All Questions
Tagged with homotopy-theory stable-homotopy
70
questions with no upvoted or accepted answers
31
votes
0
answers
849
views
The central insight in the proof of the existence of a class of Kervaire invariant one in dimension 126
I understand from a helpful earlier MO question that the techniques leading to the celebrated resolution of the Kervaire invariant one problem in the other candidate dimensions yield no insight on ...
- 2,984
25
votes
0
answers
635
views
Chromatic Spectra and Cobordism
I apologize in advance, if some of the things I've written are incorrect.
The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...
- 855
20
votes
0
answers
346
views
Homotopic version of Freyd's AT category observations
Freyd was the first to formalize a striking comparison between abelian categories and topoi, showing that their exactness properties can be jointly captured by the axioms of AT categories, and the ...
- 713
15
votes
0
answers
526
views
How well-defined is $\bar\kappa$ in the stable $20$-stem?
The $2$-completed stable $20$-stem $\pi_{20}(S)_2$ is cyclic of order $8$.
Mimura and Toda (1963, Lemma 15.4) mr=157384 show the existence of a class $\bar\kappa_7 \in \pi_{27}(S^7)$ whose stable ...
- 9,173
14
votes
0
answers
306
views
Does virtual Morava K-theory have an Eilenberg-Moore spectral sequence?
In a recent question, Tim Campion was interested in analyzing the Morava $K$–theory of a space $X$ by dissecting the space into connective and coconnective parts: $$X(m, \infty) \to X \to X[0, m].$$ ...
- 6,258
12
votes
0
answers
157
views
Known obstruction for efficient computation of Stable homotopy groups?
Computation of stable homotopy groups (for example of sphere) is hard, but still, not as hard as unstable ones.
For unstable homotopy groups there are some results showing that there cannot be ...
- 40.8k
11
votes
0
answers
324
views
$\Gamma$-sets vs $\Gamma$-spaces
I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set.
For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, ...
- 40.8k
11
votes
0
answers
205
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What are examples of spectra whose mod 2 cohomology contain A//A(n)?
Let $//$ denote the Hopf algebra quotient. We know that:
$$HF_{2}^*(ko) \simeq A//A(1)$$
$$HF_2^*(tmf) \simeq A//A(2)$$
By Hopf invariant one, we know there is no $X$ such that $HF_2^*(X) \simeq A//...
- 3,529
11
votes
0
answers
450
views
How many $E_\infty$-ring structures are there on the complex cobordism spectrum?
There is a well known $E_\infty$-ring structure on the complex cobordism spectrum $MU$ coming from the fact that $MU$ is a Thom spectrum over the infinite loop space $BU$. Here is my question: Is it ...
- 628
11
votes
0
answers
257
views
Trouble with Stable Equivariant Profinite Homotopy Theory
I've heard that there are some problems in developing a good formalism for stable equivariant homotopy theory (either from the spectral mackey functors perspective or from the orthogonal spectra ...
- 111
9
votes
0
answers
309
views
Are there non-obvious finite $E_\infty$ ring spectra?
I see two "obvious" classes of nonzero finite $E_\infty$ ring spectra $R$:
$R = \Sigma^\infty_+ (S^1)^{\times n}$
$R = D\Sigma^\infty_+ X$ ($X$ a finite space)
Questions:
Are there any others?
In ...
- 62.6k
9
votes
0
answers
316
views
Dualizable objects in homotopy category of chain complexes
The proposition 1.9 from "Duality, Trace and Transfer" by Dold and Puppe states that:
Given a commutative ring $R$, a chain complex of $R$-modules is strongly dualizable in $Ho(Ch(R))$, the homotopy ...
user95283
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
9
votes
0
answers
329
views
Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?
The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it (...
- 16.7k
8
votes
0
answers
391
views
+50
Descent vs effective descent for morphisms of ring spectra
Define a homomorphism $\varphi : A \to B$ of commutative discrete rings or commutative ring spectra to be a (effective) descent morphism if the comparison functor from $\mathsf{Mod}_A$ to the category ...
- 403