Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with homotopy-theory
Search options answers only
not deleted
user 360
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.
54
votes
Accepted
Non-examples of model structures, that fail for subtle/surprising reasons?
Here is a classical example.
Let CDGA be the category of commutative differential graded algebras over a fixed ground field k of characteristic $p$. Weak equivalences are quasi-isomorphisms, fibrati …
- 52.7k
44
votes
Why is the definition of the higher homotopy groups the "right one"?
I think that obstruction theory is one of the most important reasons to study homotopy groups. If you are interested in studying the possible homotopy classes of maps $X \to Y$ of spaces where $X$ has …
38
votes
What is the 31st homotopy group of the 2-sphere?
My apologies for updating this very old question.
As already mentioned, the 31st homotopy group of $S^2$ is the same as the 31st homotopy group of $S^3$. Serre's mod-C theory shows that this is a fini …
- 52.7k
35
votes
Accepted
Topological Langlands?
I would cautiously venture that there is not really something we could call a topological Langlands program to outsiders at this point. My objection is to the final word - we don't really know what w …
- 52.7k
34
votes
Accepted
What is the relationship between connective and nonconnective derived algebraic geometry?
Here is an example of a nonconnective $E_\infty$ ring spectrum which, I think, illustrates a key problem. (A more extensive discussion of this phenomenon occurs in Lurie's DAG VIII and in a paper by B …
- 52.7k
33
votes
Accepted
Why should I care about topological modular forms?
One of the closer connections to geometric topology is likely from invariants of manifolds. The motivating reason for the development of topological modular forms was the Witten genus. The original …
- 52.7k
32
votes
Accepted
What happened to the last work Gaunce Lewis was doing when he died?
Lewis wrote, but never published, a very influential paper setting foundations for the multiplicative theory for Mackey functors. The paper is called "The Theory of Green functors" and Doug Ravenel's …
- 52.7k
32
votes
Accepted
Endomorphism ring spectrum of the Eilenberg-MacLane spectrum
No, $A$ is not an $H\Bbb Z$-algebra.
Suppose $R$ is an $H\Bbb Z$-algebra. Then the category of left $R$-modules is $H\Bbb Z$-linear: for any $R$-modules $M$ and $N$, the function spectrum $F_R(M,N)$ …
- 52.7k
32
votes
Accepted
"Dirty" proof that Eilenberg-MacLane spaces represent cohomology?
I'd suggest looking up some basic material on obstruction theory. There, you generally find classification of maps $X \to Y$ with domain a CW-complex in terms of cohomology groups $H^s(X;\pi_t(Y))$. …
- 52.7k
30
votes
Accepted
homotopy and (co)filtered limits
This is not true, for two distinct reasons.
The first is that the inverse system of spaces may not behave well homotopy-theoretically. If $X_n = [n, \infty) \subset \Bbb R$, then the limit of $\dots …
- 52.7k
26
votes
What are some toy models for the stable homotopy groups of spheres?
My favorite warmup example to the stable homotopy groups of spheres is the following differential graded algebra.
Let $A$ have the underlying ring
$$
\Bbb Z[y] \otimes \Lambda[x],
$$
a ring with a po …
- 52.7k
25
votes
Accepted
Eilenberg-Mac lane spaces and a generalization
Assuming $m > n$, there is a method for classifying such spaces using a technique from the Postnikov tower. Namely, such a space has a map $X \to K(G,n)$ inducing an isomorphism on $\pi_n$, and if we …
- 52.7k
24
votes
Accepted
can a common mortal understand why the affine line is not smooth in brave new algebraic geom...
I think the answer to your question is "yes". Toen-Vezzosi go over this in Proposition 2.4.15, but here is some version of why.
Away from characteristic zero, there's a sharp difference between bein …
- 52.7k
23
votes
What are the uses of the homotopy groups of spheres?
Here are four applications, but some of them are cheating. I'd like to give more examples involving homotopy groups of more general spaces, but will try not to.
(Tagged cw - feel free to add.)
The …
23
votes
Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
The question used the phrase "still needed." This is a very loaded term, and the answer that you give will depend very strongly on how you interpret it.
If, as Dylan does, we interpret this as askin …