The Loop Homology Algebra of Spheres and Projective Spaces

Abstract

In [3] Chas and Sullivan defined an intersection product on the homology H * (LM)of the space of smooth loops in a closed, oriented manifold M.In this paper we will use the homotopy theoretic realization of this product described by the first two authors in [2] to construct a second quadrant spectral sequence of algebras converging to the loop homology multiplicatively, when M is simply connected. The E2 term of this spectral sequence is H * (M;H *(ΩM)where the product is given by the cup product on the cohomology of the manifold H * (M)with coefficients in the Pontryagin ring structure on the homology of its based loop space H *(ΩM)We then use this spectral sequence to compute the ring structures of H * (LS n)and H * (L $$ {H_*}\left( {L\mathbb{C}{\mathbb{P}^n}} \right). $$ )