A note on the string topology BV-algebra for $S^2$ with ${\mathbb Z}_2$ coefficients

Abstract

Luc Menichi showed that the BV algebras on $H_\bullet(L S^2;{\mathbb Z}_2)[-2]$ coming from string topology and the one on $HH^\bullet(H^\bullet( S^2;{\mathbb Z}_2), H^\bullet( S^2;{\mathbb Z}_2))$ using Poincaré duality on $H^\bullet( S^2;{\mathbb Z}_2)$ are not isomorphic. In this note we show how one can obtain the string topology BV algebra on Hochschild cohomology using a Poincaré duality structure with higher homotopies. This Poincaré duality (with higher homotopies) on cohomology is induced by a local Poincaré duality (with higher homotopies) on the cochain level.