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On certain homotopy actions of general linear groups on iterated products

Ran Levi, Stewart 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...

On co-H-spaces.

G. Mislin, Peter Hilton, J. Roitberg (1978)

Commentarii mathematici Helvetici

On shape

Ralph Fox (1972)

Fundamenta Mathematicae

Postnikov invariants of H-spaces

Dominique Arlettaz, Nicole Pointet-Tischler (1999)

Fundamenta Mathematicae

It is known that the order of all Postnikov k-invariants of an H-space of finite type is finite. This paper establishes the finiteness of the order of the k-invariants k m + 1 ( X ) of X in dimensions m ≤ 2n if X is an (n-1)-connected H-space which is not necessarily of finite type (n ≥ 1). Similar results hold more generally for higher k-invariants if X is an iterated loop space. Moreover, we provide in all cases explicit universal upper bounds for the order of the k-invariants of X.

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