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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).
41
votes
0
answers
1k
views
Homotopy type of TOP(4)/PL(4)
It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(4)\ …
- 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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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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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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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 …
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21
votes
1
answer
1k
views
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 embeddi …
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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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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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17
votes
2
answers
2k
views
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 …
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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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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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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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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 …
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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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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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