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Results tagged with gt.geometric-topology
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user 4362
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).
62
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14
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12k
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What are some of the big open problems in 3-manifold theory?
From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot …
- 35.5k
10
votes
Topological Classification of Four-Manifolds
Suppose you can classify all open 4-manifolds. In particular you can classify all manifolds of the form $M^4 - pt$ where $M^4$ is a closed 4-manifold, and consequently you can classify all closed 4-m …
- 29.2k
0
votes
Accepted
Generalizations of Hopf-Rinow theorem
It seems that the answer to 1 is no: The Hopf-Rinow Theorem is false in infinite Dimensions. If you add the assumption that $M$ is a locally compact length space then the answer is "yes" by Theorem 2 …
- 29.2k
1
vote
Which automorphisms on $H_{1}(M^{3})$ are induced by homotopy equivalences?
Since you expressed interest in the hyperbolic case: it follows from the Mostow rigidity theorem and the Hurewicz theorem that an isomorphism $\phi \colon H_1(M) \to H_1(M)$ is induced by an isometry …
- 29.2k
13
votes
Accepted
Spin structures on $S^1$ and Spin cobordism
As Fabian pointed out in the comments, you have to be more careful about how you trivialize $SO(D^2)$. I'm going to use the standard coordinates $(x,y)$ on $\mathbb{R}^2$ (note that these are not glo …
- 29.2k
8
votes
What should be taught in a 1st course on smooth manifolds?
I think there are two ways to approach a first course on manifolds: one can focus on either their geometry or their topology.
If you want to focus on geometry, then I think Anton Petrunin's suggestio …
- 29.2k