Recall, if G is a discrete abelian group and c(G) is its character group, that is,
c(G) = Hom(G, R/Z),
then Pontrjagin duality provides an isomorphism
Hq(X; c(G)) ≍ c(Hq(X; G)).
The aim of this talk is to describe, without proofs, how this duality can be incorporated into generalized homology and cohomology theories and spectra.
Regarding spectra, we work in Boardman's graded homotopy category of spectra Sh* (). We denote the morphisms of degree q (f:A → SqB) by
If A is a spectrum and X is a CW complex or a spectrum, Aq (X) and Aq(X) denote the homology and cohomology of X with coefficients in A, respectively.
Let g be the category of discrete abelian groups. We ignore the topology on c(G) so that c: g → g. Recall c takes exact sequences into exact sequences and direct sums into direct products. Hence for any spectrum A, c(A* ()) is an additive cohomology theory on the category of all CW complexes. Therefore by , there is a spectrum A′ and a natural equivalence
tA: (A′)*() ≍ c(A*()).
The main results of this talk concern the relation between A and A′ We do not know how to adequately deal with the natural topology on c(A*()). Inconsequence, most of our results on A′ require that we assume πi(A) is finite for all i.
Let S be the sphere spectrum. Choose a spectrum c(S) and a natural equivalence
tS: c(S)*() ≍ c(S*()).
We identify c(S)0(S0) with R/Z via the isomorphism:
c(S)0(S0) ≍ tS c(S0(S0)) ≍ c(Z) = R/Z.