Structure of $A(\infty)$-algebra and Hochschild and Harrison cohomology

Abstract

Stasheff's $A(\infty)$-algebra $(M,\{m_i:\otimes^iM\to M, i=1,2,3,...\})$ in fact is a DG-algebra $(M,m_1,m_2)$ with not necessarily associative product $m_2$ but this nonassociativity is measured by higher homotopies $m_{i>2}$. Nevertheless such structure arises in the strictly associative situation too, namely in the homology algebra $H(C)$ of a DG-algebra $C$ with free $H_i(C)$-s, particularly in the cohomology algebra $H^*(X,\Lambda)$ of a topological space $X$. It is clear that the $A(\infty)$-algebra $(H^*(X,\Lambda),\{m_i\})$ carries more information than the cohomology algebra $H^*(B,\Lambda)$. Naturally arises a question when this structure is degenerate, that is when an $A(\infty)$-algebra $(M, \{m_i\})$ is isomorphic to one with higher operations $m_i, i\geq 3$ trivial? In this paper we introduce the obstructions for such degeneracy. Namely, operations $\{m_i\}$ we interpret as Hochschild twisting cochain $m=m_3+m_4+..., m_i\in C^n(M,M)$ satisfying $\delta m=m\smile_1m$ where $\smile_1$ is Gerstenhabers product in $C^*(M,M)$. Using the generalized product $f\smile_1(g_1,...,g_k)$ we define perturbations of Hochschild twisting cochains (i.e. of $A(\infty)$ structures) and in particular prove that if for a graded algebra $(M,\mu)$ all Hochschild cohomologies $Hoch^{n,2-n}(M,M)=0$ for $n\geq3$ then any $A(\infty)$-algebra structure $\{m_i\}$ on $M$ with $m_1=0, m_2=\mu $, is degenerate.