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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).
6
votes
Does every triangulable manifold have a vertex-transitive triangulation?
This is to supplement Ian's answer and get examples in all dimensions $\ge 3$.
Let $M={\mathbb H}^n/\Gamma$ be a compact hyperbolic $n$-manifold; suppose that $f\in Homeo(M)$ is a homeomorphism of pri …
- 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
5
votes
What are orbifolds with corners?
Orbifolds with corners are defined by the same axioms as manifolds with corners and ordinary orbifolds: A topological $n$-dimensional orbifold with corners is a topological space $X$ (2nd countable an …
- 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
4
votes
Accepted
"Dimension" of discrete subgroups of infinite covolume in Lie groups
First of all, since you are not assuming finite generation, you should at least assume that $\Gamma$ is virtually torsion-free. (Otherwise, you need to work rationally and reprove Whitehead's lemma, i …
- 31.2k
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
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
1
vote
Are pseudo-Anosov foliations dense?
I am sure this was known earlier (check Ivanov's book, most likely, it is there), but you can refer to the following theorem of Lindenstraus and Mirzakhani (see their paper Ergodic theory of the space …
- 31.2k
9
votes
Does any smooth orbifold can be triangulated by orbi-simplex(triangulation of orbifolds)
You have to be a bit more specific about the meaning of a "triangulation" for orbifolds. Assuming that you just want to triangulate the underlying space, the claim follows from
C. T. Yang, "The tria …
- 31.2k
3
votes
Proper Group action on a metric space
Since this old question does seem to be of some interest, here is a proof. First of all, here is a useful (but not well-known) definition.
Definition. Let $X$ be a Hausdorff 1st countable topologica …
- 31.2k
2
votes
Topology of manifolds and transition functions
There is a (dated) book which attempts to do something along the lines of what you asked:
D. Sundararaman, MODULI, DEFORMATIONS AND CLASSIFICATIONS OF COMPACT COMPLEX MANIFOLDS, Research Notes in Ma …
- 31.2k
10
votes
Spaces that are finitely covered by manifolds
Indeed, C. Thomas, 3-Manifolds and PD(3)-groups, in "Novikov conjectures", Vol. 2, gives a reference to Swan's example. Namely, there is a finite complex $X$ which is a homotopy 3-sphere, a free $S_3 …
- 31.2k
8
votes
Accepted
Uniqueness theorem for conformal mapping
This is a form of the following topological question: Does the boundary value uniquely determine an immersion of the closed disk into the plane, up to precomposition with a diffeomorphism of the disk. …
- 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
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