On exterior \(A_ n\)-spaces and modified projective spaces (Q1894985)

Stasheff introduced \(A_n\)-spaces as \(H\)-spaces which satisfy an \(n\)-th order higher homotopy associativity condition. This paper shows that for a simply connected \(A_n\)-space whose cohomology is an exterior algebra on \(k\) odd dimensional generators there is a space \(Y\) and a map \(\varepsilon : \Sigma X \to Y\) such that there is a subalgebra of \(H^*(Y; Z/p)\) which is a truncated polynomial algebra and \(\varepsilon^*\) induces an \({\mathcal A}_p\)-module isomorphism between the indecomposable module in this polynomial algebra and the indecomposable module \(QH^*(X;Z/p)\). The result generalizes earlier work in that it does not assume that the exterior generators in \(X\) are \(A_n\)-primitive.