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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).
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
5
votes
Accepted
Residual finiteness and a gluing problem
Thurston never finished his project, hence, we cannot know for sure what exactly did he have in mind in this part of the diagram. Here is what we know:
Fundamental groups of good compact 3-dimensiona …
- 12.3k
2
votes
Accepted
Uniqueness of a properly convex projective domain divisible by a group
Take a discrete rank $n$ free abelian subgroup $\Gamma$ of the group of diagonal matrices in $SL(n+1, \mathbb R)$ with positive diagonal entries. The group $\Gamma$ will preserve each open coordinate …
- 12.3k
2
votes
Can a hyperbolic three-manifold have 𝑛 toric boundary components?
As for conformal moduli of the tori (more precisely, Teichmuller parameters) that appear: It is hard to tell, afaik, there is no explicit description. We know that these will be elements of $\bar{{\ma …
- 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
4
votes
Accepted
Books for learning branched coverings
Montesinos wrote several papers defining the meaning of branched coverings and proving basic properties(not just between manifolds, but for general topological spaces):
Montesinos-Amilibia, José María …
- 12.3k
2
votes
Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it n...
Here is a proof in the 2-dimensional case. It does not generalize in higher dimensions and more ideas would be needed. Reading R.Berlanga "A mapping theorem for topological sigma-compact manifolds", C …
- 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
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
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
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
3
votes
If the universal cover of a manifold is spin, must it admit a finite cover which is spin?
As promised, here is my solution based on the Davis trick. First, there is a very general construction of PL aspherical 4-manifolds (it also works in higher dimensions). Start with a finite aspherical …
- 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
3
votes
Accepted
Is the intersection of two distinct sufficiently small metric spheres always empty, a point ...
Let $C_\alpha$ denote the $n$-dimensional closed Euclidean solid cone with the cone angle $\alpha\in (0, \pi)$. Let $X_\alpha$ be the metric space obtained by gluing two copies $C^\pm_\alpha$ of $C_\a …
- 12.3k
1
vote
Hyperbolic 3-manifolds inside algebraic varieties
Let $S$ be a complete hyperbolic surface of finite area, $f: S\to S$ be a pseudo-Anosov homeomorphism which lies in a torsion-free finite index subgroup $\Gamma$ of the mapping class group $Mod_S$ (fo …
- 12.3k