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Results tagged with gt.geometric-topology
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user 184
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).
23
votes
0
answers
684
views
Do most manifolds have symmetries? or not?
Let us say that a (closed, connected) manifold has a symmetry if it admits a non-trivial action by a finite group. Note that I am not asking the action to be free. So for example rotating the 2-sphere …
- 27.5k
10
votes
2
answers
647
views
Homotopy equivalent smooth 4-manifolds which are not stably diffeomorphic?
Recall that two 4-manifolds $M$ and $N$ are stably diffeomorphic if there exist $m,n$ such that
$$M \#_n (S^2 \times S^2) \cong N \#_n (S^2 \times S^2).$$
That is, they become diffeomorphic after taki …
- 27.5k
17
votes
Accepted
How aggressive is the fibrant replacement of $\mathrm{Bord}_n$?
The completeness condition is not really about making things invertible which weren't already. It is about where the information about invertible morphisms is stored.
We can already see this with $(\i …
- 27.5k
18
votes
Does $M^o=N^o$ imply that $\partial M = \partial N$?
(Marc Kegel posted his answer just before I posted this. I will leave it because perhaps this helps elaborate some of the points)
No, this is not true in general.
Here is an example of what can ha …
- 27.5k
6
votes
TQFT and Mapping Class Groups
The following paper answers your question precisely, I think: arxiv:1408.0668
Specifically Theorem 1.3 (which is elaborated on in Section 4) describes precisely what additional data you must specify …
- 27.5k
28
votes
Accepted
Nilpotence of the stable Hopf map via framed cobordism
Answer Summary
Let $\eta$ be the framed 1-manifold which is the Lie group framing on the circle and let $\nu$ be the Lie group framing on $S^3 = Spin(3)$. I am probably going to conflate these framed …
- 27.5k
40
votes
Accepted
Classifiying sphere eversions
Answer Summary
The fundamental group of the space of immersions of $S^2$ into $\mathbb{R}^3$ is
$$ \pi_1 Im(S^2, \mathbb{R}^3) \cong \mathbb{Z}/2 \times \mathbb{Z}$$
This means that there are infini …
- 27.5k
27
votes
2
answers
2k
views
Are the mapping class groups of manifolds finitely presentable?
The mapping class group of a manifold is the group $\pi_0 Diff(M)$ of components of the diffeomorphism group. There are several variations: oriented manifolds and orientation preserving diffeomorphism …
- 27.5k
6
votes
Reference request: gluing manifolds along pieces of boundary
This was a bit too long for a comment, so I am posting it as an answer. You are sort of asking two things:
How to turn your manifolds M and S into an appropriate manifold with corners together with …
- 27.5k
9
votes
Accepted
Spin TQFT's in dimensions (1+1)
This is covered in Moore and Segal "D-branes and K-theory in 2D topological field theory". In particular on around page 16 there is a characterization analogous to "1+1 TQFTs = Commutative Frobenius a …
- 27.5k
3
votes
1
answer
547
views
Is the Action of the mapping class group transitive on embedded arcs?
Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The gro …
- 27.5k