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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
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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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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7
votes
Explicit constructions of K(G,2)?
This is a comment relating the other answers more than anything else.
Following are three isomorphic simplicial abelian groups which are Eilenberg-Maclane spaces $K(G,n)$.
The result of applying th …
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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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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 …
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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{ …
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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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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)\ …
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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 …
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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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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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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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