The connective KO theory of the Eilenberg-MacLane space K(Z/2,2)

TL;DR

This paper computes the connective KO-homology and cohomology of the Eilenberg-MacLane space K(Z/2,2) using the Adams spectral sequence, revealing duality relations and new insights into Stiefel-Whitney classes in Spin manifolds.

Contribution

It provides the first detailed computation of ko_*(K(Z/2,2)) and ko^*(K(Z/2,2)), establishing duality and deriving new results on Stiefel-Whitney classes.

Findings

Identified duality between ko-homology and cohomology groups.

Computed explicit ko-homology and cohomology groups for K(Z/2,2).

Deduced new properties of Stiefel-Whitney classes in Spin manifolds.

Abstract

We compute ko_*(K(Z/2,2)) and ko^*(K(Z/2,2)), the connective KO-homology and -cohomology of the mod 2 Eilenberg-MacLane space K(Z/2,2), using the Adams spectral sequence. The work relies heavily on work done several years earlier for the (complex) ku groups by the author and W.S.Wilson. We illustrate an interesting duality relation between the ko-homology and -cohomology groups. We deduce a new result about Stiefel-Whitney classes in Spin manifolds.