Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions Yuming Xiao 1∗ and Yiming Long2† 1 School of Mathematics, Sichuan University, Chengdu 610064, People’s Republic of China 2 Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China June 30, 2018 Abstract In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre spectral se- quence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the res- onance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of such distinct closed geodesics is finite. arXiv:1503.07006v1 [math.GT] 24 Mar 2015 Key words: Chas-Sullivan theory, Leray-Serre spectral sequence, closed geodesics, real pro- jective spaces, non-simply connected, Morse theory, resonance identity AMS Subject Classification: 58F05, 58E10, 37J45, 53C22, 34C25 1 Introduction and the main results In this paper, we study the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions RP 2n+1, which are the typically oriented 2n+1 and non-simply connected manifolds with the fundamental group π1(RP ) = Z2. Then we ∗Partially supported by the Funds for Young Teachers of Sichuan University. e-mail:
[email protected] †Partially supported by NSFC Grant 11131004, MCME, LPMC of MOE of China, Nankai University and BCMIIS of Capital Normal University.