1-manifolds
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There is no non-trivial free finite group actions on contractible 1-manifolds. | There is no non-trivial free finite group actions on contractible 1-manifolds. | ||
− | On circle if a finite group $G$ has a free action than $G$ is cyclic. Any cyclic group has a linear | + | On circle if a finite group $G$ has a free action than $G$ is cyclic. Any finite cyclic group has a linear |
− | free action on $S^1$. Any free action of a cyclic group on $S^1$ is conjugate to a linear action. | + | free action on $S^1$. Any free action of a finite cyclic group on $S^1$ is conjugate to a linear action. |
Any periodic orientation reversing homeomorphism $S^1\to S^1$ is an involution (i.e., has period 2). | Any periodic orientation reversing homeomorphism $S^1\to S^1$ is an involution (i.e., has period 2). | ||
+ | It is conjugate to a symmetry of $S^1$ against its diameter. | ||
+ | |||
Any non-trivial periodic homeomorphism of a connected 1-manifold with a fixed point is an involution | Any non-trivial periodic homeomorphism of a connected 1-manifold with a fixed point is an involution | ||
reversing orientation. | reversing orientation. | ||
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A finite group acting effectively on $S^1$ is either cyclic or dihedral, and the action is conjugate to a linear one. The standard actions of cyclic and dihedral groups on circle are provided by the symmetry groups of | A finite group acting effectively on $S^1$ is either cyclic or dihedral, and the action is conjugate to a linear one. The standard actions of cyclic and dihedral groups on circle are provided by the symmetry groups of | ||
regular polygons. | regular polygons. | ||
+ | |||
+ | A non-trivial finite group acting effectively on $\Rr^1$ or $[0,1]$ 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 $\Rr_+$. | ||
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Revision as of 02:24, 28 October 2009
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 08:32, 18 July 2013 and the changes since publication. |
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 neighborhood homeomorphic either to line or 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.
2 Examples
- Real line
- Half-line
- Circle
- Closed interval
3 Classification
3.1 Reduction to classification of connected manifolds
Any manifold is homeomorphic to the disjoint sum of its connected components.
A connected component of a 1-manifold is a 1-manifold.
Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic.
3.2 Topological classification of connected 1-manifolds
Theorem 3.1. Any connected 1-manifold is homeomorphic to one of the following 4 manifolds:
- real line
- 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 closed 1-manifold is homeomorphic to .
- Any connected compact 1-manifold with non-empty boundary is homeomorphic to .
- Any connected non-compact 1-manifold without boundary is homeomorphic to .
- Any connected non-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
Proofs of the results above are elementary. The core of them is the following simple
The theorems above solve the topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many Topology textbooks manage not to mention this fundamental result.
If we enlarge the collection of spaces by getting rid of the Hausdorff property, then the number of topological types of connected spaces becomes uncountable. Indeed, one can take two copies of line and identify an open set in one of them with its copy in the other one by the identity map. The result satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom.
3.5 Corollaries
3.5.1 Homotopy classification of 1-manifolds
Each connected 1-manifold is either contractible, or homotopy equivalent to circle.
3.5.2 0-manifolds cobordant to zero
A compact 0-manifold 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:
a map is a homeomorphism iff it is a continuous monotone bijection.
4 Invariants
As follows from the classification theorems,
- the number of connected components,
- compactness of a connected component,
- and the number of boundary points of a connected component
play fundamental role in topology of 1-manifolds.
Homotopy invariants are extremely simple. All homology and homotopy groups of dimensions are trivial.
Tangent bundles of 1-manifolds are trivial.
5 Further discussion
5.1 Orientations
Orientation of a 1-manifold can be interpreted via linear or cyclic orderings of their points.
An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold.
An orientation of a connected closed 1-manifold is a cyclic order on the set of its points such that the topology of this cyclic order coincides with the topology of the 1-manifold.
An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation).
Any 1-manifold admits an orientation.
Half-line 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 admit a triangulation. A triangulation of a non-compact connected 1-manifold is unique up to homeomorphism.
A compact 1-manifold has non-homeomorphic triangulations, but they are easy to classify up to homeomorphism. On circle the topological type of a triangulation is defined by the number of 1-simplices. This number can take any integral value . 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 diameter there is a standard model for the inner metric space. 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 . For with diameter this is . For with diameter this is .
An inner metric on a connected 1-manifolds defines a unique smooth structure on the manifold, namely, the smooth structure induced by the isometry to the corresponding standard model from the list above.
5.4 Mapping class groups
Recall that the mapping class group of a manifold 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 .
A reversing orientation homeomorphism cannot be isotopic to the identity. For an auto-homeomorphism of a connected 1-manifold this is the only obstruction to being isotopic to the identity. Therefore , while .
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.
Groups of auto-homeomorphisms of , and 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 is no non-trivial free finite group actions on contractible 1-manifolds. On circle if a finite group has a free action 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 chnages the number of connected components by 1. An index 1 surgery on a closed 1-manifold, which does not preserve any orientation, preserves the number of connected components.
5.8 Connected sums
The notion of connected sum is defined for 1-manifolds, but it does not work in the same way as for manifolds of higher dimensions.
Recall that a manifold of dimension is a connected sum of manifolds and if there are -disks and and a homeomorphism such that is homeomorphic to (that is to the result of attaching of to by with the image of interior of the identified disks and removed). One writes , however the topology of may depend not only on and , but also on , and . The dependence disappears if all connected components of are homeomorphic to each other, all connected components of are homeomorphic to each other, and a homeomorphism reversing orientation can be extended to a homeomorphism , or a homeomorphism reversing orientation can be extended to a homeomorphism . For a collection of manifolds satisfying these conditions, a connected sum can be considered as operation on topological types.
The very term connected sum is compromised in dimension 1. Indeed, a connected sum
of connected 1-manifolds may be not connected. For example a connected sum of two
copies of is a disjoint sum of two copies of .
Moreover, connected sum, as an operation on topological types of 1-manifolds, is not well-defined. Indeed, both disjoint sums and can be presented as a connected sum of two copies of .
6 References
This page has not been refereed. The information given here might be incomplete or provisional. |