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Results tagged with gt.geometric-topology
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user 1450
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).
24
votes
Accepted
Homotopies of triangulations
The standard name for this type of relation between two structures on $X$ is concordance rather than homotopy. If two structures on $X$ are isotopic (with the respect to the appropriate homeomorphism …
- 56.6k
5
votes
Accepted
What kind of 3-manifolds arise has hypersurfaces in R^4?
A simple construction that bears on the narrow version of John's question: If $M$ is a closed $n$-manifold that embeds in $\mathbb{R}^{n+1}$ (which can only happen if $M$ is orientable), then $M \tim …
- 56.6k
2
votes
Recognizing regular neighborhoods
You're not really forced to leave the smooth category at an earlier stage than before. There is a good pseudogroup for the category of "smooth manifolds with transverse submanifolds". If $r$ is smal …
- 56.6k
24
votes
Are there any very hard unknots?
There are really two questions here: (1) Can you an untangle any unknot with relatively little work, say a polynomial number of geometric moves of some kind? (2) Given a knot, can you quickly figure …
- 56.6k
11
votes
Rugged manifold
To expand on Ryan's comment: A handle decomposition of a manifold is a restricted type of quotient of a disjoint union of handles. An $n$-dimensional $k$-handle is by definition the manifold $B^k \t …
- 56.6k
12
votes
Singular chains in Spivak's Calculus on Manifolds
It's clearly an elementary oversight that, on the other hand, doesn't matter for the real development of the material. Yes, the chain $c$ ought to be at least $C^1$ for the pullback to be continuous …
- 56.6k
15
votes
Classification of homology 3-spheres?
On the other hand, there is no particular classification of hyperbolic homology 3-spheres, much less hyperbolic links in homology 3-spheres, other than in general terms that they all come from hyperbo …
- 56.6k
4
votes
Knots that unknot in a manifold
Ryan gives a very nice answer in dimension 3, leaving the higher-dimensional case of the question open. I can't discuss the question with as much authority as I would like, but I can start to piece t …
- 56.6k
8
votes
Conformal embedding of Riemann surfaces into 3-space
I have thought about this question before, but at the moment I can't remember links or references. Nonetheless, many years ago I thought of a sketch of an argument that should eventually work to prov …
- 56.6k
67
votes
Accepted
Poincaré Conjecture and the Shape of the Universe
In Einstein's theory of General Relativity, the universe is a 4-manifold that might well be fibered by 3-dimensional time slices. If a particular spacetime that doesn't have such a fibration, then it …
- 56.6k
32
votes
Accepted
Why is Casson's invariant worth studying?
Wikipedia's description of the Casson invariant gives the first important reason to study it. As an invariant that comes from the $\text{SU}(2)$ representation variety of $\pi_1(M)$, it reveals in pa …
- 56.6k
6
votes
Topological version of Bogomolov’s question
There is strong circumstantial evidence that the question came from a personal conversation with Bogomolov. First, Gromov's paper has more than 50 references, but he doesn't cite Bogomolov at all. S …
- 56.6k
1
vote
Hurwitz Encoding
I'd like to add two remarks to David Speyer's great explanation of the Hurwitz encoding of a branched covering. A key question first studied by Hurwitz is when two branched coverings are "the same", …
- 56.6k
1
vote
Accepted
Better term for a (simplicial) contractible plane continuum
In the end, we (Joel Kamnitzer, his student Bruce Fontaine, and I) agreed on the term "diskoid". In the abstract, the word "cactus" seemed too clever by half. When I actually wrote it into the paper …
1
vote
Orientation of a "glued"-manifold
I agree that the difficulty in the question is that you are relying on the homological definition of an orientation of a manifold. As Ryan implies in the comments, the solution is undergraduate-level …
- 56.6k