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

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

2005-09-28

arxiv.org/abs/math/0509667v2

Journal of Topology, vol.1 part 2 (2008), 391-408

Mathematics

Geometric Topology

21 pages, 2 figures final version published in Journal of Topology

Scientific paper

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M_1 \to M_2 be a homotopy equivalence of closed, oriented n -manifolds. Then the induced equivalence, Lf : LM_1 \to LM_2 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 's.

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