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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).
3
votes
Why is complex projective space triangulable?
Although I am more than a decade late to the discussion, here is an alternate approach to an explicit triangulation of $\mathbb{C}P^n$.
(1) A simplicial set $X$ is a natural generalization of a simpli …
- 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 …
- 1,446
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 …
- 35.5k
6
votes
Accepted
Casson's invariant and the trivial connection contribution to witten's 3-manifold invariant
The Casson invariant is not the same sum or integral over connections that you would derive from the perturbative expansion Cherns-Simons quantum field theory at all flat connections. There is more t …
- 35.5k
8
votes
Accepted
Does Lackenby's polynomial bound on knot moves imply polynomial mixing in "Quantum Money Fro...
Thanks to HJRW2 for the flattering invitation here, and I will give an answer, but it might be not all that deep. In fact I haven't been on MO much lately; maybe I should visit it more.
I don't see …
- 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 …
- 50.8k
29
votes
Accepted
Manifolds are paracompact
Theorem: A countable atlas of charts for a Hausdorff $n$-manifold $M$ can be refined to a locally finite atlas. In fact, each chart only needs to be trimmed.
Proof: Let $U_1,U_2,\ldots$ be the chart …
- 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
17
votes
Intuition behind Alexander duality
I read Alexander duality as saying "If you have any type of doughnut in a sphere, then the outside must have handles or islands that fill the doughnut's holes." The proof that Ryan outlines exactly m …
- 56.6k
9
votes
Accepted
Generalization of Moise's theorem
In fact it helps immensely that $M$, $X$, and $Y$ are all Riemannian, so much so that the question is both true and not at all a generalization of Moise's theorem. Instead, you are looking for a smo …
- 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
22
votes
Accepted
Riemannian metrics on non-paracompact manifolds
On the contrary, the long line does not have a Riemannian metric. Every countable subset of the long line has a least upper bound, so if it were Riemannian then a geodesic ray in the long direction w …
- 56.6k
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
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
10
votes
Utility of virtual knot theory?
My view of virtual knot theory: An R-matrix is a tensor with four indices that, in a natural sense, satisfies the Reidmeister relations. Actually it is enough to consider the third Reidemeister rela …
- 56.6k