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4
votes
0answers
207 views

Gluing Local Spectra

Suppose we have four spectra: $E_1$, $E_2$, $X_1$ and $X_2$ where $X_i$ is $E_i$ local and $L_{E_1\wedge E_2}X_1\simeq L_{E_1\wedge E_2}X_2$. Of course what this really means (I think), if we're ...
2
votes
1answer
128 views

The Bousfield Class of the Infinite Wedge of Telescopes of Finite Spectra

The spectrum $T(n)$ which is the telescope of a finite spectrum of type n along its self-map, has a unique Bousfield class $\langle T(n)\rangle$ which only depends on $n$. It is also known, from ...
2
votes
3answers
620 views

Fracture Squares of Bousfield Localizations of Spectra

Suppose I have a spectrum $X$ and two homology theories $E$ and $F$. If I look at the Bousfield localizations, $L_E$, $L_F$, $L_{E\vee F}$ and $L_{E\wedge F}$, do I have a homotopy pullback square ...
7
votes
0answers
243 views

Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?

The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it ...
1
vote
2answers
237 views

Relationship of Bousfield Classes of Morava K-theories

Suppose we have $\langle K(n)\rangle$ and $\langle K(n-1) \rangle$ for some fixed prime $p$. do we know whether or not $\langle K(n) \rangle \geq \langle K(n-1) \rangle$ or $\langle K(n-1) \rangle ...
3
votes
3answers
312 views

Bousfield Classes

This question has a few parts: 1) Is the Bousfield class of $\langle E\rangle$ the class of $E$-acyclics, i.e. $\langle E\rangle=\left\{ X:E\wedge X=0\right\}$ or is it the class of spectra which are ...
4
votes
1answer
527 views

A Model for the Moore Spectrum of $\mathbb{Z}_{(p)}$

Is it true that the Moore spectrum for the group $\mathbb{Z}_{(p)}$ can be constructed by smashing $\mathbb{S}$ with $q^{-1}\mathbb{S}$ for each $q\neq p$ (here both $q$ and $p$ are primes). It seems ...
7
votes
2answers
557 views

Absence of Maps Between p-local and q-local spectra

Suppose $X$ and $Y$ are spectra (or homotopy classes thereof) such that $X$ is p-local and $Y$ is q-local, for primes $p\neq q$. Is it indeed true then, and if so how would one show that ...
0
votes
0answers
535 views

The stable-homotopy-homology-theory

Hi Is there a way to stabilise relative homotopy groups into giving the stable-homotopy-homology-functor? The fact that the homotopy excision theorem holds for exactly the same kind of pair that ...
17
votes
2answers
685 views

Homotopy type of tensors of Moore spectra

I would like to hear what's known about the homotopy type of smash products of mod-$p^j$ Moore spectra, for $p$ an odd prime. First, here is what I'm specifically interested in: there is a short ...
7
votes
0answers
367 views

Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem

I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...
4
votes
1answer
421 views

Formal Group Laws on Ring Spectra?

Given a (graded) ring $R$, to define a formal group law it is equivalent to define a ring homomorphism $\phi:L\to R$ where $L$ is Lazard's ring. Is there any notion of defining a formal group law on ...
6
votes
4answers
865 views

do spectra have diagonal maps?

Topological spaces have diagonal maps $X \rightarrow X \times X$ and $X \rightarrow X \wedge X$, and suspension spectra also have diagonal maps $\Sigma^\infty X \rightarrow \Sigma^\infty(X \wedge X) ...
2
votes
1answer
545 views

Dedekind Spectra

Is there a class of ring spectra that corresponds to and/or extends the class of Dedekind rings from traditional algebra? Is there a notion of "ring of integers" of a ring spectrum? Additionally, is ...
14
votes
3answers
1k views

What are the best known results for the stable homotopy groups of spheres?

There are a number of proposed ways to compute the stable homotopy groups of spheres. One can rather peculiarly consider stable (co)homotopy of an Eilenberg Maclane spectrum as a generalised ...
22
votes
3answers
2k views

Modern Source for Spectra (including Ring Spectra)

I am looking for a modern introduction to Spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what has been ...
9
votes
0answers
235 views

stable homotopy of BG_+?

