String homology, and closed geodesics on manifolds which are elliptic spaces

Jones, John¹ ; McCleary, John²

¹Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom
²Mathematics Department, Vassar College, Poughkeepsie, NY 12604, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 5, pp. 2677-2690

Résumé

Let M be a closed, simply connected, smooth manifold. Let Fp be the finite field with p elements, where p > 0 is a prime integer. Suppose that M is an Fp–elliptic space in the sense of Félix, Halperin and Thomas (1991). We prove that if the cohomology algebra H∗(M, Fp) cannot be generated (as an algebra) by one element, then any Riemannian metric on M has an infinite number of geometrically distinct closed geodesics. The starting point is a classical theorem of Gromoll and Meyer (1969). The proof uses string homology, in particular the spectral sequence of Cohen, Jones and Yan (2004), the main theorem of McCleary (1987), and the structure theorem for elliptic Hopf algebras over Fp from Félix, Halperin and Thomas (1991).

Détail

DOI : 10.2140/agt.2016.16.2677

Classification : 55P50, 55P35, 55T05, 58E10
Keywords: string homology, closed geodesics

Bibliographie

[1] W Ballmann, G Thorbergsson, W Ziller, Closed geodesics and the fundamental group, Duke Math. J. 48 (1981) 585 | DOI

[2] V Bangert, N Hingston, Closed geodesics on manifolds with infinite abelian fundamental group, J. Differential Geom. 19 (1984) 277

[3] R Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956) 171 | DOI

[4] M Chas, D Sullivan, String topology,

[5] R L Cohen, J D S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002) 773 | DOI

[6] R L Cohen, J D S Jones, J Yan, The loop homology algebra of spheres and projective spaces, from: "Categorical decomposition techniques in algebraic topology" (editors G Arone, J Hubbuck, R Levi, M Weiss), Progr. Math. 215, Birkhäuser (2004) 77 | DOI

[7] Y Félix, S Halperin, J M Lemaire, J C Thomas, Mod p loop space homology, Invent. Math. 95 (1989) 247 | DOI

[8] Y Félix, S Halperin, J C Thomas, Gorenstein spaces, Adv. in Math. 71 (1988) 92 | DOI

[9] Y Félix, S Halperin, J C Thomas, Elliptic Hopf algebras, J. London Math. Soc. 43 (1991) 545 | DOI

[10] Y Félix, S Halperin, J C Thomas, Elliptic spaces, II, Enseign. Math. 39 (1993) 25

[11] Y Félix, J C Thomas, M Vigué-Poirrier, Rational string topology, J. Eur. Math. Soc. 9 (2007) 123 | DOI

[12] D Gromoll, W Meyer, Periodic geodesics on compact riemannian manifolds, J. Differential Geometry 3 (1969) 493

[13] S Halperin, M Vigué-Poirrier, The homology of a free loop space, Pacific J. Math. 147 (1991) 311 | DOI

[14] J Mccleary, On the mod p Betti numbers of loop spaces, Invent. Math. 87 (1987) 643 | DOI

[15] J Mccleary, W Ziller, On the free loop space of homogeneous spaces, Amer. J. Math. 109 (1987) 765 | DOI

[16] J Mccleary, W Ziller, Corrections to : “On the free loop space of homogeneous spaces” [Amer. J. Math. 109 (1987), 765–781], Amer. J. Math. 113 (1991) 375 | DOI

[17] J W Milnor, J C Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965) 211 | DOI

[18] B Ndombol, J C Thomas, A contribution to the closed geodesic problem, J. Pure Appl. Algebra 214 (2010) 937 | DOI

[19] A L Onishchik, Transitive compact transformation groups, Mat. Sb. 60 (102) (1963) 447

[20] M Vigué-Poirrier, D Sullivan, The homology theory of the closed geodesic problem, J. Differential Geom. 11 (1976) 633