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Results tagged with gt.geometric-topology
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user 1573
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).
23
votes
0
answers
1k
views
Boundaries of noncompact contractible manifolds
It is known that a manifold $B$ bounds a compact contractible topological manifold if and only if $B$ is a homology sphere. The "only if" direction follows by excising a small ball in the interior of …
- 29.1k
22
votes
0
answers
622
views
Smooth thickenings of non-smoothable manifolds
It is known that any closed topological manifold is homotopy equivalent to an open smooth manifold.
Question 1. What can be said about the smallest
dimension of a smooth manifold
that is homotopy e …
- 29.1k
20
votes
2
answers
1k
views
Characteristic classes for block bundles
Block bundles are PL analogs of vector bundles, see e.g. Rourke-Sanderson's
article
in Bulletin of AMS, 1966. There is a classifying space $B\widetilde{PL}_q$ for rank $q\ $ block bundles, which is …
- 29.1k
20
votes
3
answers
2k
views
Homotopy groups of spaces of embeddings
Let $\mathrm{Emb}(M, N)$ be the space of smooth embeddings of a closed manifold $M$ into a manifold $N$ equipped with smooth compact-open topology.
Question 1. Are there conditions ensuring that t …
- 29.1k
17
votes
1
answer
654
views
Groups with finitely generated center
Does every group with a finite classifying space have finitely generated center?
Remarks:
If $G$ is a finitely generated group with infinitely generated center $Z(G)$,
then the quotient $G/Z(G)$ i …
- 29.1k
16
votes
0
answers
642
views
Approximating homeomorphisms of 2-disk by diffeomorphisms
Any homeomorphism of a compact surface can be approximated by diffeomorphisms.
Is there a parametrized version of this result, where the parameter space is an $n$-disk?
In other words, if $S$ is a com …
- 29.1k
14
votes
3
answers
1k
views
Irreducible homology 3-spheres that bound smooth contractible manifolds
Some examples of irreducible homology 3-spheres that bound smooth contractible 4-manifolds are listed in the comment to problem 4.2 in Kirby's problem list, and all of them happen to occur among the …
- 29.1k
12
votes
1
answer
466
views
Finitely presented group in which every element is conjugate to its square
Does there exist a nontrivial finitely presented group in which every element is conjugate to its square? Is this an open problem?
Motivation: Jahren proved in [Geom Dedicata (2010)] that if $M$ is a …
- 29.1k
12
votes
2
answers
1k
views
Geometrization for 3-manifolds that contain two-sided projective planes
Often when people write about the geometrization conjecture they assume (for simplicity) that the manifold is orientable. I never seriously thought of non-orientable 3-manifolds, but while reading Mor …
- 29.1k
11
votes
2
answers
2k
views
Isometry classification of spherical space forms
A spherical space form is a compact Riemannian manifold of constant sectional curvature $1$, or equivalently, the quotient of the unit sphere by a finite group of orthogonal transformations that have …
- 29.1k
9
votes
1
answer
467
views
What is the normalizer of the circle in the diffeomorphism group of the 2-sphere?
What is the normalizer of $SO(2)$ in $\mathrm{Diff}(S^2)$?
Remarks:
We let $SO(2)$ act on $S^2$ via the rotation about the $z$-axis.
It is immediate that each element of the normalizer must map an …
- 29.1k
9
votes
1
answer
909
views
Diffeomorphism group of the 2-sphere with $C^0$ topology
What is known about the homeomorphism (or homotopy) type of the group of $C^\infty$ diffeomorphisms of $S^2$ equipped with $C^0$ topology?
The group is the image of the inclusion $\mathrm{Diff}(S^2)\ …
- 29.1k
9
votes
2
answers
828
views
Hyperbolic $3$-manifold groups that embed in compact Lie groups
Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple conne …
- 29.1k
9
votes
0
answers
281
views
Extending smooth triangulation
Can one always extend a smooth triangulation from a smooth submanifold $S$ to the ambient manifold $M$? (For simplicity both $S$ and $M$ are compact without boundary). Is the extension possible when $ …
- 29.1k
8
votes
1
answer
696
views
Hyperbolic 3-manifolds with no geometrically finite structure
Does there exist a compact hyperbolic 3-manifold $M$ that is not diffeomorphic to a geometrically finite hyperbolic manifold? If yes, can such $M$ have incompressible boundary?
I think the answer sho …
- 29.1k