User eric peterson - MathOverflow

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

2012-10-06T20:29:58Z

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 the sphere begins with $E_2$-page $$E_2^{*, *} = \operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, BP_*)$$ and converges to $\pi_* \mathbb{S} \otimes_{\mathbb{Z}} \mathbb{Z}_{(p)}$.

There are a variety of cool periodicities visible in this $E_2$-page, which we can organize via the following secondary spectral sequence. There is an ascending chain of $(BP_* BP)$-invariant ideals for $BP_*$ given by $I_r = (p, v_1, \ldots, v_{r-1})$, connected to one another by the short exact sequences $$0 \to BP_* / I_r^\infty \to v_r^{-1} BP_* / I_r^\infty \to BP_* / I_{r+1}^\infty \to 0.$$ The quotient $BP_* / I_r^\infty$ is thought of as the closed substack of the moduli of formal groups detected by the ideal sheaf corresponding to $I_r$ together with its formal neighborhood inside the parent stack. Applying $\operatorname{Ext}$ and stringing the resulting long exact sequences together, one arrives at the (trigraded) chromatic spectral sequence (CSS): $$E_1^{r, *, *} = \operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, v_r^{-1} BP_* / I_r^\infty) \Rightarrow \operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, BP_*).$$ Much of the fun in chromatic homotopy theory after this point comes from identifying the groups in this $E_1$-page as other sorts of things, like certain group cohomologies.

Shifting gears somewhat, Bousfield localization at the Johnson-Wilson $E(r)$-theories and the Morava $K(r)$-theories is meant to perform the same organization at the level of homotopy types. The spectra $E(\infty)$ and $E(0)$ correspond to $BP$ and to $H\mathbb{Q}$ respectively, so the sequence of localization functors $L_{E(r)}$ are meant to interpolate between rational homotopy theory and the sort of homotopy theory visible to the $p$-local Adams-Novikov spectral sequence.

There are two ways to study these functors as $r$ increases. First, there there is a natural map $L_{E(r)} X \to L_{E(r-1)} X$. Its homotopy fiber detects the difference between these two spectra, denoted $M_r X$ and called the $r$th monochromatic layer of $X$. Second, there is a pullback square, dubbed chromatic fracture: $$\begin{array}{ccc} L_{E(r)} X & \to & L_{K(r)} X \\ \downarrow & & \downarrow \\ L_{E(r-1)} X & \to & L_{E(r-1)} L_{K(r)} X. \end{array}$$ In both of these situations, you can hope to inductively study the filtering spectra $L_{E(r)} X$ by studying the "filtration layers", which are either $M_r X$ or $L_{K(r)} X$ depending upon your approach.

My question is: How exactly do these two approaches connect to the chromatic spectral sequence?

I suspect that the CSS for $L_{E(R)} X$ looks like the CSS for $X$, after quotienting out the information in $r$-degrees $r > R$. I also suspect that the CSS for one of the two of $M_R X$ and $L_{K(R)} X$ looks like that for $L_{E(R)} X$, after additionally quotienting out the information in $r$-degrees $r < R$. However, I can't seem to make the pieces line up. For instance, Prop. 7.4 of Hopkins, Mahowald, and Sadofsky's Constructions of elements in Picard groups suggests that this description holds for $L_{K(R)} \mathbb{S}$, as that statement matches their Adams-Novikov spectral sequence converging to $\pi_* L_{K(R)} \mathbb{S}$ --- just as one would expect from a collapsing chromatic spectral sequence. On the other hand, the bottom corner of the fracture square is of the form $L_{E(R-1)} L_{K(R)} X$, and this description seems to say that its CSS is empty, which doesn't sound right.

I'd appreciate someone setting me straight about this. Thanks!

Answer by Eric Peterson for K(r)-localization and monochromatic layers in the chromatic spectral sequence

2013-03-16T02:44:40Z

I have at least a partial answer to my question. It's fairly complicated, and pieces of it are written down in a variety of places, so I'm going to do what I can to be thorough. Before we do anything involving spectral sequences at all, it will turn out to be useful to have a certain pair of families of $BP_* BP$-comodules at our disposal, defined by the formulas $$N_r^s = BP_* / \langle p, \ldots, v_{r-1}, v_r^\infty, \ldots, v_{r+s-1}^\infty \rangle,$$ $$M_r^s = v_{r+s}^{-1} BP_* / \langle p, \ldots, v_{r-1}, v_r^\infty, \ldots, v_{r+s-1}^\infty\rangle = v_{r+s}^{-1} N_r^s.$$ In fact, these formulas even make sense on the level of spectra, since $N_0^0$ can be taken to be $BP$, $M_r^s$ appears as a mapping telescope built out of $N_r^s$, and there are cofiber sequences (/ short exact sequences) $$N_r^s \xrightarrow{\cdot v_{r+s}^\infty} M_r^s \to N_r^{s+1},$$ $$N_r^r \xrightarrow{\cdot v_r} N_r^r \to N_{r+1}^{r+1}.$$ The $BP_* BP$-comodules are recovered by taking homotopy groups.

The most fundamental of all the spectral sequences in play was brought up by Drew in the comments above. The chromatic tower is a tower of fibrations $$\cdots \to L_{E(n+1)} \mathbb{S}^0 \to L_{E(n)} \mathbb{S}^0 \to \cdots \to L_{E(0)} \mathbb{S}^0,$$ and the fibers of these maps define the monochromatic layers. Applying $\pi_*$ to the diagram produces a spectral sequence of signature $$\pi_* M_r \mathbb{S}^0 \Rightarrow \pi_* \mathbb{S}^0_{(p)}.$$ To study this spectral sequence, there are two reasonable-sounding things to do involving the homology theory $BP_*$:

Apply $BP_*$ to the chromatic tower diagram and study the resulting spectral sequence.
Use the $BP$-Adams spectral sequence to compute $\pi_* M_n \mathbb{S}^0$ from $BP_* M_n \mathbb{S}^0$.

These both turn out to be relevant, and they both rest upon a certain input, computed by Ravenel. Namely, he shows how to compute $BP_* L_{E(r)} \mathbb{S}^0$ and the surrounding pieces:

Theorem 6.2 (Ravenel, Localizations with respect to certain periodic cohomology theories): The going-around maps $N_0^{s+1} \to \Sigma N_0^s$ compose to give a map $\Sigma^{-s-1} N_0^{s+1} \to N_0^0 = BP$. The cofiber of this map can be identified as $$\Sigma^{-s-1} N_0^{s+1} \to BP \to L_{E(s)} BP.$$ Moreover, the rotated triangle $BP_* \to \pi_* L_{E(s)} BP \to \pi_* \Sigma^{-s} N_0^{s+1}$ is split short exact. (There's an exception in the bottom case, where $BP_* L_{E(0)} \mathbb{S}^0 = BP_* \otimes \mathbb{Q}$.)

