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− | As follows from the Theorems \ref{thm:classification} and \ref{thm:characterisation} above, the following | + | As follows from the Theorems \ref{thm:classification} and \ref{thm:characterisation} above, the following invariants |
− | * the | + | * the number of connected components, |
− | * the | + | * the compactness of each connected component, |
− | * and | + | * and the number of boundary points of each connected component |
− | + | determine the topological type of a 1-manifold. | |
The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$ are trivial. | The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$ are trivial. |
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Contents
|
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.
Specific properties of 1-manifolds can be related to the fact that the topological structure in a 1-manifold is defined by linear or cyclic ordering of points.
For other expositions about -manifolds, see [Gale1987] and also [Fuks&Rokhlin1984, Chapter 3, Section 1.1.16-19].
2 Examples
- The real line:
- The half-line:
- The circle:
- The closed interval:
3 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.
3.4 Remarks
The proofs of Theorems 3.1 and 3.2 above are elementary. The core of them is the following simple
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.
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.
3.5 Corollaries
3.5.1 Homotopy classification of 1-manifolds
Each connected 1-manifold is either contractible, or homotopy equivalent to circle.
3.5.2 0-manifolds cobordant to zero
A compact 0-manifold bounds a compact 1-manifold iff the number of points in is even.
3.5.3 Smooth structures
Any 1-manifold admits a smooth structure.
If smooth 1-manifolds and are homeomorphic, then they are also diffeomorphic. Moreover,
any homeomorphism can be approximated in the -topology by a diffeomorphism.
Technically this can be considered as a corollary of the following simple theorem whose proof is as an exercise.
Theorem 3.4. A map is a homeomorphism iff it is a continuous monotone bijection.
4 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.
The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions are trivial.
The tangent bundles of 1-manifolds are trivial.
5 Further discussion
5.1 Orientations
An orientation of a 1-manifold can be interpreted via linear or cyclic orderings of their points.
An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold.
An orientation of a connected closed 1-manifold is a cyclic order on the set of its points such that the topology of this cyclic order coincides with the topology of the 1-manifold.
An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation).
Any 1-manifold admits an orientation.
The half-line does not admit a 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.
5.2 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.
5.3 Inner metrics
Recall that a metric on a path-connected space is said to be inner if the distance between any two points is equal to the infimum of lengths of paths connecting the points, and that the length of a path 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.
5.4 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 the identity. For an auto-homeomorphism of a connected 1-manifold this is the only obstruction to being isotopic to the identity. Therefore
and
5.5 Homotopy types of groups of auto-homeomorphisms
The group contains as a subgroup, which is its deformation retract. Similarly, the group of auto-homeomorphisms of isotopic to identity contains as a subgroup, which is its deformation retract.
The groups of auto-homeomorphisms of , and which are isotopic to identity are contractible.
Thus for each connected 1-manifold the group of homeomorphisms isotopic to identity is homotopy equivalent to .
5.6 Finite group actions
There are no non-trivial free finite group actions on contractible 1-manifolds. 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.
Any periodic orientation reversing homeomorphism is an involution (i.e., has period 2). It is conjugate to a symmetry of against its diameter.
Any non-trivial periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.
A finite group acting effectively on 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.
A non-trivial finite group acting effectively on or is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
There is no non-trivial action of a finite group in .
5.7 Surgery
Any 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary.
If given two 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen preserving the orientation (this means that the corresponding cobordism is an oriented 2-manifold and its orientation induces on the boundary the given orientation on one of the 1-manifolds and the orientation opposite to the given one on the other 1-manifold).
An index 1 surgery preserving orientation on closed 1-manifold 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.
5.8 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
6 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
7 External links
- The Encylopedia of Mathematics article on one-dimensional manifolds.
- The Encylopedia of Mathematics article on lines.
- The Wikipedia page about curves.
''any homeomorphism $X \to Y$ can be approximated in the $C^0$-topology by a diffeomorphism.'' Technically this can be considered as a corollary of the following simple theorem whose proof is as an exercise. {{beginthm|Theorem}} A map $\Rr\to\Rr$ is a homeomorphism iff it is a continuous monotone bijection. {{endthm}}
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.
Specific properties of 1-manifolds can be related to the fact that the topological structure in a 1-manifold is defined by linear or cyclic ordering of points.
For other expositions about -manifolds, see [Gale1987] and also [Fuks&Rokhlin1984, Chapter 3, Section 1.1.16-19].
2 Examples
- The real line:
- The half-line:
- The circle:
- The closed interval:
3 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.
3.4 Remarks
The proofs of Theorems 3.1 and 3.2 above are elementary. The core of them is the following simple
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.
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.
3.5 Corollaries
3.5.1 Homotopy classification of 1-manifolds
Each connected 1-manifold is either contractible, or homotopy equivalent to circle.
3.5.2 0-manifolds cobordant to zero
A compact 0-manifold bounds a compact 1-manifold iff the number of points in is even.
3.5.3 Smooth structures
Any 1-manifold admits a smooth structure.
If smooth 1-manifolds and are homeomorphic, then they are also diffeomorphic. Moreover,
any homeomorphism can be approximated in the -topology by a diffeomorphism.
Technically this can be considered as a corollary of the following simple theorem whose proof is as an exercise.
Theorem 3.4. A map is a homeomorphism iff it is a continuous monotone bijection.
4 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.
The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions are trivial.
The tangent bundles of 1-manifolds are trivial.
5 Further discussion
5.1 Orientations
An orientation of a 1-manifold can be interpreted via linear or cyclic orderings of their points.
An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold.
An orientation of a connected closed 1-manifold is a cyclic order on the set of its points such that the topology of this cyclic order coincides with the topology of the 1-manifold.
An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation).
Any 1-manifold admits an orientation.
The half-line does not admit a 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.
5.2 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.
5.3 Inner metrics
Recall that a metric on a path-connected space is said to be inner if the distance between any two points is equal to the infimum of lengths of paths connecting the points, and that the length of a path 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.
5.4 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 the identity. For an auto-homeomorphism of a connected 1-manifold this is the only obstruction to being isotopic to the identity. Therefore
and
5.5 Homotopy types of groups of auto-homeomorphisms
The group contains as a subgroup, which is its deformation retract. Similarly, the group of auto-homeomorphisms of isotopic to identity contains as a subgroup, which is its deformation retract.
The groups of auto-homeomorphisms of , and which are isotopic to identity are contractible.
Thus for each connected 1-manifold the group of homeomorphisms isotopic to identity is homotopy equivalent to .
5.6 Finite group actions
There are no non-trivial free finite group actions on contractible 1-manifolds. 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.
Any periodic orientation reversing homeomorphism is an involution (i.e., has period 2). It is conjugate to a symmetry of against its diameter.
Any non-trivial periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.
A finite group acting effectively on 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.
A non-trivial finite group acting effectively on or is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.
There is no non-trivial action of a finite group in .
5.7 Surgery
Any 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary.
If given two 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen preserving the orientation (this means that the corresponding cobordism is an oriented 2-manifold and its orientation induces on the boundary the given orientation on one of the 1-manifolds and the orientation opposite to the given one on the other 1-manifold).
An index 1 surgery preserving orientation on closed 1-manifold 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.
5.8 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
6 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
7 External links
- The Encylopedia of Mathematics article on one-dimensional manifolds.
- The Encylopedia of Mathematics article on lines.
- The Wikipedia page about curves.