Vol. 28, No. 2, 1969

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Nilpotency class of a map and Stasheff’s criterion

Cheong Seng Hoo

Vol. 28 (1969), No. 2, 375–380

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 (ef) where e: Y ΩΣY is the adjoint of 1ΣY .

Mathematical Subject Classification
Primary: 55.40
Received: 23 January 1968
Published: 1 February 1969
Cheong Seng Hoo