Applying the octahedral axiom to the pair $\Sigma^{-s-1} N_0^{s+1} \to \Sigma^{-s} N_0^s \to BP$ and then applying $BP_*$ -homology gives the calculation $$BP_* M_s \mathbb{S}^0 = \Sigma^{-s} M_0^s.$$

Now we can address 1. and 2.:

If we delete the boring $BP_*$ summands in $BP_* L_{E(n)} \mathbb{S}^0$ , then the exact couple coming from applying $BP_*$ -homology to the chromatic tower just falls apart into a string of short exact sequences of $BP_* BP$-comodules. Now, we know that $H^{*, *} N_0^0$ is the input to the $BP$-Adams spectral sequence computing $\pi_* \mathbb{S}^0$, and applying $H^{*, *}$ to this diagram of short exact sequences of $BP_* BP$ -modules yields a spectral sequence of signature $$E_1^{r, *, *} = H^{*, *} M_0^r \Rightarrow H^{*, *} N_0^0.$$ This is the usual chromatic spectral sequence, as stemming from algebraic considerations.
Applying the $BP$-Adams spectral sequence to compute $\pi_* M_r \mathbb{S}^0$ begins with the computation of $\operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, BP_* M_r \mathbb{S}^0)$ , which we now know to be isomorphic to $H^{*, *} M_0^r$ . (In fact, this spectral sequence is supposed to collapse for $p \gg n$, e.g., $p \ge 5$ for $n = 2$.)

Another part of this whole story is how the $K(r)$-local sphere plays into this picture. Now, there is also an Adams-type spectral sequence computing $\pi_* L_{K(r)} \mathbb{S}^0$ , and it has signature $$\operatorname{Ext}^{*, *}_{\Gamma}(K(n)_*, K(n)_*) \Rightarrow \pi_* L_{K(n)} \mathbb{S}^0,$$ where $\Gamma = K(n)_* \otimes_{BP_*} BP_* BP \otimes_{BP_*} K(n)_*$ . Morava's change of rings theorem states that the map $M_r^0 \to K(n)_*$ induces an isomorphism between the sheaf cohomology groups $H^{*, *} M_r^0$ and the $\operatorname{Ext}$-groups in the Adams-type spectral sequence.

The difference, then, between the monochromatic sphere and the $K(r)$-local sphere is recorded in the index swap $M_0^r$ and $M_r^0$ --- i.e., whether the generators below $v_r$ are taken to be zero or to be torsion. The difference in these two situations is of course itself recorded as a spectral sequence: the inclusions $$v_{r+s}^{-1} BP_* / \langle p, \ldots, v_{r-1}, v_r^j, v_{r+1}^\infty, \ldots, v_{r+s-1}^\infty \rangle \xrightarrow{\cdot v_r} v_{r+s}^{-1} BP_* / \langle p, \ldots, v_{r-1}, v_r^{j+1}, v_{r+1}^\infty, \ldots, v_{r+s-1}^\infty \rangle$$ all have cofiber $M_{r+1}^{s-1}$, regardless of choice of index $j$. Applying $H^{*, *}$ to the string of inclusions (extended to cofiber sequences) yields the $v_r$-Bockstein spectral sequence, of signature $$H^{*, *} M_{r+1}^{s-1} \otimes \mathbb{F}_p[v_r] / v_r^\infty \Rightarrow H^{*, *} M_{r}^{s}.$$ So, there is a length $r$ string of $v_*$ -Bockstein spectral sequences beginning with $H^{*, *} M_r^0$ and concluding with $H^{*, *} M_0^r$.

Some things not included in this answer are:

What happens when analyzing the chromatic tower of spaces other than the sphere spectrum?
What is the relevance of the corner space $L_{E(r-1)} L_{K(r)} \mathbb{S}^0$ in terms of the chromatic spectral sequence?
What parts of this story can be made sense of mutatis mutandis when replacing $BP$ with other spectra in the same family, like $E(r)$? My expectation (as the comments reveal) is that there should be an analogue of the algebraic chromatic spectral sequence for $E(R)$-homology, which is the truncation of the usual one for $r \le R$. (In October I even thought I knew how to prove this, but I've since forgotten. This is the least interesting question of the bunch.)

Answer by Eric Peterson for What is the universal deformation of the formal additive group $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$?

2012-11-27T19:46:58Z

$\DeclareMathOperator{\Ext}{Ext} \newcommand{\G}{\hat{\mathbb{G}}} \DeclareMathOperator{\Maps}{Maps} \renewcommand{\phi}{\varphi}$ The analysis of the infinitesimal deformation space of the Honda formal groups $H_n$ uses three calculations which govern the existence of square-zero deformations. These can be phrased in terms of certain $\Ext$ groups, and here's how I think of them: we want to study the deformation of a formal group $G_0$ over a ground ring $R_0$ along a square-zero infinitesimal deformation $I \to R \to R/I = R_0$ to a formal group $G$ over $R$. Because $I$ is square-zero and the only formal group law truncated at degree $2$ is the additive formal group, we think of $G$ as sitting in an extension $I \otimes \G_a \to G \to G_0$ sort of "over" the original extension of $R$-modules. So, to study these extensions (their existence, their uniqueness, so on), we want to calculate some kind of $\Ext$ groups of the form $\Ext^*(G_0; \G_a)$.

This is exactly what Lubin and Tate do in their paper. Fix a pair of formal groups $F$ and $G$; then there's a simplicial object $BF = B_*(*, F, *)$ associated to $F$ via its group structure, and the cosimplicial object $\operatorname{Maps}(BF, G)$ should be a thing whose cohomology $H^*_{LT}$ computes $\Ext^{*-1}(F; G)$. Trying to work out exactly what this means, you'll find the following descriptions of the first few groups (written in coordinates, but this is inessential):

$$H^1_{LT}(F; G) = \{\phi: F \to G \mid \phi(x) - \phi(x + y) + \phi(y) = 0\} = \operatorname{Hom}(F, G),$$ $$H^2_{LT}(F; G) = \frac{\{\phi: F^2 \to G \mid \phi(x, y) - \phi(x, y + z) + \phi(x + y, z) - \phi(y, z) = 0\}}{\{\delta^1 \phi \mid \phi : F \to G, \delta^1 \phi(x, y) = \phi(x) - \phi(x + y) + \phi(x)\}},$$ $$H^3_{LT}(F; G) = \frac{\left\{\phi: F^3 \to G \middle| \begin{array}{c}\phi(x, y, z) - \phi(x, y, z + w) + \phi(x, y + z, w) - \\ \phi(x + y, z, w) + \phi(y, z, w) = 0\end{array}\right\}}{\{\delta^2 \phi \mid \phi : F^2 \to G, \delta^1 \phi(x, y) = \phi(x, y) - \phi(x, y + z) + \phi(x + y, z) - \phi(y, z)\}}.$$

General facts about infinitesimal deformation theory then tell you the utility of these groups: $H^1_{LT}$ tracks automorphisms of the square-zero deformation space, so it tells you whether you should expect the deformation space to be a scheme or some sort of stack; $H^3_{LT}$ tracks the obstruction to the existence of square-zero deformations; and $H^2_{LT}$ tracks the available square-zero extensions, in the sense that when $H^1_{LT} = 0$, its dimension will tell you the dimension of the tangent space of the deformation space.

