Subject: Palmieri's questions From: Tornike Kadeishvili Date: Fri, 1 Oct 2004 09:17:24 -0400 (GMT) To: Don Davis Here are some answers to John Palmieri's questions. Question 1. There are many E_\infty operads, among them nice and small representativesthes - the surjection operad and the Barratt-Eccleles operad. The "largeset" E-infty operad can be expressed in "functional" terms as "all functorial operations". Al l E_\infty operads must contain Steenrod \cup, \cup_1,\cup_2, operations (and much more else), while A_\infty is responsible just for \cup (may be nonassociative and m_i-s measure the nonassociativity), so I do not think that E_infty can be viewed as A_\infty with extra condition (except rationbal case). Question 3. The construction of A(\infty)-algebra structure in homology of a dga is in T. Kadeishvili, On the Homology Theory of Fibrations, Russian Math. Surveys, 35, 3, 1980, 231-238. Very briefly the construction is folloving. Let A be a dga and H=H(A). Let f_1:H\to A be a cycle choosing homomorphism (H is asumed free!). This f_1 is not multiplicative in general but f_1(ab)-f_1(a)f_1(b) is homological to zero, so we can construct a homomorphism f_2:H\otimes H\to A s.t. df_2(a,b)= f_1(ab)-f_1(a)f_1(b). Then the expression f_1(a)f_2(b,c)+f_2(ab,c)+f_2(a,bc)+f_2(a,b)f_(c) (up to signes) is a cycle and we define m_3(a,b,c) as it's class. It is a Hochschild cocycle. If a,b,c is a Massy tripple, i.e. ab=0 and bc=0 then the two middle terms vanish and we obatin m_3(a,b,c)\in . This is mentioned in the above cited paper. Note that m_3 is not defined uniquelly (the freedom in choosing of f_1, f_2), but the resulting A_\infty algebra structure is uniqie up to A_infty isomorphism. All possible m_3-s fulfill . If A is commutative then f_2 can be choosed to be symmetric f_2(a,b)=f_2(b,a) and this implies that m_3 vanishes on shuffles, so it is a Harrison cocycle. It leads to C_\infty algebra structure on H. The complete construction of C_\infty-algebra in homology of a commutative dga is in T. Kadeishvili, A( )-algebra Structure in Cohomology and Rational Homotopy Type, Pros. of Tbil. Mat. Inst., 107, 1993, 1-94, (Russian) . Note that such a structure on H^*(X,Q) completelly determines the rationa homotopy type of X. Question 2. I think the situation is following: if a A is an algebra over an E-infty operad E then H(A) will be an algebra over some other (larger) E_\infty operad E'. I am shure that at last H(A) is an algebra over the huge E'=\Omega B E, the composition of cobar and bar constructions of E. Tornike Kadeishvili