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 not deleted
user 2039
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.
4
votes
Derivators (in English)
A theory of a very similar flavour can be found in a preprint of Jens Franke.
- 12.1k
18
votes
Accepted
Does an H-space have at most one delooping?
Another example is $S^3$. If I am not mistaken, there are exactly $12$ H-space structures on $S^3$. Indeed, we can consider the long exact sequence
$$[S^4\vee S^4, S^3] \to [S^6, S^3] \to [S^3\times S …
- 12.1k
8
votes
2
answers
408
views
Localizing Model Structures
I came along the following question while trying to understand and apply some ideas of Dugger's article Universal Homotopy Theories.
Suppose, we are given a nice model category $\mathcal{C}$, say le …
- 12.1k
16
votes
1
answer
2k
views
Group Completions and Infinite-Loop Spaces
Let $X$ be a commutative H-space. A group completion is an H-map $X\to Y$, where $Y$ is another H-space, such that
$\pi_0(Y)$ is a group
The Pontrjagin ring $H(Y; R)$ is the localization of the Pont …
- 12.1k
12
votes
Historical transition from classical homotopy to modern homotopy theory
I am surely not a historian of topology, but I might try a few words.
That the usual literature concerning model categories is quite far away from traditional homotopy as presented in Whitehead's clas …
9
votes
(Homotopy theory) When does the 2 of 3 property not imply 2 of 6?
Here is an example, which I got from a discussion with Karol Szumilo (and maybe he got it from Cisinski?).
Consider the notion of a cofibration category, which means essentially that you have weak e …
- 12.1k
8
votes
Colimits of cofibrations and homotopy colimits
In general, this is certainly not true. Take for example a space $X$ with an action by a group $G$. As a group acts by isomorphisms, it acts in particular by cofibrations. But the map $X/G \to X_{hG}\ …
- 12.1k
8
votes
Is there a good way to understand the free loop space of a sphere?
The next best thing to the knowledge of a CW-structure is probably the knowledge of the integral homology (not just the Betti numbers). Perhaps Ziller was the first to compute it in The Free Loop Spac …
- 12.1k
35
votes
What non-categorical applications are there of homotopical algebra?
As soon as you do serious homotopy theory in a context outside topological spaces, a formalism for abstract homotopy theory is very helpful. Let's give a few examples:
1) Waldhausen's algebraic K-the …
53
votes
Accepted
Spheres with the same homotopy groups
It is a result from Serre's thesis that for $n\geq 3$ and a prime $p$, the first $p$-torsion in $\pi_*S^n$ occurs precisely for $* = n+2p-3$. This shows that $(m,n) = (2,3)$ is the only pair of (edit: …
- 12.1k
38
votes
Do we still need model categories?
One nice feature of model categories is that you can speak also of the non-bifibrant objects (which is not longer possible, once you passed to the corresponding infinity-category). A few examples wher …
- 12.1k
13
votes
1
answer
2k
views
Homotopy limits of quasi-categories
Quasi-categories (or $\infty$-categories, as they are often called) are a very convenient setting for doing abstract homotopy theory. One of their amazing features is the following: Given a diagram of …
- 12.1k
4
votes
Accepted
Do trivial homotopy groups imply existence of boundary preserving homotopies?
Let me first talk only about continuous maps. Your question becomes equivalent to asking whether every map from $\partial (M\times I) \to N$ (for $I$ the interval) can be extended to $M\times I$. This …
- 12.1k
6
votes
Example s.t. the unbased loop-space is not $\Omega X \times X$
The unbased loop space is also known as the free loop space and is often also denoted by $LX$ or $\Lambda X$. If you compute the homology of $LX$ (which might be difficult in general) you will often s …
- 12.1k
7
votes
Finiteness of stable homotopy groups of spheres
Let me phrase a proof of Serre's computation of the rational stable homotopy groups of spheres as stably as I can:
For every spectrum $X$, we can define its rationalization $X_{\mathbb{Q}}$ as the ho …
- 12.1k