Anick's space and Hopf invariants
type: Item
instance of: scholarly article
title: Anick's space and Hopf invariants
zbMATH Open document ID: 1350.55013
author: Stephen Theriault
published in: Topology and its Applications
publication date: 2015-11-16T00:00:00Z
review text: In this paper the author initiates a new approach to a 1978 conjecture of Cohen, Moore and Neisendorfer that the fiber \(W_n\) of the double suspension \(S^{2n-1}\to\Omega^2S^{2n+1}\) at odd primes \(p\) is equivalent to the double loops on \(T^{2np+1}(p)\), where \(T^{2m+1}(p^r)\) is the so-called Anick space appearing in the fiber sequence \[ S^{2m-1}\to T^{2m+1}(p^r)\to\Omega S^{2m+1} \] whose connecting map is degree \(p^r\). By analyzing the properties of all the spaces in sight (e.g., their mod-\(p\) homology), the desired equivalence \(W_n\simeq \Omega^2T^{2np+1}(p)\) can be shown to follow from the existence of an \(H\)-space map \(\Theta:\Omega T^{2n+1}(p)\to\Omega^2 P^{2np+1}(P)\) that lifts \(\Omega H_{AG}:\Omega T^{2n+1}(p)\to\Omega T^{2np+1}(p^2)\) along \(\Omega\omega:\Omega^2 P^{2np+1}(p)\to\Omega T^{2np+1}(p^2)\), where \(\omega\) is a standard construction and \(H_{AG}\) is a ``Hopf invariant''-type map constructed by Anick and Gray that induces a homology epimorphism. Motivated by this, the author produces a first approximation to the lift \(\Theta\) by considering the \((2np-1)\)-skeleton of \(T^{2n+1}(p)\) (denoted \(A_{p-1}\)). More precisely, the author's main theorem is the construction of a composite of \(H\)-space maps \[ \Omega A_{p-1}\to\Omega S^{2np-1}\to\Omega^2 P^{2np+1}(p) \] emanating from \(\Omega A_{p-1}\) that behaves like the lift \(\Theta\). A key ingredient in the proof is the James construction on \(S^{2n}\). The author's method not only takes a step forward toward a complete proof of the Cohen-Moore-Neisendorfer conjecture, but also gives a road map for how such a complete proof might be obtained: namely, by producing lifts out of higher skeletons \(A_{p^k-1}\) of \(T^{2n+1}(p)\).
reviewed by: Donald M. Larson
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55Q40
zbMATH DE Number: 6509442
zbMATH Keywords: Anick's fibration
zbMATH Keywords: Hopf invariant
MaRDI profile type: Publication
full work available at URL: j.topol.2015.09.045
OpenAlex ID: W2178258878
cites work: Q4693770
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DOI: 10.1016/J.TOPOL.2015.09.045
description: scientific article; zbMATH DE number 6509442