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Results tagged with homotopy-theory
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user 39747
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
Why is $bo$ not flat?
Let me me write $ko$ for connective $KO$, since I'm more used to that. If $ko_*ko$ were a flat $ko_*$-module, then by basechange to $H\mathbb{Z}$, $H\mathbb{Z}_*ko$ would be a flat $\mathbb{Z}$-module …
- 10.8k
7
votes
Accepted
Infinite suspension is cotangent complex
It is not, for example note that if this were true, the case $A=B$ would give you that $L_{B/B}$ is the infinite suspension of $\mathrm{id}: B\to B$. But that's the absolute cotangent complex $L_B$, w …
- 10.8k
14
votes
Accepted
Direct limits in homotopy category
This is the classical $\mathrm{lim}^1$ phenomenon: While $\mathrm{Map}(X,Y) = \operatorname{lim}\mathrm{Map}(X_n,Y)$ (a homotopy limit), on homotopy classes it is not true that $[X,Y] \cong \operatorn …
- 10.8k
7
votes
Accepted
Reference for homotopy groups of filtered homotopy colimits
Here's one way to get this out of the literature:
By [Lurie, Higher topos theory, Prop. 5.3.3.3], for filtered $I$ we have that the colimit functor $\mathrm{Fun}(I,\mathcal{S})\to \mathcal{S}$ commute …
- 10.8k
5
votes
Accepted
Realization of a constant simplicial anima
I don't think this is true as written, for example there are anima $X$ which are not loop spaces of anything (e.g. $X=S^2$). One can also directly see that $\mathrm{ev}_n$ is corepresented by $\Delta^ …
- 10.8k
3
votes
LS category of 4-manifolds with free fundamental group
Since we have settled on an argument in the comments, let me post it as an answer.
We have to show that a closed $4$-manifold with nontrivial free $\pi_1(M)$ does not have $\mathrm{cat}(M)=1$. Indeed, …
- 10.8k
7
votes
Accepted
Proof of the equivalence of spectra $(\mathbb{S}^{-1} \otimes \mathbb{S}^{-1})_{h \Sigma_2} ...
This is probably not quite what you were looking for, but since you said you'd also be happy to see any proof, here is one using a modern perspective on Thom spectra of virtual bundles:
The flip actio …
- 10.8k
13
votes
Accepted
Does there exist a Bousfield localization of the category of spectra which makes the sphere ...
EDIT: I'll leave the old answer up, see below. In the meantime, Maxime Ramzi and I have thought this through and came up with a fun general argument.
Claim. Let $L: \mathrm{Sp}\to \mathrm{Sp}$ be a Bo …
- 10.8k
6
votes
Accepted
Filtered homotopy colimits of spectra
This is a partial answer, not addressing the relation to homotopy colimits in a model category presenting $\mathcal{C}$ (but I think there should be some reasonable statement there, the slogan is cert …
- 10.8k
12
votes
Accepted
Why isn't the anchor map in Lurie's "Rotation Invariance in Algebraic K-Theory" zero?
It's zero as a map of filtered spectra, but it is considered as a map of $\mathbb{A}$-bimodules.
- 10.8k
3
votes
Accepted
(Lower) homotopy groups from triangulations
Being a manifold or the dimension restriction $k\leq n$ doesn't matter, the following applies to finite simplicial complexes in general:
As others have explained, if the fundamental group is not finit …
- 10.8k
7
votes
The center of $\mathbf{hTop}$
Here's a partial answer constraining $\alpha_{S^1}$. First of all, I think my comment shows that (at least if we take $\operatorname{hTop}$ to be the homotopy category of CW complexes), it is possible …
- 10.8k
5
votes
Accepted
Reference for the equivalence between chain complexes and sequential diagrams in a stable $\...
I believe https://arxiv.org/abs/2109.01017 does what you want! The description of coherent chain complexes used there is a bit different than what you suggest, but they look equivalent at first glance …
- 10.8k
10
votes
In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for co...
Not in general, no. The problem is that animated functors play well with colimits, and homotopy groups play better with limits. However, if your functors $\mathcal{A}\xrightarrow{F}\mathcal{B}\xrighta …
- 10.8k
1
vote
Can the loops in the definition of the fundamental group be considered injective?
For the new version of the question (where you allow to replace the space by a homotopy equivalent one) the answer is now "yes": just replace every $X$ by $\lvert\operatorname{Sing}(X)\rvert$. This is …
- 10.8k