A note on the string topology BV-algebra for $S^2$ with $Z_2$ coefficients

Abstract

Luc Menichi showed that the BV algebras on $H^\bullet(LS^2;Z_2)[-2]$ coming from string topology and the one on $HH^\bullet(H^\bullet(S^2;Z_2),H^\bullet(S^2;Z_2))$ using Poincar\'e duality on $H^\bullet(S^2;Z_2)$ are not isomorphic. In this note we show how one can obtain the string topology BV algebra on Hochschild cohomology using a Poincar\'e duality structure with higher homotopies. This Poincar\'e duality (with higher homotopies) on cohomology is induced by a local Poincar\'e duality (with higher homotopies) on the cochain level.