On the homotopy invariance of string topology

Let M be a closed, oriented, n-manifold, and LM its free loop space. In [C-S1] a commutative algebra structure in homology, H∗(LM), and a Lie algebra structure in equivariant homology H 1 ∗ (LM), were defined. In this paper we prove that these structures are homotopy invariant for simply connected manifolds in the following sense. Let f : M1 → M2 be a homotopy equivalence of simply connected closed 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’s.