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Results tagged with homotopy-theory
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user 12166
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.
8
votes
Accepted
Quasi-equivalent vs. homotopy equivalent functors in $A_\infty$ categories
As you expect, the answer is negative in general, although counterexamples are relatively recent. You can find a counterexample in:
Alberto Canonaco y Paolo Stellari, «Non-Uniqueness of Fourier–Mukai …
- 15.2k
7
votes
What are some good examples of spectral sequences which degenerate after the first nontrivia...
Any DG algebra $A$ over a field has a minimal model, which is a minimal $A_\infty$-algebra $(H,m_3,m_4,\dots)$. It consists of a graded algebra $B=H^*(A)$ equipped with multi-linear operations of degr …
4
votes
Accepted
Left Proper model structure on the category of non-symmetric operads in chain complexes
Let $k$ be a commutative ring and let $\mathrm{Ch}(k)$ be the category of non-negatively graded chain complexes of $k$-modules. We endow it with the projective model structure. Weak equivalences are q …
- 15.2k
11
votes
Accepted
The second stable homotopy group
I love this question! I've enjoyed thinking of it. Below, I show why the sequence splits always.
Let $X\to Y=K(H_1(X,\mathbb{Z}/2),1)$ be the map inducing the identity in $H_1(-,\mathbb{Z}/2)$. By nat …
- 15.2k
3
votes
Accepted
Computing the homotopy type of $B\operatorname{Aut}(K(G,1))$ using a fibration sequence: why...
This argument is rather elementary. Maybe we should later move this to MathStackExchange. Anyway:
As mentioned above in comments, $K(G,1)$ is not a topological monoid and $K(G,2)$ doesn't exist, so th …
- 15.2k
3
votes
Spaces homotopy dominated by $S^2 \times S^2\times S^2$
You're actually right. The cohomology algebra of $X=S^2\times S^2\times S^2$ with coefficients in $\mathbb{Z}$ is $H^*(X)=\mathbb{Z}[x,y,z]/(x^2,y^2,z^2)$ with $|x|=|y|=|z|=2$. The cohomology algebra …
- 15.2k
5
votes
Which homotopy 2-types are H-spaces?
I just wanted to add to Tyler Lawson's answer that all the maps $\beta\colon K(G,1)\rightarrow K(A,3)$ ($G$ and $A$ abelian and no action of $G$ on $A$) satisfying his additivity condition are loop ma …
- 15.2k
15
votes
Accepted
Cohomology theories for spaces vs cohomology theories for spectra
The category of cohomology theories on pointed CW-complexes is not equivalent to the stable homotopy category. The latter projects onto the former, and this projection induces a bijection on isomorphi …
- 15.2k
23
votes
Accepted
Homology of the fiber
As usual, there's no loss of generality in assuming that $f$ is the inclusion of a subspace $X\subset Y$, replacing $Y$ with the homotopy equivalent mapping cylinder of $f$ if necessary. By your assum …
- 15.2k
10
votes
Connectivity of suspension-loop adjunction
Indeed, as John Klein shows, the map is $(2k+2)$-connected. Let me offer an alternative proof of the fact that, for $X$ a $k$-connective spectrum, $k\geq 0$, the homomorphism $\pi_i\Sigma^\infty\Omega …
- 15.2k
5
votes
Accepted
When does the projective model structure on functors exist?
I found it in Hirschhorn's book. It's in 11.6 Diagrams in a cofibrantly generated model category. The answer is always, provided K
is cofibrantly generated (Theorem 11.6.1). For him (as well as for Ho …
- 15.2k
4
votes
How stable is the top cell of a Lie group?
Let me just give the naive answer. The homotopy category of $n$-connected $(n+k)$-dimensional CW-complexes is stable for $k<n-1$, by the Freudenthal suspension theorem etc. So the $(\dim G+3)$-fold su …
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4
votes
Accepted
isotopy equivalence (topological meaning) between $CW$-complexes
Since $M$ is a closed manifold and $gf$ is homotopic to the identity, it must be surjective, otherwise it wouldn't preserve the fundamental class mod 2. In particular $g$ is surjective. Your isotopy c …
- 15.2k
17
votes
Accepted
Does every (co)homology functor (in particular, stable homotopy) factor through chain comple...
No (I mean, not in a triangulated way), otherwise any generalized homology theory of a mod 2 Moore space would be 2-torsion, but this is not true for mod 2 stable homotopy groups (it's well known that …
- 15.2k
5
votes
When are (weak) homotopy equivalence testable on open covers?
Let me offer sufficient conditions in both cases. They follow from the existence of the following two left proper model structures on the category of topological spaces, and the well-known gluing lemm …
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