So, Lubin and Tate go about computing these three things in the case that $G_0 = H_n$ is the height $n$ Honda formal group. They know that $\operatorname{Hom}(H_n, \G_a) = 0$, and so $H^1_{LT} = 0$. They also compute that $H^3_{LT} = 0$, so they know that the deformation space they're computing is actually a formal variety --- there are never any kind of conditions imposed on their generators $u_i$ to ensure the existence of a deformation. The only thing left is to determine the dimension of the formal variety, and they compute $\dim \Ext^1(H_n, \G_a) = n-1$.

You ask what happens when we replace $H_n$ with $\G_a$ and instead study $H^*_{LT}(\G_a; \G_a)$. The computation of these cohomology groups is worked out in the COCTALOS notes, and also in Theorem 4.3 and Section 8.4 of a project I worked on: [link]. Grinding through the homological algebra, you'll find that the group $H^n_{LT}$ consists of the polynomials of homogeneous degree $n$ in the dual Steenrod algebra, after assigning the degrees $|\xi_*| = 1$ , $|\tau_*| = 1$ , and $|P_*| = 2$ .

All of these things are woefully $\infty$-dimensional. I don't actually know what that means for the existence of your object --- and so this isn't a proper answer --- but I do know that it isn't anywhere as simple as the case Lubin and Tate analyze.

Construction of Morava E-theory

2010-08-22T01:40:26Z

I'm wrapping up a summer project that involved a computation in Morava $E$-theory. As background knowledge I had to look into how the Johnson-Wilson theories $E(n)$ and Morava $K$-theories were constructed. This was manageable since I'd been part-way down that road already and there's lots of support in, for example, the form of Hopkins' course notes. Then, in early May I spent some time digging around for a construction of Morava $E$-theory, which led me to some conclusions:

The words "Morava $E$-theory" don't determine what object you're talking about; there's a whole bunch of slightly different Morava $E$-theories.
People frequently conflate Morava $E$-theory with (completed) Johnson-Wilson theory. One source even claimed (without citation) that one was a finite free module over the other, and that I therefore shouldn't worry about the difference.

In the end, I didn't need to really know much about $E_n$ beyond a couple formal properties to work out the broad strokes of my computation, so I let the whole thing slide and pretended there existed a spectrum that did what I'd hoped.

However, I'm now getting to a point where understanding what I'm actually doing would be valuable. In decreasing order of importance, can someone provide a reference that...

... constructs a family of Morava $E$-theories. Any family would be a start! I am particularly interested in one with a coefficient ring of the form $\mathbb{Z}_p[\![v_1, \ldots, v_{n-1}]\!][v_n^{\pm 1}]$ .
... illustrates that $\mathrm{spf}\,(E_n)^* \mathbb{C}P^\infty$ is the universal deformation of $\mathrm{spf}\,K(n)^* \mathbb{C}P^\infty$ to formal groups over formal spectra of complete, local rings with residue field $\mathbb{F}_p[v_n^{\pm 1}]$. The remark above about the comparison between Johnson-Wilson theory and Morava $E$-theory made me particularly uncomfortable in this respect; it's not clear to me that the formal group associated to Johnson-Wilson theory should be thought of as the universal deformation of the Honda formal group. Clearing that up would be nice too.
... also shows that the reduction of the universal deformation to the "mod $p$" case exists as a spectrum, and the reduction map exists as a map of spectra. That is, there is a complex-oriented, structured ring spectrum $E_n/p$ with coefficient ring $\mathbb{F}_p[\![v_1, \ldots, v_{n-1}]\!][v_n^{\pm 1}]$ whose associated formal group is the universal deformation of the Honda formal group to complete, local rings of characteristic $p$.
... also shows that the reduction of the universal deformation modulo the $n$^th power of its maximal ideal exists as a spectrum, and the reduction map exists as a map of spectra.
... demonstrates this fact about $E_n$ being a finite free module over $E(n)$, at least for an appropriate interpretation of the symbol "$E_n$".

It may not be the case that points 3 and 4 are even true, but I'm hopeful. Still, surely this is all catalogued somewhere!

Answer by Eric Peterson for Reference on the question mark cell complex

2012-07-21T21:47:21Z

This is hardly a "canonical example", but one place I've seen the question mark complex $Q$ in action is in Hovey and Sadofsky's paper Invertible spectra in the $E(n)$-local stable homotopy category. There they compute that the even part of the Picard group of the $E(1)$-local stable category at $p = 2$ is generated by (the $E(1)$-localization of) $Q$. As a warning, the exposition at the tail of the paper where this appears is a bit skeletal; you'll be working out the details of the computations and digging up references on your own.

But, in turn, they do cite Hopkins' Minimal atlases of real projective spaces for some facts about $Q$, which I have not read but looks to be useful. He seems to call the complex $N_2$ in section 7, for reasons he explains pertaining to the cohomology of $bo\langle i \rangle$.

Proofs of Bott periodicity

2009-12-13T20:48:03Z

K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of the Bott periodicity theorem for topological complex K-theory that I've found in the literature:

Bott's original proof used Morse theory, which reappeared in Milnor's book Morse Theory in a much less condensed form.
Pressley and Segal managed to produce the homotopy inverse of the usual Bott map as a corollary in their book Loop Groups.
Behrens recently produced a novel proof based on Aguilar and Prieto, which shows that various relevant maps are quasifibrations, therefore inducing the right maps on homotopy and resulting in Bott periodicity.
Snaith showed that $BU$ is homotopy equivalent to $CP^\infty$ once you adjoin an invertible element. (He and Gepner also recently showed that this works in the motivic setting too, though this other proof relies on the reader having already seen Bott periodicity for motivic complex K-theory.)
Atiyah, Bott, and Shapiro in their seminal paper titled Clifford Modules produced an algebraic proof of the periodicity theorem. EDIT: Whoops x2! They proved the periodicity of the Grothendieck group of Clifford modules, as cdouglas points out, then used topological periodicity to connect back up with $BU$. Wood later gave a more general discussion of this in Banach algebras and Bott periodicity.
Atiyah and Bott produced a proof using elementary methods, which boils down to thinking hard about matrix arithmetic and clutching functions. Variations on this have been reproduced in lots of books, e.g., Switzer's Algebraic Topology: Homotopy and Homology.
A proof of the periodicity theorem also appears in Atiyah's book K-Theory, which makes use of some basic facts about Fredholm operators. A differently flavored proof that also rests on Fredholm operators appears in Atiyah's paper Algebraic topology and operations on Hilbert space.
Atiyah wrote a paper titled Bott Periodicity and the Index of Elliptic Operators that uses his index theorem; this one is particularly nice, since it additionally specifies a fairly minimal set of conditions for a map to be the inverse of the Bott map.
Seminaire Cartan in the winter of '59-'60 produced a proof of the periodicity theorem using "only standard techniques from homotopy theory," which I haven't looked into too deeply, but I know it's around.

