Homotopy commutativity of \(H\)-spaces with finitely generated cohomology (Q2731953)

An \(H\)-space which is \(p\)-complete is called a mod \(p\) \(H\)-space. The authors classify all simply connected homotopy associative and homotopy commutative mod 3 \(H\)-spaces \(X\) whose mod 3 cohomology is finitely generated as an algebra. For mod 2 \(H\)-spaces \(X\) a similar classification has been proved by \textit{M. Slack} [A classification theorem for homotopy commutative \(H\)-spaces with finitely generated mod 2 cohomology rings, Mem. Am. Math. Soc. 449 (1991; Zbl 0755.55003)]. A \(C_n\)-space is a monoid whose multiplication \(X\times X\to X\) is an \(A_n\)-map in the sense of Stasheff. The second main result is the classification of connected \(C_p\)-spaces, \(p\) an odd prime, whose mod \(p\) cohomology is finitely generated as an algebra. For the proofs the authors investigate the multiplicative behavior of the natural map \(X\to F_A (X)\), where \(F_A\) is the composite of the \(A\)-localization of Dror Farjoun, \(A\) a space, and the \(p\)-completion functor of Bousfield and Kan. Combining this with results of Broto and Crespo reduces the classification to the case of a finite \(X\), which is either known or established in the paper.