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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.
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
22
votes
Accepted
Is $[X, \_]$ a homology theory?
This holds only for compact objects (i.e. finite CW spectra), since it is easy to see that additivity fails otherwise (the other axioms of homology theories are satisfied). The usual way to obtain a h …
- 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
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
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
12
votes
Accepted
Motivation for the definition of complex orientable cohomology theory
As you wrote, complex orientability can be characterized by the cohomology of $\mathbb{C}P^\infty$: $E$ is complex orientable if $E^*(\mathbb{C}P^\infty)$ splits according to the cell structure of $\m …
- 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
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
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
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
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
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
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
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 …
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