A∞-Algebras and the Cyclic Bar Complex (original) (raw)

This paper arose from our use of Chen's theory of iterated integrals as a tool in the study of the complex of S 1-equivariant differential forms on the free loop space LX of a manifold X (see [2]). In trying to understand the behaviour of the iterated integral map with respect to products, we were led to a natural product on the space of S 1-equivariant differential forms Ω(Y)[u] of a manifold Y with circle action, where u is a variable of degree 2. This product is not associative but is homotopy associative in a precise way; indeed there is whole infinite family of "higher homotopies". It turns out that this product structure is an example of Stasheff's A ∞-algebras, which are a generalization of differential graded algebras (dgas). Using the iterated integral map, it is a straightforward matter to translate this product structure on the space of S 1-equivariant differential forms on LX into formulas on the cyclic bar complex of Ω(X). Our main goal in this paper is to show that in general, the cyclic bar complex of a commutative dga A has a natural A ∞-structure and we give explicit formulas for this structure. In particular, this shows that the cyclic homology of A has a natural associative product, but it is a much stronger result, since it holds at the chain level. Thus, it considerably strengthens the results of Hood and Jones [3]. We also show how to construct the cyclic bar complex of an A ∞-algebra, and in particular define its cyclic homology. As hinted at in [2], this construction may have applications to the problem of giving models for the S 1 × S 1-equivariant cohomology of double loop spaces LL(X) of a manifold and, since the space of equivariant differential forms on a smooth S 1-manifold Y is an A ∞-algebra, to the problem of finding models for the space of S 1 × S 1-equivariant differential forms on LY. Although the methods that we use were developed independently, they bear a strong resemblance with those of Quillen [6]. Finally, we discuss in our general context the Chen normalization of the cyclic bar complex of an A ∞-algebra. This is a quotient of the cyclic bar complex by a complex of degenerate chains which is acyclic if A is connected, and which was shown by Chen to coincide with the kernel of the iterated integral map in the case A = Ω(X). This normalization is an important tool, since it allows us to remove a large contractible sub-complex of the cyclic bar complex. The first two sections of this paper are devoted to generalities concerning coalgebras and A ∞algebras; a good reference for further background on coalgebras is the book of McCleary [5]. The cyclic bar complex of an A ∞-algebra is constructed in Section 3, the A ∞-structure on the cyclic bar complex of a commutative dga in Section 4, and we discuss Chen normalization in Section 5. All our algebra will be carried out over a fixed coefficient ring K; in fact nothing will be lost by thinking of the case where K is the integers Z. In particular, all tensor products are taken over K unless explicitly stated otherwise. We will make use of the sign-convention in the category of Z 2-graded K-modules, which may be phrased as follows: the canonical map S 21 from V 1 ⊗ V 2 to V 2 ⊗ V 1 is defined by S 21 (v 1 ⊗ v 2) = (−1) |v1||v2| v 2 ⊗ v 1. *In the preprint of [2], the maps m andm are exchanged, for which we beg the reader's forgiveness.