The stable-homotopy tag has no usage guidance.

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### Adams Spectral Sequence for Triangulated Categories

We have the Adams SS with
$$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$
where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients.
I was wondering if there is a SS for ...

**20**

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**0**answers

621 views

### 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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votes

**1**answer

211 views

### Stable homotopy of classifying space for nilpotent groups

Let $BG$ denote the classifying space of a (discrete) group and $BG_+$ its disjoint union with a point.
Question: What is known about the stable homotopy groups $\pi^S_*(BG_+)$ ?
If $G$ is finite ...

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votes

**1**answer

268 views

### When was the word “stable” first used to describe stable homotopy theory?

The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean one of two things:
Homotopy groups stabilize after taking suspensions ...

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**2**answers

156 views

### Homology exponents for $QX$

We say that a space $X$ has a homology $p$-exponent if some power of $p$ annihilates the $p$-torsion in $H_\ast(X;\mathbb{Z})$.
I am interested in the homology exponents of the free infinite loop ...

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192 views

### “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-theory", namely to ...

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407 views

### Truncations of E_infinity algebras

In section 4.1 of Lurie's DAG VIII, he implies the existence of an $E_\infty$-ring spectrum $A$ such that the coconnective truncation $\tau_{\leq 0} (A)$ does not admit the structure of an ...

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914 views

### What is to tmf as KR is to KO?

The $E_\infty$-ring spectrum $KU$ of complex K-theory carries a canonical involution induced from complex conjugation of complex vector bundles. The homotopy fixed points of this $\mathbb{Z}_2$-action ...

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203 views

### Units of a ring spectrum

Is there a good notion of the spectrum of units $R^\ast$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$?
A standard definition (see section 1.2 in http://arxiv.org/abs/0810.4535) seems ...

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**1**answer

387 views

### Are these two notions of “dualizable” spectra equivalent?

A spectrum $X$ is dualizable if the natural map $$Map(X,\mathbb S) \wedge X \rightarrow Map(X,X)$$ is an equivalence of spectra. This is equivalent to having evaluation and coevaluation maps in the ...

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**1**answer

351 views

### Stable moduli interpretation of $\mathbb{R}\mathrm{P}^\infty_{-1}$

I attended a talk recently which closed with the following tantalizing facts: there is a naturally occurring map of spectra $$K(\mathbb{S}) \to \Sigma \mathbb{C}\mathrm{P}^\infty_{-1},$$ which can be ...

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**1**answer

331 views

### localizing subcategories of $HF_p$-local spectra

This entire question takes place in the $HF_p$-local category of $p$-local spectra, i.e. the essential image of $HF_p$-localization on the stable homotopy category. $HF_p$ itself is in there, and of ...

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**2**answers

289 views

### Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper?
Since weak ...

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**1**answer

1k views

### Double coset formulas for Orthogonal groups [Solved]

According to Madsen-Brumfiel "Evaluation of the Transfer and the Universal Surgery Classes" Inventiones mathematicae 32 (1976): 133-170 Theorem 3.11, we can compute
the composition
...

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**1**answer

325 views

### Is there a $K(0)$-local Rezk logarithm?

If $R$ is a $K(n)$-local $E_\infty$-algebra, then a construction of Rezk gives a natural transformation
$$ \mathfrak{gl}_1(R) \to R,$$
by using the equivalence (arising from the Bousfield-Kuhn ...

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**1**answer

2k views

### Why not a Roadmap for Homotopy Theory and Spectra?

MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up ...

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600 views

### Must a weak homotopy equivalence induce an isomorphism between stable homotopy groups?

I'm confused by the following question:
$f:X\to Y$ is a weak homotopy equivalence, that is $f_*:\pi_*(X)\to \pi_*(Y)$ is an isomorphism for any dimensional homotopy groups. However, for the stable ...

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**1**answer

267 views

### The first element in the stable homotopy of a $K(\mathbb{Z}/2, n)$

The first element in the stable homotopy groups of a $K(\mathbb{Z}/2, n)$ (which is outside the range of the Freudenthal suspension theorem) is $\pi_{2n} K(\mathbb{Z}/2, n) \simeq \mathbb{Z}/2$. In ...

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286 views

### 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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166 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 ...

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136 views

### Formal n-buds from BU(n) rather than SU(n)

It's known, from Ravenel's green book, as well as other sources, that we build formal group laws over a ring from n-buds, where an n-bud is essentially a truncated formal group law (sometimes called a ...

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2k views

### What is known about the sum x^{n^2}/n?

It follows from a general theorem of Honda that the formal group with the logarithm
$$
x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots
$$
has integer coefficients. I became interested in it because its ...

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**1**answer

530 views

### Category of motivic spectra

When the survey Axiomatic Stable Homotopy, Neil Strickland, 2004 was written the category of motivic spectra was not investigated from the point of view of axiomatic stable homotopy, as considered ...

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151 views

### Are finite (levelwise) homotopy limits of spectra homotopy invariant?

I found an easy proof that the (levelwise) homotopy limit of a pointwise equivalence of finite diagrams of orthogonal spectra is an equivalence, without assuming that the spectra in the diagrams are ...

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510 views

### On triangulated categories of pro-objects

Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?
I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) ...

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**1**answer

220 views

### Toda brackets and factorisation of a sequence of spectra

I've found a paper of Spanier's (Higher Order Operations) where he uses the theory of "carriers" to study $n$-th order operations. The set-up is rather general; for example a particular case defines ...

