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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.
41
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0
answers
1k
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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
30
votes
4
answers
3k
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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 …
- 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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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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17
votes
2
answers
2k
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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
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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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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14
votes
2
answers
2k
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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 …
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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 …
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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 …
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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 …
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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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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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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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