Higher homotopy commutativity of \(H\)-spaces and homotopy localizations
A space is an \(AC_n\)-space if it is an \(H\)-space such that the multiplication satisfies both the higher homotopy associativity of the \(n\)-th order and the higher homotopy commutativity of the \(n\)-th order. The main theorem of the paper under review is stated as follows: Let \(p\) be a prime. Then a connected \(AC_p\)-space such that its mod \(p\) cohomology is finitely generated as an algebra over the mod \(p\) Steenrod algebra is mod \(p\) homotopy equivalent to a Postnikov \(H\)-space. Here an \(H\)-space is called Postnikov if its homotopy groups are finitely generated over the \(p\)-adic integers \(\mathbb Z_p^{\wedge}\) and vanish above some dimension. This result is a generalization of Corollary 7.4 of [\textit{N. Castellana, J. A. Crespo} and \textit{J. Scherer}, Invent. Math. 167, No.~1, 1--18 (2007; Zbl 1109.55005)], which is just the case that \(p=2\). To show the above theorem, the author first shows that if \(X\) is an \(AC_n\)-space for some \(n\geq1\), then for any space \(A\), the \(A\)-localization \(L_A(X)\) of \(X\) is an \(AC_n\)-space, and the universal map \(\phi_X\colon X \to L_A(X)\) is an \(AC_n\)-map. As a corollary of the main theorem, the author proves that the \(K(n)_*\)-localization of \(X^{\wedge}_p\) is mod \(p\) homotopy equivalent to the \(\Sigma^nB\mathbb Z/p\)-localization of \(X^{\wedge}_p\) for a connected \(AC_p\)-space \(X\) with finitely generated cohomology algebra over the mod \(p\) Steenrod algebra, where \(K(n)_*\) is the Morava \(K\)-homology theory. In particular, \(X^{\wedge}_p\) is \(K(n)_*\)-local if and only if \(\Omega X^{\wedge}_p\) is \(B\mathbb Z/p\)-local. It is also proved that if a connected \(A_p\)-space for an odd prime \(p\) admits an \(AC_n\)-structure with \(n>(p-1)/2\) and if its mod \(p\) cohomology is finitely generated as an algebra over the mod \(p\) Steenrod algebra, then the space is mod \(p\) homotopy equivalent to a finite product of \((S^1)^{\wedge}_p\)'s, \((\mathbb CP^{\infty})^{\wedge}_p\)'s and \(B\mathbb Z/p^i\)'s \((i\geq1)\) provided that the Steenrod operation \(\mathcal P^j\) for \(j\geq1\) acts trivially on the indecomposable module of its mod \(p\) cohomology.
scientific article; zbMATH DE number 5366234