The geometry of the James-Hopf maps

Nicholas J. Kuhn

Vol. 102 (1982), No. 2, 397–412

DOI: 10.2140/pjm.1982.102.397

Abstract

Snaith’s splitting of the suspension spectrum of the space Ω^kΣ^kX, for X path connected, into the wedge of the suspension spectra of spaces denoted D_k,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

Projection onto the q-th wedge summand and adjunction yield James-Hopf maps j_q : Ω^kΣ^kX → QD_k,qX, where QY = Ω^kΣ^kY .

In this paper I study various compatibility relations which hold among the j_q as X is replaced by ΣⁿX. In particular, I show that, for k > n, the iterated evaluation map 𝜀_n : ΣⁿΩ^kΣ^kX → Ω^k−nΣ^kX is naturally compatible with the stable splittings of these two spaces. This is done by exhibiting maps δ_k,n : ΣⁿD_k,qX → D_k−n,qΣⁿX making the following diagram of suspension spectra homotopy commute:

In certain cases, the maps δ_k,n are then identified as standard projection maps. Consequences are then discussed.

Mathematical Subject Classification 2000

Primary: 55P35

Secondary: 55P47

Milestones

Received: 10 December 1980

Published: 1 October 1982