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1 October 2009 Loop products and closed geodesics
Mark Goresky, Nancy Hingston
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Duke Math. J. 150(1): 117-209 (1 October 2009). DOI: 10.1215/00127094-2009-049

Abstract

The critical points of the length function on the free loop space Λ(M) of a compact Riemannian manifold M are the closed geodesics on M. The length function gives a filtration of the homology of Λ(M), and we show that the Chas-Sullivan product Hi(Λ)×Hj(Λ)*Hi+jn(Λ) is compatible with this filtration. We obtain a very simple expression for the associated graded homology ring GrH*(Λ(M)) when all geodesics are closed, or when all geodesics are nondegenerate. We also interpret Sullivan's coproduct (see [Su1], [Su2])) on C*(Λ) as a product in cohomology Hi(Λ,Λ0)×Hj(Λ,Λ0)Hi+j+n1(Λ,Λ0) (where Λ0=M is the constant loop). We show that is also compatible with the length filtration, and we obtain a similar expression for the ring GrH*(Λ,Λ0). The nonvanishing of products σ*n and τn is shown to be determined by the rate at which the Morse index grows when a geodesic is iterated. We determine the full ring structure (H*(Λ,Λ0),) for spheres M=Sn, n3

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Mark Goresky. Nancy Hingston. "Loop products and closed geodesics." Duke Math. J. 150 (1) 117 - 209, 1 October 2009. https://doi.org/10.1215/00127094-2009-049

Information

Published: 1 October 2009
First available in Project Euclid: 15 September 2009

zbMATH: 1181.53036
MathSciNet: MR2560110
Digital Object Identifier: 10.1215/00127094-2009-049

Subjects:
Primary: 53C22 , 55N45 , 58E05 , 58E10
Secondary: 53C , 55N , 57R

Rights: Copyright © 2009 Duke University Press

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Vol.150 • No. 1 • 1 October 2009
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