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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).
34
votes
Accepted
Does a *topological* manifold have an exhaustion by compact submanifolds with boundary?
Since topological manifolds of dimension $\le 3$ are smoothable, the question is about manifolds of dimension $\ge 4$. Kirby and Siebenmann proved for $n\ge 6$ that every topological $n$-manifold admi …
- 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
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
19
votes
Accepted
Proper discontinuity and existence of a fundamental domain
I will assume that you are interested in group actions on connected manifolds: In the case of more general spaces it is not even completely clear what a fundamental domain means since an element of fi …
- 31.2k
15
votes
Accepted
orbit space of a topological manifold
The answer is no. Bing constructed a space $X$ (called the dogbone space) so that $X$ is not a manifold, but $X\times R$ is homeomorphic to $R^4$. In particular, $M^4=X\times S^1$ is a $4$-manifold (s …
- 31.2k
15
votes
Accepted
Can the SL_2 character variety of a three-manifold be nonreduced?
There is actually an old (ca 1986) example of nonreduced $SL(2, {\mathbb C})$-representation scheme of a 3-manifold group. Take an oriented Seifert manifold $M$ which fibers over the $S^2(3,3,3)$ orbi …
- 31.2k
14
votes
Accepted
Do quasi convex hyperbolic subgroups remain quasi convex after adding redundant generators?
First, one should decide what quasiconvexity (qc) means in the context of subgroups $H\subset G$, where $G$ is merely a semihyperbolic group, e.g., a RAAG. (I am assuming that generating sets are fixe …
- 31.2k
13
votes
Accepted
Growth of Poincaré duality groups
The question is well beyond of what is currently known about Poincare duality groups and groups of intermediate growth. The only known case is of PD(2) groups, since they are virtually surface groups. …
- 31.2k
12
votes
Accepted
A terminological question concerning orbifolds.
Take a look at Thurston's notes (Chapter 13 of his "The geometry and topology of 3-manifolds", available at the MSRI's site: http://library.msri.org/books/gt3m/). Thurston calls these points singular …
12
votes
Accepted
Hyperbolic 3 manifold with trivial deformation of flat conformal structures
Let $M$ be a compact hyperbolic 3-manifold and let $C(M)$ be the moduli space of flat conformal structures on $M$. The hyperbolic metric on $M$ defines a distinguished point $h\in C(M)$. One says tha …
- 31.2k
12
votes
History of the notion of $(G,X)$-structure
Here is a collection of remarks on the history; when and if I have more time, I will add more detail.
The ideas go back to 19th century (Poincare and others) who studied 2nd order holomorphic ODEs …
- 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
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
11
votes
Accepted
Representation varieties of 3-manifold groups in $\mathrm{SL}(n,\mathbb{C})$
These are difficult questions and very little in general is known about this. Mostly, what's known is ad hoc results for specific classes of manifolds (say, take some surgeries on 2-bridge knots...). …
- 31.2k
11
votes
Accepted
Counterexamples to analogue of Cannon conjecture in higher dimensions
There are various compact manifolds of negative curvature which are not homnotopy-equivalent to closed hyperbolic manifolds: Locally symmetric ones (complex hyperbolic, etc) as well as Gromov-Thurston …
- 31.2k