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Journal of Topology 2008 1(2):391-408; doi:10.1112/jtopol/jtn001
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© 2008 London Mathematical Society

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

Ralph L. Cohen

Department of Mathematics, Stanford University, Stanford, CA 94305, USA ralph@math.stanford.edu

John R. Klein

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA klein@math.wayne.edu

Dennis Sullivan

Mathematics Department, SUNY, Stony Brook, NY 11794, USA dennis@math.sunysb.edu


  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 Formula , 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.

Received February 5, 2007.

2000 Mathematics Subject Classification 55N45, 55R80

All three authors were partially supported by grants from the NSF.


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