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Results tagged with gt.geometric-topology
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user 21684
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).
4
votes
Accepted
If H is a homeomorphism from $R^{n}$ to itself, and P and H(P) are compact polyhedra, is H(...
The answer is: Yes, this can happen. Suppose that $P, Q$ are $k$-dimensional polyhedra in ${\mathbb R}^n$ which are homeomorphic but not PL homeomorphic. If $2k+2\le n$ (and you can always increase th …
- 31.2k
11
votes
Accepted
Is $\mathrm{Diff}_0(S_g)$ torsion-free?
Here is a proof that $Homeo_0(S)$ is torsion-free for every compact hyperbolic surface $S$. With more analytic assumptions on homeomorphisms one can get the same conclusion for noncompact hyperbolic s …
- 31.2k
4
votes
Mechanisms generating free subgroups of Artin braid groups
This is a very extended comment to your question.
Your question does not have "obvious yes or no" answer; in part, this is because you asked 3 somewhat different questions:
"Can one explicitly de …
- 31.2k
9
votes
Accepted
Characterization of cocompact group action
I agree with @Bugs that some extra assumptions are needed, although I do not have a counter-examples either.
Here is an argument assuming that $X$ is locally compact. For each $y\in Y=X/G$ choose (a …
- 31.2k
8
votes
When are Brieskorn Manifolds Homeomorphic?
This is by no means a complete answer but, rather, a DIY suggestion:
Let $B$ be a $2k+1$-dimensional Brieskorn manifold. Then $B$ is $k-1$-connected.
C.T.C. Wall wrote in "Classification problems in …
- 31.2k
4
votes
Accepted
construct Seifert fibration on mapping torus of surface with monodromy a periodic mapping class
Just use the suspension flow of the periodic diffeomorphism $f: S\to S$ in the periodic mapping class. Then all flow lines will be periodic (i.e., circles) and you are done; the base will be the quoti …
- 31.2k
7
votes
Accepted
Classification of symplectic surfaces
This is a bit late answer to an old MO question, but Moser's theorem was generalized to open manifolds in
R. Greene and K. Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact ma …
- 31.2k
11
votes
Accepted
When is a classifying space a topological manifold?
Here is a more detailed answer.
Theorem. $K(G,1)$ is homotopy-equivalent to a (textbook) topological manifold if and only if $G$ is countable and has finite cohomological dimension (over ${\mathbb Z …
- 31.2k
5
votes
Accepted
Poincaré dodecahedron space
You can read the book by Seifert and Threlfall "A textbook of topology", pages 223-225: They
start by writing down a presentation of $\pi_1$ of the Poincare homology sphere (by reading off generators …
- 31.2k
4
votes
Accepted
Example of dynamical system $M$ such that $M \rightarrow \mathbf{R}\backslash M$ is not loca...
The answer is negative. Take, for instance, the irrational foliation of the flat 2-torus by geodesics with the obvious ${\mathbb R}$ action via translations along leaves.
Note that every fiber bundl …
- 31.2k
2
votes
Mapping class between coverings of Riemann surfaces
Let $F$ be a finite group, let $\pi_g$ be the fundamental group of the closed oriented genus $g$ surface $S_g$ and let $Mod_g$ denote the mapping class group of $S_g$. Consider the action of $Mod_g$ o …
- 31.2k
6
votes
How to specify a compact topological 4-manifold with a finite amount of data
I happened to discuss this very same question with Fico last December. Maybe he is the one who was asking you.
What's written below makes sense in any dimension, but dimension 4 is the most interest …
- 31.2k
10
votes
Accepted
Why does not a closed 3-manifold modelled on SL(2,R) admit a metric of nonpositive curvature?
If you read our paper a bit further, you will find that on page 348 we mention that this result is due to Eberlein and give a reference to his 1982 paper.
More precisely, he proves a more general theo …
- 31.2k
19
votes
Accepted
Diffeomorphisms of finite order not in the image of a circle action
Such examples exist in dimension 5, they are contained in the paper by Cameron Gordon "On the higher-dimensional Smith conjecture", Proc. London Math. Soc. (3) 29 (1974), 98–110. Namely, Gordon prove …
- 31.2k
25
votes
Are topological manifolds homotopy equivalent to smooth manifolds?
For every $n\ge 4$ there exists a closed aspherical topological $n$-manifold $N$ which is not homotopy-equivalent to a PL manifold. Furthermore, $\pi_1(N)$ is a CAT(0) group. This is a theorem of Dav …
- 31.2k