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Results tagged with gt.geometric-topology
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user 317
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
33
votes
Which manifolds are homeomorphic to simplicial complexes?
I don't know about dimension 4, but for high dimensions this is a well-known open problem. I don't think much progress has been made on it for a while. I recommend Ranicki's lecture notes from Siebe …
- 44.8k
30
votes
Accepted
Are the mapping class groups of manifolds finitely presentable?
These things can be pretty wild. For instance, for $n \geq 5$ the mapping class group of the $n$-torus is not even finitely generated : it is a split extension of $\text{GL}_n(\mathbb{Z})$ by an infi …
- 44.8k
28
votes
Accepted
Realizing symmetric groups by diffeomorphisms
I'm going to reverse the roles of $n$ and $d$ (since otherwise I will screw things up in this answer). If $M^n$ is a connected $n$-dimensional smooth manifold, then there is no section of the map $Di …
- 44.8k
27
votes
Accepted
A function composed with itself produces the identity
Yes. Observe first that $f$ can be first extended to an involution of $\mathbb{R}^3$ and then to an involution $F : S^3 \rightarrow S^3$ of the one-point compactification of $\mathbb{R}^3$. A classi …
- 44.8k
26
votes
Accepted
Is there a smooth manifold which admits only rigid metrics?
The answer to the question in the first sentence is "yes". Let $M$ be a hyperbolic 3-manifold whose isometry group is trivial. Then by Theorem 1.1 of
Farb, Benson; Weinberger, Shmuel Hidden symmetr …
- 44.8k
26
votes
Thurston's "tinker toy" problem
It was published here:
M. Kapovich, J. Millson, Universality theorems for configuration spaces of planar linkages,
Topology 41 (2002), no. 6, 1051–1107.
- 44.8k
25
votes
Accepted
Mapping Class Groups of Punctured Surfaces (and maybe Billiards)
1) Let me start by dealing with punctures and higher genus mapping class groups.
Aside from a few low-genus cases, there is no easy description of the mapping class group. As you said, the mapping c …
- 44.8k
24
votes
Thurston's 24 questions: All settled?
They have all been resolved in some fashion with the exception of problem 23, which asserts that the volumes of all hyperbolic 3-manifolds are not rationally related. We know basically nothing about …
- 44.8k
23
votes
Accepted
Thurston geometries in dimension 4
The 4-dimensional geometries were classified in the unpublished thesis of Filipkiewicz, which is available here.
- 44.8k
21
votes
Accepted
Manifolds with two coordinate charts
I'll only discuss the first question (EDIT: Actually, I address the second question at the end). As Agol pointed out in the comments, for $n \geq 5$ this is an easy consequence of Newman's 1966 proof …
- 44.8k
20
votes
Accepted
Do the following set of Dehn twists generate the mapping class group?
No. If they did, then they would still generate the mapping class group of the closed surface that results from gluing a disc to the boundary component. However, in that surface they all commute wit …
- 44.8k
17
votes
Accepted
Homotopy type of the plane minus a sequence with no limit points
Yes. More generally, if $X$ is a proper closed subset of $\mathbb{R}^2$, then every path component $M$ of $\mathbb{R}^2 \setminus X$ is homotopy equivalent to a wedge of circles (observe that $X$ mig …
- 44.8k
16
votes
Beyond an intro to topological graph theory...
I wouldn't describe it as an "area" of math, exactly, but there are certainly mathematicians who study embeddings of graphs into surfaces and related objects. Let me recommend two sources to learn mo …
- 44.8k
16
votes
Compactification theorem for differentiable manifolds ?
As everyone has said, the answer is "no". You have to make assumptions to ensure that the "ends" of your manifold are sufficiently simple. It appears hard to find using the refs Paul posted, but the …
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16
votes
Accepted
Who proved that two homotopic embeddings of one surface in another are isotopic?
First, for simple closed curves, this was known long before Freedman-Hass-Scott. For closed surfaces, it was first proved by Baer in
Baer, R., Kurventypen auf Flächen. J. reine angew. Math., 156 (19 …
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