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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 or -manifold.
The material presented below is not well represented in the literature. This happened, probably, because it is considered too simple. It was difficult to find appropriate references. The proofs are really elementary, but use ideas that are barely applicable in similar high-dimensional situations. That's why most the results below are provided with proofs.
Besides the adjacent field of dynamics, there is no research activity related to the topology of 1-manifolds. The dynamical topics have been left outside the scope of this text. In particular, no infinite group actions are considered, although actions of finite groups on 1-manifolds are on this side of the natural borderline, and a complete account of them is presented.
2 Construction and examples
2.1 Examples of connected 1-manifolds
- The real line:
- The half-line:
- The circle:
- The closed interval:
If is an -manifold and is its boundary (i.e., the set of points of that do not have neighborhoods homeomorphic to the Euclidean space ), then the quotient of the disjoint union of two copies of by the identity map is an -manifold with empty boundary, called the double of and denoted by .
This operation is well defined up to homeomorphism. It gives a natural embedding of a manifold with boundary into a manifold without boundary (i.e., with empty boundary) and allows one to reduce many problems about manifolds with boundary to problems about manifolds without boundary.
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 -manifold is homeomorphic to one of the following four 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
- Any connected non-compact -manifold without boundary is homeomorphic to .
- Any connected non-compact -manifold with non-empty boundary is homeomorphic to .
- Any connected closed -manifold is homeomorphic to .
- Any connected compact -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 desire. 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 188.8.131.52-19] and [Gale1987]. In the case of 1-manifolds without boundary, the proofs are based on the following simple lemmas:
Lemma 3.3. Any connected -manifold covered by two open sets homeomorphic to is homeomorphic either to or to .
Under assumptions of Lemma 3.3, the 1-manifold is homeomorphic to iff the intersection of two open sets is connected, and it is homeomorphic to iff the intersection consists of two connected components (the intersection cannot have more than two components).
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 .
From the topological classification for 1-manifolds without boundary, the classification for 1-manifolds with non-empty boundary is obtained using the doubling operation, see Section 2.2 above.
3.5 Corollary: homotopy classification
Theorem 3.5. Each connected -manifold is either contractible, or homotopy equivalent to circle.
This follows immediately from Theorem 3.1.
3.6 Corollary: Cobordant 0-manifolds
Theorem 3.6. A compact -manifold bounds a compact -manifold iff the number of points in is even.
Corollary 3.7. Two compact -manifolds are cobordant iff their numbers of points are congruent modulo .
3.7 Characterizing connected 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 3.8. (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 3.9. (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 .
4 Orders and orientations
4.1 Interval topology
Most properties specific for 1-manifolds can be related to the fact that the topological structure on a connected 1-manifold is defined by linear or cyclic ordering of its points.
Open intervals form a base of the standard topology on . This way of introducing a topological structure can be applied in any (linearly) ordered set (though in a general linearly ordered set one should include into the base, together with open intervals , also open rays and ). On and , the standard topology is induced from the standard topology on , and can be described in terms of the order.
Theorem 4.1. Every connected non-closed -manifold admits exactly two linear orders defining its topology.
Proof. A linear order on a set is encoded in the system of rays for .
By Theorem 3.2, a connected 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. For these orders, the rays and are defined by the topology: they are just the connected components of .
For any other linear order defining the same topology on , the rays and are open and intersect the connected components and of in disjoint open sets. By connectedness of and , one of them coincides with , the other with . Hence, coincides with one of the standard orders, either with , or .
An orientation of a 1-manifold can be interpreted via linear orderings on its open subsets homeomorphic to or . An orientation of or is nothing but one of the two linear orders defining the topological structure.
Relation to the general homological definition of orientation. Recall that in high dimensional situations orientation of an open set is defined as a coherent choice of generators in homology groups for . In our case and the group is generated by a homology class of a singular cycle consisting of a single singular 1-simplex , which is an embedding with . There are two generators: one is represented by a monotone increasing , another, by a monotone decreasing . A choice of linear order on allows one to distinguish one of the generators of : namely, the one represented by which is monotone increasing with respect to the order chosen on .
For extending the notion of orientation to a general 1-manifold, one needs to globalize the idea of linear order. It can be done in several ways.
For example, due to the topological classification, one can restrict to just four model 1-manifolds: , , and . For , and , 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 order one can rely on cyclic orders that define the topology. However, this is a bit awkward, as cyclic orders are more cumbersome than usual linear orders.
There is a more conceptual approach, which imitates the classical definition of orientations of differentiable manifolds, but relies, instead of coordinate charts, on local linear orders.
Let be a 1-manifold. A local order of is a pair consisting of an open set homeomorphic to or and a linear order on defining the topology on . Two local orders , are said to agree if on any connected component of the orders and induce the same order.
