All Questions
14
questions
4
votes
1
answer
201
views
Reference for the equivalence between chain complexes and sequential diagrams in a stable $\infty$-category
Lurie's $\infty$-categorical Dold-Kan Correspondence relates simplicial objects and sequential diagrams in a stable $\infty$-category. Is there any reference for an equivalence to a category of ...
- 728
14
votes
1
answer
344
views
The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$
$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...
- 10.3k
5
votes
1
answer
393
views
Thom spectra, tmf, and Weierstrass curve Hopf Algebroid
Let $X(4)$ be the Thom spectrum associated to $\Omega SU(4) \to \Omega SU \simeq BU$. Since $X(4)$ is a homotopy commutative ring spectrum, for any spectrum Y we can construct a resolution
$$
Y \wedge ...
- 141
6
votes
0
answers
215
views
When $\Sigma^{\infty}Y^{\wedge}_p\simeq (\Sigma^{\infty} Y)^{\wedge}_p$?
When studying the stable homotopy of $BG^{\wedge}_p$, with $G$ a finite group, authors know that this abuse of notation is not dangerous because $\Sigma^{\infty}BG^{\wedge}_p$ and $(\Sigma^{\infty}BG)^...
- 795
5
votes
1
answer
568
views
Topological Hochschild homology using equivariant orthogonal spectra
In the Hesselholt-Madsen paper "On the K-theory of finite algebras over Witt vectors of perfect fields", the authors develop some results concerning the Topological Hochschild homology (THH) of ...
- 255
10
votes
3
answers
569
views
Iterated free infinite loop spaces
Let $Q$ denote $\Omega^\infty\circ \Sigma^\infty$ the free infinite loop space functor. Given some space $X$, we see that $QX$ carries all the stable homotopy information about $X$. Naturally I wanted ...
- 101
22
votes
2
answers
6k
views
References and resources for (learning) chromatic homotopy theory and related areas
What references and resources (e.g. video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)?
5
votes
1
answer
202
views
A question regarding generalized cohomology and spectra : proof of $E^{\ast}(S)\otimes\mathbb{R} = H^{\ast}(S;\pi_{\ast}E\otimes \mathbb{R})$
I asked a question on m.se about generalised cohomology and spectra. Not having received any specific answer I attempted to draw more attention by offering a bounty. But I still could not get any help....
- 709
4
votes
0
answers
766
views
Access to a classic reference of Dold-Puppe
There is an old reference that I am unable tofind. It is Dold-Puppe´s communication at a conference. More concretely, it is cited as:
A. Dold, D. Puppe: Duality, trace and transfer. Proceedings of the ...
- 2,811
9
votes
0
answers
368
views
Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?
First of all, what I want to ask is slightly more elaborate than what stands in the title (hence the subquestion).
I am telling this since as it is, the title contains a meaningful question, but it ...
- 17.8k
22
votes
1
answer
1k
views
Why do homotopy theorists care whether or not $BP$ is $E_\infty$?
I have often heard that it is not known whether or not the Brown-Peterson spectrum $BP$ is an $E_\infty$-ring spectrum. Though I see that this is a somewhat natural question to ask, I have often ...
- 628
15
votes
2
answers
1k
views
$RO(G)$-graded homotopy groups vs. Mackey functors
Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work $\infty$-categorically.
Also, I've looked through other similar MO questions, but I didn't find ...
- 6,049
5
votes
0
answers
275
views
Moore spectra are not E-infinity (oldest known proof)
Fix a prime $p$. Let $M_p(i)$, the $i$-th Moore spectrum at the prime $p$, be the cofiber of the map
$$ S^0 \overset{p^i}\longrightarrow S^0 $$
where $S^0$ be the sphere spectrum. In the Mathoverflow ...
- 2,023
2
votes
0
answers
184
views
Quillen functors and stable model categories
Are there any books or papers where I can find some general statements and methods for working with Quillen functors that are not equivalences (and not localizations)? In particular, I would like to ...
- 16.7k