The homotopy invariance of the string topology loop product and string bracket

Let Mn be a closed, oriented, n-manifold, and LM its free loop space. In [Chas and Sullivan, ‘String topology’, Ann. of Math., to appear] a commutative algebra structure in homology, H*(LM), and a Lie algebra structure in equivariant homology H * S 1 , were defined. In this paper, we prove that these structures are homotopy invariants in the following sense. Let f:M1 → M2 be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equivalence, Lf:LM1 → LM2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds true for any generalized homology theory h* that supports an orientation of the Mi.