**17**

votes

**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 ...

**19**

votes

**2**answers

3k views

### 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 ...

**5**

votes

**2**answers

910 views

### 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 ...

**5**

votes

**1**answer

132 views

### 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 ...

**6**

votes

**0**answers

152 views

### 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 ...

**8**

votes

**0**answers

162 views

### 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 ...

**9**

votes

**1**answer

375 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 ...

**13**

votes

**1**answer

351 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 ...

**9**

votes

**1**answer

855 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 ...

**3**

votes

**3**answers

583 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 ...

**12**

votes

**0**answers

404 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:
...

**5**

votes

**2**answers

178 views

### 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 ...

**3**

votes

**0**answers

174 views

### 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 ...

**3**

votes

**1**answer

322 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 ...

**6**

votes

**1**answer

228 views

### 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.

**11**

votes

**1**answer

424 views

### 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
...

**23**

votes

**4**answers

3k views

### 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 ...

**2**

votes

**1**answer

270 views

### 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 ...

**36**

votes

**2**answers

2k views

### 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. ...

**3**

votes

**1**answer

303 views

### 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 ...

**11**

votes

**1**answer

850 views

### Connection of X(n) spectra to formal group laws

In the proof of the Nilpotence Theorem, or at least in Ravenel's account of it in his Orange Book, a sequence of spectra are used, denoted $X(n)$ with $X(0)=\mathbb{S}$ and and $X(\infty)=MU$ such ...

**2**

votes

**1**answer

204 views

### Compact MU or BP Modules

Is there a classification of the compact MU or BP modules in any category of spectra? Can the periodicity theorem be finagled to give a MU-module structure on finite spectra?

**10**

votes

**1**answer

394 views

### Smashing localizations in the category of spectra

Let $E$ be a spectrum. Then $E$ determines an idempotent localization functor $L_E: \mathrm{Sp} \to \mathrm{Sp}$ sending each spectrum to its $E$-localization.
The functor $L_E$ generally does not ...

**4**

votes

**2**answers

533 views

### Computation of stable homotopy groups of $RP^2$

I would like to compute the first few stable homotopy groups of $RP^2$.
I first thought to use the Atiyah-Hirzebruch Spectral Sequence, (see Davis & Kirk, pg. 242). Here is what I computed for ...

**3**

votes

**1**answer

261 views

### If a t-truncation of the unit object in a stable homotopy category is a ring object up to homotopy, can it be lifted to a ring spectrum? What about the Postnikov t-truncations of the sphere spectrum?

Let $S$ be the unit object in a monoidal stable homotopy category $SH$ (we demand that the multiplication $S\times S\to S$ is commutative and associative on the level of spectra, and not just up to ...

**3**

votes

**1**answer

135 views

### Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq 0$?

It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion.
So I wonder ...

**4**

votes

**0**answers

366 views

### Good morphisms of distinguished triangles: can Neeman's method be applied to the motivic stable homotopy category?

It is well known that non-uniqueness of a cone for a morphism in a triangulated category $C$ makes constructing exact functors (of triangulated categories) a difficult task. In section 3 of his "Some ...

**11**

votes

**2**answers

433 views

### The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?

I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one.
For the algebraic cobordism theory ...

**34**

votes

**3**answers

1k views

### Lambda-operations on stable homotopy groups of spheres

The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable ...

**6**

votes

**1**answer

460 views

### Is a wedge of spheres an $E_\infty$ ring spectrum?

The wedge sum $\bigvee_{k \in 2 \mathbb{Z}} S^{k}$ is an $A_\infty$-ring spectrum: the connective cover is the free $A_\infty$-ring on the sphere $S^2$, if I'm not mistaken, and then one inverts the ...

**0**

votes

**0**answers

98 views

### Periodicity Theorem for D(R)

For a derived category of a Noetherian ring (or perhaps more generally), can we talk about a Periodicity Theorem? We have Thick Subcategory Theorems and Nilpotence Theorems (HPS 91) for D(R), and in ...

**7**

votes

**1**answer

289 views

### Monoidal Model Categories with Suspension Functor

This is basically just me trying to find out what such categories are called, and where they are written about. If I think of some model category of spectra being a "stabilization" of some model ...

**7**

votes

**1**answer

324 views

### On the natural (bigraded) homotopy groups of a simplicial object in a model category

$\def\mc{\mathcal} \def\sm{\wedge}$
This question stems from the Goerss-Hopkins paper Moduli Problems for Structured Ring Spectra. Let me begin by attempting to summarize the relevant framework -- ...

**2**

votes

**0**answers

122 views

### Bousfield Complements of Interesting Spectra

For a spectrum $X$, Bousfield constructs a spectrum (which is only well-defined up to Bousfield equivalence) $aX$, which he shows satisfies some nice properties, like $\langle a^2X\rangle=\langle ...

