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Results tagged with gt.geometric-topology
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user 78
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).
9
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3
answers
3k
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Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vect...
A version of the "hairy ball" theorem, due probably to Chern, says that the Euler-characteristic of a closed (i.e. compact without boundary) manifold $M$ can be computed as follows. Choose any vector …
- 54.6k
5
votes
1
answer
538
views
What is the Heegaard Floer Homology of a connect sum of $S^2 \times S^1$s?
There are many conjecturally-equivalent three-manifold Floer homologies, of which my understanding is the most-computable is Heegaard Floer homology.
What is the (Heegaard) Floer homology of a c …
- 54.6k
4
votes
2
answers
539
views
Is every group object in TopMan a Lie group?
Recall that a Lie group is a group object in the category of C∞ manifolds.
If I have a group object in the category of topological manifolds, can I necessarily equip it with a smooth structure so tha …
- 54.6k
16
votes
2
answers
771
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Are there results from gauge theory known or conjectured to distinguish smooth from PL manif...
My question begins with a caveat: I sometimes spend time with topologists, but do not consider myself to be one. In particular, my apologies for any errors in what I say below — corrections are encou …
- 54.6k
5
votes
1
answer
687
views
What is the Euler characteristic of a mapping space?
Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. Is it there a formula for th …
- 54.6k
4
votes
1
answer
352
views
When does a VBLA induce an isomorphism on Lie algebroid cohomology?
This question is geared towards the experts, so I will only briefly gloss the definitions. Everything I say is in the category of finite-dimensional smooth manifolds, and whenever I say "$\mathbb Z$- …
- 54.6k
94
votes
4
answers
15k
views
Can every manifold be given an analytic structure?
Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy …
- 54.6k
19
votes
1
answer
1k
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How difficult is Morse theory on stacks?
The title is a little tongue-in-cheek, since I have a very particular question, but I don't know how to condense it into a pithy title. If you have suggestions, let me know.
Suppose I have a Lie gro …
- 54.6k
10
votes
2
answers
700
views
When does an even-dimensional manifold fiber over an odd-dimensional manifold?
Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with oriented fibers)?
For ex …
- 54.6k
5
votes
0
answers
166
views
In cell-decomposed manifolds, how easy is it to arrange for the tubular neighborhood of a di...
Suppose that you have decomposed a manifold $M$ into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category, all the better). …
- 54.6k
24
votes
5
answers
3k
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Can surfaces be interestingly knotted in five-dimensional space?
It's possible this question is trivial, in which case it will be answered quickly. In any case, I realized that it's a basic question the answer to which I should know but do not.
Everybody loves kn …
- 54.6k
6
votes
1
answer
440
views
Are framed manifolds cubulatable?
Let's say an $n$-manifold is cubulated if it is glued out of cubes $[0,1]^n$ in a way that looks locally like the standard cubulation of $\mathbb R^n$. For instance, the face $[0,1]^{k-1} \times \{1\} …
- 54.6k
18
votes
1
answer
515
views
How aggressive is the fibrant replacement of $\mathrm{Bord}_n$?
Lurie (On the Classification of Topological Field Theories), with some corrections by Calaque and Scheimbauer (A note on the $(\infty,n)$-category of cobordisms), famously constructed a symmetric mono …
- 54.6k
11
votes
1
answer
334
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Reference request: sheaves on the site of d-manifolds
I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for thes …
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