Subject: Re: 2 responses to Karoubi Date: Wed, 4 Apr 2001 16:35:47 +0300 (EET DST) From: Samson Saneblidze To: Don Davis Dear Don, Here is my response to Karoubi's question for distribution. Thank you Samson Saneblidze ************************************************************************* On Mon, 2 Apr 2001, Don Davis wrote: > Two responses to Karoubi's question..........DMD > ____________________________ > > Subject: Re: Karoubi question > Date: Mon, 2 Apr 2001 16:13:15 +0100 (WETDST) > From: "L. Menichi" > > > Subject: spaces with polynomial cohomology > > Date: Sat, 31 Mar 2001 17:13:04 +0200 > > From: Max Karoubi > > > > I "discovered" recently the following fact : let k be an arbitrary > > commutative ring and X a space such that its k-cohomology H*(X) is a > > polynomial algebra with a countable set of generators (viewed as a > > DGA with 0-differential). Then the DGA of k-cochains C*(X) is related > > to H*(X) by a zigzag sequence of 2 quasi-isomorphisms of DGA's. > > The proof of this fact is quite easy and I presume it is hidden > > somewhere in the litterature. Does anybody know a reference ? > > > Max Karoubi > > For the finite case, this is due to Halperin and Stasheff > "Differential algebra in its own rite" Theorem 9 > For the countable case, this is due to Munkholm > "The Eilenberg-Moore spectral sequence and strongly homotopy > multiplicative > maps" 7.2 and Lemma 7.3 > This is summarized in the book of McCleary second edition > (Proposition 8.21 and Theorem 8.22) > > Luc Menichi > _________________________________ > Subject: Re: Karoubi question > Date: Mon, 02 Apr 2001 08:27:53 -0500 > From: Clarence Wilkerson > > I think there is related material in the Gugenheim-May memoir from > 1974 . I seem to recall that that in their applications, the prime > 2 was a little different ( a cup i condition), and I'm not sure they > considered a countable number of generators. > > Clarence Wilkerson > > > It seems in most generality when such a phenomenon happens for a dga A the conditions are that there must be \cup_1 -operation related with the multiplication on A by the Hirsch formula ( only a little part of what is meant under the homotopy G (Gerstenhaber) algebra structure). This is used in the proof of Theorem 5.6, ch.II (S. Saneblidze, Perturbation and obstruction theories in fibre spaces, Proc. A. Razmadze Meth. Inst. 1994). Samson Saneblidze