From: "aniceto" <aniceto@agt.cie.uma.es>
Subject: More about co-H-spaces
Date: Fri, 24 Mar 2000 12:50:29 +0100

Hola!

 Concerning the two questions of Jianzhong Pan and the interesting answers
given by M. Arkowitz, J. Klein and T. Goodwillie, here isa couple of more
general results:

1) As noted by J. Klein $XbX$ is the join of $\Omega X$ with itself.  It is
an easy exercise to prove that in general, one has the following: Given
f:X--->Y and g:Y--->B Then (conn stands for connectivity)

conn f \joint g is greater or equal than
min( conn f, conn g)+ min(conn X, conn Y, conn A, conn B)+1ç

2) Recall that co-H spaces are just spaces of Lusternik-schnirelmann
category 1. Then, generalizing results of Arkowitz and Golasinski-Klein,
Marek and I proved
that given a space
of category less or equal than n, then the spaces of the homology
decomposition can be taken to be of category
 less or equal than n. Moreover tha maps of the homology decomposition can
be taken to be n-B-maps (this is the analogous of being co-H-maps, see the
paper by Octav Cornea "Lusternik-Schnirelmann categorical sections", Ann.
Sci. Ec. NOr. Sup. 28 (1995))