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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.
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 …
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10
votes
Accepted
Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
I have no idea at the moment where to find a reference for the specific result you seek. However, it can be deduced from the following fact: topological manifolds (paracompact and Hausdorff) are absol …
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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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7
votes
Accepted
Does the signature admit a homotopy coherent refinement?
[Since my comment above appears to have been helpful, I am repeating it here.]
I must admit I am unfamiliar with L-theory. Nevertheless, I came across a recent article on the arXiv which is related: …
- 6,210
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}}$$\newcommand …
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23
votes
Accepted
Is the counit of geometric realization a Serre fibration?
$\newcommand{\real}[1]{\left\lvert #1 \right\rvert}$$\newcommand{\Sing}[1]{\operatorname{Sing}(#1)}$$\newcommand{\counit}{\epsilon}$$\newcommand{\To}{\longrightarrow}$$\newcommand{\proj}{\mathrm{proj} …
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6
votes
Is the space of diffeomorphisms homotopy equivalent to a CW-complex?
[Edit: Allen Hatcher posted an answer while I was writing this one. Both answers seem to use similar ideas. I will leave my answer here anyway.]
$\newcommand{\Diff}{\operatorname{Diff}}$$\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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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 …
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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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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 …
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8
votes
Serre Spectral Sequence of Representations
Personal comment: I feel that the two previous answers may be together creating some confusion on the subject of the question. I wish to address that with my answer, which would be more suited as a co …
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4
votes
Homotopy Equivalences and Induced Correspondences between Fibre Bundles
Here is a quick argument which proves homotopy equivalence directly. First, the pullback of a homotopy equivalence along a Hurewicz fibration is again a homotopy equivalence. Further, by the well-know …
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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 …
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 …
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