infinite cyclic group, if the connected component of containing is homeomorphic to circle, and trivial otherwise.
5.3 Homology invariants
Let be a 1-manifold with a finite number of connected components. By Theorem 3.1, it is homeomorphic to a disjoint union
of some numbers of copies of , , and :Then is a free abelian group of rank equal to the number of all connected components of and is a free abelian group of rank equal to the number of closed (compact without boundary) components of .
Relative homology groups: is a free abelian group of rank equal to the number of connected components of without boundary; is a free abelian group of rank equal to the number of compact components of . So,
Numbers , , , and and the topological type of can be recovered from the ranks of these groups.
Above by homology we mean homology with compact support. The homology with closed support (Borel-Moore homology):
The Poincare duality is an isomorphism between usual cohomogy (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So
A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over has non-trivial monodromy, then all the homology groups are trivial.
5.4 Tangent bundle invariants
The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial.
6 Additional structures
6.1 Triangulations
Any 1-manifold admits 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 . Similarly, the topological type of a triangulation of is defined by the number of 1-simplices, which can take any positive integral value.
6.2 Smooth structures
Any 1-manifold admits a smooth structure.
If smooth 1-manifolds and are homeomorphic, then they are also diffeomorphic. Moreover,
Theorem 6.1. Any homeomorphism between two smooth 1-manifolds can be approximated in the -topology by a diffeomorphism.
Proof. By Theorems 4.4 and 4.5, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points.
6.3 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 in a metric space with metric is .
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 with metric is .
For each value of the diameter there is a standard model for the inner metric space. For the four homeomorphism types of connected 1-manifolds these standard models are as follows.
- For with diameter this is .
- For with diameter this is .
- For a circle with inner metric of diameter this is the circle of radius on the plane with the inner metric.
- For with diameter this is .
An inner metric on a connected 1-manifold 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.
7 Constructions
7.1 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 changes 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.
7.2 Connected sums
The notion of connected sum is defined for 1-manifolds, but the connectivity of the outcome is different in dimension 1 compared to other dimensions. Indeed term connected sum can be misleading in dimension 1 since a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of is a disjoint sum of two copies of .
Note that connected sum is only a well defined operation on oriented manifolds and one has to be careful with the orientations. For example
8 Groups of self-homeomorphisms
8.1 Mapping class groups
Recall that the mapping class group of a manifold is the quotient group of the group of all homeomorphisms by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of is .
An orientation reversing homeomorphism cannot be isotopic to an orientation preserving homeomorphism. For auto-homeomorphisms of a connected 1-manifold this is the only obstruction to being isotopic:
Theorem 8.1. Any two auto-homeomorphisms of a connected 1-manifold that are either both orientation preserving, or both orientation reversing are isotopic.
This is a corollary of the following two obvious lemmas.
Lemma 8.2. On rectilinear isotopy. Let be one of the following 1-manifolds: , , or . Let be two monotone bijections that are either both increasing or both decreasing. Then the family with consists of monotone bijections (and hence is an isotopy between and ).
Lemma 8.3. Let be two bijections that either both preserve or both reverse the standard cyclic order of points on . Let and coincide at . Then and are isotopic via the canonical isotopy which is stationary at and is provided on the complement of by stereographic projections and the rectilinear isotopy from Lemma 8.2 of the corresponding self-homeomorphisms of .
Corollary 8.4. and
Remark. All the statements in this section remains true, if everywhere the word homeomorphism is replaced by the word diffeomorphism and is replaced by .
8.2 Homotopy types of groups of self-homeomorphisms
The group contains as a subgroup, which is its deformation retract. It follows from Lemma 8.3. More precisely, for each point , Lemma 8.3 provides a deformation retraction .
Similarly, the group of self-homeomorphisms of isotopic to identity contains as a subgroup, which is its deformation retract.
The groups of self-homeomorphisms of , and which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 8.2 applied to and an arbitrary .
