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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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1answer
124 views

Derivator of localizations of spectra

Since my question is very specific let me introduce the context. I am trying to apply derivator theory to stable homotopy theory. Theorem 3 of the following preprint by Franke https://pdfs....
13
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1answer
533 views

Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?

It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence. This statement is ...
8
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1answer
263 views

Motivic cohomology is universal with respect to what (co)homology theories?

I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem? ...
14
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1answer
401 views

Spectra with “finite” homology and homotopy

As known, any non-trivial finite spectrum $X$ can not have non-zero homotopy groups $\pi_i(X)$ only for finite number of $i$. As I understand, the same is true for any spectrum $X$ with finitely ...
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0answers
107 views

Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrum

This is a following up question of Sphere spectrum, Character dual and Anderson dual. What are the differences and the significances of the following: (1). Homotopy classes of maps from a Thom ...
8
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2answers
454 views

Sphere spectrum, Character dual and Anderson dual

The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres. However, could you help me to appreciate the mathematical meanings of the following: What is the significance of ...
9
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0answers
99 views

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 ...
9
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0answers
218 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 ...
9
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1answer
410 views

Spectral and derived deformations of schemes

I'd like to understand how ordinary schemes deform or lift to spectral and derived schemes in two basic examples as well as what the structure of the space of deformations in general is. Let $S = (X, ...
5
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0answers
104 views

The $E_2$-page of the May spectral sequence

I recently started to read about May spectral sequence, which converge to the $E_2$ term of the classical ASS. At the prime $2$, this is a spectral sequence with $E_1$ page a polynomial algebra on ...
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0answers
45 views

Proving a property of tame spectra

Let $U$ be a universe. Let $D$ be tame spectra, and let $f:E_1\rightarrow E_2$ be a map of spectra that is a spacewise homotopy equivalence. It is supposed to hold that $f^*: h\mathcal{S}U(D,E_1) \...
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0answers
114 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 ...
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0answers
205 views

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 ...
13
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1answer
291 views

Stable Cohomotopy as $K \mathbb{F}_1$

Various classical results suggest that stable cohomotopy may usefully be regarded as being the algebraic K-theory over the "field with one element" $\mathbb{F}_1$: $$ K \mathbb{F}_1 \;\simeq\; \...
7
votes
2answers
285 views

Connectivity of suspension-loop adjunction

Let $X$ be a $k$-connected spectrum for $k \in \Bbb{Z}$. I want to deduce how connected the counit of $(\Sigma^\infty, \Omega^\infty)$- adjunction is, that is, how connected is the map $$ \Sigma^\...
4
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1answer
159 views

How are p-primary parts determined for odd p?

When looking at surveys of computations of the homotopy groups of spheres there is a common theme. All the odd primary parts are thrown away. How are odd primary part calculations done in relation ...
11
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1answer
354 views

Homotopy orbits, spectra and infinite loop spaces

Let $X$ be an (naive) $O(n)$-spectrum (I'm choosing to work with orthogonal spectra). I've recently come across the following results, $$(S^{n-1} \wedge X)_{hO(n)} \simeq X_{hO(n-1)}$$ and $$\Omega^...
12
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0answers
169 views

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 ...
4
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0answers
263 views

A question on the slice filtration and the slice of the motive of the projective space

In the following $k$ is an algebraically closed field of characteristic $0$. Consider the category $SH(k)$ (the Morel-Voevodsky stable motivic homotopy category). By the work of Voevodsky (see for ...
2
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1answer
239 views

Basic questions on spectra

I have a basic question on Voevodsky's stable homotopy category of spectra $\mathbf{SH}(S)$, where $S$ is a finite dimensional noetherian scheme. Let $E$ be an $\Omega$-spectrum and $\varphi \colon ...
8
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1answer
203 views

Reference for: $p$-primary component of $\pi^S_k$ is $\Bbb Z_p$ when $k=2l(p-1)-1$

I remember coming across this result some time ago but I am having trouble finding a reference for it. It goes something like this: Let $p$ be a(n odd?) prime, then the $p$-primary component of $\...
4
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1answer
142 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....
2
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2answers
181 views

Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data

Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step ...
5
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2answers
357 views

Ring structures on algebraic K-theory spectrum, and its non-connective counterpart

I have a few naive questions on the algebraic K-theory spectrum construction, but whose answers I couldn't figure out using the internet. I'm mostly interested in the case of a commutative ring, but I ...
16
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1answer
494 views

Idempotent ring spectrum

Is there a lot of ring spectrum which are idempotent in the sense that the multiplication map $R \wedge R \rightarrow R$ is an equivalence ? The sphere spectrum $\mathbb{S}$ and the $0$ spectrum are ...
10
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0answers
195 views

Comonadicity of spaces over spectra?

As connective spectra are equivalent to group-like $E_{\infty}$ algebras in spaces, the $\infty$-category of connective spectra is monadic over the $\infty$-category of spaces though the usual $\Sigma^...
5
votes
0answers
97 views

Realizing the 0-th Postnikov truncation of a spectrum in the category of orthogonal/symmetric spectra

Suppose $E$ is a connective spectrum, then there exists a natural map in the stable homotopy category $\mathcal{SHC}$, $E \rightarrow P_0 E$, called the $0$-th Postnikov truncation, which is ...
3
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0answers
187 views

Access to a classic reference of Dold-Puppe

There is an old reference that I am unable to easily find. It is Dold-Puppe´s communication at a conference. More concretely, it is cited as: A. Dold, D. Puppe: Duality, trace and transfer. ...
4
votes
1answer
171 views

Spectrifications of spacewise homotopy equivalences

One can show that spectrifications of maps between $\Sigma$-cofibrant prespectra that are spacewise homotopy equivalences are homotopy equivalences of spectra. I'm interested in the following: Does ...
12
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2answers
464 views

Status of Thomason's idea for a symmetric monoidal model of stable homotopy - from his last paper

In 1995, Robert Thomason published “Symmetric monoidal categories model all connective spectra” in TAC. On page 2, he argues that symmetric monoidal categories are more convenient than “May’s ...
9
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1answer
337 views

When is the Thom spectrum of a virtual vector bundle effective?

