Questions tagged [stable-homotopy]
Stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor.
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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 ...
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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 ...
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Calabi-Yau cohomology?
My question here is going to be this -- but I'll give a bit of background to explain myself in a moment:
What has been done/what results are available on Calabi-Yau cohomology in degree $n \geq 3$ (...
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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 ...
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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 ...
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"topological" Ochanine genus?
The Witten genus has famously been lifted to the string orientation of tmf ("topological Witten genus"). For the Ochanine genus, I am aware of a lift to a "spin orientation of Tate K-...
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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: $$X(m, \infty) \to X \to X[0, m].$$ ...
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Uniqueness of connected cover of Morava K-theory
Let $k(n)$ denote the connected cover of Morava $K$-theory $K(n)$ at the prime $2$ and in particular $n=2$. It is known that $$ H^*(k(n)) = A//E(Q_n), $$
where $A$ is the Steenrod algebra and $Q_n$ is ...
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How to see the quaternionic hopf map generates the stable 3-stem?
I am looking for a direct proof that the quaternionic hopf map generates (after suspension) the 3rd stable homotopy group of spheres. There are some related MO questions, for example:
third-stable-...
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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 $\mathbb{E}_\infty$-...
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How does quotienting by a finite subgroup act on the framed-cobordism class of a group manifold?
Let $G$ be a connected simple connected compact Lie group, and $\Gamma \subset G$ a finite subgroup. Then (the underlying manifold of) $G$ can be framed by right-invariant vector fields, and this ...
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A geometric interpretation of the odd-primary Kervaire elements
Let $\Omega^\mathrm{fr}_\ast \cong \pi_\ast S$ denote the graded ring of cobordism classes of framed manifolds, which, by the Pontryagin-Thom construction, is isomorphic (as a graded ring) to the ...
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A tensor product for triangulated categories?
Many triangulated categories which show up in mathematics, such as derived categories of various sorts, arise as the homotopy category of a stable $\infty$-category.
Stable $\infty$-categories give ...
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Refinement of concept of support of a module
My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...
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What is missing in the current constructions of pure and mixed motives?
Yo! Maybe this question is too broad, so maybe it should be community wiki? In summary, I want to known all the known comparisons between all the constructions of pure and mixed motives and what make ...
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Lattices and stable homotopy groups of spheres
The number $65520$ arises in two very different scenarios:
It occurs in the formula for the theta series of the Leech lattice:
$$ \Theta_{\Lambda_{24}}(q) = 1 + \sum\limits_{m=1}^{\infty} \dfrac{...
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Chromatic blueshift and Tate cohomology
Let $R$ be an $L_n$-local ring spectrum. Then one knows that the Tate construction $R^{tC_p}$ (with respect to the trivial $C_p$-action on $R$) is $L_{n-1}$-local; this "blueshift" result is ...
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Chromatic Homotopy Theory and Physics
Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...
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Known obstruction for efficient computation of Stable homotopy groups?
Computation of stable homotopy groups (for example of sphere) is hard, but still, not as hard as unstable ones.
For unstable homotopy groups there are some results showing that there cannot be ...
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What are examples of spectra whose mod 2 cohomology contain A//A(n)?
Let $//$ denote the Hopf algebra quotient. We know that:
$$HF_{2}^*(ko) \simeq A//A(1)$$
$$HF_2^*(tmf) \simeq A//A(2)$$
By Hopf invariant one, we know there is no $X$ such that $HF_2^*(X) \simeq A//...
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How many $E_\infty$-ring structures are there on the complex cobordism spectrum?
There is a well known $E_\infty$-ring structure on the complex cobordism spectrum $MU$ coming from the fact that $MU$ is a Thom spectrum over the infinite loop space $BU$. Here is my question: Is it ...
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Trouble with Stable Equivariant Profinite Homotopy Theory
I've heard that there are some problems in developing a good formalism for stable equivariant homotopy theory (either from the spectral mackey functors perspective or from the orthogonal spectra ...
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How does the HHR Norm functor interact with the cotensor over $G$-spaces?
Let $N_H^G$ be the norm functor from orthogonal $H$-spectra to orthogonal $G$-spectra. We know the category of orthogonal $G$-spectra $\mathcal{S}_G$ is enriched over the category of based $G$-spaces $...