Thus for each connected 1-manifold the group of homeomorphisms isotopic to identity is homotopy equivalent to .
9 Finite group actions
Consider an action of a finite group on a 1-manifold .
9.1 Free actions
For any point , its orbit is a finite set and has an invariant neighborhood whose connected components are disjoint open sets, each of them contains exactly one point of , and either all the components are homeomorphic to , or all homeomorphic to .
If the action is free, then the orbit space is a 1-manifold and the natural projection is a covering.
Therefore the theory of coverings gives a simple classification of free finite group actions on 1-manifolds.
A contractible 1-manifold has no non-trivial covering. Thus, if a free finite group action on 1-manifold has a contractible orbit space , then is a disjoint union of copies of and permutes these copies. In particular, there is no non-trivial free group action on a connected 1-manifold having contractible orbit space.
Coverings with connected are in one-to-one correspondence with subgroups of finite indices of . For each index there is one subgroup, and hence one covering. The total space is homeomorphic to , and the covering is equivalent to . In the corresponding action, the group is cyclic of order , it acts on by rotations.
In this classification of free finite group actions on connected 1-manifolds, the orbit space plays the main role. However, it is easy to reformulate it with emphasis the 1-manifold on which the group acts. This is done in the next two theorems.
Theorem 9.1. There is no non-trivial free finite group action on a contractible 1-manifold.
Theorem 9.2. If a finite group acts freely on the circle than is cyclic. Any finite cyclic group has a linear free action on . Any free action of a finite cyclic group on is conjugate to a linear action.
9.2 Asymmetry of a half-line
Theorem 9.3. There is no non-trivial action of a finite group in $\Rr_+$.
Proof. We will prove that the only homeomorphism $h:\Rr_+\to\Rr_+$ of finite order is the identity. Obsreve, first that any homeomorphism $\Rr_+\to\Rr_+$ preserves the only boundary point $0\in\Rr_+$. Assume that $h$ is a homeomorphism $\Rr_+\to\Rr_+$ of finite order $m$, and there exists $a\in\Rr_+$ such that $h(a)=b\ne a$. Then $h([0,a])=[0,b]$. Without loss of generality, we may assume that $b<a$ (otherwise just replace $h$ by $h^{-1}$).
Then $h([0,a]=[0,b]\subset[0,a]$ and $a\not\in h([0,a])$. By the assumption, $h^m(a)=a$. On the other hand, $h^m(a)\in h^m([0,a])\subset h^{m-1}([0,a])\subset\dots \subset h([0,a])\not\ni a$.
</wikitex>
9.3 Actions on line and segment
Theorem 9.4. The only orientation preserving homeomorphism of finite order is the identity.
Proof. Fix a homeomorphism (say, define it by formula ). Consider a homeomorphism . It preserves orientation (since preserves orientation). So, it is a monotone increasing bijection of finite order. It can be extended to by letting . The extended homeomorphism has the same finite order. But by Theorem 9.3 any such homeomorphism is the identity.
Theorem 9.5. Any orientation reversing homeomorphism of finite order is of order two. It is conjugate to the symmetry against a point.
Proof. An orientation reversing homeomorphism is a monotone decreasing bijection. Consider the function . It is also monotone decreasing bijection and hence there exists a unique such that , that is .
The homeomorphism maps each connected component of to a connected component of . The connected components are open rays and . If each of them is mapped to itself, then defines a homeomorphism of a finite order of the closed rays and . Then by Theorem 9.3, is identity, which contradicts to our assumption. Thus, and . Then preserves the rays, and, by Theorem 9.3, is identity. Thus has order two.
Choose a homeomorphism . Define function by formula . It's a homeomorphism. Together, and form a homeomorphism . As easy to check, .
Theorem 9.6. A non-trivial finite group acting effectively on is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
Proof. As follows from Corollaries 9.4 and 9.5, any non-trivial element of the group is an orientation reversing involution. We have to prove that the group contains at most one such element. Assume that there are two orientation reversing homeomorphisms, and of the line . Their composition preserves orientation. Since it belongs to a finite group, it has finite order. By Theorem 9.4, it is identity. So, and hence . But . Therefore and .