Let $G$ be an abelian group (if this helps, let's say also finite). Let $BG_+$ be the classifying space together with a disjoint base point. What are the stable homotopy groups $$\pi^s_m (BG_+) := ...
6
votes
1answer
416 views

Stable homotopy theory of orbifolds

Is there a notion of stable homotopy, spectrum, or a stable homotopy category which corresponds to orbifolds and orbispaces, in the same way that classical stable homotopy theory corresponds to ...
14
votes
2answers
1k views

Are $G$-spectra the same as modules over a “group ring spectrum”?

Let $G$ be a finite group (maybe this will also work when $G$ is compact, or something, but to be safe we'll let it be finite). I imagine it's quite natural to ask: is the category of $G$-spectra ...
83
votes
4answers
3k views

What do the stable homotopy groups of spheres say about the combinatorics of finite sets?

The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way: $\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$ ...
0
votes
1answer
314 views

Understanding a proof in Adams' Stable Homotopy and Gen. Coh

[Question cross posted on stack-exchange] I'm slowly working through Part III of the book, and I'm scratching my head a bit while reading the proof of Lemma 3.2 (here reproduced): Let $X, A$ be a ...
7
votes
1answer
588 views

How should I think of the $\infty$-category of spectra?

I've seen a bunch of definitions of spectra in the literature, and the fanciest seems to be the $(\infty, 1)$-category of spectra obtaining by "stablizing" the higher category of spaces, as in DAG I. ...
1
vote
1answer
187 views

Is there a map of spectra implementing the inverse of the Thom isomorphism?

In the top answer to the question "Is there a map of spectra implementing the Thom isomorphism?" it is explained (with a reference to Rudyaks book) that from a rank $r$ vector bundle $\mu:V\to X$, a ...
4
votes
2answers
400 views

Reference request: Spec A_* is the automorphism group of the additive formal group law

Dear all, I'm seeking a reference for a claim made in lecture 8 of Jacob Lurie's chromatic homotopy theory notes (http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf). More particularly, ...
12
votes
3answers
1k views

Stable homotopy category and the moduli space of formal groups

The usual disclaimer applies: I'm new to all this stuff, so be gentle. It seems like the spectrum, as defined by Balmer, of the stable homotopy category of finite complexes is something like ...
6
votes
2answers
345 views

Does a finite suspension spectrum make a space finite?

Suppose that $X$ is a space whose suspension spectrum $\Sigma_+^\infty(X)$ is dualizable in the stable homotopy category. I believe this is equivalent to saying that $\Sigma_+^\infty(X)$ is (weakly) ...
10
votes
2answers
782 views

How is the differential in complex cobordism defined?

This is my first MO question...hopefully it's not a bad one... Background: As a stable homotopy theorist, I like to think of complex cobordism $MU$ as a ring spectrum. If I needed to get my hands ...
11
votes
6answers
1k views

Why are equivariant homotopy groups not RO(G)-graded?

I know very little about the fancy equivariant stable homotopy category, so I apologize if this question is silly for one reason or another, but: I think that stable homotopy, in the non-equivariant ...
3
votes
1answer
413 views

Complex orientation of the Adams Summand

First lets fix a prime $p$ (I really care about $p=2$ but would be happy to know about other primes as well). When localized at a prime the spectrum $ku$ (Complex connective K-theory) splits as a ...
9
votes
4answers
713 views

Computing squaring operations in the Adams spectral sequence

This question is about the classical Adams spectral sequence. Squaring operations are defined on its $E_2$ term. I'd like to know how to compute some of the non-trivial operations, such as $Sq^2 ( ...
29
votes
4answers
2k views

Integral cohomology (stable) operations

There have been a couple questions on MO, and elsewhere, that have made me curious about integral or rational cohomology operations. I feel pretty familiar with the classical Steenrod algebra and its ...
26
votes
1answer
820 views

Early stabilization in the homotopy groups of spheres

Thanks to Freudenthal we know that $\pi_{n+k}(S^n)$ is independent of $n$ as soon as $n \ge k+2$. However, I was looking at the table on Wikipedia of some of the homotopy groups of spheres and noticed ...
8
votes
1answer
486 views

Technology for various models of spectra

There are a couple different models for spectra, or constructions of the categories of spectra that have the desired properties (homotopically and otherwise). The construction of the Categories of ...
16
votes
2answers
1k views

Is there a map of spectra implementing the Thom isomorphism?

