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 1 There 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.
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
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 .