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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.
7
votes
Given $f: X \to Y$, $g: X \to Z$, when does it exists $h: Y \to Z$ such that $hf \simeq g$?
By replacing $f: X\to Y$ by a CW inclusion you can attack this with classical obstruction theory. I think the obstructions will lie in relative cohomology groups with local coefficients, $H^{n+1}(Y,X; …
- 10.8k
14
votes
Simplest example of non-trivial Toda bracket in spaces
The definition you've most likely encountered is the following:
For maps $W\xrightarrow{f} X \xrightarrow{g} Y \xrightarrow{h} Z$, such that adjacent maps compose to $0$, $g$ extends to a map on th …
- 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
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
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
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
24
votes
Accepted
Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an i...
No and no. For an explicit counterexample to 1. (which is also a counterexample to 2.) take the map $\mathbb{R}P^2\to \mathbb{R}P^{\infty}$.
- 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
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
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
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
16
votes
What is the first interesting matric Toda bracket in the stable homotopy of the sphere?
I know this post is quite old, but in case you are still interested, or anyone else is, I thought about sharing my recent thoughts about the topic. After all, this is the second result on "matric toda …
- 10.8k
7
votes
Accepted
$(B\mathbb Z/p^{\infty})^{\wedge}_p\rightarrow (BS^1)^{\wedge}_p$ induced by inclusion is an...
Think of the Prüfer group as $\mathbb{Q}/\mathbb{Z}$ and of $S^1$ as $\mathbb{R}/\mathbb{Z}$. Here $\mathbb{Q}$ is discrete, $\mathbb{R}$ has its usual topology and is contracible. The map $\mathbb{Q} …
- 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