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.

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