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Results tagged with homotopy-theory
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user 43054
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.
9
votes
Example of a non-$\infty$-category whose homotopy category is a groupoid
Since the homotopy category depends only on the 2-skeleton of the simplicial set, the easiest thing to do is to take the 2-skeleton of an ∞-groupoid. For example, let $E$ be the nerve of the contracti …
- 16.5k
16
votes
Accepted
Algebraic topology and homotopy theory with simplicial sets instead of topological spaces
It depends on what you mean by "all results". Of course results regarding manifolds or vector bundles do not admit statements completely internal to the world of simplicial sets (although most of them …
- 16.5k
5
votes
Accepted
The contravariant mapping space represented by a homotopical classifying space (e.g. BG)
Let $G$ be a topological group and $X$ be a paracompact Hausdorff topological space. For simplicity let us assume that $G$ has the homotopy type of a CW complex, although a lot of this answer does not …
- 16.5k
11
votes
Accepted
Roadmap for L-Theory
I apologize for the self promotion -- I hope the content of this answer can be useful anyway...
My favourite introduction to L-theory is Lurie's notes on Algebraic L-theory and surgery (warning: aggr …
- 16.5k
3
votes
Accepted
(Algebraic) cobordism and the rank function
Let me first write what happens for classical cobordism. You are basically asking whether the map $\operatorname{MU}\to H\mathbb{Z}$ factors through the projection $\operatorname{ku}\to H\mathbb{Z}$. …
- 16.5k
6
votes
Accepted
Action of fundamental group on homotopy fiber
(This answer is written in a model-independent fashion -- translate to your favourite formalism).
For every path $\gamma:[0,1]\to B$ you get an isomorphism in the homotopy category $X_{\gamma0}\xrigh …
- 16.5k
9
votes
Accepted
Contractible chain complex from non-contractible space
These are known as acyclic spaces (note that since $\tilde C_*(X)$ is a bounded below complex of projectives, it being contractible is equivalent to its homology being trivial).
There's an extensive …
- 16.5k
3
votes
Accepted
Pullbacks and fibers in the $\infty$-category of spaces
Well, I guess I can write as an answer what I wrote as a comment.
Any pullback square where $C$ is not discrete will yield a counterexample. For simplicity let $B=G=\ast$ and $C=S^1$. Then $E=H=\Omeg …
- 16.5k
12
votes
Accepted
How to construct the Moore spectrum?
What you're missing is that $[\mathbb{S},\mathbb{S}]=\mathbb{Z}$. Let now $R$ be the infinite matrix of integers representing $\rho$. Note that since $\rho$ takes value in $\bigoplus_{I_1}\mathbb{Z}\s …
- 16.5k
16
votes
Accepted
What is the relationship between spectral sequences and obstruction theory?
This is a partial answer, but every obstruction theory (in some precise sense) provides you with a spectral sequence (in fact several). Let me clarify what do I mean with obstruction theory. All this …
- 16.5k
6
votes
Accepted
Homotopy of group actions
It's enough to pick a contractible manifold $M$ with two non-homotopic actions. For example, let us pick $M=\mathbb{R}^2$ with two actions of $S^1$, the trivial one and the one given by rotations. The …
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7
votes
Accepted
Intuition - difference between Moore spectrum and Eilenberg-Mac Lane spectrum
I'm not sure I can answer to the general question, but I can explain why the Moore spectrum $\mathbb{S}/p$ shows up in the discussion of Bousfield localizations.
This is just the cofiber of multiplic …
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8
votes
Accepted
Precise reference for the equivalence of $E_n$ algebras and locally constant factorization a...
The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra
Theorem 5.4.5.9. Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Composition with the map
$$\mathrm{Disk}(M) …
- 16.5k
33
votes
Accepted
Why is Voevodsky's motivic homotopy theory 'the right' approach?
(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in).
I'm going to try with a very naive answer, although I'm not sure I understan …
- 16.5k
9
votes
Accepted
Reference request for K-Theory linearization
I claim that for every $A_\infty$-space $A$, there is a canonical $A_\infty$-ring structure on $\Omega^\infty\Sigma^\infty_+A$.
First, $\Sigma^\infty_+$ from spaces to spectra is symmetric monoidal. …
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