Nancy Hingston : Geodesics and the structure of the free loop space #1/3

Abstract: The Chas-Sullivan product is a product on the homology of the free loop space LM of a compact oriented manifold M, the loop homology of M. If we fix a metric on M, the critical points of the length function on LM are the closed geodesics on M in the given metric. Morse theory gives a link between the geometry of closed geodesics, and the algebraic structure on the loop homology M given by the Chas-Sullivan product.I will "review" Morse theory, then introduce the Chas Sullivan product and the “dual” cohomology product, and the fundamental inequality for the "minimax" critical level cr(X) of a homology class X on LM: cr(X∗Y) ≤ cr(X) + cr(Y).There is a dual inequality for loop cohomology (with opposite sign!): cr(x∗y) ≥ cr(x) + cr(y).I will discuss what the geometry of geodesics tells us about algebraic structure on LM, and what algebra tells us about geometry. Some theorems from the past are naturally stated in terms of loop products. For manifolds such as spheres and projective spaces for which there is a metric with all geodesics closed, the loop homology and cohomology rings are nontrivial, and closely linked to the geometry. For spheres we obtain significant (and so far mysterious) constraints on the critical levels of loop homology classes: for any fixed metric on a sphere there are positive constants A and B so that for each homology class X we have A deg(X) - B ≤ cr(X) ≤ A deg(X) + B.References:[1] R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171-206.[2] M. Chas and D. Sullivan, String topology, preprint, math.GT/9911159 (1999).[3] R. L. Cohen, J. D. S. Jones, and J. Yan. The loop homology algebra of spheres and projective spaces. In Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), volume 215 of Progr. Math., pages 77--92. Birkh¨auser, Basel, 2004.[4] M. Goresky and N. Hingston. Loop products and closed geodesics. Duke Math. J., 150(1):117--209, 2009.[5] N. Hingston and A. Oancea. The space of paths in complex projective space with real boundary conditions. arXiv: 1311.7292, 2013.[6] N.Hingston and H-B Rademacher, Resonance for loop homology of spheres. J. Differential Geom. 93 (2013), no. 1, 133--174.[7] W. Klingenberg, Lectures on Closed Geodesics, Grundlehren der mathematischen Wissenschaften 230, Springer Verlag, Berlin, 1978.[8]. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, Princeton N.J., 1963.