Denote by the set of all local orders of . An orientation on is a map such that for any and the restrictions of and to any connected component of coincide if and do not coincide if .
Obvious Lemma 4.2. Let be a collection of open sets in a -manifold X homeomorphic to and let for any open set homeomorphic to or there exist such that is connected. If each is equipped with a linear order defining the topology on such that the local orders and agree for any , then there exists a unique orientation on such that for any . Moreover any orientation on comes from such coherent linear orders on all elements of .
Theorem 4.3. On any connected -manifold there exists exactly two orientations.
Proof. If is a non-closed connected 1-manifold, then for satisfying the hypothesis of Lemma 4.2 we can take a collection consisting of a single element . If is closed connected 1-manifold, then for we can take the collection of complements of single points. Then the intersection for any consists of two connected components homeomorphic to . We can choose stereographic projections as homeomorphisms of them to in such a way that the transition mapping from one of these charts to another one is . It is monotone increasing on each of the rays and . Therefore the orders obtained on complements of points via these stereographic projections from the standard order on agree with each other and we can apply Lemma 4.2.
Corollary 4.4. Any -manifold admits an orientation. If a -manifold consists of connected components, then it admits exactly orientations.
Theorem 4.5. 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 direction 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 4.5.
- 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 maps an orientation of to some orientation, which is either or (as a connected 1-manifold, has just these two orientations). Whether maps to itself or to the opposite orientation does not depend on : it maps to iff it maps to . We say that is orientation preserving if it maps any orientation of to itself and orientation reversing if if it maps any orientation of to the opposite orientation.
The half-line does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to admits an orientation reversing map. Thus, is chiral and connected 1-manifolds non-homeomorphic to are amphicheiral.
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.
5.1 Basic 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 Homology groups
The low dimensional homology groups of 1-manifolds are presented in the following table:
|Homology group \ 1-manifold|
It is clear from the last column of this table that the absolute and relative homology groups of dimensions 1 and 2 determine the topological type of 1-manifold. Of course, the basic invariants considered above do the job in a more elementary way.
Above by homology we mean homology with compact support. For homology with closed support (Borel-Moore homology) see the following table:
|Borel-Moore homology group \ 1-manifold|
An orientation of a 1-manifold gives rise to the fundamental class of , which belongs to . Cap-product by this class defines various versions of the Poincaré duality isomorphisms between usual cohomology (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So there are isomorphisms
A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over with fibre has non-trivial monodromy, then all the homology groups are trivial.
5.3 Euler characteristics
The Euler characteristic of is 1. The Euler characteristic of is 0.
Any closed 1-manifold has Euler characteristic 0. The Euler characteristic of any compact 1-manifold is half the number of its boundary points.
When it comes to non-compact spaces, the notion of Euler characteristic becomes ambiguous. A non-compact space has many Euler characteristics. The notion of Euler characteristic depends on the properties that one wants to preserve. If one wants to have invariance under homotopy equivalences, then the additivity would be lost. If one wants to keep additivity, then it is not the alternating sum of Betti numbers. If one wants to keep topological invariance and additivity, then , similarly .
For not necessarily compact 1-manifolds probably most useful is the Euler characteristic which is defined as the alternating sum of the ranks of homology groups with closed support (Borel-Moore homology). It is additive and a topological invariant. The only property which may confuse a person with purely compact experience is that it is not homotopy invariant. With this Euler characteristic, and .
5.4 Other homotopy invariants
The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions are trivial. The fundamental group is an infinite cyclic group, if the connected component of containing is homeomorphic to circle, and trivial otherwise.
5.5 Tangent bundle invariants
The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial.
6 Additional structures
A triangulation of a 1-manifold is a locally finite cover of by subspaces homeomorphic to , any two of which have disjoint interiors and at most one common point. The subspaces are assumed to be equipped with affine structure or, rather, with homeomorphisms to . The subspaces are called edges or -simplices, the images of the endpoints of are called vertices or -simplices.
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 the 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 Metrics and intrinsic metrics
Recall that any manifold is metrizable. On a connected manifold, a metric defining its topology can be replaced by an intrinsic metric defining the same topology. (Recall that a metric on a path-connected space is said to be intrinsic 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 .)
A connected 1-manifold with an intrinsic 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 and each homeomorphism type of a connected 1-manifold with intrinsic metric there is a unique standard model:
Theorem 6.1. A -manifold with intrinsic metric and diameter is isometric to
- with the metric induced from the standard metric on and any ;
- with the metric induced from the standard metric on and any ;
- the circle with the intrinsic metric defined by the lengths of paths on the circle measured in and any ;
- with the metric induced from the standard metric on and any .
The isometries to the standard models can be constructed using distances from some points. For example, for a connected 1-manifold homeomorphic to equipped with an intrinsic metric, the distance to the only boundary point defines a canonical isometry .
