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Results tagged with homotopy-theory
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user 184
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.
11
votes
Accepted
Diffeomorphisms and homotopy equivalences sliced over BO(n)
I wanted to say I think this is a great question, though phrasing things in terms of stacks might scare off some of the people who can best answer this question. I think in general understanding the …
- 27.5k
40
votes
Accepted
Classifiying sphere eversions
Answer Summary
The fundamental group of the space of immersions of $S^2$ into $\mathbb{R}^3$ is
$$ \pi_1 Im(S^2, \mathbb{R}^3) \cong \mathbb{Z}/2 \times \mathbb{Z}$$
This means that there are infini …
- 27.5k
7
votes
1
answer
867
views
Whitehead Products without Base Points?
Let $(X, x_0)$ be a pointed space. Then we can define the homotopy groups $\pi_i(X, x_0)$ for $i \geq 1$. They are abelian groups for $i \geq 2$. It is well-known that the fundamental group $\pi_1(X, …
- 27.5k
11
votes
1
answer
599
views
Do h-coequalizers and coproducts give all h-colimits?
It is well known that if a category has all coequalizers and all (small) coproducts then in fact it has all (small) colimits. More important is the proof which shows that every colimit can be built by …
- 27.5k
22
votes
4
answers
2k
views
Functorial Whitehead Tower?
The Whitehead tower of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice spac …
- 27.5k
4
votes
Base change for category objects in topological spaces
There are a couple different versions of the geometric realization of simplicial spaces. There is the literal one, which is badly behaved in general. Then there are better realizations, e.g. the "fat" …
- 27.5k
21
votes
Accepted
Is super-vector spaces a "universal central extension" of vector spaces?
This will be a little imprecise, but hopefully if you need a precise result like this you can fill in the details.
First of all Vect has not only the symmetric monoidal structure but also the direct …
- 27.5k
7
votes
Accepted
Arcwise-connectedness generalized to higher connectivity?
No, there is no generalization to "n-arcwise connected" that you ask for.
Take $X= \mathbb{R}^3$. This space is as nice a space as you could ever hope for. It is also contractible, so in particular …
- 27.5k
9
votes
1
answer
316
views
Framed version of the "copants bordism"?
The "pants" bordism in dimension n is a bordism which goes from $S^n \sqcup S^n$ to $S^n$ witnessing the connected sum operation - equivalently by attaching 1-handle to the trivial bordism, equivalent …
- 27.5k
8
votes
Accepted
A step in Lurie's treatment of $L$-theory
Here is a way to see it which might constitute "just unfolding the definitions" (at least if one were sitting in on the class at Harvard when it was being taught).
Set $Z(T) = Y(T^c)$, (compliment tak …
- 27.5k
5
votes
2
answers
415
views
Connectivity after Geometric Realization?
Suppose that I have a map of simplicial spaces,
$ f: X_* \to Y_*$,
and that I know that the map on zero spaces $f_0: X_0 \to Y_0$ is n-connected. Can I conclude anything about the connectivity of th …
- 27.5k
11
votes
1
answer
1k
views
Why does the internal singular simplicial space realize to the same thing as the discrete si...
There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations (a …
- 27.5k
11
votes
2
answers
1k
views
Is the geometric realization of a level-wise weak equivalence a weak equivalence?
For the purposes of this question a topological space will mean a compactly generated weak Hausdorff space, though I am actually somewhat flexible on what category of topological spaces we use. I woul …
- 27.5k
12
votes
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dime...
As requested I am writing this as an answer.
No there are spaces with vanished homology which are not homotopy equivalent to finite CW-complexes.
For example if $G$ is an acyclic group, then the cl …
- 27.5k
20
votes
2
answers
1k
views
How many model categories have the same weak equivalences?
There are many situations which arise where one might consider different Model categories with the same underlying category. For example in (left) Bousfield localization you start with a model categor …
- 27.5k