Unstable splitting of \(V(1) \wedge V(1)\) and its applications (Q2575550)

In [Ann. Math. (2) 109, 121--168 (1979; Zbl 0405.55018)], \textit{F. R. Cohen, J. C. Moore} and \textit{J. A. Neisendorfer} found a product decomposition for the loop spaces of mod \(p\) Moore spaces \(P^n(p)\). Two key elements for the proof are (1) a Hurewicz theorem for the functors \([P^n(p), -]\), and (2) a Samelson product \([P^m(p),-]\otimes [P^n(p),-]\to [P^{m+n}(p),-]\). In the nilpotence and periodicity theory, the mod \(p\) Moore spaces are complexes of type \(1\), and -- in that context -- denoted \(V(0)\). They fit into a family of complexes \(V(n)\) for all \(n\), so it is natural to inquire whether the results of [loc. cit.] could be extended to \(V(n)\). In the paper under review, the author concentrates on \(V(1)\); he writes \(V^n\) for the cofiber of an Adams map \(A: P^{n-1}\rightarrow P^{n-2p+1}\). The main results are (1) a Hurewicz-type theorem for the functors \([V^n, -]\) and (2) an unstable splitting \(V^m \wedge V^n \simeq V^{n+m} \vee\) (other stuff). The equivalence of (2) yields a map \(V^{m+n}\rightarrow V^m\wedge V^n\), which is used to construct natural Samelson products \([V^m,-]\otimes [V^n,-]\rightarrow [ V^{m+n}, -]\); basic algebraic properties of these products are verified. Recently, \textit{B. Gray} [Trans. Am. Math. Soc. 340, No.~2, 595--616 (1993; Zbl 0820.55005)] has worked to establish a framework for generalizing the work of Cohen, Moore and Neisendorfer. The author suggests that the results of the current paper could be applied to this program.