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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86 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 ...

**6**

votes

**2**answers

390 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 ...

**8**

votes

**0**answers

88 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 ...

**8**

votes

**0**answers

205 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 ...

**8**

votes

**1**answer

381 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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**0**answers

99 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 ...

**1**

vote

**0**answers

43 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) \...

**8**

votes

**0**answers

99 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 ...

**8**

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**0**answers

194 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**

votes

**1**answer

277 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

**2**answers

268 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**

votes

**1**answer

158 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**

votes

**1**answer

344 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**

votes

**0**answers

158 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**

votes

**0**answers

250 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**

votes

**1**answer

233 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**

votes

**1**answer

197 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**

votes

**1**answer

140 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**

votes

**2**answers

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**

votes

**2**answers

321 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**

votes

**1**answer

490 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**

votes

**0**answers

187 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

**0**answers

95 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**

votes

**0**answers

170 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

**1**answer

170 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**

votes

**2**answers

458 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**

votes

**1**answer

332 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

**2**answers

475 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

**2**answers

416 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 ...

**9**

votes

**2**answers

452 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**

votes

**1**answer

292 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

**1**answer

153 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

**1**answer

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

**1**answer

239 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

**1**answer

432 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→...

**7**

votes

**0**answers

123 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 ...

**7**

votes

**0**answers

174 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 ...

**10**

votes

**0**answers

149 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//...

**4**

votes

**0**answers

200 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

**3**answers

192 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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votes

**0**answers

80 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**

votes

**2**answers

699 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**

votes

**0**answers

191 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

**1**answer

543 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 ...

**4**

votes

**0**answers

107 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**

votes

**2**answers

576 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 ...

**29**

votes

**1**answer

1k views

### Stable homotopy theory and physics

At various points in my life, I have held the following beliefs:
1) Stable homotopy theory is "easy" rationally, and "interesting" integrally.
2) The spectrum of topological modular forms (TMF) is ...

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votes

**0**answers

137 views

### stable (?) model category of simplicial monoids

If $\mathcal{C}$ is the category of commutative unitary monoids, one can endow the category of simplicial objects in $\mathcal{C}$, $s\mathcal{C}$, with the structure of a cofibrantly generated model ...

**8**

votes

**2**answers

361 views

### Cohomology theories as colimits

I am looking for examples of cohomology theories that can be written as (filtered, or another nice class of) colimits of "simpler" functors, i.e. which $\{h^n : {\bf Top}^2 \to {\bf Ab}\}_n$ are such ...

**4**

votes

**1**answer

421 views

### Functorial construction of (“pre”-)spectral sequences? (Or - what is the “higher structure” underlying spectral sequences?)

Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the ...