Coherent homomorphisms of H-spaces
by
Andrei V. Prasolov
Department of Mathematics, University of Tromsø, N-9037 Tromsø, Norway

0. Let X and Y be H-spaces, associative up to coherent homotopies. We define a space HOM_ass( X, Y) of homomorphisms up to a coherent homotopy X --> Y, and constructs a spectral sequence

(0)E2st ===> \pi-s-t( HOMass( X, Y)) .

A natural mapping ₍₀₎j: HOM_ass( X, Y) --> Map _* (BX, BY) is constructed where BX and BY are the deloopings of X and Y respectively. For a number of cases the morphism ₍₀₎j is investigated and appears to be a homotopy equivalence.

1. Let now X and Y be homotopy everything H-spaces, such that the multiplication in both spaces is associative and commutative up to coherent homotopies. We define a space HOM_asscomm( X, Y) similar to the space HOM_ass( X, Y) above. The coherent homotopies involved should interact with the coherent homotopies of associativity and commutativity. One constructs also a spectral sequence

(1)E2st ===> \pi-s-t( HOMasscomm(X, Y) ) .

A natural mapping

(1)j:HOMasscomm( X, Y) --> MapSpectra(Sp( X) , Sp( Y) )

is considered where Sp( X) and Sp( Y) are the corresponding \Omega-spectra that infinitely deloops X and Y. For a number of cases ₍₁₎j appears to be a homotopy equivalence.

2. Finally, let X and Y be C-spaces where C is an operad in the sense of J. P. May. We construct a space HOM_C( X, Y) and a spectral sequence ₍₂₎E₂^st ===> \pi_-s-t( HOM_C( X, Y) ). For some cases, the terms ₍₂₎E₂^st are calculated.

Date received: August 25, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caje-56.