Maps between iterated loop spaces (Q806102)

An obstruction theory is developed which facilitates the computation of Dyer-Lashof operations in the homology of iterated loop spaces. If \(f:X\to K\) is an \(h_{n-1}\)-map between \(H_ n\)-spaces, then the \(h_ n\)-deviation of f, \(h_ n(f)\), lies in the group \([X\wedge X,\Omega^ nK]\). It satisfies that \(h_ n(f)=0\) if and only if f is an \(h_ n\)- map. If K is a mod 2 Eilenberg-MacLane space, then \(h_ n(f)\) is related directly to the Dyer-Lashof operations in \(H_*(X)\). Examples of nontrivial \(h_ n(f)\) are given.