Hochschild Cohomology of the Cohomology Algebra of Closed Orientable Three- Manifolds

Abstract

Let F be a field of characteristic other than 2. We show that the zeroth Hochschild cohomology vector space HH0(A) of a degree 3 graded commutative Frobenius F-algebra A = iAi, where we will always assume A0 = F, is isomorphic to the direct sum of the degree 0, 2 and 3 graded components and the kernel of a certain natural evaluation map ιμ : A1 Λ2(A1). In particular, this holds forA = H∗(M; F) the cohomology algebra of a closed orientable 3-manifoldM. In Theorem A of [1], Charette proves the vanishing of a certain discriminantΔassociated to a closed orientable 3-manifold L with vanishing cup product 3-form. It turns out that if we could show that HH2,−2(A) = 0for A = H∗(L;C), we would have found a more elementary proof of this part of Charette’s theorem. We show that for any β 3, the degree 3 graded commutative Frobenius algebra A with μA = 0and dim(A1) = β satisfiesHH2,−2(A) = 0. Thus Charette’s theorem is not simplified.