A derived homotopy product

Abstract

l~ In this article I introduce and investigate the properties of a homotopy product [ SA, X] x [B, X] -~ [A;g,B, X] which is defined if B is an h-coloop and which is the Whitehead product if B is a suspension: the construction gives two new methods of defining the Whitehead product in the latter case. Each simply connected CW complex having a comultiplication is an h-coloop. There are therefore spaces S3w e 2"+1 which are h-coloops but are not homotopy associative ([1]), whilst S3v S 15, for example, has h-cogroup structures which are not suspensions (4.3 of [2]): the new homotopy product is defined when B is either of these spaces. Let A and B have nondegenerate base points and let B be an h-coloop, then the sequence B ~ A x B/A x , & AN, B p is a split short exact sequence of h-coloops. Let ~: AxB/Ax *-~SAvAxB/Ax , be the coaction in the cofibre sequence A ~ A  B ~ A x B/A x �9 -,.-., then there is a unique class of maps co: ANB-~SAvB satisfying co q+p~(1 v p) 0: A x B/A x �9 --~ SAv B. The class of co is the universal example for the cooperator product [ SA, X] x [B, X] --, [ANB, X] . Functoriality, deviation from bilinearity, and the relation of the product with the operation rct(X ) x [B, X] --~ [B, X] of the fundamental group are investigated in w 3. In the case where B is an h-cogroup I define a semidirect sum h-cogroup structure on SAv A x B/A x �9 and hence obtain a short exact sequence of h-cogroups SA v B-* SA v A x B/A x * (*'q) , A)Yd, B.