Nilpotency class of a map and Stasheff’s criterion

Cheong Seng Hoo

Vol. 28 (1969), No. 2, 375–380

DOI: 10.2140/pjm.1969.28.375

Abstract

Let f : X → Y be a map and let e : ΣΩX → X be the map whose adjoint is 1_Ωx. Then we prove the following results.

Theorem 1. nil f ≦ 1 if and only if fe∇ : ΣΩX ∨ ΣΩX → Y can be extended to ΣΩX × ΣΩX.

Theorem 2. Let X be an H′-space. Then nil f ≦ 1 if and only if f∇ : X ∨X → Y can be extended to X × X.

Theorem 3. nil f = nil (fe).

Theorem 1 may be regarded as an extension of Stasheff’s criterion for a loop space to be homotopy-commutative. These theorems may all be regarded as extensions of Stasheff’s criterion in various ways. We also discuss the duals of these results. Theorem 3 dualises, but the others do not. A sample result in the dual situation is

Theorem. conil f ≦ Σw cat (e′f) where e′ : Y → ΩΣY is the adjoint of 1_ΣY .

Mathematical Subject Classification

Primary: 55.40

Milestones

Received: 23 January 1968

Published: 1 February 1969