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