Now, for my question: the proofs of the periodicity theorem that make use of index theory are in some vague sense appealing to the existence of various Thom isomorphisms. It seems reasonable to expect that one could produce a proof of Bott periodicity that explicitly makes use of the facts that:

The Thom space of the tautological line bundle over $CP^n$ is homeomorphic to $CP^{n+1}$.
Taking a colimit, the Thom space of the tautological line bundle over $CP^\infty$ (call it $L$) is homeomorphic to $CP^\infty$.
The Thom space of the difference bundle $(L - 1)$ over $CP^\infty$ is, stably, $\Sigma^{-2} CP^\infty$. This seems to me like a route to producing a representative of the Bott map. Ideally, it would even have good enough properties to produce another proof of the periodicity theorem.

But I can't find anything about this in the literature. Any ideas on how to squeeze a proof out of this -- or, better yet, any ideas about where I can find someone who's already done the squeezing?

Hope ~~this isn't~~ less of this is nonsense!

-- edit --

Given the positive response but lack of answers, I thought I ought to broaden the question a bit to start discussion. What I was originally looking for was a moral proof of the periodicity theorem -- something short that I could show to someone with a little knowledge of stable homotopy as why we should expect the whole thing to be true. The proofs labeled as elementary contained too much matrix algebra to fit into parlor talk, while the proofs with Fredholm operators didn't seem -- uh -- homotopy-y enough. While this business with Thom spaces over $CP^\infty$ seemed like a good place to look, I knew it probably wasn't the only place. In light of Lawson's response, now I'm sure it isn't the only place!

So: does anyone have a good Bott periodicity punchline, aimed at a homotopy theorist?

(Note: I'll probably reserve the accepted answer flag for something addressing the original question.)

Homotopy type of tensors of Moore spectra

2012-01-10T06:36:49Z

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 exact sequence \[0 \to \mathbb{Z} \xrightarrow{p^j} \mathbb{Z} \to \mathbb{Z}/p^j \to 0.\] Tensoring this short exact sequence against your favorite group $G$ yields an exact sequence \[\cdots \to G \xrightarrow{p^j} G \to G \otimes \mathbb{Z}/p^j \to 0,\] which exhibits $G \otimes \mathbb{Z}/p^j \cong G / p^j G$ as $G$ with its $p^j$-divisible part stripped out.

Moore spectra play a related role in homotopy theory: they are defined by the cofiber sequence \[S \xrightarrow{p^j} S \to M(p^j).\] Smashing through with any spectrum $X$ gives the new cofiber sequence \[X \xrightarrow{p^j} X \to X \wedge M(p^j),\] and chasing this around shows that the homotopy group $\pi_n X \wedge M(p^j)$ is a mix of $\pi_n X / p^j(\pi_n X)$, as one would expect, together with the $p^j$-torsion of $\pi_{n-1} X$, which is new and different. So, though $X \wedge M(p^j)$ is sometimes written $X / p^j$, and though this notation suggests a useful analogy, this isn't exactly true, and we have to be careful about things we expect to follow from the algebraic setting.

The specific algebraic fact I'm interested in is that the composition of the tensor functors $- \otimes \mathbb{Z}/p^j$ and $- \otimes \mathbb{Z}/p^i$ for $j > i$ has a reduction: \[- \otimes \mathbb{Z}/p^j \otimes \mathbb{Z}/p^i \cong - \otimes \mathbb{Z}/p^i.\] The exact translation of this statement to Moore spectra and the smash product is not true --- one can, for instance, compute the reduced integral homology of $M(p^i) \wedge M(p^j)$ to see that there are too many cells around for it to be equivalent to $M(p^i)$ alone. However, the same homology calculation suggests something related: there is an abstract isomorphism between the reduced homology groups of $M(p^j) \wedge M(p^i)$ and those of $M(p^i) \wedge M(p^i)$. This is, of course, also true for groups; it is indeed the case that $\mathbb{Z}/p^i \otimes \mathbb{Z}/p^j \cong \mathbb{Z}/p^i \otimes \mathbb{Z}/p^i$. This is what I want to know:

For $j > i$, is $M(p^j) \wedge M(p^i)$ homotopy equivalent to $M(p^i) \wedge M(p^i)$?

If this is not true, I'm willing to throw in some extra qualifiers. For instance, is the situation improved if we work $K(n)$-locally? Is it true only when $j \gg i$? What if we additionally restrict to $j \gg i \gg 0$?

This specific question aside, I am also interested in any and all known features of the homotopy type of $M(p^i) \wedge M(p^j)$ --- any favorite fact you have that would help me get a grip on them. I'm also specifically interested in variants of the above question for generalized Moore spectra: can anything similar be said about those?

Answer by Eric Peterson for Is there a high-concept explanation of the dual Steenrod algebra as the automorphism group scheme of the formal additive group?

2011-12-10T04:55:06Z

Given two spectra $E$ and $F$, how might we get a handle on the contents of the spectrum $E \wedge F$? One thing we could try is to produce an interesting map involving $E \wedge F$ as its source or target that relates to things we already understand. Let's assume complex orientations for $E$ and $F$, so maps $u: MU \to E$ and $v: MU \to F$, which we combine to get a map $MU \wedge MU \to E \wedge F$. The homotopy of the spectrum $MU \wedge MU$ carries the universal example of a formal group law isomorphism, and so the map we constructed selects an isomorphism of the formal group laws associated to the composite orientations $MU \xrightarrow{u} E \xrightarrow{\eta_F} E \wedge F$ and $MU \xrightarrow{v} F \xrightarrow{\eta_E} E \wedge F$. In some cases we are lucky enough to produce an isomorphism of the formal groups of $E$ and of $F$, like with your computation of interest when $E = F = H\mathbb{F}_2$: $$\operatorname{Spec} \pi_* H\mathbb{F}_2 \wedge H\mathbb{F}_2 = \operatorname{Aut}(\hat{\mathbb{G}}_a).$$ This is definitely not going to happen in general, since the homotopy of the smash product can be either too complicated or not complicated enough, or the composite orientations might not compare well with the originals --- they might be "damaged" in some way through pushforward. In the case of the dual Steenrod algebra it is in some sense a statement about spread-out-ness of the objects involved. But, as far as I know, this is as deep an explanation as you can get at present.

I'm told that something similar happens in the odd primary case, but involves automorphisms of the formal additive "supergroup". I have no idea how it works, though, and is probably generally related to my poor understanding of graded commutativity and odd dimensional phenomena in general. Definitely it is mentioned in brief in COCTALOS; searching on 'super' will bring it right up if you want to read a few sentences more.

