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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: $\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$ ...
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 ...
33
votes
3answers
819 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 ...
26
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. ...
25
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 ...
23
votes
1answer
717 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 ...
22
votes
4answers
2k 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 ...
22
votes
4answers
987 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 ...
22
votes
1answer
765 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 ...
21
votes
1answer
1k 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 ...
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 ...
20
votes
1answer
974 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 ...
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 ...
18
votes
0answers
464 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 ...
17
votes
1answer
813 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 ...
17
votes
0answers
439 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 $n \geq 3$ ...
16
votes
2answers
578 views

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 ...
16
votes
2answers
624 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 ...
16
votes
2answers
471 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} & ...
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: ...
15
votes
1answer
364 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 ...
14
votes
2answers
750 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 ...
14
votes
1answer
550 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 ...
14
votes
0answers
356 views

Refinement of concept of support of a module

My rings are commutative and noetherian. The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...
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 ...
13
votes
2answers
473 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 ...
13
votes
1answer
708 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 ...
13
votes
1answer
520 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 ...
12
votes
3answers
979 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 ...
12
votes
2answers
862 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 ...
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. ...
12
votes
2answers
369 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 ...
12
votes
1answer
279 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 ...
12
votes
1answer
303 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 ...
11
votes
3answers
810 views

Classifying triangulated structures on a graded category

I know of several results to the effect that two triangulated categories are equivalent categories (usually one coming from algebra and one coming from topology). However, it's never been clear to me ...
11
votes
3answers
733 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 ...
11
votes
2answers
581 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 ...
11
votes
1answer
537 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 ...
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 ...
11
votes
1answer
275 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 ...
11
votes
1answer
607 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 ...
11
votes
2answers
787 views

Exotic spheres and stable homotopy in all large dimensions?

Given that the kervaire invariant one problem has been solved in (almost) all dimensions....my question is whether there exists an exotic sphere in all sufficently lagre dimensions? Given the ...
11
votes
0answers
293 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: ...
10
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 ...
10
votes
1answer
214 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 ...
10
votes
1answer
351 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 ...
10
votes
0answers
170 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 ...
9
votes
1answer
324 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 ...
9
votes
1answer
207 views

Verdier localization for stable $\infty$-categories

Verdier localization is one of the more intuitive ways to localize a triangulated category, "killing" a suitable class of objects via a functor which is universal with respect to this property. I ...
9
votes
1answer
367 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 ...