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Results tagged with differential-topology
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user 21095
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
17
votes
Accepted
Manifolds with homeomorphic interiors
Gjergji Zaimi's answer gives a strong positive conclusion: the product of the boundaries with $\mathbb{R}$ are necessarily homeomorphic. I just want to add a couple of explicit examples illustrating t …
- 6,210
1
vote
Codimension zero embeddings and diffeomorphism groups
[The following is an elaboration of my comment above, in response to Igor Belegradek's inquiries. Due to the typographical limitations of comments, I am posting it as an answer.]$\DeclareMathOperator{ …
- 6,210
40
votes
Converse of Poincaré-Hopf theorem
$\newcommand{\ZZ}{\mathbb{Z}}$$\newcommand{\CC}{\mathbb{C}}$A simple counter-example is given by $M = \CC P^3$.
Recall first that the cohomology ring of $\CC P^3$ is a truncated polynomial algebra:
$ …
- 6,210
18
votes
Accepted
Cancellation law for $M^n\times \mathbb R= N^n\times \mathbb R$.
[Edit: I have added some details and a more explicit example by Milnor.]
I will present a couple of examples verifying the conditions required in the question.$\newcommand{\RR}{\mathbb{R}}
\newcomman …
- 6,210
10
votes
Accepted
Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
I have no idea at the moment where to find a reference for the specific result you seek. However, it can be deduced from the following fact: topological manifolds (paracompact and Hausdorff) are absol …
- 6,210
34
votes
1
answer
4k
views
Strong Whitney embedding theorem for non-compact manifolds
$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong …
- 6,210