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Results tagged with manifolds
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user 21095
A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.
21
votes
1
answer
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isotopy inverse embeddings vs. diffeomorphisms
I would like to find an example, if one exists, of manifolds $M$ and $N$ with embeddings $f:M\to N$ and $g:N\to M$ such that $f\circ g$ and $g\circ f$ are both isotopic (i.e. homotopic through embeddings … You may assume that the manifolds have no boundary, but I would also be interested in compact examples. …
- 6,210
18
votes
Accepted
Cancellation law for $M^n\times \mathbb R= N^n\times \mathbb R$.
closed, smooth manifolds of dimension greater than 3, then $M\times\RR$ is diffeomorphic to $N\times\RR$. … The article "Contact manifolds with symplectomorphic symplectizations" states the result for smooth manifolds as corollary 2.5. [The proof described there applies also to topological manifolds.] …
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15
votes
Accepted
homotopy type of embeddings versus diffeomorphisms
Personal comment: It seems the discussion in this question finally led me to understand how to modify Agol's argument to answer the present question. In fact, my motivation when asking that question a …
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17
votes
2
answers
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homotopy type of embeddings versus diffeomorphisms
Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask a further question a …
- 6,210
10
votes
Accepted
Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
However, it can be deduced from the following fact: topological manifolds (paracompact and Hausdorff) are absolute neighbourhood retracts, and thus have the homotopy type of CW-complexes. … For the specific case of compact manifolds, an elementary proof is given in the appendix of Hatcher's book Algebraic topology (see corollary A.12 there).
We can now prove the result you state. …
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11
votes
Distinct manifolds with the same configuration spaces?
I will present an example involving only (non-compact) manifolds without boundary. As far as I know, the analogous problem for closed manifolds is wide open. … A few consequences of this are:
An embedding $V \to W$ between contractible $n$-manifolds without boundary induces a homotopy equivalence on all ordered and unordered configuration spaces. …
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