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Results tagged with homotopy-theory
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user 4177
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
an explicit weak equivalence between $B{\mathbb G}_m$ and ${\mathbb P}^\infty$
OK, here's a first attempt, with a small gap.
The ind-scheme $\mathbb{P}^\infty$ is the quotient of the ind-scheme $\mathbb{A}^\infty \setminus\lbrace 0\rbrace = \lim_{n\to \infty}\mathbb{A}^n\setmin …
- 35.4k
8
votes
The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres
Ronnie and collaborators' HHvK theorems are essentially all for crossed complexes and similar. These are, as I'm sure you're aware, partially linearised homotopy types. In particular, for a simply con …
- 35.4k
7
votes
Accepted
What's the name of this flavor of n-category?
Ronnie Brown has a related idea, contained in this article:
Moore hyperrectangles on a space form a strict cubical omega-category
arXiv
discussed briefly here at the nLab.
If you are instead …
- 35.4k
2
votes
Functorial Whitehead Tower?
If we have a functorial Postnikov tower of a pointed space X, the tower inherits a basepoint. Then take the tower over the Postnikov tower which is pointwise the path fibration. This is pointwise a fi …
- 35.4k
7
votes
Accepted
How can I visualize the nontrivial element of $\pi_4(S^3)$ and $\pi_5(S^3)$ ?
(This is a bit late, but I hope you find it interesting!)
Here's smooth representation of the generator of $\pi_4(Sp(1))$ (and so the same homotopy group of $S^3$ and $SU(2)$). Consider $S^4 = \mathb …
- 35.4k
8
votes
2
answers
647
views
Is there an interesting definition of a category of test categories?
Given a pair of test categories $C_1$ and $C_2$ (in the sense of Grothendieck - weak or strict or otherwise), has anyone defined an interesting notion of morphism between them? Or are ordinary functor …
- 35.4k
11
votes
Accepted
Why is the string group not a Lie group?
The result is that a compact, connected simple Lie group $G$ has $\pi_3(G) = \mathbb{Z}$. Simple covering space or subgroups arguments should get you to $\mathrm{SO}(n)$ which is all that matters. For …
- 35.4k
10
votes
Why is the string group not a Lie group?
To follow up, there is now an infinite-dimensional Lie group model of String:
Thomas Nikolaus, Christoph Sachse, Christoph Wockel, A Smooth Model for the String Group, Int. Math. Res. Not. 16 (2013) …
- 35.4k
7
votes
Analogue to covering space for higher homotopy groups?
In my 2010 PhD thesis I carried out a little bit of this program in a pedestrian way for $n=2$, and constructed a functor from suitably locally connected topological spaces to topological groupoids, w …
- 35.4k
4
votes
Examples of Brown (co)fibration categories that are not Quillen model categories?
Consider the category $Cat(S)$ of internal categories in a finitely complete category $S$ equipped with a Grothendieck pretopology $J$. For $S$ and $J$ satisfying certain properties, then there is a Q …
1
vote
Base change for category objects in topological spaces
If $Z \to X$ admits local sections over a numerable open cover, then $|N(Z,W)| \to |N(X,Y)|$ is a homotopy equivalence (say if $X$ is paracompact), not just a weak equivalence. This boils down to a le …
- 35.4k
4
votes
A conceptual proof that local fibrations over paracompact spaces are global fibrations?
This is not really an answer, but is just some points that may help towards a conceptual understanding:
Let $p:E\to B$ be your map which is locally a Hurewicz fibration.
Since $B$ is paracompact, nu …
- 35.4k
7
votes
3
answers
448
views
Conclusion of Hurewicz for $H_3$ without vanishing fundamental group?
Fix a space $X$, which I want to assume is a manifold. Under the assumption of simple-connectivity, Hurewicz's theorem tells us that
$$
\pi_3(X)\to H_3(X,\mathbb{Z})\qquad (*)
$$
is surjective, hence …
- 35.4k
5
votes
3
answers
568
views
Natural examples of finite dimensional spaces with interesting 2-type
Riemann surfaces provide interesting examples of 1-types - interesting as they have roles in diverse areas. However, apart from 2-dimensional lens spaces, I can't readily bring to mind natural example …
- 35.4k
7
votes
1
answer
355
views
Detecting homotopy nontriviality of an element in a torsion homotopy group
I have a map, constructed geometrically, $S^4 \to S^3$. I suspect that it is a representative for the generator $\eta_3\in \pi_4(S^3) \simeq \mathbb{Z}_2$, but I am not 100% sure ($\eta_3$ is defined …
- 35.4k