14
votes
4answers
387 views

Fibrations and Cofibrations of spectra are “the same”

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is: "For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
10
votes
0answers
269 views

From the perspective of bordism categories, where does the ring structure on Thom spectra come from?

To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
5
votes
1answer
212 views

endomorphisms of modules over symmetric ring spectra

I have a probably very basic question about modules over symmetric ring spectra: Let $R$ be a commutative symmetric ring spectrum and let $M$ and $N$ be module spectra over $R$. Moreover, let ...
10
votes
1answer
226 views

Is there a non-zero ghost map between finite suspension spectra?

A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map. Not every ghost map ...
5
votes
1answer
313 views

Dennis trace map K----> THH

I have some questions about Dennis trace map in algebraic K-Theory. I was wondering if there is some conceptual way to look at this map $K(-)\rightarrow THH(-)$ (natural transformation from K-Theory ...
23
votes
4answers
1k views

What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?

I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra. In the former, one assigns to ...
10
votes
1answer
377 views

The homotopy of universal Thom spectrum

Let $S^0_p$ be the $p$-adic sphere spectrum. Let $GL_1(S^0_p)$ be the set of unit componen of $\Omega^{\infty}S^0_p$. For any map $ X \to BGL_1(S_p^0)$ we get a Thom spectrum call it $Mf$. Now ...
2
votes
0answers
152 views

Generator of $\pi_3(SU(4))$ in Mimura-Toda

In this paper of Mimura and Toda, tables are given for low-dimensional homotopy groups of $SU(3)$, $SU(4)$ and $Sp(2)$. As far as I understand it, Theorem 6.1 gives the generator of $\pi_3(SU(4))$ as ...
5
votes
1answer
227 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 ...
0
votes
0answers
55 views

Cohomology operations over general rings [duplicate]

If $X$ is a topological space and $R$ is a commutative ring, then the singular cohomology groups $H^*(X,R)$ support cohomology operations coming from the homology of symmetric groups. If $R = ...
9
votes
1answer
324 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 ...
6
votes
2answers
265 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 ...
20
votes
1answer
1k 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 ...
2
votes
0answers
153 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 ...
4
votes
2answers
426 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) ...
16
votes
2answers
504 views

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} & ...
17
votes
1answer
834 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
2k 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
566 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 ...
12
votes
1answer
311 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 ...
3
votes
0answers
151 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 ...
6
votes
1answer
228 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
1answer
275 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 -- ...
22
votes
1answer
774 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
0answers
171 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 ...
12
votes
2answers
879 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
267 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 ...
6
votes
1answer
426 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 ...
13
votes
1answer
522 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 ...
11
votes
1answer
370 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 ...
4
votes
1answer
228 views

How to construct maps between (co)fibre sequences in a stable $\infty$-category?

Fix a stable $\infty$-category $\mathcal{C}$ and two (co)fibre sequences $a \rightarrow b \rightarrow c$ and $x \rightarrow y \rightarrow z$ in $\mathcal{C}$. Now suppose we are given a map $a ...
15
votes
1answer
365 views

Is the 4-line of the E_2 term of the classical Adams spectral sequence known?

In other words: What is $\mathrm{Ext}_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$? If the 4-line is not known, how much is known about it? Here, $\mathcal{A}$ is the 2-primary Steenrod ...
6
votes
2answers
367 views

Chromatic convergence of E(n)-localized homotopy categories

Given the Chromatic Convergence Theorem, can we state this globally as some convergence in the category of categories? That is, we have the subcategories of finite spectra in each stable homotopy ...
2
votes
1answer
123 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 ...
6
votes
0answers
222 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 ...
0
votes
0answers
500 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 ...
16
votes
2answers
627 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 ...
5
votes
4answers
741 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
520 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 ...
13
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 ...
20
votes
3answers
1k 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 ...
69
votes
3answers
2k 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
134 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 ...
3
votes
1answer
356 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 ...
26
votes
4answers
1k 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 ...
15
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: ...
42
votes
6answers
4k 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
311 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 ...
4
votes
2answers
577 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 ...
4
votes
1answer
277 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 ...