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Results tagged with gt.geometric-topology
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user 39654
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).
22
votes
When does a group act effectively and holomorphically on some Riemann surface?
Donu's answer is correct but amounts to killing a fly with a gun shot: Greenberg proves a harder result than the one needed for the problem.
Theorem. Let $G$ be a countable group. Then there exists a …
- 12.3k
17
votes
Is there a continuous partition of space into circles?
Yes, there is a topological foliation of $\mathbb R^3$ by smooth circles. A foliation by topological circles was constructed in by Vogt in
Vogt, Elmar, A foliation of ${\mathbb{R}}^3$ and other punctu …
- 12.3k
13
votes
Accepted
Can you cover a genus a billion hyperbolic surface with 15 balls?
Your conjecture is false. Every nonorientable closed connected surface of negative Euler characteristic, admits a hyperbolic metric such that the surface is covered by 3 embedded disks. Hence, for eac …
- 12.3k
12
votes
Accepted
Extending diffeomorphisms
The answer is positive and follows from Corollary 2 in
Palais, Richard S., Extending diffeomorphisms, Proc. Am. Math. Soc. 11, 274-277 (1960). ZBL0095.16502.
(A caveat: Palais is not entirely clear ab …
- 12.3k
11
votes
Accepted
Quantitative word problem for 3-manifold groups
Suppose that $M$ is a compact irreducible 3-manifold.
Assume that $M$ is neither a Nil nor a Sol-manifold. Then $G=\pi_1(M)$ is automatic, which implies that $G$ has quadratic Dehn function and the w …
- 12.3k
10
votes
Accepted
All non-compact simply connected $2$-manifolds with boundary
Here is one proof, using the Uniformization Theorem. This proof will be easier in the setting of the "Primer" since the authors are considering universal covering spaces of complete hyperbolic surface …
- 12.3k
9
votes
Accepted
Conformal covers of all degrees
Here is a partial answer: If there is such a conformal manifold $M$ of dimension $n\ge 2$, then $M$ admits a flat metric. The reason is that the sequence of conformal covering maps $\phi_k: M\to M$ ca …
- 12.3k
9
votes
Accepted
Examples of the Thurston geometries with transitive Lie group action
This is an answer to questions 7 and 8 (I have to say, having 8 questions in one post is way too much for my taste):
Suppose that $M$ is a finite-volume quotient of $H^3$ or a compact quotient of $H^ …
- 12.3k
8
votes
Accepted
Existence of a geometric structure on a solid torus
It all depends on your definition of a "geometric manifold."
One definition would require the existence of a complete finite volume locally homogeneous Riemannian metric (from Thurston's list of eigh …
- 12.3k
8
votes
Accepted
Are there simplicial spheres with "non-geometric symmetries"?
The answer is negative. Already in dimension 4 there are fake real-projective spaces, which are smooth 4-manifolds homotopy-equivalent but not homeomorphic to $RP^4$. These correspond to smooth free i …
- 12.3k
8
votes
Accepted
Proper action on product manifold
First, let's formulate the question properly:
Given a topological space $X$, define be
$$
d(X):=\sup \{ n: X~ \hbox{is homeomorphic to} ~Y\times {\mathbb R}^n\}.
$$
Lemma. The following quantities a …
- 12.3k
8
votes
Can every manifold be given an analytic structure?
Let me correct the misconception appearing in Greg's answer, which also made its way into the nLab article real analytic space:
"Anyway, the result is much harder than what Whitney did, which is a ve …
- 12.3k
8
votes
Accepted
Ergodicity of action of finite index subgroups in the boundary
Let $X$ be a Riemann surface of class $P_G$ (i.e. which carries a Green function) but is Liouville (i.e. admits no nonconstant bounded harmonic functions). One way to construct these is to take a $\m …
- 12.3k
6
votes
An approach to showing hyperbolic groups are CAT(0)
This approach is quite hopeless for several reasons. First of all, let me try to make sense of what you wrote.
You write:
$\Delta 𝐺$ factors as a hyperbolic space ${\mathcal H}𝐺$ which is quasi-is …
- 12.3k
5
votes
Virtually large groups of small rank (related to 3-manifolds)
The question stems from a misinterpretation of Theorem 1.1 in the paper by Boileau and Zieschang. Theorem 1.1 excludes a fair number of cases, in particular, it does not apply to (totally oriented) c …
- 12.3k