Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions

Abstract

In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre spectral sequence 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 resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of such distinct closed geodesics is finite.

The spaces of non-contractible closed curves in compact space forms

“…Recently Xiao and Long studied the topological structure of non-contractible loop spaces for odd-dimensional projective spaces computing, in particular, the equivariant cohomology with Z 2 -coefficients of the path spaces [18] and together with Duan applied these results to proving the existence of at least two geometrically distinct non-contractible closed geodesics for irreversible bumpy Finsler metrics on RP 3 [7].…”

Resonance identity and multiplicity of non-contractible closed geodesics on FinslerRPn

“…Indeed, we can make a reduction by (3.10) (with n = 1 and k = 2) so that only one irrational number is rest. The uniformly distribution mod one in Number theory then enables the authors in [14] to find some l ∈ N such that the Betti numberβ 2l = 1 which contradicts to the topological structure of the noncontractible loop space on RP 3 obtained in [42]. However when one tries to use such a means to deal with higher dimensional RP 2n+1 , more irrational numbers are rest to be controlled simultaneously for larger k. What is even worse, those irrational numbers may be linearly dependent over Q.…”

“…Motivated by [39] and [42], in section 2 of this paper we obtain the resonance identity for the non-contractible closed geodesics on RP n by using rational coefficient homology for any n ≥ 2 regardless of whether n is odd or not.…”

The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

“…In particular, if Γ = Z 2 , then S n /Γ is the n-dimensional real projective space RP n . Motivated by the works [44], [12] and [25] about closed geodesics on Finsler RP n , and based on Taimanov's work [39] on rational equivariant cohomology of non-contractible loops on S n /Γ, this paper is concerned with the multiplicity of closed geodesics on Finsler S n /Γ. Let (M, F ) be a Finsler manifold and ΛM be the free loop space on M defined by ΛM = γ : S 1 → M | γ is absolutely continuous and 1 0 F (γ,γ) 2 dt < +∞ , endowed with a natural structure of Riemannian Hilbert manifold on which the group S 1 = R/Z acts continuously by isometries (cf.…”

“…Motivated by the studies on simply connected manifolds, in particular, the resonance identity proved by Rademacher [33], and based on Westerland's works [42], [43] on loop homology of RP n , Xiao and Long [44] in 2015 investigated the topological structure of the non-contractible loop space and established the resonance identity for the non-contractible closed geodesics on RP 2n+1 by use of Z 2 coefficient homology. As an application, Duan, Long and Xiao [12] proved the existence of at least two distinct non-contractible closed geodesics on RP 3 endowed with a bumpy and irreversible Finsler metric.…”