Anick's space and Hopf invariants
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 \(T^{2n+1}(p)\).
scientific article; zbMATH DE number 6509442
scientific article