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On certain homotopy actions of general linear groups on iterated products

Ran LeviStewart Priddy — 2001

Annales de l’institut Fourier

The n -fold product X n of an arbitrary space usually supports only the obvious permutation action of the symmetric group Σ n . However, if X is a p -complete, homotopy associative, homotopy commutative H -space one can define a homotopy action of GL n ( p ) on X n . In various cases, e.g. if multiplication by p r is null homotopic then we get a homotopy action of G L n ( / p r ) for some r . After one suspension this allows one to split X n using idempotents of 𝔽 p GL n ( / p ) which can be lifted to 𝔽 p GL n ( / p r ) . In fact all of this is possible if X is an H -space...

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