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Fields in Stable Homotopy Theory
It is known that the only "fields" in stable homotopy theory, after localizing at a prime $p$, are Eilenberg-Mac Lane spectra for fields and the Morava K-theories (this is true in a few senses: these ...
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stable homotopy of BG_+?
Let $G$ be an abelian group (if this helps, let's say also finite). Let $BG_+$ be the classifying space together with a disjoint base point. What are the stable homotopy groups
$$\pi^s_m (BG_+) := ...
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What is the original source for the Goerss-Hopkins-Miller-Lurie theorem on tmf?
The central basic theorem of topological modular forms states that the structure sheaf of $\widehat{\mathcal{M}}_{ell}$ lifts to a sheaf of complex-oriented $E_{\infty}$-rings whose formal groups are ...
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Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$
First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
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$\Gamma$-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, ...
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Stable homotopy type of $BG^{\wedge}_p$ in algebraic terms
In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:
$\mathbb{F}_p\mathrm{Inj}(P,G)\cong \mathbb{F}_p\mathrm{Inj}(P,H)$ as $\mathbb{F}_p\mathrm{...
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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 ...
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Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$
I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...
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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 ...
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How to show that a spectrum X is not Chromatically Complete
There are some criteria which tell us when a spectrum $X$ is chromatically complete (it's the homotopy limit of its chromatic tower):
It has to be p-local and finite, according to the chromatic ...
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Chromatic Completion of Suspension Spectra and affine results
There is the Chromatic Convergence Theorem by Hopkins and Ravanel which states that the homotopy inverse limit of the chromatic tower of a finite spectra $X$ is $X$.
Let's call any spectra with this ...
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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 ...
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t-structures on the tensor product of stable $\infty$-categories
It is a matter of checking definitions that the tensor product of presentable $\infty$-categories restrict to a tensor product between stable presentable $\infty$-categories; I was wondering if there ...
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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 (...
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Worst-case complexity of calculating homotopy groups of spheres
Is the best known worst-case running time for calculating the homotopy groups of spheres $\pi_n(S^k)$ bounded by a finite tower of exponentials? How high is a tower? Does $O(2^{2^{2^{2^{n+k}}}})$ ...
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What is the Balmer spectrum of the p-complete stable homotopy category?
When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...
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What can I say about an $E_\infty$ ring spectrum with an odd invertible element?
I have an $E_\infty$ ring spectrum $R$. I suspect it is trivial, but I'm not sure. What I do know is that there is an $R$-linear equivalence $R \simeq \Sigma R$. Unless I am very confused, this ...
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Bringing cohomology recipes from algebra to topology?
In algebra, cohomology theories are often defined very roughly like this. Start with a scheme $X$ you want to study. Form a category-with-extra-structure (a site) $\mathcal{C}$ whose objects are, ...
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Did the Goerss-Hopkins manuscript "Multiplicative stable homotopy theory" ever appear?
A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant. It was ...
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Parametrized cancelations in stable Morse theory
Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...
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Fibrations of orthogonal G-spectra and fixed points
There are at least two fixed point functors that characterize stable equivalences of orthogonal G-spectra: the geometric fixed points and the naive fixed points of a fibrant replacement.
Is this true ...
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Explicit framed null bordism realizing $\eta\nu =0$ in stable homotopy group of spheres
There are many standard results in the stable homotopy group of spheres (or equivalently framed bordism groups), about which I would like to acquire better geometric understanding. For example I ...
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Homotopy theory of differential objects
In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
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A confusion about geometric fixed points via spectral Mackey functors and smashing localisations
Let $G$ be a finite group and $N$ a normal subgroup. One of the modern ways to construct the $\infty$-category of $G$-spectra is as product-preserving spectral presheaves $\text{Sp}^G = \text{Fun}^{\...
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Example of a tensor triangulated category with two different monoidal t-structures?
What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures?
While I'm at it: is there an example of a tensor ...
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Funtoriality of twisted K-theory
I posted this question on math.stackexchange, but received no answer there.
In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until ...
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Reference request: complex K-theory as a commutative ring spectrum
Does anyone know of a point-set level model for complex K-theory as a commutative ring spectrum?
For real $K$-theory
I know of "A symmetric ring spectrum representing KO-theory" by Michael Joachim (...
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