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

**1. A non-example**

To start with, here’s a spectrum which does *not* work: . This is a natural choice because

On the other hand, from the cofiber sequence

we obtain a cofiber sequence

which shows in particular that

in particular, its is isomorphic to , 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 on pointed spaces, or on spectra. An example of a homology theory

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

one has an exact sequence

Given an *injective* abelian group, we can dualize this. Take for example ; we find that for any cofiber sequence as above, there is an exact sequence

because the functor is *exact*. In particular, the functor given by (that is, the dual to stable homotopy) defines a cohomology theory. Applying the Brown representability theorem yields:

Theorem 1There is a spectrum with the property that, for any spectrum , we have a natural isomorphism

The spectrum 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 , it defines a homology theory

and thus a dual cohomology theory

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

as an unwinding of the adjunction (the analog in stable homotopy theory of the tensor-hom adjunction) shows. This definition gives a natural map

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

**3. **

The claim is that is an example of a spectrum such that . Note that

while for . In particular, 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.

In order to see this, we will show first that

In fact, the spectrum clearly has -torsion homotopy groups (namely, the mod homology groups of ). It therefore suffices to show that . Here we consider the Moore spectrum : this is the cofiber of the multiplication by map , and is consequently a torsion spectrum.

But is self-dual under Spanier-Whitehead duality, up to suspension, so that is a suspension of . We are reduced to showing that

But

because it is a theorem that there are no nontrivial maps from to a finite spectrum. (A nice proof can be found in Ravenel’s “Localization with respect to periodic homology theories” paper.) Since 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 -acyclic (i.e., which smash to zero with ) is closed under homotopy colimits and (de)suspensions. It follows that if it contains each , it contains each , and each for a finitely generated torsion group. It follows that the category contains for an *arbitrary* torsion group. Now we have

where each has only torsion homotopy groups and is concentrated in : it is thus an iterated extension of spectra of the form , torsion. We find

As another example, the -local version of provides an example of a spectrum whose Bousfield class is strictly smaller than that of .

January 3, 2013 at 1:00 am

Dear Akhil,

In the final displayed equation of section 1, is there a wedge square missing on the left-hand side?

Regards,

Matt

January 3, 2013 at 9:17 am

Thanks for the correction!