The homology of the James-Hopf maps
If X is a path-connected space, then \(\Omega^ n\Sigma^ nX\) is stably equivalent to a bouquet \(\vee_{q\geq 0}D_{n,q}X\) [\textit{D. S. Kahn}, Lect. Notes Math. 658, 206-214 (1978; Zbl 0403.55008); \textit{V. Snaith}, J. Lond. Math. Soc., II. Ser. 7, 577-583 (1974; Zbl 0275.55019)]. Adjoint this stable equivalence to obtain a map \(j:\quad \Omega^ n\Sigma^ nX\to \prod_{q\geq 0}QD_{n,q}X.\) Project to the qth factor to obtain \(j_ q:\quad \Omega^ n\Sigma^ nX\to QD_{n,q}X.\) The author gives extensive information on the behavior of the map \(j_ q\) in mod-2 homology whenever q is a power of 2. The author gives several applications of these computations. The Kahn- Priddy theorem follows immediately [\textit{D. S. Kahn} and \textit{S. Priddy,} Bull. Am. Math. Soc. 78, 981-987 (1972; Zbl 0265.55009)]. Non- desuspension results follow: Let \(\tilde j_ q:\quad \Sigma^\infty \Omega^ n\Sigma^ nX\to \Sigma^\infty D_{n,q}X\) denote the adjoint of \(j_ q\). If \(q=2^ t\), \(2^ a+1\leq n\leq 2^{a+1}\), and \~ȷ\({}_ q\) desuspends to \(\Sigma^ d\Omega^ n\Sigma^ nX\to \Sigma^ dD_{q,n}X,\) then \(d>2^{a+t+1}-2^{a+1}+n-1\). This generalizes a result for \(n=2\) in the (unpublished) Northwestern thesis of P. Kirley. If \(X=S^{2^ bj+1}\), \(2^ b\geq n\), and \(q=2\), the same results were subsequently proven for an arbitrary stable map \(f:\quad \Sigma^\infty \Omega^ n\Sigma^ nX\to \Sigma^\infty D_{n,q}X\) which induces an epimorphism in homology [the reviewer and \textit{M. Mahowald}, Contemp. Math. 12, 81-90 (1982; Zbl 0512.55009)]. Also in a subsequent paper [Math. Proc. Camb. Philos. Soc. 92, 467-483 (1982; Zbl 0515.55005)] the author uses these sorts of homological calculations to prove G. W. Whitehead's conjecture for the symmetric product spectra \(Sp^ n(S)\) where S denotes the sphere spectrum. The author's methods are to study the map \(j_*\) together with the interplay of two natural types of homology operations in \(H_*(\prod_{q\geq 0}QD_{n,q}X)\). In an appendix, the author includes a proof of Tsuchiya's mixed Adem formula concerning these two types of homology operations. Information about \(j_{q*}\) is then obtained by projecting \(\prod_{q\geq 0}QD_{n,q}X\) to \(QD_{n,q}X\).
stable decomposition of the n-fold loop space of an n-fold suspension
Whitehead's conjecture for the symmetric product spectra
scientific article; zbMATH DE number 3844638