On the loop homology of complex projective spaces (Q2880700)

Let \(LM\) be the space of continuous maps from the circle into a manifold \(M\) of dimension \(d\). By introducing string topology, Chas and Sullivan have defined new operations on \(H_*(LM)\). They have proved that the shifted homology \({\mathbb H}_*(LM) = H_{*+d}(LM;F)\) with coefficients in a commutative ring \(F\) has a structure of BV algebra. A fundamental problem in string topology is then to compare this BV algebra with the BV-Hochschild algebra \(H\! H^*(C^*(M;F), C^*(M;F))\). Working over a field Cohen and Jones proved that they are isomorphic as graded algebras.NEWLINENEWLINEOn the other hand, for \(F=\mathbb Z\), L. Menichi proved that the two BV algebras are not isomorphic. The present work extends the comparison between the two BV structures. The authors prove that with integer coefficients there is an isomorphism of BV algebras for \(M= \mathbb CP^{2n}\), but never for \(M=\mathbb CP^{2n+1}\).