All Questions
8 questions
33
votes
1
answer
1k
views
Nilpotence of the stable Hopf map via framed cobordism
The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...
- 28.1k
49
votes
4
answers
7k
views
Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
I'm looking for an elegant proof that any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$.
- 1,709
9
votes
2
answers
641
views
Künneth formulas/theorem for bordism groups and cobordisms?
We are familiar with Künneth theorem:
The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain ...
- 10.5k
7
votes
1
answer
260
views
Bordism for oriented triangulable manifolds without smooth differentiable structures
We know the bordism group for oriented smooth differentiable structures such as $\Omega_d^{SO}$ that requires the special orthogonal group structure on the tangent bundle $TM$ of manifold $M$.
$$\...
- 10.5k
7
votes
1
answer
470
views
Twisted spin bordism invariants in 5 dimensions
[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance!
The spin $G$-bordism invariant can be twisted in the way that ...
- 10.5k
5
votes
0
answers
161
views
Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$
This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post.
I had discussed my computation of
$$
\Omega_5^{...
- 10.5k
4
votes
1
answer
192
views
Version of pseudo-isotopy $\neq$ isotopy for $(n+1)$-framings
Let $M$ be a closed $n$-manifold and $\varphi$ be a self-diffeomorphisms of $M$.
There is a bordism from $M$ to itself given by $M\times [0,1]$ with the identification $M \cong M \times \{0\}$ induced ...
- 890
0
votes
1
answer
376
views
Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]
We know that
framing structure means the trivialization of tangent bundle of manifold $M$.
string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
- 447