Vol. 102, No. 2, 1982

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The geometry of the James-Hopf maps

Nicholas J. Kuhn

Vol. 102 (1982), No. 2, 397–412

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 ΣX denotes the suspension spectrum of a space X then this can be restated as

Σ∞ ΩkΣkX  ∼=   Σ ∞Dk,qX.

Projection onto the q-th wedge summand and adjunction yield James-Hopf maps jq : ΩkΣkX QDk,qX, where QY = lim
→ΩkΣkY .

In this paper I study various compatibility relations which hold among the jq as X is replaced by ΣnX. In particular, I show that, for k > n, the iterated evaluation map 𝜀n : ΣnΩkΣkX ΩknΣkX is naturally compatible with the stable splittings of these two spaces. This is done by exhibiting maps δk,n : ΣnDk,qX Dkn,qΣnX making the following diagram of suspension spectra homotopy commute:

ΣnΩk ΣkX   ∼=    ∨ ΣnDk,qX
↓ 𝜀n        ∨  ↓ δk,n     .
Ωk− nΣnX   ∼=     Dk−n,qΣnX

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

Mathematical Subject Classification 2000
Primary: 55P35
Secondary: 55P47
Received: 10 December 1980
Published: 1 October 1982
Nicholas J. Kuhn
Department of Mathematics
University of Virginia
Kerchof Hall PO Box 400137
Charlottesville VA 22904-4137
United States