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

Abstract

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 M_i.