Higher homotopy commutativity of \(H\)-spaces with finitely generated cohomology.
In this paper, generalizing results of \textit{C. Broto} and \textit{J. A. Crespo} [Topology 38, No. 2, 353--386 (1999; Zbl 0927.55017)], \textit{J. A. Crespo} [Structure of \(\operatorname{mod} p\) \(H\)-spaces with finiteness conditions. (Cohomological methods in homotopy theory. Proceedings of the Barcelona conference on algebraic topology (BCAT), Bellaterra, Spain, June 4-10, 1998. Basel: Birkhäuser) Prog. Math. 196, 103--130 (2001; Zbl 0990.55004)], the author studies simply connected \(\operatorname{mod} p\) \(H\)-spaces such that the \(\operatorname {mod} p\) cohomology \(H^{*}(X;\mathbb{Z}\diagup p)\) is finitely generated as an algebra. The main theorem shows that if X is a simply connected \(A_{n}\)-space such that the \(\operatorname{mod} p\) cohomology \(H^{*}(X;\mathbb{Z}\diagup p)\) is finitely generated as an algebra, then X is the total space of a principal \(A_{n}\)-fibration with base a simply connected finite \(A_{n}\)-space and fiber a finite product of \(\mathbb{C} P^{\infty}\)'s. As an application of this result, it is shown that if X is a simply connected quasi \(C_{p}\)-space such that the \(\operatorname{mod} p\) cohomology \(H^{*}(X;\mathbb{Z}\diagup p)\) is finitely generated as an algebra, then \(X\) is homotopy equivalent to a finite product of \(\mathbb{C} P^{\infty}\)'s.
scientific article; zbMATH DE number 1818987