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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).
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
6
votes
1
answer
873
views
Recognizing regular neighborhoods
In a Riemannian manifold consider two compact smooth submanifolds $S$, $S^\prime$ that intersect transversely. It seems intuitively obvious that for a sufficiently small number $r$, the union of $r$- …
- 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
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
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
5
votes
1
answer
361
views
Whitney sum formula for topological Pontryagin classes
Is there a Whitney sum formula for topological rational Pontryagin classes? I thought the answer is yes, but now I cannot find a reference. Is it even true? The PL case would also be of interest.
- 29.1k
4
votes
0
answers
103
views
Convex subsets of infinite dimensional spaces up homeomorphism
Let $C$ be a convex, infinite-dimensional, non-locally-compact subset of a separable Frechet space.
If $C$ is a closed subset (or more generally, if $C$ is completely metrizable), then it is known t …
- 29.1k
4
votes
0
answers
78
views
Implicit function theorem for PL maps
Let $K$ be a PL triangulation of a closed manifold and $f: K\to\mathbb R^k$ be a PL map. Equivalently, $f$ is a map that becomes linear on every simplex after subdividing. Suppose $v$ is a vertex in t …
- 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
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
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
8
votes
0
answers
296
views
Spaces that never separate the Hilbert cube
I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.
Any finite dimensional space has this prop …
- 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
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
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