Journal of Topology

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

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 , 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.

Received February 5, 2007.

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2000 Mathematics Subject Classification 55N45, 55R80

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

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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions Google Scholar Articles by Cohen, R. L. Articles by Sullivan, D. Social Bookmarking          What's this?