Remark: My question is valid in the classic setting of the stable homotopy category of spectra of CW-complexes. An answer on that setting will also be valid. Denote as $SH(X)$ Voevodsky's stable ...
8
votes
2answers
482 views

Realizing $\mathcal{A}(2)//\mathcal{A}(1)$ by a finite spectrum

Let $\cal A$ denote the mod 2 Steenrod algebra. Can the $\mathcal{A}(2)$-module structure on $\mathcal{A}(2)//\mathcal{A}(1)$ be enriched to an $\cal A$-module structure? If so, is there a finite ...
9
votes
2answers
429 views

Cotangent complex of perfect algebra over a perfect field

Let $A$ be a perfect $\kappa$-algebra over a perfect field $\kappa$ of positive characteristic $p$. Then the algebraic (= classical) cotangent complex $L_{A/\kappa}^{\operatorname{alg}}$ is known to ...
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2answers
460 views

“Standard arguments” in Mahowald's eta_j paper

In “A new infinite family in $_{2}\pi^S_*$" (1976), Mark Mahowald constructs elements $\eta_j \in \pi_{2^j}(S^0)$ for $j \neq 2$ which come from permanent cycles in the Adams Spectral Sequence that ...
6
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1answer
294 views

Crafting Suspension Spectra

There is a theorem by Hopkins and Smith which states that for every $n > 0$ there is an ideal $I_n = (v_0^{k_0}, \dots, v_n^{k_n})$ such that there exist a spectrum $X_n$ with the following ...
5
votes
1answer
155 views

Homology of a limit of spectra + Cofiber

I have a countable sequence of finite suspension spectra $X_i$, whose $BP$-homology is a $BP_*(BP)$-comodule. Let's assume $BP_*(X_i) = \Sigma^{d_i} BP_* / (v_0^{k_0}, \dots v_i^{k_i}),$ for some $d_n$...
5
votes
1answer
257 views

Map between homology of spectra

Let $X$ be a suspension spectra whose $BP$-homology is infinitely generated ($BP_*(X) = \Sigma^d BP_*/I$, where $I$ has the form $I=(v_0^{i_0}, \dots , v_n^{i_n})$ such that the homology is a $BP_*(BP)...
3
votes
1answer
240 views

Studying the limit of a sequence of spectra knowing their BP-Homology

QUESTION EDITED: There was a mistake, the spectrum i had written before didn't even exist, so a big thanks to the people who made me notice that in the comments. Let $X_n$ be the spectrum such that $...
9
votes
1answer
437 views

Are cofibrant commutative S-algebras flat?

Let $R$ be a cofibrant commutative $S$-algebra (in the sense of Elmendorf-Kriz-Mandell-May; they call them "$q$-cofibrant") and $A$ be a cofibrant commutative $R$-algebra. Does $A\wedge_R-:RMod→...
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0answers
125 views

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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0answers
181 views

Atiyah duality with coefficients and boundary

Looking at Atiyah's paper "Thom complexes", I find two statements of the Atiyah duality theorem: Proposition 3.2 Let $X$ be a compact smooth manifold with boundary $Y$ and tangent bundle $\tau$. Then ...
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0answers
150 views

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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0answers
204 views

Stable Dold-Kan correspondence

There exists a Quillen equivalence between $HRModSpectra$ (homotopy category of ring spectra over Eilenberg-MacLane spectra $EM(R)$, where $R$ is a commutative ring, with stable model structure) and $...
4
votes
3answers
195 views

Morphisms of formal group laws $\,F_a \rightarrow F_m\,$ and $\,F_m\to F_m$

While studying cohomology theories on the stable homotopy setting, I have come up with the following basic question: Consider the additive formal group law, $F_a$, and the multiplicative formal group ...
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0answers
81 views

Defining structure maps of spectra by lifting from the homotopy category

Voevodsky's original definition of the algebraic $K$-theory spectrum, $KGL$, was given as follows: The component spaces were fibrant replacements of the infinite Grassmannian $BGL$. The structure ...
17
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2answers
727 views

Stable homotopy type theory?

This is a combined question, strictly speaking I am asking three questions concerning, respectively, homotopy type theory, stable homotopy theory and Yetter-Drinfeld modules. But I believe in the ...
8
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0answers
193 views

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 ...
8
votes
1answer
560 views

The universal property of the unseparated derived category

In Appendix C of his book in progress Spectral Algebraic Geometry, Lurie defines the unseparated derived category $\check{{\cal D}}({\cal A})$ (see Definition C.5.8.2 loc.cit) associated to a ...
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0answers
111 views

Truncation and connected cover of spectra

Let $X$ be an $n$-connective spectrum for some $n\in \mathbb{Z}$. Is then $[X, Y] = [X, Y\langle n\rangle]$ for all spectra $Y$, where $Y\langle n\rangle$ denotes the $n$-connective cover of $Y$? ...
10
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2answers
617 views

Connective spectra and infinite loop spaces

It seems to be standard that connective spectra are "the same" as infinite loop space. However, I do not understand the reason why the associated spectrum is connective. For me, an infinite loop ...