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Results tagged with homotopy-theory
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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
votes
Is $S^1\vee S^1$ an Eilenberg-Mac Lane Space to a Homotopy Purist?
This is not really an answer. The inclusion of homotopy $1$-types into homotopy types has a left adjoint $\tau_{\le 1}$ given by truncation, and hence preserves homotopy limits (e.g. products). The qu …
- 118k
6
votes
Eilenberg-Mac lane spaces and a generalization
One way to think about how to distinguish, if not classify, such spaces is by homotopy operations. In the same way that cohomology operations are natural transformations between cohomology functors, h …
- 118k
6
votes
Whitehead for maps
Another class of examples comes from group cohomology: if $G$ is any group with interesting higher cohomology $H^n(G, A), n \ge 2$ then there are non-nullhomotopic maps $BG \to B^n A$ for some $n \ge …
- 118k
2
votes
A Dold-Thom style construction of a cohomology class from a sphere bundle
The answer to your second question is that you need to pick a basepoint in $S^n$. Recall that Dold-Thom tells you that $\pi_i$ of the infinite symmetric product recovers the reduced homology of a conn …
- 118k
25
votes
Why is it so hard to compute $\pi_n(S^n)$?
Even when computing $\pi_k(S^n), k < n$ all of the hard work, as far as I can tell, comes from showing that continuous maps behave reasonably up to homotopy; there is no difficulty once you show, what …
- 118k
7
votes
An abstract nonsense proof of the Hurewicz theorem
Here are some thoughts; I don't know if they'll add up to a satisfying answer. Let $X$ be $(n-1)$-connected. We have a Hurewicz map $\pi_n(X) \to H_n(X)$ given by applying $H_n$ to maps $S^n \to X$ an …
- 118k
15
votes
Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
This is not much of an answer, but it might help. The category of simplicial sets should be thought of as the free cocompletion of $\Delta$. In other words, it's what you get if you freely take colimi …
- 118k
5
votes
When is the quotient by an $n$-fold loop space an $m$-fold loop space?
One way to get $G/H$ from the map $f : H \to G$ is to first deloop it, getting $Bf : BH \to BG$, and then take homotopy fibers, getting a fiber sequence
$$H \to G \to G/H \to BH \to BG.$$
This sugge …
- 118k
5
votes
Accepted
Mayer-Vietoris sequence for twisted R-homology
Here is a sketch. First, here is the argument for untwisted homology that I want to base the argument for twisted homology off of. Every categorical thing I say below is $\infty$-categorical by defaul …
- 118k
6
votes
Accepted
Convergence of a sum with the ranks of homotopy groups
Suppose $F$ is a simply connected finite CW complex. Then it's known that exactly one of the following two things is true:
$F$ is rationally elliptic: its rational homotopy groups are finitely gener …
- 118k
9
votes
What do loop groups and von Neumann algebras have to do with elliptic cohomology?
There are various programs (which start with Segal's survey, I believe), all of which I know nothing about, to interpret elliptic cohomology classes in terms of von Neumann algebras, loop group repre …
- 118k
22
votes
What are Homotopy rings good for?
The rationalization of this ring can be understood in a very nice way, as follows. Suppose for simplicity that $X$ is simply connected. Then we can define its rational homotopy groups
$$\pi_n(X, \math …
- 118k
10
votes
Accepted
Homotopy of quivers
Once upon a time I noticed roughly this but didn't know what to do with it. I would rephrase as follows.
Any category $C$ has an associated "category algebra" $k[C]$ spanned by the morphisms of $C$ wh …
- 118k
15
votes
Why the Dold-Thom theorem?
This won't involve any geometry, but here is a model-independent description of the situation as I understand it. I will not prove anything. The very short summary is that
The infinite symmetric …
- 118k
59
votes
Accepted