Chromotopy

Devinatz--Hopkins--Smith I

I’ve been chewing my way through Devinatz, Hopkins, and Smith’s landmark paper from the early 1980s, in which they prove this amazing fact that the complex cobordism spectrum is exactly the right tool for thinking about nilpotence and periodicity in stable homotopy theory, previously conjectured by Ravenel. An important maxim of homotopy theory is that Free Objects Are Complicated, which is really what sets algebraic topology apart from algebra – in algebra, free objects are relatively easily understood objects and other algebraic structures can be systematically studied in terms of subquotients of free things. This is emphatically not the case in homotopy theory; the most basic invariant in homotopy theory are the homotopy groups, the homotopy groups of the spheres are only known through a very narrow range, and we only arrived at those through sixty years’ worth of incredible work. The most basic structural fact about the stable homotopy groups of spheres come with a ring structure. (Note: we’ll work stably whenever necessary without comment. For the uninitiated, working stably effectively means that we have a desuspension functor available to use.)

A famous result of Nishida from the early 70s demonstrated that every element of $\pi_n \mathbb{S}$ is nilpotent for $n > 0$. To paraphrase Hopkins, the ring structure on $\pi_* \mathbb{S}$ can be understood in three ways: the ring structure lives at the spectrum level on $\mathbb{S}$, but is also given by composition of maps, or alternatively by the smash product of maps. D-H-S proves that all three of these approaches generalize up from the sphere spectrum, in the following precise senses:

Theorem 1

• For a ring $R$, the stable MU-Hurewicz map $h_{MU}: \pi_* R \to MU_* R$ given by smashing the unit $\eta: \mathbb{S} \to MU$ with R produces nilpotent elements.
• For F a finite spectrum, if a map $f: F \to X$ induces the zero map in MU-homology, then f is smash nilpotent.
• Let $\cdots \to X_n \xrightarrow{f_n} X_{n+1} \xrightarrow{f_{n+1}} X_{n+2} \to \cdots$ by a sequence of spectra such that $X_n$ is (mn+b)-connected for some m and b. If each $f_n$ induces the zero map in MU-homology, then the homotopy colimit of the tower is contractible.

It’s fairly immediate that the statement about smash nilpotency implies the statement about the stable MU-Hurewicz map; some categorical formalities and a good definition of the smash product are all that’s needed. The tower statement is considerably more tricky, and I won’t address it here – instead, let’s work through why the smash nilpotence theorem is true.

To start, we can replace $f: F \to X$ by a map $f': \mathbb{S} \to DF \wedge X$, where DF is the Spanier-Whitehead dual of the finite spectrum F, so we may as well work with f’ from the start and assume $F = \mathbb{S}$. Furthermore, we can replace X with a James-like construction $\bigvee_{n=0}^\infty X^{\wedge n}$, which is a ring under concatenation. By investigating the definition of the stable MU-Hurewicz map we see it suffices to show

Reduction 1 : If we have a connective, associative ring R of finite type and an element $\alpha \in \pi_* R$ in the kernel of the stable MU-Hurewicz map $\pi_* R \to MU_* R$, then $\alpha$ is smash nilpotent.

The next thing to do is to interpolate between MU and $\mathbb{S}$ so that we can handle this problem inductively. MU is defined as the Thom spectrum of BU, BU is weakly equivalent to $\Omega SU$ by the Bott periodicity map, and $SU$ comes with a sort of filtration by the spaces $SU(n)$. So, define $X(n)$ to be the Thom spectrum of the bundle classified by

$$\Omega SU(n) \to \Omega SU \xrightarrow{\sim} BU.$$

The natural maps $X(n) \to X(n+1) \to MU$

are compatible with the ring structures, and MU is in fact the limit of the X(n) tower.

With these rings in hand and the observation that $X(1) = \mathbb{S}$, we make the last meaty reduction:

Reduction 2: Let R be a connective, associative ring of finite type, and let $\alpha \in \pi_* R$. If $X(n+1)_* \alpha$ is nilpotent, then so is $X(n)_* \alpha$.

So we’ve gone from having to make a gigantic leap from MU to $\mathbb{S}$ to having to make a bunch of significant leaps from X(n+1) to X(n). There’s a well-studied fibration

$$SU(n) \to SU(n+1) \xrightarrow{p} S^{2n+1},$$

and the space $\Omega S^{2n+1}$ is also well studied, so let’s make use of some of the available technology given to us by history. For any space X, we may produce the free monoid JX on X, and James famously found that JX was weakly equivalent to $\Omega \Sigma X$. The space JX comes with a filtration $J_r X$ consisting of words in the monoid of length at most r, and we can make use of this filtration to build stepping stones between the X(n). Namely, set

$$F'_k = \mathrm{holim}(J_k S^{2n} \to \Omega S^{2n+1} \xleftarrow{\Omega p} \Omega SU(n+1),$$

and then let $F_k$ be the Thom spectrum of the composite

$$F'_k \to \Omega SU(n+1) \to \Omega SU \to BU;$$

we think of $F_k$ as being the part of $X(n+1)$ sitting over the kth stage of the filtered James construction. The $F_k$ clearly do give an exhaustive filtration of $X(n+1)$, but even more is true – this construction is sufficiently natural that the $F_k$ are in fact $X(n)$-modules, as $F_0 = X(n)$.

The proof of the section reduction now rests on these stepping stones $F_k$, which we’ll go through in two later posts. We will show (and explain)

First half : If $X(n+1)_* \alpha$ is nilpotent, then $F_{p^k-1} \wedge \alpha^{-1} R$ is contractible for sufficiently large k.

Second half : $F_{p^{k+1}-1}$ is Bousfield-equivalent to $F_{p^k-1}$, powering the induction down to $F_0 = X(n)$ that we need.

… but that’s enough for now.

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