Higher homotopy commutativity of \(H\)-spaces with finitely generated cohomology.
type: Item
instance of: scholarly article
title: Higher homotopy commutativity of \(H\)-spaces with finitely generated cohomology.
zbMATH Open document ID: 1053.55005
author: Yusuke Kawamoto
published in: Pacific Journal of Mathematics
publication date: 2002-10-22T00:00:00Z
review text: 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.
reviewed by: Corina Mohorianu
Mathematics Subject Classification ID: 55P45
zbMATH DE Number: 1818987
zbMATH Keywords: \(A_n\)-space
zbMATH Keywords: \(C_p\)-space
MaRDI profile type: Publication
full work available at URL: pjm.2002.204.145
OpenAlex ID: W2019100338
DOI: 10.2140/PJM.2002.204.145
description: scientific article; zbMATH DE number 1818987