-- edit --

Your mentioning of $\mathbb{R}\mathrm{P}^\infty$ is somewhat separate. The reason the Steenrod operations show up there is that $\mathbb{R}\mathrm{P}^\infty$ is a $B\mathbb{Z}/2$ and $\mathbb{Z}/2 = \Sigma_2$ is a symmetric group permuting the inputs to a cup product. It's an altogether different miracle that this recipe for constructing cohomology operations exhausts all of $H\mathbb{F}_2$'s.

Multiplicativity in the descent spectral sequence

2011-09-02T00:52:53Z

For a homotopy sheaf $\mathcal{F}$ of ring spectra over some space (/ site / whatever) $X$ with a cover $U_i$, we can build a "descent spectral sequence" with signature $$E^1_{p, q} = \pi_{p+q} \mathcal{F}\left(\coprod_{|I| = q} U_I \right) \Rightarrow \pi_{p+q} \mathcal{F}(X).$$ This comes about by building the simplicial object associated to the cover, applying the sheaf to get a cosimplicial object, and then instantiating a filtration-type spectral sequence given by looking at varying truncations of totalizations --- the standard spectral sequence for a cosimplicial object. The $E^2$-page of the spectral sequence can be identified with Cech cohomology, and so the spectral sequence is meant to provide an intermediary between that homological object and the homotopy-sensitive information in the sheaf of spectra. This construction is natural enough that...

...a refinement of the covering induces a map of spectral sequences. Just like limiting Cech cohomologies over cover refinements gives sheaf cohomology, limiting Cech descent spectral sequences gives a descent spectral sequence with $E^2$-page described by sheaf cohomology.
...a map of sheaves induces a map of spectral sequences.

This construction doesn't really use the fact that $\mathcal{F}$ takes values in rings. I feel that this must appear in the spectral sequence, that we should expect some kind of multiplicativity --- maybe one that mixes the products of the ring spectra and the Cech product.

So, my question is: Is there a multiplicative structure in any of these spectral sequences? What is its signature? How might I compute with it? Better yet, are there examples of other people computing with it that I can read about?

For what it's worth, I'm interested most in starting with a sheaf of ring spectra $\mathcal{F}$ and computing $\mathcal{F}(X)^*(Y)$ for my favorite space $Y$ by augmenting $\mathcal{F}$ to $F(\Sigma^\infty_+ Y, \mathcal{F})(U) := F(\Sigma^\infty_+ Y, \mathcal{F}(U))$ and working with that. This comes with a map $F(\Sigma^\infty_+ (Y \times Y), \mathcal{F}) \to F(\Sigma^\infty_+ Y, \mathcal{F})$, which is in the vein of the usual construction of the cup product.

(There are a lot of details I'm eliding past, like conditional convergence, $\lim^1$ problems, when my augmented guy is actually a homotopy sheaf, when the analogy with sheaf cohomology can be made, so on and so forth. I don't think I've said anything not true in the nicest of settings, which is where I'd like to start learning, at least. Sorry if this is off-putting.)

Answer by Eric Peterson for What does Rng^{op} look like?

2011-02-09T00:46:14Z

Here is a too-serious answer to your question, along with answers to a couple questions I think you should be asking:

The category you're interested in, as noted by others, is the category of coalgebras / corings, which is emphatically not the opposite category of rings --- but we're going to see exactly what's different between the two in the nicest case. To start things off, here's a definition of a coalgebra so that we're all on the same page: an $R$-coalgebra is an $R$-module $A$ together with morphisms $\Delta: A \to A \otimes_R A$ and $\epsilon: A \to R$ satisfying

Coassociativity: $(\Delta \otimes \operatorname{id}) \circ \Delta = (\operatorname{id} \otimes \Delta) \circ \Delta$,
Counitality: $(\epsilon \otimes \operatorname{id}) \circ \Delta = (\operatorname{id} \otimes \epsilon) \circ \Delta = \operatorname{id}$ after identifying $R \otimes_R A$ and $A \otimes_R R$ with $A$.

Coalgebras are familiar objects in algebraic topology, and you've already found the biggest source of them. Suppose you have a cohomology theory $E$ whose coefficient ring is a (graded) field. For any space $X$, the constant map $X \to \mathrm{pt}$ induces a map in homology $E_* X \to E_*$ which serves as the counit for $E_* X$. Toward a comultiplication, we have maps $E_* X \xrightarrow{\Delta} E_* (X \times X) \leftarrow E_* X \otimes_{E_*} E_* X$ , but asking for the right-hand map to have an inverse is the same as asking for a Kunneth isomorphism. This only happens under restrictions on $X$ or restrictions on $E$ --- such as when $E_*$ is a field, which is one reason we requested that.

Now, let's lay our cards on the table and just announce some dualities we see in front of us:

There's your opposite category $\mathsf{Rings}^{op}$, which is dual to rings in the sense that $(\mathsf{Rings}^{op})^{op}$ is equal to $\mathsf{Rings}$.
In the language of the comments below, coalgebras are Eckmann-Hilton duals of algebras. This is really a statement about how we produced the definition above: we took the definition for an algebra, and we flipped all the arrows around.
There's also the notion of linear algebraic duals: $V^\vee = \operatorname{Hom}_k(V, k)$. To avoid some very serious technicalities, we'll want to work in the nicest, most familiar setting possible: modules of finite rank over a ground ring that's a field.

Now, we would like to compare these three ideas. There is another category of interest floating around: the category of $k$-algebras has an associated category of presheaves $\widehat{\mathsf{Algebras}_k}$ $=$ $\operatorname{Functors}(\mathsf{Algebras}_k, \mathsf{Sets})$ which receives a map $\mathsf{Algebras}_k^{op} \to \widehat{\mathsf{Algebras}_k}$ described by the left side of the $\operatorname{Hom}$-functor: $X \mapsto \operatorname{Hom}(X, -)$. This assignment, called the Yoneda embedding, is a functor into a cocomplete category which is an equivalence onto its image and whose image is codense --- these are consequences of the Yoneda lemma. That the target of the Yoneda embedding is cocomplete makes it a much nicer category to play around in, and so it's worth considering what this embedding's use is.

I claim there's a relation between the category of $k$-coalgebras and $\widehat{\mathsf{Algebras}_k}$ . Again, to make linear algebra behave nicely, we need to encode finiteness restrictions into our setup, and to make that happen we'll turn to "$k$-formal schemes". The classical $\operatorname{Spec}$ construction in algebraic geometry also gives a contravariant functor off the category of $k$-algebras which is an equivalence onto its image. Rather than fussing with what a Zariski spectrum is, since we're just playing around with categories, I will instead take the Yoneda embedding to be my definition of $\operatorname{Spec}$ and the presheaf category to be something dimly, vaguely, sorta like the category of schemes. Representable presheaves (i.e., those in the image of $\operatorname{Spec}$) are called affine schemes. Plenty of constructions from algebraic geometry transfer almost without comment; for instance, defining $\mathbb{A}^1 = \operatorname{Spec} \mathbb{Z}[x]$, we recover the functor $\mathcal{O}(X) = \operatorname{Hom}_{\widehat{\mathsf{Algebras}_k}}(X, \mathbb{A}^1)$, which in the case of an affine $X = \operatorname{Spec}(A)$ gives $\mathcal{O}(\operatorname{Spec} A) \cong A$.

