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Noetherian loop spaces. (English) Zbl 1229.55011
This paper studies properties of loop spaces having Noetherian mod \(p\) cohomology, paying attention to differences between such loop spaces and those with finite mod \(p\) cohomology; here \(p\) is any prime number.
More precisely, a \(p\)-Noetherian group is a triple \((X,BX,e)\) such that the cohomology \(H^*(X; \mathbb F_p)\) is a Noetherian algebra (it is finitely generated as an algebra), \(BX\) is a \(p\)-complete space, and \(e: X\rightarrow \Omega BX\) (where \(\Omega BX\) is the space of loops on \(BX\)) is a weak homotopy equivalence. In this setting, \(X\) is called the Noetherian loop space, and \(BX\) is referred to as the classifying space of \(X\). In particular, \(p\)-compact groups in the sense of W. Dwyer and C. Wilkerson [Ann. Math. (2) 139, No. 2, 395–442 (1994; Zbl 0801.55007)] are \(p\)-Noetherian groups.
The authors study the cohomology of the classifying space \(BX\) and show that it is as small as expected. More precisely, they prove that for any \(p\)-Noetherian group \((X,BX,e)\), \(H^*(BX; \mathbb F_p)\) is finitely generated as an algebra over the Steenrod algebra and, in addition, the module of indecomposable elements, \(QH^*(BX; \mathbb F_p)=\tilde H^*(BX; \mathbb F_p)/\tilde H^*(BX; \mathbb F_p)\cdot \tilde H^*(BX; \mathbb F_p)\), belongs to \(\mathcal U_1\) of the Krull filtration, \(\mathcal U_1\subset \mathcal U_2\subset \cdots\), of the category \(\mathcal U\) of unstable modules over the Steenrod algebra (\(\mathcal U_0\) consists of the locally finite unstable modules and contains \(QH^*(X; \mathbb F_p)\); that is, \(QH^*(BX; \mathbb F_p)\) lies only one stage higher than where \(QH^*(X; \mathbb F_p)\) lies).
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
13A02 Graded rings
13E05 Commutative Noetherian rings and modules
55P20 Eilenberg-Mac Lane spaces
55P60 Localization and completion in homotopy theory
55S10 Steenrod algebra
55T10 Serre spectral sequences
57T10 Homology and cohomology of Lie groups
Full Text: DOI
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