Duality in BP<n> (co)homology

TL;DR

This paper establishes a duality isomorphism in BP<n>-cohomology for locally finite E_*(X), linking cohomology to a shifted and dualized version of homology, inspired by prior work on ku-homology of K(Z/2,2).

Contribution

It proves a new duality theorem connecting BP<n>-cohomology and homology via Pontryagin duality for locally finite modules.

Findings

Established an isomorphism E^*(X) = (E_*(Sigma^{D+n+1}X))^V under certain conditions.

Linked BP<n>-cohomology to shifted and dualized homology modules.

Motivated by and extending previous work on ku-homology of Eilenberg-MacLane spaces.

Abstract

Let E=BP<n> denote the Johnson-Wilson spectrum, localized at p. It is proved that if E_*(X) is locally finite, then there is an isomorphism of right E_*-modules E^*(X) = (E_*(Sigma^{D+n+1}X))^V, where D=Sum |v_i| and M^V=Hom(M,Q/Z) is the Pontryagin dual. This result was motivated by work of the author and W.S.Wilson regarding the 2-local ku-homology and -cohomology of the Eilenberg-MacLane space K(Z/2,2).