A scheme $X$ will be called finite if it is $\operatorname{Spec}$ of an algebra of finite dimension as a $k$-module. These, too, are in ample supply in algebraic topology. If $X$ is a compact pointed space, then the algebra $H^* X$ will be finite in the sense we need. Of course, algebraic topology gets done on more than compact spaces, so we need to broaden our perspective a little bit: we can ask instead that $X$ be compactly generated, so that if $X_\alpha$ denotes the collection of compact subsets of $X$ directed by inclusion, we then have $X = \operatorname{colim} X_\alpha$. In the case that $X$ is a CW-complex, it is sufficient to take $X_\alpha$ to be just the finite subcomplexes of $X$. We might then be interested in the scheme $X_E := \operatorname{colim} \operatorname{Spec} E^* X_\alpha$. Such a scheme which occurs as the colimit of a directed system of finite $k$-schemes is called a $k$-formal scheme.

So now we have a suitable notion of a 'finite' scheme that still captures all our interesting (and frequently large) cohomology rings. Here's the comparison I promised early on: the functor taking a formal scheme $\operatorname{colim} \operatorname{Spec} A_i$ to the $k$-coalgebra $\operatorname{colim} A_i^\vee$ is an equivalence of categories. There's almost nothing to say because the sea of definitions we've made take care of most everything, but there is one key lemma: given a $k$-coalgebra, you need to know that you can write it as the colimit of finite $k$-coalgebras. The exact lemma that gets used is: if $E$ is a finite dimensional subspace of a $k$-coalgebra $A$, then there exists a subcoalgebra $F \subseteq A$ which is finite as a $k$-vector space and which contains $E$ (i.e., $E$ can be finitely enlarged so that it becomes closed under comultiplication). If you push around elements a bit you'll see that that's the case (and that this requires working over a ground field). Then, straight after, here's the second big assertion: the assignments $X \mapsto E_* X$ and $X \mapsto X_E$ are equivalent under the above equivalence of categories.

Now, this construction is delicate, and the limitations are not meant to be taken lightly! You need a ring structure on the underlying (co)homology theory to even dream of having products, you need Kunneth isomorphisms to make sense of the coalgebra structure, you need a coefficient field to have good linear-algebraic duality, and there are still potential problems with supercommutativity that we haven't addressed. But, when all the stars align and God smiles on us, this is what the coalgebra structure on homology is supposed to mean: it's another presentation of the formal scheme associated to the (perhaps more familiar) ring structure on cohomology.

To counterbalance that caveat, that's not to say that this point of view is not immensely useful. Here are some sample applications:

All of chromatic homotopy: $\mathbb{C}\mathrm{P}^\infty$ ($ = B\mathrm{U}(1) = \mathrm{Gr}_1$) carries the structure of an $H$-group, and there are a whole sea of cohomology theories $E$, called complex-orientable, for which $\mathbb{C}\mathrm{P}^\infty_E$ is (noncanonically) isomorphic to $\hat{\mathbb{A}}^1 = \operatorname{colim} \operatorname{Spec} E^*[x] / \langle x^n \rangle$, an object which behaves a lot like an infinitesimal one-dimensional Lie group. This "formal Lie group" carries an immense amount of information about the cohomology theory $E$, and the "space" of available formal Lie groups carries an immense amount of information about stable homotopy theory as a whole.
There are a variety of partial theorems in the following spirit: if $E$ and $F$ are complex oriented cohomology theories and $F$ is represented by the spaces $F_k$ in the sense that $F^k X = [X, F_k]$, then $\bigoplus_k E_* F_k$ behaves like $\operatorname{Hom}(\mathbb{C}\mathrm{P}^\infty_E, \mathbb{C}\mathrm{P}^\infty_F)$ . Goerss has shown this when $E = H\mathbb{F}_p$ and $F$ satisfies a certain condition on $F_* \Omega^2 S^3$ (which is also satisfied for a class of complex-oriented spectra called Landweber exact). (Addendum: Goerss spends a lot of time setting up a "super" version of Dieudonne modules, which is meant to address in part issues with supercommutativity ignored/avoided here.)
Armed with an ample supply of Kunneth isomorphisms, Ravenel and Wilson, the progenitors of the idea above, computed these coalgebras in the cases where $E$ and $F$ ranged in $K(n)$ (Morava $K$-theory), $E(n)$ (Johnson-Wilson theories), $BP$ (Brown-Peterson theory), and $H\mathbb{Z}/p^j$ (singular theories / Eilenberg-Mac Lane spaces). For instance, one can define the free (supercommutative) algebra on a group object in the category of $k$-formal schemes, and it turns out that $H_*(K(\mathbb{Z}/p^j, q); \mathbb{F}_p)$ is the free alternating algebra in this sense on the formal group scheme $B\mathbb{Z}/p^j_{H\mathbb{F}_p}$ . A similar statement can be made for $K(n)_* K(\mathbb{Z}/p^j, q)$.
The above ideas have stable versions too: the homology of spectra $E_* F$ is to be thought of as the scheme of isomorphisms $\operatorname{Iso}(\mathbb{C}\mathrm{P}^\infty_E, \mathbb{C}\mathrm{P}^\infty_F)$, and so, for instance, the dual of the Steenrod algebra, an object of classical interest, can be thought of as $\mathcal{O}$ of the automorphisms of a particular formal Lie group $\hat{\mathbb{G}}_a$ . Relatedly, the statement that the dual Steenrod algebra coacts on the homology coalgebras $H_* X$ is straightened out in the category of formal schemes by saying that $\operatorname{Aut} \hat{\mathbb{G}}_a$ acts on the formal scheme $X_{H\mathbb{F}_2}$ . This has also received classical interest, though not in this language: the final part of Thom's thesis on calculations in the real bordism ring amount to showing that this $\operatorname{Aut} \hat{\mathbb{G}}_a$ action on the homology of the real bordism spectrum is free.
Finally, this is an example of a broader phenomenon: often the schemes associated to rings have enlightening interpretations in moduli theoretic terms. A great many classical objects in stable homotopy theory lead double lives in this framework as moduli spaces, and the geometry of the moduli space frequently informs us on how the original topological objects behave.

Anyway, this is all to say that you should definitely care deeply about the coalgebra structure on homology, as it's one way to get into formal schemes, where everything is interesting and magical and great.