6.3 Smooth structures
Theorem 6.2. Any -manifold admits a smooth structure of any class .
Proof. A smooth structure can be induced by the isometry to the corresponding standard model from Theorem 6.1 above.
If smooth 1-manifolds and are homeomorphic, then they are also diffeomorphic. Moreover,
Theorem 6.3. Any homeomorphism between two smooth -manifolds can be approximated in the -topology by a diffeomorphism.
Proof. By Theorems 4.5 and 4.6, 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.
Any compact 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary.
If two compact 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen to preserve 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 a closed 1-manifold changes the number of connected components by 1. An index-1 surgery on a 1-manifold, which does not preserve every orientation, preserves the number of connected components. In particular, if an index-1 surgery on a connected 1-manifold does not preserve an orientation, then its result is a connected 1-manifold.
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, the 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 -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 -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 homeomorphisms
Similarly, the group of self-homeomorphisms of isotopic to the identity contains as a subgroup, which is its deformation retract.
The groups of self-homeomorphisms of , and which are isotopic to the 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 the identity is homotopy equivalent to .
9 Finite group actions
9.1 The invariant intrinsic metric
Theorem 9.1. For any effective action of a finite group on a -manifold , there exists an intrinsic metric on invariant under this action.
Proof. Such a metric is created in three steps.
- Choose any metric on defining the topology.
- Symmetrize : define for any the number . This is a metric and it is invariant under the action.
- Take the intrinsic metric induced by .
Since an intrinsic metric on a 1-manifold defines an isometry onto one of the standard models, this gives a equivariant homeomorphism to one of the standard connected 1-manifolds with intrinsic metric and action of by isometries.
9.2 Free actions
If the action of a finite group on a 1-manifold 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 there is one subgroup with index , and hence one -fold covering. The total space is homeomorphic to , and the covering is equivalent to . In the corresponding action, the group is cyclic of order and 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 on the 1-manifold on which the group acts. This is done in the next two theorems.
Theorem 9.2. There is no non-trivial free finite group action on a contractible -manifold.
Theorem 9.3. 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.
9.3 Asymmetry of a half-line
Theorem 9.4. There is no non-trivial action of a finite group on .
Proof. We will prove that the only homeomorphism of finite order is the identity. Observe first that any homeomorphism must map a boundary point to a boundary point, and hence it preserves the only boundary point . By Theorem 9.1, there exists an intrinsic metric invariant under . Therefore, preserves the distance from the boundary point. However each point on is uniquely defined by its distance to the boundary point.
9.4 Actions on line and segment
Theorem 9.5. 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. Since it is increasing, it can be extended to by letting . The extended homeomorphism has the same finite order. But by Theorem 9.4 any such homeomorphism is the identity.
Theorem 9.6. Any orientation reversing homeomorphism of finite order has order two. It is conjugate to the reflection in a point.
Proof. An orientation reversing homeomorphism is a monotone decreasing bijection. Consider the function . It is also a 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 9.4, is the identity, which contradicts our assumption. Thus, and . Then preserves the rays, and, by Theorem 9.4, is the identity. Thus has order two.
Choose a homeomorphism . Define function by the formula . It's a homeomorphism. Together, and form a homeomorphism . As is easy to check, .
Theorem 9.7. A non-trivial finite group acting effectively on is a cyclic group of order . The action of the non-unit element is conjugate to the reflection in a point.
Proof. As follows from Corollaries 9.5 and 9.6, 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 9.5, it is the identity. So, and hence . But . Therefore and .
Corollary 9.8. A non-trivial finite group acting effectively on is a cyclic group of order . The action of the non-unit element is conjugate to the reflection in 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 a 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 an auto-homeomorphism of . The interior is homeomorphic to .
9.5 Actions on the circle
Theorem 9.9. Any periodic orientation reversing homeomorphism is an involution (i.e., has period ). It is conjugate to a reflection of in its diameter.
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 a homeomorphism is 2.
Consider the complement of a fixed point. The restriction of the homeomorphism to this complement satisfies the conditions of Theorem 9.6, which gives the required result.
Theorem 9.10. 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 9.5 would conclude that it is the identity and hence the whole homeomorphism is the 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 9.3.
Theorem 9.11. A finite group acting effectively on is either cyclic or dihedral, and the action is conjugate to a linear one and extends to the standard (linear) action of the orthogonal group . The standard actions of cyclic and dihedral groups on the circle are provided by the symmetry groups of regular polygons.
Proof. By Theorem 9.1, for a finite group action on there exists an invariant intrinsic metric. By Theorem 6.1, with this metric is isometric to the circle defined in the plane by an equation . Any isometry of this circle belongs to .
