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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.
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
18
votes
Accepted
What is the free symmetric monoidal $\infty$-category on one object?
Yes, it is the same as $\mathbb{F}$.
As John Baez points out, it is the same as the free symmetric monoidal $\infty$-groupoid on one object. (This can also be seen by playing around with the adjoints …
- 27.5k
7
votes
Accepted
Categorical models for truncations of the sphere spectrum
I don't understand what you mean about the "directed sphere" so will focus on the other questions.
The free Picard $n$-category on one object has a description as a bordism $n$-category. Specifically …
- 27.5k
20
votes
Accepted
For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty...
I like to think of $EG$ and $BG$ in terms of configuration spaces.
The space $BG$ can be identified with the following configuration space. It consists of configurations of finitely many points in the …
- 27.5k
10
votes
Accepted
Classifying Space of a Group Extension
Yes. The principal bundles are the same and your guess that $BA$ is an abelian group is exactly right. A good reference for this story, and of Segal's result that David Roberts quotes, is Segal's pape …
- 6,023
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
6
votes
Accepted
Space of sections of a fibration under weak homotopy equivalence
This is not true in general, unless you assume the base is sufficiently nice (eg a CW-complex). Here is a counter-example.
Let $B = \mathbb{Q}$, the rationals with its topology as a subspace of the …
- 27.5k
68
votes
Accepted
Is there an accepted definition of $(\infty,\infty)$ category?
One thing that might interest you is my result with Clark Barwick which gives an axiomatiation + uniqueness result for the homotopy theory of higher categories:
arXiv:1112.0040
(i.e. $(\infty,n)$-cat …
- 22.3k
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
8
votes
Accepted
Understanding model independently the equivalence of two ways of obtaining homotopy types fr...
Here is an argument, which is basically Denis Nardin's comment.
To have a model independent proof you need model independent definitions of the hocolim and of the localization. You can define them …
- 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
5
votes
Accepted
Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections
Here is one way of proving the conjecture is true in general, using the modern method of weak factorization systems.
A weak factorization system has at its core two classes of maps the left class an …
- 27.5k
16
votes
Accepted
Super-cobordisms
There are a number of technical issues with making what you describe precise, for example: what precisely is a supermanifold with boundary? how can you glue/compose bordisms? etc. I am going to ignore …
- 27.5k
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
11
votes
Accepted
I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-categ...
First, as Rune pointed out in the comments, his paper with David Gepner gives a very general approach to your wish list. However to make it so general that it applies to arbitrary monoidal $(\infty,1) …
- 27.5k