The geometry of the JAMES-HOPF maps

Abstract

Snaith’s splitting of the suspension spectrum of the space ΩkΣkX for X path connected, into the wedge of the suspension spectra of spaces denoted Dk, qX, has been of considerable interest to homotopy theorists in recent years. If Σ∞X denotes the suspension spectrum of a space X then this can be restated as Σ∞X ≅ Vq≧1 Σ∞ Dk, qX Projection onto the qth wedge summand and adjunction yield James-Hopf maps jq: ΩkΣkX → QDk, qX, where QY = lim→ #x03A9;kΣkY. In this paper I study various compatibility relations which hold among the jq as X is replaced by ΣnX. In particular, I showthat, for k > n, the iterated evaluation map εn:ΣnΩkX → Ωk − nΣkX is naturally compatible with the stable splittings of these two spaces. This is done by exhibiting maps δk, n: ΣnDk, qX → Dk‒n, q,ΣnX making the following diagram of suspension spectra homotopy commute. In certain cases, the mapsδk, n are then identified as standard projection maps. Consequences are then discussed. © 1982, University of California, Berkeley. All Rights Reserved.