All Questions
Tagged with at.algebraic-topology differential-topology
503
questions
8
votes
1
answer
1k
views
Topological degree of polynomial maps.
The $\mathbb{Z}_2$ topological degree of a (non-constant) polynomial in one variable, clearly, coincides with its degree as a polynomial, $\mod 2$.
Consider further a polynomial self-mapping $F$ on ...
- 58.1k
9
votes
3
answers
696
views
references / general idea of kervaire invariant problem
There's a workshop at MSRI in a couple months on the Kervaire invariant problem that I'd really like to attend. I saw Hopkins speak about it a while back without understanding much of the talk, but I'...
- 6,049
17
votes
5
answers
3k
views
Homologically trivial submanifolds
Unuseful prequel
Let $M$ be a (compact, oriented, differentiable) manifold. Before knowing anything about homology theory a naif but clever mathematician may want to measure the holes in $M$ by the ...
- 14.6k
17
votes
2
answers
2k
views
Is a conceptual explanation possible for why the space of 1-forms on a manifold captures all its geometry?
Let $M$-be a differentiable manifold. Then, suppose to capture the underlying geometry we apply the singular homology theory. In the singular co-chain, there is geometry in every dimension. We look at ...
- 3,669
24
votes
1
answer
5k
views
Do "surjective" degree zero maps exist?
Is there a map $f\colon X \to Y$ of closed, connected, smooth and orientable $n$-dimensional manifolds such that the degree of $f$ is 0 but $f$ is not homotopic to a non-surjective map?
Added: The ...
- 2,560
13
votes
3
answers
3k
views
Contractible manifold with boundary - is it a disc?
I'm sure this is standard but I don't know where to look. Let $M$ be a contractible compact smooth $n$-manifold with boundary. Does it have to be homeomorphic to $D^n$? What about diffeomorphic?
[...
- 32.2k
20
votes
1
answer
2k
views
Every Manifold Cobordant to a Simply Connected Manifold
I am wondering if it is true that every compact, connected, oriented manifold is cobordant to a simply connected manifold.
I believe that some sort of surgery will do the trick. Roughly speaking, I ...
- 2,694
39
votes
4
answers
7k
views
Two kinds of orientability/orientation for a differentiable manifold
Let $M$ be a differentiable manifold of dimension $n$. First I give two definitions of Orientability.
The first definition should coincide with what is given in most differential topology text books, ...
- 7,382