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**1**answer

267 views

### Construction of Thom-Spectrum for G_2-Structures

The motivation to this question is the paper of Crowley and Nordstrøm "A New Invariant of $G_2$-Structures". I am trying to find a homotopy theoretic interpretation of the following geometric ...

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**1**answer

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### Does Wolbert's derived equivalence between $E_*^R$-local $R$-modules and $R_E$-modules come from a Quillen equivalence?

Let $R$ be a ring spectrum (in the world of EKMM $S$-modules) and let $E$ be a smashing $R$-module. Denote by $R_E$ the $E_*$-localization of $R$. By a theorem of Wolbert (Theorem 2 in Classifying ...

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### Conceptual explanation for the relationship between Clifford algebras and KO

Recall the following table of Clifford algebras:
$$\begin{array}{ccc}
n & Cl_n & M_n/i^{*}M_{n+1}\\
1 & \mathbb{C} & \mathbb{Z}/2\mathbb{Z} \\
2 & \mathbb{H} & ...

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### KK-theory as a stable infinity-category and KU Mod

The category KK of bivariant operator K-theory (or possibly its E-theory variant) ought to be the homotopy category of something at least close to a stable infinity-category; notably in that it ...

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**1**answer

1k views

### The cell structure of Thom spectra

I would like to understand the cell structure of integrally oriented Thom spectra. A Thom spectrum over a space $X$ is something you can build from a stable spherical bundle, which is classified by a ...

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### Open Problems in Algebraic Topology and Homotopy Theory

Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been ...

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### Homotopy limit-colimit diagrams in stable model categories

It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book Model Categories that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument ...

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**1**answer

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### Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator?

This should be obvious but I'm not seeing it:
The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\_)$ preserves ...

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### Schwede's DB spectra and MU

In Stefan Schewede's paper Formal groups and stable homotopy of commutative rings, he introduces $\Gamma$-rings (ring spectra) $DB$ for any commutative ring $B$ such that the the set of 1-dimensional ...

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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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**1**answer

357 views

### Thom isomorphism's effect on module structure of n-oriented spectra

This question is specifically related to the spectra $X(n)$ used in Devinatz, Hopkins and Smith's proof of the nilpotence conjectures, but any general answer in terms of the Thom isomorphism would ...

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334 views

### Does the signature admit a homotopy coherent refinement?

Cobordism genera can often be refined to $E_\infty$-orientations in the sense of Ando-Blumberg-Gepner-Hopkins-Rezk:
1) the mod 2 Euler characteristic $MO\to H\mathbb{F}_2$;
2) the $\widehat A$-genus ...

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**1**answer

788 views

### Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum?

The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum.
In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO).
On the other hand, MU and ...

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votes

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493 views

### Mayer-Vietoris Sequence for Arbitrary Bicartesian Square of Spectra

Can anyone tell me if there is a Mayer-Vietoris sequence for an arbitrary homotopy pushout (hence homotopy pullback) of spectra and an arbitrary (co)homology theory. If this comes from some easy way ...

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334 views

### 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:
...

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### Filtration on Smash Product of Cofibers

I have seen some similar questions to this one on here recently, so I hope this isn't redundant. Basically, suppose I have two cofiber sequences of spectra (or perhaps just work in some general ...

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### In the cohomology of Thom spectrum over LoopS^{2} and p-adic characteristic classes

Let $T$ denote the thom spectrum over $\Omega S^{2}$ defined by the map
$1+3: \Omega S^{2} \to BG_{3}$
where $1 +3$ is a unit in $3$-adics.
Here $G_{3}$ is the unit component of ...

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**1**answer

313 views

### Connection between complex orientations and R-orientations for a ring spectrum R?

We have a well defined notion of complex orientation for a spectrum (coh. theory) $E$, that is, we have a class $x_E\in \tilde{E}^2(\mathbb{C}P^\infty)$ which restricts to identity along the inclusion ...

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### Adams-Novikov spectral sequence at p = 2

Does anyone know of any computer calculations of the E2-term of the Adams-Novikov spectral sequence at p=2?
I'd love to get my hands on this data.

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### Computation of [ HZ/4, HZ/4]

I am trying to compute $ [\mathbb{HZ}/4,\mathbb{HZ}/4 ]$ the mod 4 Steenrod Algebra. For some reason, I need to work it out till dimension 6 or so. My approach is to use the cofiber sequence
...

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### What is a simplicial commutative ring from the point of view of homotopy theory?

Let $k$ be a field. There are two natural categories to consider:
The category of simplicial commutative $k$-algebras.
The category of connective $E_\infty$ $k$-algebras (i.e., chain complexes of ...

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**1**answer

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### Counterexamples to Smallness of Harmonic Spectra

It is a theorem of Neil Strickland's that the category of harmonic spectra (i.e. the category of $p$-localized spectra localized at the infinite wedge of Morava K-theories) has no small objects. That ...

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### Are spectra really the same as cohomology theories?

Let $E \to F$ be a morphism of cohomology theories defined on finite CW complexes. Then by Brown representability, $E, F$ are represented by spectra, and the map $E \to F$ comes from a map of spectra. ...

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### Are filtered colimits of weak-equivalences of spectra again weak-equivalences?

Hi, I have a question on weak-equivalences of spectra.
More precisely, I wonder whether filtered colimits of weak-equivalences of spectra are again weak-equivalences of spectra. Here, spectra are in ...