Let f : X → Y be a map
and let e : ΣΩX → X be the map whose adjoint is 1Ωx. 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
1ΣY .
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