A Morse theoretic description of string topology

Ralph L. Cohen and Matthias Schwarz

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Submission date: 19. Sep. 2008
Pages: 32
published in: New perspectives and challenges in symplectic field theory / M. Abreu ... (eds.)
Providence, R.I. : American Mathematical Society, 2009. - P. 147 - 172
(CRM proceedings & lecture notes ; 49) 
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Abstract:
Let M be a closed, oriented, n-dimensional manifold. In this paper we give a Morse theoretic description of the string topology operations introduced by Chas and Sullivan, and extended by the first author, Jones, Godin, and others. We do this by studying maps from surfaces with cylindrical ends to M, such that on the cylinders, they satisfy the gradient flow equation of a Morse function on the loop space, LM. We then give Morse theoretic descriptions of related constructions, such as the Thom and Euler classes of a vector bundle, as well as the shriek, or umkehr homomorphism.