2002/10/21 by Tornike Kadeishvili
Mathematics · #math.AT #msc:55S30 #msc:55S35 #msc:16E40
published as Proc. of A. Razmadze Math. Inst., 91 (1988), 20-27
arxiv created 2002/10/21 · arxiv updated 2009/11/30
Stasheff's A(∞)-algebra (M,\mi:⊗iM→ M, i=1,2,3,...\) in fact is a DG-algebra (M,m1,m2) with not necessarily associative product m2 but this nonassociativity is measured by higher homotopies mi>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 Hi(C)-s, particularly in the cohomology algebra H^*(X,Λ) of a topological space X. It is clear that the A(∞)-algebra (H^*(X,Λ),\mi\) carries more information than the cohomology algebra H^*(B,Λ). Naturally arises a question when this structure is degenerate, that is when an A(∞)-algebra (M, \mi\) is isomorphic to one with higher operations mi, i≥ 3 trivial? In this paper we introduce the obstructions for such degeneracy. Namely, operations \mi\ we interpret as Hochschild twisting cochain m=m3+m4+..., mi∈ Cn(M,M) satisfying δm=m\smile1m where \smile1 is Gerstenhabers product in C^*(M,M). Using the generalized product f\smile1(g1,...,gk) we define perturbations of Hochschild twisting cochains (i.e. of A(∞) structures) and in particular prove that if for a graded algebra (M,μ) all Hochschild cohomologies Hochn,2-n(M,M)=0 for n≥3 then any A(∞)-algebra structure \mi\ on M with m1=0, m2=μ, is degenerate.