A well known theorem in algebraic topology relates the (co)homology of the Thom space $X^\mu$ of a orientable vector bundle $\mu$ of dimension $n$ over a space $X$ to the (co)homology of $X$ itself: ...
8
votes
2answers
719 views

Does the category of topological symmetric spectra satisfy the monoid axiom ?

In the paper "Symmetric spectra"by Hovey, Smith and Shipley, they say that they don't know if the monoid axiom holds for topological symmetric spectra. This paper was written in 1998 so I am wondering ...
49
votes
6answers
5k views

third stable homotopy group of spheres via geometry?

It is ''well-known'' that the third stable homotopy group of spheres is cyclic of order $24$. It is also ''well-known'' that the quaternionic Hopf map $\nu:S^7 \to S^4$, an $S^3$-bundle, suspends to a ...
8
votes
1answer
387 views

Is the injective model structure on symmetric spectra Bousfield localizable?

I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on ...
5
votes
2answers
666 views

The homotopy cofiber of the smash product of two maps of spectra

It is a standard fact that smashing with a fixed spectrum $Z$ preserves cofiber sequences. So if I have a cofiber sequence $$X \xrightarrow{f} Y \rightarrow C_f$$ then there is also a cofiber ...
12
votes
3answers
843 views

Isomorphism between two universal p-typical formal group laws

EDIT: I've tried to alter the question so that its basic nature is clearer, as it's been unclear to a number of people now. At any prime p, there is a graded polynomial ring $V \cong {\mathbb ...
8
votes
1answer
856 views

A heart for stable equivariant homotopy theory

Let $G$ be a finite group. I wonder whether the following statement is true, known and written down: There is a t-structure on the stable $G$-equivariant homotopy category such that the associated ...
4
votes
1answer
295 views

Values of the multiplicative group over a ring spectrum

In his notes on elliptic cohomology, Lurie defines the multiplicative group $\mathbb{G}_m$ over a ring spectrum $A$ as $\operatorname{Spec} A[\mathbb{Z}]$. What is the value $\mathbb{G}_m(B)$ of the ...
4
votes
1answer
531 views

A Model Structure on Symmetric Monoidal Categories

The recent article found here revisits Thomason's proof that symmetric monoidal categories model all connective spectra, but stops short of showing that there is a full closed model structure on this ...
5
votes
1answer
685 views

Serre spectral sequence with spectra

A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason other than ...
13
votes
1answer
802 views

Complex orientations on homotopy

I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^*\mathbb{C}$P$^{\infty}$ or a statement about Thom classes. It ...
5
votes
1answer
628 views

Spectra and localizations of the category of topological spaces

Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces using some kind of localization combined with other categorical ...
3
votes
3answers
352 views

Are there universe-indexed spectra over simplicial sets?

In "Rings, Modules, and Algebras in Stable Homotopy Theory" Elmendorf, Kriz, Mandell and May introduce a notation of spectra indexed over an universe $\mathcal{U}$ as a collection of pointed ...
13
votes
1answer
679 views

Stable ∞-categories as spectral categories

Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C does have all finite colimits, the suspension functor on C is an ...
12
votes
3answers
1k views

complex cobordism from formal group laws?

Reading Ravenel's "green book", I wonder about his question on p.15 "that the spectrum MU may be constructed somehow using formal group law theory without using complex manifolds or vector bundles. ...
6
votes
2answers
305 views

Reference for iterated homotopy fixed points?

What are (good) references for results about iterated homotopy fixed points? That is, suppose G is a topological group acting on a space (or spectrum) X, and H is a normal subgroup of G. Then one ...
11
votes
2answers
650 views

squares in stable homotopy

I noticed that the generator of the second stable stem b is the square of the generator of the first stable stem a, in the sense that if take two copies of a and smash product them together you get b ...