Subject: Karoubi's question
Date: Thu, 5 Apr 2001 21:22:07 -0500 (CDT)
From: may@math.uchicago.edu
To: dmd1@lehigh.edu

For the record, with respect to Max Karoubi's question:

The 1974 results Clarence Wilkerson referred to are in

V.K.A.M. Gugenheim and J.P. May
On the theory and applications of differential torsion products
Memoirs AMS 142, 1974.

The results there include a quasi-isomorphism  f: C^*(X) >--> H^*(X)

(1)  of differential algebras when X=BT^n and coefficients are
in any commutative ring; here f annihilates cup-one products.

(2) of differential Hopf algebras when  X  is a not quite completely
general finite product of Eilenberg-MacLane spaces and coefficients
are in Z/2; here there result cup-1 products in  H^*(X),  and there
is an algorithm for their computation.

(3) of differential coalgebras when  X  is a finite product of
Eilenberg-MacLane spaces and coefficients are in Z/p for p odd.

(Theorems A.25 and A.26 of the Appendix give more information).

The annihilation of cup-1 products in (1) leads to a general collapse
theorem for the Eilenberg-Moore spectral sequence for many classical
and generalized homogeneous spaces. For the latter, see

F. Neumann and J.P. May
On the cohomology of generalized homogeneous spaces.
Proc. Amer. Math. Soc. To appear. (On the Hopf and K-theory archives)

Claude Schochet, in his thesis, used the algorithm in (2) to obtain
an example of a two-stage Postnikov system with non-collapsing
Eilenberg-Moore spectral sequence.

Although not used for exactly the purposes that Saneblidze indicates,
the use of cup-1 products satisfying the Hirsch formula in this context
is a main ingredient of Gugenheim-May, and was explained briefly in a
1968 announcement of the results that were published in Gugenheim-May:

J.P. May
The cohomology of principal bundles, homogeneous spaces, and
two-Postnikov systems.
Bulletin Amer. Math. Soc. 74(1968), 334-339.