**25**

votes

**1**answer

966 views

### K(r)-localization and monochromatic layers in the chromatic spectral sequence

While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.
The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for ...

**0**

votes

**0**answers

194 views

### Whitehead Theorem for Harmonic Spectra

What are the chances that, for an arbitrary $p$-local harmonic spectrum $X$, if $K(n)\wedge X\simeq\ast$ for all $n$, then $X$ is contractible? This, I believe, holds for suspension spectra and finite ...

**5**

votes

**1**answer

259 views

### Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence.

In Shipley's paper http://arxiv.org/abs/math/0209215 she proves a Quillen equivalence between the category of $H\mathbb Z$-modules and dg $\mathbb Z$-modules. So, to a chain complex $C$, she assigns a ...

**4**

votes

**0**answers

124 views

### Bousfield Lattices for which Minimal Objects Coproduct to Sphere Object

Is it known what conditions we require of a stable homotopy category to have $\langle S\rangle = \coprod\limits_{\mathbb{N}}\langle K(n)\rangle$, where $\langle K(n)\rangle$ is some minimal Bousfield ...

**6**

votes

**1**answer

349 views

### Image of J in the classical Adams Spectral Sequence

Hey all,
I know that in some versions of the Adams Spectral Sequence you can easily identify the image of $J$, and I was wondering if there was a way to identify the image of $J$ in the $E_2$ page of ...

**13**

votes

**2**answers

1k views

### Finiteness of stable homotopy groups of spheres

Since the work of Serre in the early 50's on homotopy groups of spheres, it is known that the homotopy group $\pi_k(S^n)$ is finite, except when $k=n$ (in which case the group is $\mathbb{Z}$), or ...

**2**

votes

**0**answers

290 views

### Cohomology theory associated to the spectrum BG

Hi, I've recently been interested in Stable Homotopy Theory and was reading this text to understand some basics: http://www.maths.ed.ac.uk/~aar/papers/carlmilg.pdf
Near the end of the text (p582) we ...

**4**

votes

**1**answer

375 views

### Reference on the question mark cell complex

The question mark complex is a finite spectrum whose cohomology looks like a "question mark" (when drawn as a module over the Steenrod algebra): that is, there is an element in dimension zero $a_0$, ...

**8**

votes

**2**answers

545 views

### Differentials in the Adams Spectral Sequence for spheres at the prime p=2

How does one compute the differentials in the Adams Spectral Sequence for spheres at the prime 2 in the range $13\le t-s\le 20$? There seem to be 6 nonzero differentials, and at this point I only ...

**14**

votes

**1**answer

610 views

### Is there an algebro-geometric description of $\nu$?

Motivation: According to the "chromatic" picture of stable homotopy, we should think of the moduli stack $M_{FG}$ of formal groups as a "good approximation" to the stable homotopy category (more ...

**6**

votes

**1**answer

264 views

### Is there an obvious reason why p-localization of spectra is a finite localization?

Is there an obvious reason why $p$-localization of spectra is a "finite" localization in the sense of Haynes Miller? In other words, is there an obvious reason why the localizing subcategory (of the ...

**4**

votes

**1**answer

307 views

### How do you know when something must die in the Adams Spectral Sequence for $\pi_*^s$

Hey everybody,
I think this question might be just a simple oversight on my part, but this has been bugging me a few days.
I am reading Hatcher's Spectral Sequences book, and trying to understand ...

**3**

votes

**1**answer

220 views

### Formal group laws arising from localizations of MU

This is sort of a two part question:
1) In one construction of $BP$, the Brown-Peterson spectrum, one uses this idempotent map of Quillen's, $g:MU_{(p)}\to MU_{(p)}$ and from what I can tell, the ...

**13**

votes

**1**answer

593 views

### Geometric interpretation of families in the stable homotopy groups of spheres

There are infinite families in the stable homotopy groups of spheres; many of these can be seen by looking for "periodicity" phenomena in the Adams-Novikov spectral sequence. An example is the image ...

**9**

votes

**2**answers

614 views

### Second homotopy group of the mod 2 Moore spectrum

Let $S/2$ be the mod 2 Moore spectrum (i.e. the cofiber of $2: S \to S$). Then multiplication by 2 acts nontrivially on this spectrum: the homotopy groups of $S/2$ are all $\mathbb{Z}/4$-modules by a ...

**11**

votes

**1**answer

445 views

### What is the first interesting matric Toda bracket in the stable homotopy of the sphere?

Feel free to gloss ‘interesting’ as you see fit. One way:
1. What is the lowest degree matric Toda bracket in $\pi_\ast(S)$ that doesn't contain zero?
By ‘degree’ I mean total homotopical ...