Answer by Eric Peterson for Spectral sequences: opening the black box slowly with an example

2010-11-06T06:59:33Z

This is more than you bargained for, but it's too good an opportunity to pass up plugging a couple cool and readable papers with impact beyond containing computable spectral sequences. In the 1980s Ravenel and Wilson famously used Hopf rings to compute some extraordinary homologies of a variety of families of infinite loopspaces. In the specific case where the loopspaces are Eilenberg-Mac Lane spaces, they used a bar spectral sequence (which arises as a filtration spectral sequence), together with Hopf ring information, to compute $E_* K(G, *)$ for various $E$ and $G$. One of the cool features of their work is that everything involved is explicit and identifiable.^† You might try:

Doug Ravenel and Steve Wilson, The Morava $K$-theories of Eilenberg-Mac Lane spaces, published in 1980 in the American Journal of Mathematics, vol. 102, no. 4, pages 691-748

for a mildly complicated but very rewarding example. Or, the algebras $H_*(K(\mathbb{Z}/p, *); \mathbb{Z}/p)$ (along with a million other things!) are computed in Wilson's exceptionally nice book

Steve Wilson, Brown-Peterson Homology: An Introduction and Sampler, published in 1980, no. 48 in the CBMS series of conference notes,

which uses all the same machinery as the Morava $K$-theory paper but employs singular homology and tells you about some algebras which you already understand. The familiarity of these two things will probably ease digestion of the ideas.

^† -- Well, this is kind of a lie, since they argue the existence or nonexistence of some of their differentials by knowing what the $E^\infty$ page must look like together with some kind of sparseness of the $E^2$ page. However, the way they build the spectral sequence does actually give you a formula for the differentials, and it's certainly possible, if difficult, for you to make the relevant calculations once you read their argument so you know whereabouts to look. At small primes (and small heights, in the Morava K-theory paper), this is probably even accessible.

Identifying the stacks in Devinatz-Hopkins-Smith

2010-09-29T06:10:14Z

I read the Devinatz-Hopkins-Smith proof of the nilpotence conjectures last year, and while I followed along sentence to sentence I don't think I understood much of the motivating reasons for why what they did was a sensible mode of proof. I intend to go back and figure some of these conceptual pieces out; this question is a step in that direction.

The proof of smash nilpotence is, on the face of it, accomplished by a sequence of interpolations. The original statement is:

(Smash nilpotence:) If a map off a finite spectrum induces the zero map on $MU$-homology, then the map is in fact smash nilpotent.

Without any hassle, they reduce this to the following statement:

(Special case of Hurewicz nilpotence:) Suppose $\alpha$ is an element of the homotopy $\pi_* R$ of an associative ring spectrum $R$ of finite type. If $\alpha$ is in the kernel of the map $\pi_* R \to MU_* R$ induced by smashing $R$ with the unit map $S \to MU$, then $\alpha$ is nilpotent.

To address this question, they interpolate between the sphere spectrum and $MU$ in two ways. First, they produce a sequence of spectra $X(n)$, each given by the Thom spectrum associated to the composite $\Omega SU(n) \to \Omega SU \to BU$, sort of a restriction of the Bott map. The colimit of the $X(n)$ is $MU$, and hence $X(N)_* \alpha$ is zero for some sufficiently large $N$, where $\alpha$ is considered as a map $S^t \to R$. Since $X(1)$ is the sphere spectrum, the new goal is to show that nilpotence in $X(n+1)$-homology forces nilpotence in $X(n)$-homology for any $n$, then to walk down from $X(N)$ to the sphere spectrum. To move between $X(n+1)$ and $X(n)$, they interpolate between these spectra by pulling $X(n+1)$ apart using the filtered James construction; this results in an increasing sequence of $X(n)$-module spectra $F_{n, k}$ satisfying $F_{n, 0} \simeq X(n)$ and converging to $X(n+1)$ in the limit.

The rest of the argument falls into two pieces:

1) If $\alpha: S^t \to R$ is nilpotent in $X(n+1)$-homology then it induces the zero map in $F_{n,p^k-1}$-homology for some large $k$ --- that is, our argument continues somewhere in the approximating tower between $X(n+1)$ and $X(n)$.

2) If it induces the zero map in $F_{n, p^k-1}$-homology it also induces the zero map in $F_{n, p^{k-1}-1}$-homology.

To prove part 1, they investigate the $X(n+1)$-based Adams spectral sequence $$\mathrm{Ext}^{*, *}_{X(n+1)_* X(n+1)}(X(n+1)_*, X(n+1)_* F_{n, p^k-1} \wedge R) \Rightarrow (F_{n, p^k-1})_* R.$$ The key is the existence of vanishing lines in these spectral sequences, where the slope of the vanishing line can be made small by making $k$ large. In order to establish these vanishing lines, they perform a sequence of approximations, finishing with spectral sequences with the following $E_2$ terms: $$\mathrm{Ext}^{*, *}_{\mathbb{F}_p[b_n]}(\mathbb{F}_p, \mathbb{F}_p\{1, \ldots, b_n^{p^k-1}\}),$$ $$\mathrm{Ext}^{*, *}_{\mathbb{F}_p[b_n^{p^k}]}(\mathbb{F}_p, \mathbb{F}_p).$$

My question is: Is there a geometric interpretation for the stacks associated to the Hopf algebroids above? ( --or any of the other Hopf algebroids involved which I haven't listed.) Or: what's the geometric content of Part II of D-H-S?

For instance, it's well-known that there's a spectral sequence computing the homotopy groups of real K-theory whose input corresponds to the moduli stack of quadratics and translations. This stack is supposed to parametrize the available multiplicative groups over some non-algebraically closed field, which provides some geometric insight into the problem. I'd like to know if there's some kind of geometry that corresponds to the controlling stacks that sit at the bottom of the D-H-S argument.

Note: Of course, a positive answer to this question as phrased might not mean much. The process that D-H-S uses to reduce to this much smaller Ext calculation is an extremely lossy one with the very clear intention of just getting at the existence of a vanishing line. The geometry of these bottom stacks may have very little to say about the geometry of the stacks we started with.

Answer by Eric Peterson for Why is homology not (co)representable?

2009-10-24T15:40:38Z

While it's true that there are lots of internal things that a corepresentable homology functor wouldn't support, I think it's also enlightening to see that you wouldn't get the nice sorts of dualities that homology and cohomology theories have. After all, we've already agreed that cohomology theories ought to be somehow representable, so maybe we should start there. Instead of using stable maps $X \to E_n$ to produce $n$-degree $E$-cohomology classes of $X$, you can think of these instead as elements in the stable homotopy groups $\pi_{-*}^S F(X, E)$ , where $F(X, E)$ denotes the function spectrum of maps $X \to E$. This presentation makes the right choice for defining $E$-homology somehow much more obvious: the functor $F(X, -)$ has an adjoint, called the smash product (this is the whole point of the smash product -- it plays the role of "tensor product" for spaces!), and so for homology we think about maps $S^n \to E \wedge X$ instead. That homology and cohomology are not (usually) exact duals in a linear algebraic sense is somehow measuring the twist introduced by this adjunction. This does actually turn out to be the right definition for homology; (extraordinary) homology theories in the traditional sense are in fact modeled by functors of the form $\pi_*^S (E \wedge -)$.

