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Results tagged with homotopy-theory
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user 11546
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.
30
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Is the $H$-space structure on $S^7$ associative up to homotopy?
It is not. See Theorem 1.4 of this paper by I.M. James (Trans. AMS 84 (1957), 545-558).
In particular, there exists no homotopy associative multiplication on $S^n$ unless $n=1$ or $n=3$.
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0
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0
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Whitehead Theorem for Harmonic Spectra
What are the chances that, for an arbitrary $p$-local harmonic spectrum $X$, if $K(n)\wedge X\simeq\ast$ for all $n$, then $X$ is contractible? This, I believe, holds for suspension spectra and finite …
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0
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0
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D(R) versus Ho(HR)?
Given an algebraic ring, how is its derived category related to the homotopy category of HR modules?
Thanks. This is essentially a reference request, since I know there may be a lot (or nothing) to …
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1
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Localization at Infinite Wedges of K-theories or BP
I mentioned that I proved this in the comment above but am "answering" just for closure. A link to the proof is here:
http://chromotopy.org/?p=1110
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9
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answer
289
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Essential maps of spectra which are null when localized at any prime
There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of th …
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5
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1
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274
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Localization at Infinite Wedges of K-theories or BP
This is basically a reference request. Does anyone know if the structure of the homotopy category of spectra (or maybe just the model, i.e. w/o the homotopy, category), localized at infinite wedges o …
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3
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1
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The Bousfield Class of the Infinite Wedge of Telescopes of Finite Spectra
The spectrum $T(n)$ which is the telescope of a finite spectrum of type n along its self-map, has a unique Bousfield class $\langle T(n)\rangle$ which only depends on $n$. It is also known, from Rave …
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10
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Geometric Interpretations of Homotopy Theoretical Constructions
In homotopy theory there are lots of nice constructions that seem designed to have some effect on the homotopy of a space, i.e. completing, localizing, and taking various homotopy (co)limits. It seems …
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7
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2
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Coherent MU_*-Modules
It is proven by Thom that for a finite cw-complex $X$, its $MU$-homology, which, in honor of the authors I'm currently reading, I'll denote by $\Omega_\ast^U(X)$, is a coherent module over $\Omega_\as …
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7
votes
1
answer
771
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Thom isomorphism from the ABGHR perspective
In ABGHR Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is equiv …
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Thom Spectra and Hopf-Galois Extensions of Ring Spectra
So this can definitely be done. It took me a while to figure out all the details, but in the end it's not so conceptually complex.
The basic idea is that if you've got a fibration $F\overset{i}\to E …
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5
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0
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(Co)homology of a directed space with coefficients in a commutative monoid
This is essentially a reference request, or a request for an explanation of why this cannot be done in a useful or interesting way (i.e. an explanation of why no such reference exists!).
If I have a d …
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2
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250
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Invariant Ideals in Split Hopf Algebroids
Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following:
An ideal $J\subset S$ is invariant under the action of the group $\mathrm{ …
- 10.5k
4
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Accepted
Interaction of Grothendieck Construction with Coherent Nerve
Answering this question took me some time. First of all, Liang Ze Wong and I had to write down a version of the enriched Grothendieck construction that worked for these purposes, as Tamaki's construct …
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4
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Accepted
Is the Thom diagonal co-$E_\infty$?
The answer to this is yes, it is co-$E_\infty$. Let $\iota\colon BGL_1(R)\to Mod_R$ be the inclusion. Since colimit is left adjoint to the strong monoidal diagonal functor, it's oplax monoidal. Note t …
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