11
votes
2answers
338 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 ...
14
votes
0answers
303 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 ...
1
vote
0answers
153 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 ...
9
votes
1answer
285 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 ...
9
votes
1answer
309 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 ...
12
votes
1answer
209 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 ...
6
votes
1answer
612 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 ...
11
votes
1answer
252 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 ...
19
votes
1answer
858 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 ...
13
votes
2answers
450 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 ...
5
votes
1answer
243 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 ...
3
votes
0answers
202 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 ...
2
votes
0answers
110 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 ...
5
votes
1answer
330 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 ...
3
votes
0answers
114 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 ...
4
votes
2answers
416 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) ...
5
votes
1answer
190 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 ...
5
votes
1answer
228 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 ...
15
votes
1answer
730 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 ...
18
votes
2answers
1k 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
2answers
477 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 ...
7
votes
0answers
124 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
1answer
332 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 ...
11
votes
1answer
288 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 ...
6
votes
1answer
538 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 ...
2
votes
3answers
329 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 ...
11
votes
0answers
281 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
2answers
161 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
0answers
141 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
1answer
296 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 ...
5
votes
1answer
184 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.
10
votes
1answer
327 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 ...
22
votes
4answers
1k 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 ...
25
votes
2answers
1k 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. ...
2
votes
1answer
211 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 ...
10
votes
1answer
507 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 ...
8
votes
1answer
307 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
2answers
356 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
1answer
208 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
1answer
122 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 ...
32
votes
3answers
732 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
1answer
356 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 ...
7
votes
1answer
267 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 -- ...
21
votes
1answer
734 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 ...
5
votes
1answer
234 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 ...
6
votes
1answer
276 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 ...
12
votes
2answers
786 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
0answers
259 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
1answer
343 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 $a_0$, ...
14
votes
1answer
544 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 ...