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Results tagged with homotopy-theory
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user 32022
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.
3
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Geometric realization of simplicial spaces and finite limits
To avoid leaving this question open:
Assuming we work in the category of compactly generated spaces, geometric realization commutes with pullbacks.(It's crucial that we use the compactly generated pr …
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1
vote
Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories i...
If everything takes place in the category of compactly generated spaces, it holds $$\pi_0(BC)=\pi_0(obC)/\tilde{},$$ where two path components in the object space get identified, if there are objects …
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12
votes
1
answer
712
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Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant
I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize.
All manifolds are closed, smooth and have dimensions $n\ge 5$.
The Atiyah-Shapiro-Bott-Orientation gives …
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11
votes
4
answers
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Topological Grothendieck Construction
Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and $x\i …
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The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete s...
Here's a direct way of seeing it, without using the Quillen equivalence to simplicial sets equipped with the Joyal model structure:
The cofibrant objects in the Bergner model structure are the 'simpl …
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5
votes
$\pi_{2n-1}(\operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not tor...
For $n=1$, the answer to your question is negative, as explained by Gregory Arone in the comments.
In the cases $n\neq 1,2,4$, there is the following easy argument:
The long exact sequence of the fib …
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8
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1
answer
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Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces
In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following.
There exist $\Sigma$-free operads $\mathcal{ …
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14
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2
answers
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Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a CW- …
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35
votes
2
answers
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Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop ...
This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather t …
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