Non-simply connected \(H\)-spaces with finiteness conditions (Q2725001)

The main result of the article is the following structure theorem:NEWLINENEWLINENEWLINETheorem: Let \(X\) be a connected mod \(pH\)-space such thatNEWLINENEWLINENEWLINE(F1) \(H^*(X; \mathbb{F}_p)\) is of finite type;NEWLINENEWLINENEWLINE(F2) \(H^*(X; \mathbb{F}_p)\) has finite transcendence degree;NEWLINENEWLINENEWLINE(F3) the module of indecomposables \(QH^*(X; \mathbb{F}_p)\) is locally finite as module over the Steenrod algebra.NEWLINENEWLINENEWLINEThen there is a principal \(H\)-fibration \(BP\to X\to F(X)\to B^2P\) where \(P\) is an abelian \(p\)-toral group and \(F\) is the composite of the \(B\mathbb{Z}/p\)-nullification functor with the \(p\)-completion functor. Hence \(X\) is a \(B\mathbb{Z}/p\)-null space, an Eilenberg-MacLane space \(K(\widehat \mathbb{Z}_p,2)\), \(K(\mathbb{Z}_{p^r},1)\) an extension of those. \(H\)-spaces with Noetherian \(H^*(X;\mathbb{F}_p)\) satisfy the assumptions of the theorem. In this case \(F(X)\) is a mod \(p\) finite \(H\)-space. For the proof the authors consider the universal cover NEWLINE\[NEWLINE\widetilde X\to X\to B\pi_1(X) \tag{*}NEWLINE\]NEWLINE They prove that \(\widetilde X\) also satisfies the finiteness conditions (F1), (F2), (F3) and then use the structure theorems for simply connected \(H\)-spaces proved by the first two authors in earlier papers. Fiberwise localization combines the \(H\)-fibration (*) with the \(H\)-fibration of the structure theorem for \(\widetilde X\). An analysis of \(BP\)-principal fibrations, which is of some separate interest, and of properties of \(\widetilde X\) establishes the result.