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Results tagged with gt.geometric-topology
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user 21095
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).
24
votes
How many distinct homeomorphism classes of lens spaces are there with a fixed p?
This is an interesting question which can be solved by applying Burnside's lemma (as suggested by Qiaochu Yuan in a comment) and some knowledge of the group of units modulo $n$. The relevant facts abo …
- 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
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. Nevertheless, the article Configuration sp …
- 6,210
3
votes
Simplicial replacements in smoothing theory
I present here a reference for Peter May's comment to Tom Goodwillie's answer in this thread. It also corroborates the comment by John Klein below the question stating that there is no obvious topolog …
- 6,210
40
votes
Converse of Poincaré-Hopf theorem
$\newcommand{\ZZ}{\mathbb{Z}}$$\newcommand{\CC}{\mathbb{C}}$A simple counter-example is given by $M = \CC P^3$.
Recall first that the cohomology ring of $\CC P^3$ is a truncated polynomial algebra:
$ …
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11
votes
Accepted
What fraction of n-point sets in the unit ball have diameter smaller than 1?
I am certainly not the best person to answer this question, as I do not have much insight to share regarding how to approach this kind of problems. My only (fairly obvious) suggestion is to estimate t …
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34
votes
1
answer
4k
views
Strong Whitney embedding theorem for non-compact manifolds
$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong …
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9
votes
Accepted
Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but whe...
$\newcommand{\set}[1]{\lbrace #1 \rbrace}$I will assume that the notation $\Sigma X$ in the question denotes the unreduced suspension of the space $X$.
Quick answer: The notion of homotopy equivalenc …
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6
votes
Is the space of diffeomorphisms homotopy equivalent to a CW-complex?
[Edit: Allen Hatcher posted an answer while I was writing this one. Both answers seem to use similar ideas. I will leave my answer here anyway.]
$\newcommand{\Diff}{\operatorname{Diff}}$$\newcommand{ …
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30
votes
4
answers
3k
views
Is the space of diffeomorphisms homotopy equivalent to a CW-complex?
Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ fo …
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5
votes
Does Euclidean space have a compact factor?
Here is a proof which uses only singular homology.$\newcommand{\RR}{\mathbb{R}}$$\newcommand{\ZZ}{\mathbb{Z}}$$\newcommand{\To}{\longrightarrow}$$\def\set#1{\lbrace#1\rbrace}$$\newcommand{\Xminusx}{X\ …
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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 …
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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 …
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11
votes
Accepted
Is there a general theory of fiber theorems?
Edit: I have added some definitions and details to my answer.
In the most general form I can find, your third question is a consequence of two results regarding cell-like maps and fine homotopy equiv …
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15
votes
What manifolds are boundaries of euclidian spaces ?
The answer is in fact given in the question Misha links to in his comment. For completeness, I wanted to give the details in a comment, but that became too long, so I turned it into this answer. The i …
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