Deconstructing Hopf spaces

Abstract

We characterize Hopf spaces with finitely generated cohomology as an algebra over the Steenrod algebra. We "deconstruct" the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. We give a precise description of homotopy commutative H-spaces in this setting.

Noetherian loop spaces

“…The most basic examples of p-Noetherian groups are p-compact groups and Eilenberg-Mac Lane spaces K(Z/p r , 1) and, K(Z ∧ p , 2). In the spirit of our deconstruction results for H -spaces ( [9]), we show that these are the basic building blocks for all p-Noetherian groups. Theorem 1.9.…”

“…For example QH * (K(Z/p, m); F p ) lies in U m−1 for any m ≥ 1 ([9, Example 2.2]). Moreover we observed in [9,Lemma 7.1] that if H * (BX; F p ) is finitely generated as an algebra over A p , then QH * (BX; F p ) must be finitely generated as a module over A p , and hence lies in U k for some k. We remark that the condition that H * (X; F p ) be a Noetherian F p -algebra is equivalent to saying that H * (X; F p ) is finitely generated as an algebra over A p and that the unstable module QH * (X; F p ) of indecomposable elements lies in U 0 . Our second result shows that the cohomology of the classifying space of a p-Noetherian group is as small as expected in terms of the Krull filtration.…”

Homotopy Exponents for Large H-Spaces

“…A connected H-space such that H * (X; F p ) is finitely generated as an algebra over the Steenrod algebra can always be seen as the total space of an H-fibration F → X → Y where Y is an H-space with finite mod p cohomology and F is a p-torsion Postnikov piece whose homotopy groups are finite direct sums of copies of cyclic groups Z/p r and Prüfer groups Z p ∞ , [3,Theorem 7.3]. This is a fibration of H-spaces and H-maps, so that we obtain another fibration F Corollary 1.3.…”

“…The methods we use are based on the deconstruction techniques of the third author in his joint work with Castellana and Crespo, [3]. Our results on homotopy exponents should also be compared with the computations of homological exponents done with Clément, [4].…”

Homology exponents for $H$-spaces

“…Therefore there exists by [CCSa,Theorem 7.3] a simply connected H-space Y = P BZ/2 X with finite mod 2 cohomology and a series of principal…”