Let ${X}$ be a connective spectrum with finitely generated homotopy groups. Then the lowest homotopy group in ${\pi_*(X \wedge X)}$ is the tensor square of the lowest homotopy group in ${\pi_*(X)}$: in particular, $X \wedge X$ is never zero (i.e., contractible). The purpose of this post is to describe an example of a nontrivial spectrum ${I}$ with ${I \wedge I \simeq 0}$. I learned this example from Hovey and Strickland’s “Morava ${K}$-theories and localization.”

1. A non-example

To start with, here’s a spectrum which does not work: ${H \mathbb{Q}/\mathbb{Z}}$. This is a natural choice because

$\displaystyle \mathbb{Q}/\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Q}/\mathbb{Z} = 0.$

On the other hand, from the cofiber sequence

$\displaystyle H \mathbb{Z} \rightarrow H \mathbb{Q} \rightarrow H \mathbb{Q}/\mathbb{Z} \rightarrow \Sigma H \mathbb{Z},$

we obtain a cofiber sequence

$\displaystyle H \mathbb{Z} \wedge H \mathbb{Q}/\mathbb{Z} \rightarrow 0 \rightarrow (H \mathbb{Q}/\mathbb{Z})^{\wedge 2} \rightarrow \Sigma H \mathbb{Z} \wedge H \mathbb{Q}/\mathbb{Z}$

which shows in particular that

$\displaystyle (H \mathbb{Q}/\mathbb{Z})^{\wedge 2} \simeq \Sigma H \mathbb{Z} \wedge H \mathbb{Q}/\mathbb{Z};$

in particular, its ${\pi_1}$ is isomorphic to ${\mathbb{Q}/\mathbb{Z}}$, not zero.

2. Brown-Comentz duality

A useful way of producing interesting spectra is the Brown representability theorem. In order to produce a spectrum, it suffices to give a cohomology (or homology, by Adams’s variant of the theorem) theory ${H^*}$ on pointed spaces, or on spectra. An example of a homology theory

$\displaystyle \mathbf{Sp} \rightarrow \mathbf{Ab}$

is given by stable homotopy ${\pi_0}$. This functor has the defining property of a homology theory: that is, given a cofiber sequence of spectra

$\displaystyle X' \rightarrow X \rightarrow X'' \rightarrow \Sigma X,$

one has an exact sequence

$\displaystyle \pi_0 X ' \rightarrow \pi_0 X \rightarrow \pi_0 X''.$

Given an injective abelian group, we can dualize this. Take for example ${\mathbb{Q}/\mathbb{Z}}$; we find that for any cofiber sequence as above, there is an exact sequence

$\displaystyle \hom( \pi_0 X'', \mathbb{Q}/\mathbb{Z}) \rightarrow \hom(\pi_0 X, \mathbb{Q}/\mathbb{Z}) \rightarrow \hom(\pi_0 X',\mathbb{Q}/\mathbb{Z})$

because the functor ${\hom(\cdot, \mathbb{Q}/\mathbb{Z})}$ is exact. In particular, the functor ${\mathbf{Sp} \rightarrow \mathbf{Ab}}$ given by ${\hom( \pi_0(\cdot), \mathbb{Q}/\mathbb{Z})}$ (that is, the dual to stable homotopy) defines a cohomology theory. Applying the Brown representability theorem yields:

Theorem 1 There is a spectrum ${I}$ with the property that, for any spectrum ${X}$, we have a natural isomorphism

$\displaystyle [X, I] \simeq \hom(\pi_0 X, \mathbb{Q}/\mathbb{Z}).$

The spectrum ${I}$ is called the Brown-Comenentz dual of the sphere. One can generalize this construction to dualizing any homology theory to a cohomology theory. That is, given a spectrum ${E}$, it defines a homology theory

