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Results tagged with homotopy-theory
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user 21095
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
15
votes
Accepted
Three questions on $\operatorname{hocolim}$
First question$\newcommand{\op}[1]{{#1}^{\mathrm{op}}}$$\newcommand{\sSet}{\mathrm{sSet}}$$\newcommand{\Grpd}{\mathrm{Grpd}}$$\newcommand{\Cat}{\mathrm{Cat}}$$\newcommand{\NN}{\mathbb{N}}$$\newcommand …
- 6,210
0
votes
(Homotopy) Y ENR and contractible subset implies Y is a retract
Observe that any retract of $\newcommand{\RR}{\mathbb{R}} \RR^n$ is necessarily a closed subspace of $\RR^n$. Assuming this necessary condition, the answer to the question is affirmative. More precise …
- 6,210
5
votes
How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there?
A rather roundabout method for computing the fundamental group of $S^n$ comes from using Kan's loop group construction as briefly described in this answer by John Klein. The basic theory of the Kan lo …
2
votes
How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there?
A stable version of Jeremy Miller's answer uses instead the Barratt-Priddy-Quillen theorem about $\Omega^\infty\Sigma^\infty S^0$ (for example, as stated in Graeme Segal's "Categories and cohomology t …
2
votes
Multisimplicial geometric realization
[This answer is mostly a long comment to Peter May's answer.]
Edit: I have corrected some arrows which were pointing the wrong way.$\newcommand{\real}[1]{\lvert #1\rvert}
\newcommand{\Map}{\operatorn …
- 6,210
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 …
- 6,210
14
votes
2
answers
2k
views
Well-pointed space which is not locally contractible
I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the inc …
- 6,210
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
23
votes
Accepted
Is the counit of geometric realization a Serre fibration?
$\newcommand{\real}[1]{\left\lvert #1 \right\rvert}$$\newcommand{\Sing}[1]{\operatorname{Sing}(#1)}$$\newcommand{\counit}{\epsilon}$$\newcommand{\To}{\longrightarrow}$$\newcommand{\proj}{\mathrm{proj} …
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12
votes
Plus construction considerations.
For convenience (at least my own) and completeness, I want to give an explanation of Tom Goodwillie's answer, as it was not obvious to me how to prove the statement he makes. I wanted to leave it as a …
- 6,210
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 …
- 6,210
11
votes
Accepted
Monoidal model category structure on a functor category.
[This should probably be a comment, since it is so short. Nevertheless, it is an answer to the question.]
The result you ask for is a consequence of proposition 2.2.15 in Sam Isaacson's Ph.D. thesis …
- 6,210
21
votes
Accepted
Example of fiber bundle that is not a fibration
$\newcommand{\RR}{\mathbb{R}}
\newcommand{\To}{\longrightarrow}
\newcommand{\id}{\mathrm{id}}$The example described in Tom Goodwillie's answer to a related mathoverflow question essentially solves thi …
- 6,210
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 …
- 6,210
13
votes
Accepted
Does the bordism homology theory satisfy the weak equivalence axiom?
This answer is simply to write the details for my comment above. It amounts to doing a little more work with homotopy equivalences, so as to carry out essentially the argument you gave in your comment …
- 6,210