Primitive obstructions in the cohomology of loopspaces

Abstract:

Let $X$ and $X’$ be $H$-spaces. If $f:\Omega X \to \Omega X’$ is an $H$-map then the obstruction to $f$ being a homotopy-commutative map is a subset $\left \{ {{c_2}(f)} \right \} \subset \left [ {\Omega X\Lambda \Omega X;{\Omega ^2}X’} \right ]$. In this paper we prove: $If[f]$ is in the image of the composition \[ \left [ {{P_{k + m}}\Omega X;X’} \right ] \to \left [ {\Sigma \Omega X;X’} \right ]\to \limits ^ \approx \left [ {\Omega X;\Omega X’} \right ],\] then $\left \{ {{c_2}(f)} \right \}$ is in the image of the composition \[ \left [ {{P_k}\Omega X\Lambda {P_m}\Omega X;X’} \right ] \to \left [ {\Sigma \Omega X\Lambda \Sigma \Omega X;X’} \right ]\to \limits ^ \approx \left [ {\Omega X\Lambda \Omega X;{\Omega ^2}X’} \right ].\] Consequently if $\alpha \in {H^n}(\Omega X;{Z_p})$ is an ${A_3}$-class in the sense of Stasheff then each element of $\left \{ {{c_2}(f)} \right \}$ is of the form $\sum {{{c’}_i}} \otimes {c''_i}$ where the ${c''_i}$ are primitive.