$\displaystyle X \mapsto E_0(X) \stackrel{\mathrm{def}}{=}\pi_0( E \wedge X)$

and thus a dual cohomology theory

$\displaystyle X \mapsto \hom(\pi_0(E \wedge X), \mathbb{Q}/\mathbb{Z})$

which we can conclude is representable by a spectrum ${cE}$, called the Brown-Comenentz dual of ${E}$. This is nothing really new: in fact,

$\displaystyle cE = \mathrm{Fun}(E, I)$

as an unwinding of the ${\wedge, \mathrm{Fun}}$ adjunction (the analog in stable homotopy theory of the tensor-hom adjunction) shows. This definition gives a natural map

$\displaystyle E \rightarrow cc E$

which is an equivalence when ${E}$ has finite homotopy groups, by ordinary Pontryagin duality for finite abelian groups.

3. ${I \wedge I}$

The claim is that ${I}$ is an example of a spectrum such that ${I \wedge I = 0}$. Note that

$\displaystyle \pi_0 I = [S, I] = \hom( \pi_0 S, \mathbb{Q}/\mathbb{Z}) = \mathbb{Q}/\mathbb{Z},$

while ${\pi_i I = 0}$ for ${i > 0}$. In particular, ${I}$ is nonconnective; its negative homotopy groups are dual to the stable homotopy groups of spheres.

The goal of this blog post is to describe the proof of:

Theorem 2 ${I \wedge I = 0}$.

In order to see this, we will show first that

$\displaystyle I \wedge H \mathbb{F}_p = 0 . \ \ \ \ \ (1)$

In fact, the spectrum ${I \wedge H \mathbb{F}_p}$ clearly has ${p}$-torsion homotopy groups (namely, the mod ${p}$ homology groups of ${I}$). It therefore suffices to show that ${I \wedge H \mathbb{F}_p \wedge S/p = 0}$. Here we consider the Moore spectrum ${S/p}$: this is the cofiber of the multiplication by ${p}$ map ${S \stackrel{p}{\rightarrow} S}$, and is consequently a torsion spectrum.

But ${S/p}$ is self-dual under Spanier-Whitehead duality, up to suspension, so that ${ I \wedge S/p }$ is a suspension of ${\mathrm{Fun}(S/p, I) = c S/p}$. We are reduced to showing that

$\displaystyle \pi_* ( H \mathbb{F}_p \wedge c (S/p)) = 0.$

But

$\displaystyle \hom(\pi_* ( H \mathbb{F}_p \wedge c (S/p)), \mathbb{Q}/\mathbb{Z}) = [H \mathbb{F}_p, cc S/p]_{-*} = [ H \mathbb{F}_p, S/p]_{-*} = 0,$

because it is a theorem that there are no nontrivial maps from ${H \mathbb{F}_p}$ to a finite spectrum. (A nice proof can be found in Ravenel’s “Localization with respect to periodic homology theories” paper.) Since ${\pi_* ( H \mathbb{F}_p \wedge c (S/p))}$ is a torsion group, it must vanish if its dual does. This proves (1).

The rest of the proof is now “formal.” The class of spectra which are ${I}$-acyclic (i.e., which smash to zero with ${I}$) is closed under homotopy colimits and (de)suspensions. It follows that if it contains each ${H \mathbb{F}_p}$, it contains each ${H \mathbb{F}_{p^n}}$, and each ${HG}$ for ${G}$ a finitely generated torsion group. It follows that the category contains ${HG}$ for ${G}$ an arbitrary torsion group. Now we have

$\displaystyle I = \varinjlim_{n} \tau_{\geq -n} I,$

where each ${\tau_{\geq -n}}$ has only torsion homotopy groups and is concentrated in ${[-n, 0]}$: it is thus an iterated extension of spectra of the form ${HG}$, ${G}$ torsion. We find

$\displaystyle I \wedge I = \varinjlim_n (\tau_{\geq -n} I ) \wedge I = \varinjlim_n 0 = 0.$

As another example, the $p$-local version of $I$ provides an example of a spectrum whose Bousfield class is strictly smaller than that of $H \mathbb{F}_p$.