| Let M be a closed, oriented manifold of dimension d, and LM the free loop space of M. In [3], Chas and Sullivan defined a Batalin-Vilkovisky structure on the loop homology H*LM inducing a Gerstenhaber structure, and a Lie algebra structure on the string homology HS1* LM . In [1], Cohen and Jones identified the loop homology H*LM with the Hochschild cohomology HH*(C*( M); C*(M)) as graded algebras; and it is expected that the natural Gerstenhaber structures, and even the BV-structures on both sides can be identified. The main theorem of this thesis is to calculate the Batalin-Vilkovisky structure of HH*C* CPn;Z ;C*CP n;Z ; and show that in the special case when M = CP1=S2 , this structure can not be identified with the BV-structure of H*LS2; Z computed by Luc Memichi in [13]. However, the induced Gerstenhaber structures are still identified in this case. As a consequence of [8], the string homology HS1* LM is identified with the negative cyclic cohomology HC*-C* M , and the Lie structures on both sides are naturally expected to be identified. As a consequence of the main theorem of this thesis, the Lie structure of HC*-C* CPn;Z as well as HC*-C* CPn;Q are calculated; and in the rational case, this Lie bracket becomes trivial and so coincides with the string bracket on the rational string homology HS1*L CPn;Q computed by Felix, Thomas and Vigue-Poirrier in [6]. |