Assume that the action contains an orientation reversing homeomorphism. Then the group acts on the set of orientations. The orientation preserving homeomorphisms form a subgroup of index two. This is a cyclic group as above. 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 reversing involution, as in Theorem 9.9 above.
By Theorem 9.1, the action is equivalent to an action consisting of isometries of the circle. Its orientation preserving part consists of rotations and is cyclic, say of order . Its elements are rotations by angles . The orientation reversing elements are reflections in diameters. There are diameters.
When , that is the group contains only 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 . It contains two reflections in diameters orthogonal to each other, the symmetry in the center of the circle and the identity.
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 Exotic relatives of 1-manifolds
As we eliminate the Hausdorff or second countability property, the theory becomes somehow weird, but many aspects survive.
10.1 First examples of non-Hausdorff 1-manifolds
1. Line with two origins. In the disjoint union of two copies identify each point of one of the copies different from the origin with its corresponding point from the second copy: . This space has a single point for each nonzero real number and two points and taking place of the origin. Each neighborhood of intersects each neighborhood of , so the space is not Hausdorff.
2. Branching line. This is also a quotient space of two copies of the real line: . This space has a single point for each real number and two points , for every non-negative . As in the line with two origins above, in this space there is only one pair of points that have no disjoint neighborhoods: and .
10.2 Spaces of leaves
At first glance, the examples above of non-Hausdorff 1-manifolds look artificial. However they appear naturally in some classical mathematical contexts. For example, the branching line is homeomorphic to the space of leaves of the following foliation of the plane . The leaves are the vertical lines with and the graphs with and arbitrary real . In other words, the branching line is homeomorphic to the quotient space of by partition into these lines and graphs. The leaves correspond to points , for non-negative , the leaves with correspond to points .
Similarly, the line with two origins can be identified with the space of leaves of a foliation of the cylinder .
Although at first glance the space of leaves of a foliation looks substantially more natural than descriptions of non-Hausdorff 1-manifolds as quotient spaces of some 1-manifolds, they are not far away from each other: in either case we deal with factorization of a nice Hausdorff space, but factorization easily gives rise to a non-Hausdorff space.
10.3 Uncountable family of non-homeomorphic connected non-Hausdorff 1-manifolds
If we enlarge the collection of spaces by eliminating 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.
10.4 Non-orientable non-Hausdorff 1-manifolds
The definition of orientation generalizes straightforwardly to non-Hausdorff 1-manifolds. Unlike usual 1-manifolds, there exist non-Hausdorff 1-manifolds that are non-orientable. For example, the quotient space of the open interval under identification for is a non-Hausdorff 1-manifold which does not admit any orientation.
10.5 Differential structures
The notion of differential structure has a natural generalization to non-Hausdorff 1-manifolds. Let be a non-Hausdorff 1-manifold, that is a second countable topological space such that each point of has a neighborhood homeomorphic either to or . Let be either a natural number, or . A sheaf of real valued functions on is called a differential structure of class or just -structure if it satisfies the following condition: for any open set , a section and a function of class (i.e., a function having the first continuous derivatives , if is finite and derivatives of all natural orders if ), the composition belongs to .
If each point has a neighborhood homeomorphic to or such that a -structure , as a sheaf, induces on a sheaf isomorphic to the sheaf on or of functions of class , then the -structure is called non-singular. A 1-manifold equipped with a non-singular -structures is a -manifold of dimension 1.
Any 1-manifold admits a non-singular -structure, and any two homeomorphic -manifolds of dimension 1 are diffeomorphic, see Section 6.3 above.
10.6 Homeomorphic, but non-diffeomorphic non-Hausdorff 1-manifolds
There exist two homeomorphic, but not diffeomorphic non-Hausdorff 1-manifolds. In order to construct such an example, take a pair of monotone decreasing sequences and on convergent to 0. There exists a homeomorphism with for all , but one can find sequences for which there is no diffeomorphism with this property. For example, this is the case for and . Take such a pair of sequences. Let and . By the identity map of attach one copy of to another and denote the result by . Similarly, in the disjoint sum of two copies of identify by the identity map the copies of and denote the result by . The homeomorphism defines a homeomorphism , on the other hand there is no diffeomorphism between and , because if one existed, it would map to (as the sets of separating points) and extend to a diffeomorphism of mapping to .
10.7 Relatives of 1-manifolds without countable base
One can easily construct a topological space which satisfies all the conditions of the definition of a 1-manifold except second countability by taking the disjoint sum of an uncountable number of copies of (or any other 1-manifold). There are more interesting examples, which are connected. Most famous of them is the long line.
10.8 Sheaf of germs of functions on a 1-manifold
The sheaf of germs of continuous functions on a 1-manifold is locally homeomorphic to or . So, it satisfies one condition (out of three) of the definition of a 1-manifold. The sheaf of germs of differentiable functions on a 1-manifold has a natural differential structure.
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