This construction has a number of attractive features -- for instance, it means that we can (under some flatness and ringy conditions) think about "homology cooperations" associated to a spectrum, and they look like $E_* E$, a pleasant mirror of cohomology operations living in $E^* E$. We also always get a pairing $E^* X \times E_* X \to E_* E$ of cohomology and homology classes that lands in homology operations, by composing as $S^n \to E \wedge X \to E \wedge E$. (This pairing even gets used occasionally, though I'd be hard-pressed to come up with an obvious citation.)

For the most familiar homology theory, singular homology with $\mathbb{Z}/p$-coefficients, this flatness business does hold, the operations and cooperations even turn out to be $\mathbb{Z}/p$-vector space duals, and the coaction and action line up in the way you'd expect from Milnor's work.

(This belongs as a comment on Lawson's answer, I think, but it looks like I'm too new here to make that happen.)

Answer by Eric Peterson for when cup product is a zero homomorphism

2010-08-04T15:17:31Z

13.66 in Switzer's Algebraic Topology: Homotopy and Homology. The idea is to use the fact that $\Sigma X$ decomposes into two copies of $CX$, say $A$ and $B$, glued along the common boundary of $X$. For any two cohomology classes $x$ and $y$ in $\tilde{E}^* \Sigma X$, you can uniquely pull $x$ back to a class $x'$ on the relative pair $(\Sigma X, A)$ and $y$ back to a class $y'$ on $(\Sigma X, B)$. Cupping is natural w.r.t the two relative inclusions $i_A: (\Sigma X, {x_0}) \to (\Sigma X, A)$ and $i_B: (\Sigma X, {x_0}) \to (\Sigma X, B)$, and so you get the calculation $x \smile y = i_A^*(x') \smile i_B^*(y') = i^*(x' \smile y')$ , where $i: (\Sigma X, {x_0}) \to (\Sigma X, \Sigma X)$ is another relative inclusion and $x' \smile y'$ a class on the pair $(\Sigma X, \Sigma X)$ --- but that guy has trivial reduced cohomology.

Comment by Eric Peterson

2013-03-13T16:16:42Z

It doesn't answer what you asked, but I thought you might enjoy this answer if you hadn't seen it already: mathoverflow.net/questions/51905/… .

Comment by Eric Peterson

2013-01-15T20:47:19Z

This is also Lemma 6.2 in Toda's On spectra realizing exterior parts of the Steenrod algebra (sciencedirect.com/science/article/pii/…), which you might find helpful. (I don't know if this proof is any different from the one in the Toda reference you mention, which I don't have available.)

Comment by Eric Peterson

2012-12-22T22:06:24Z

It's maybe worth noting that that pattern doesn't continue step-by-step. Moving from $B\mathrm{Fivebrane}$ to the next stage in the Whitehead tower is not described by Pontryagin classes. This isn't a very exciting thing to point out, though; for almost identical reasons, lifting unoriented real vector bundles to oriented ones and then to spin ones is also not controlled by Pontryagin classes.

Comment by Eric Peterson

2012-12-18T18:12:23Z

@Akhil: Yes, good idea, let's chat about this in person. The comodule-spectra-for-$\Sigma^\infty_+ \Omega^2 S^{2m+1}$ stuff is precisely what I find most resistant --- I think the core of it is that I have no idea what geometric object should be attached to $H_* \Omega^2 S^{2m+1}$. Ravenel has a paper about this at math.rochester.edu/people/faculty/doug/mypapers/… (along with various others in the citations), which is intriguing but not so enlightening.

Comment by Eric Peterson

2012-12-18T04:44:27Z

This is a very nice answer. I'd make one additional remark: nevermind explaining everything away using formal groups, it's very difficult even to phrase the algebraic component of D-H-S's second half in that language. Neil Strickland has put substantial effort into this, and I've tried my hand a few times, but it's extremely stubborn. There are a lot of things in and around this proof's mechanics that I think are super interesting exactly because of this stubbornness. The proof of the nilpotence theorems is a real gem of thought that, decades later, doesn't feel completely demystified.

Comment by Eric Peterson

2012-12-17T23:49:18Z

There's Tyler Lawson's answer to this question: mathoverflow.net/questions/40432/… .

Comment by Eric Peterson

2012-12-10T18:41:40Z

If I may drag stable homotopy into it: the map assigning a stable spherical bundle to a stable vector bundle can be realized as a map $BO \to BGL_1 S$. Then, $BGL_1 S \simeq \Omega^\infty \Sigma^\infty S^0$, so in principle it should be easy to name a few such bundles: pick any element of $\pi_*^S$ which is guaranteed, for reasons of order, not to be in the image of this first map (since $\pi_m BO$ contains $2$-torsion at most). However, writing down geometric presentations of such bundles may be hard, and as these are stable statements, they will at best translate unstably only in a range...

Comment by Eric Peterson

2012-12-02T01:41:58Z

Actually, I am a little surprised that you can make this characterization in terms of binomials and trinomials in the Steenrod algebra. The statement is true, but I fear it may be an accidental isomorphism rather than an instance of a natural construction. Caveat lector.

Comment by Eric Peterson

2012-11-27T19:47:43Z

@Akhil: Rather, you mean that a formal group of height $n$ doesn't admit infinitesimal homomorphisms to the formal additive group.

Comment by Eric Peterson

2012-11-01T19:41:42Z

I don't know a reference, but I do know this has a name: ncatlab.org/nlab/show/cartesian+monoidal+category .

Comment by Eric Peterson

2012-10-12T17:13:34Z

@Tyler: I did! That morning he sent me a copy by email to make sure I didn't miss it -- good thinking on his part. In reply, I sent him a copy of the brainstorm I sent you earlier this summer, and we're both very excited about all of it.

Comment by Eric Peterson

2012-10-07T21:47:28Z

@Charles: Ahh, OK, I see some of your point. You can build an $E(R)$-CSS converging to the $E(R)$-Adams SS's $E_2$-page via the same sexseqs as above. This produces the $(r < R)$-truncated CSS, avoids my confusion by keeping $Y = S$, and points out that a crucial component is flatness/Landweber-ness of $E(R)$. You can also get the $(r = R)$-truncated CSS from the $E_R^\vee$-CSS. That's enough to know what I'll say in the notes, but I'm still curious about how or whether $M_R$ belongs in this picture. A $BP$-CSS for $M_r S$ would require some flatness of $BP_* M_r S$ that I doubt we have...?

Comment by Eric Peterson

2012-10-07T20:58:35Z

@Drew: I've opened it many times, but I still haven't properly read it; it's a very dense paper! That spectral sequence probably should have been mentioned above; it would also be nice to know what happens when using $BP_*$ rather than $\pi_*$ on the tower of fibrations $L_{E(r)} \mathbb{S} \to L_{E(r−1)} \mathbb{S}$ and whether the resulting spectral sequence is connected to the CSS.

Comment by Eric Peterson

2012-08-19T16:03:36Z

enclosing your tex in backquotes (like `$ and the mirror on the other side) is often enough help for the parser to fix its errors.

Comment by Eric Peterson

2012-07-22T14:21:50Z

Sure thing. It's a bit of a silly name, certainly easier for humans to remember than for Google.