Corollary 9.7. A non-trivial finite group acting effectively on is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
Proof. Any auto-homeomorphism of preserves the boundary and the interior of . Hence an effective finite group action on induces an action of finite group on the interior of . An auto-homeomorphism of is recovered from its restriction to the interior. Moreover, any auto-homeomorphism of has a unique extension to auto-homeomorphism of . The interior is homeomorphic to .
9.4 Actions on circle
{{beginthm|Theorem} Any periodic orientation reversing homeomorphism is an involution (i.e., has period 2). It is conjugate to a symmetry of against its diameter. </div>
Proof. Observe first that any orientation reversing auto-homeomorphism of the circle has a fixed point. One can prove this by elementary arguments, but we just refer to the Lefschetz Fixed Point Theorem: the Lefschetz number of such homeomorphism is 2.
Consider the complement of a fixed point. The homeomorphism resticted to it satisfies the conditions of Corollary 9.5, which gives the required result.
Observe that by theorems 9.5, 9.7, and 9.7 any non-identity periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.
Theorem 9.8. A periodic non-identity orientation preserving homeomorphism has no fixed point. It is conjugate to a rotation.
Proof. If if had a fixed point, then we could consider its restriction to the complement of this point, and by Theorem 9.4 would conclude that it is identity and hence the whole homeomorphism is identity.
For the same reasons, the non-identity powers of our periodic non-identity orientation preserving homeomorphism have no fixed points. Therefore, these powers form a cyclic group freely acting on . See Theorem 9.2.
Theorem 9.9. A finite group acting effectively on 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.
Proof. If all the homeomorphisms in the action preserve orientation, then by Theorem 9.8 the action is free, and the result follows from Theorem 9.2.
Assume that the action contains an orientation reversing homeomorphism. The orientation preserving homeomorphisms from the action form a cyclic subgroup as above. It is of index 2. Its complement consists of orientation reversing involutions. If the subgroup of orientation preserving homeomorphisms is trivial, then the whole group is of order 2 and the only non-trivial element is an orientation resersing involution. When the group contains two orientation preserving homeomorphisms, the whole group is the cartesian product of two cyclic groups of order 2. It is called Klein's Vierergruppe or dihedral group . If the number of orientation preserving homeomorphisms is , then the whole group is called the dihedral group . It is the symmetry group of an -sided regular polygon.
10 Relatives of 1-manifolds
10.1 Non-Hausdorff 1-manifolds
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 the disjoint union of two copies of the line and identify an open set in one of them with its copy in the other one by the identity map. The quotient space is connected and satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. In this way one can construct uncountably many pairwise non-homeomorphic spaces. To prove that they are not homeomorphic, one can use, for example, the topological type of the subset formed by those points that do not separate the space.
11 References
- [Fuks&Rokhlin1984] D. B. Fuks and V. A. Rokhlin, Beginner's course in topology. Geometric chapters. Translated from the Russian by A. Iacob. Universitext. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1984. MR759162 (86a:57001) Zbl 0562.54003
- [Gale1987] D. Gale, The Teaching of Mathematics: The Classification of 1-Manifolds: A Take-Home Exam, Amer. Math. Monthly 94 (1987), no.2, 170–175. MR1541035 Zbl 0621.57001
- [Ghys2001] E. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no.3-4, 329–407. MR1876932 (2003a:37032) Zbl 1044.37033
- [Moore1920] R. L. Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), no.3, 333–347. MR1501148 () Zbl 47.0519.08
- [Ward1936] A. J. Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198. MR1577110 Zbl 62.0693.02
12 External links
- The Encylopedia of Mathematics article on one-dimensional manifolds.
- The Encylopedia of Mathematics article on lines.
- The Wikipedia page about curves.