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A cohomology theory for \(A(m)\)-algebras and applications. (English) Zbl 0801.55004
In [J. D. Stasheff, Trans. Am. Math. Soc. 108, 275-312 (1963; Zbl 0114.394)] the notion of an \(A(m)\)-algebra was introduced in order to study homotopy associative \(H\)-spaces. In this paper, a cohomology theory of \(A(m)\)-algebras (with coefficients in an \(A(m)\)-algebra bimodule) is defined and applied to various situations found in homotopy theory. In particular, it is shown that the \(A(m)\)-cohomology of a (balanced) \(A(m)\)-algebra \(A\), denoted \(H_{(m)}^*(A;M)\), has a type of Hodge decomposition with \(H_{(m)}^{*,1}(A;M)\cong HB_{(m)}^*(A;M)\), the balanced cohomology of \(A\). The inclusion \(HB_{(m)}^*(A;M) \to H_{(m)}^*(A;M)\) is shown, for appropriate \(A\)’s, to give the canonical map from Harrison to Hochschild cohomologies, the Hurewicz map for the loop space of a manifold and the map from Lie algebra cohomology (with coefficients in the Lie algebra) to the Hochschild cohomology of the universal enveloping algebra (with coefficients in the enveloping algebra). It is also shown that this cohomology computes the ordinary cohomology of the projective space associated to a homotopy associative \(H\)-space.

MSC:
55N35 Other homology theories in algebraic topology
55P62 Rational homotopy theory
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