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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).
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
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
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
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
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
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
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
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
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
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
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