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In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serrespectral sequence to investigate the topological structure of thenon-contractible component of the free loop space on the real projective spaceswith odd dimensions. Then we apply the result to get the resonance identity ofnon-contractible homologically visible prime closed geodesics on such spacesprovided the total number of distinct prime closed geodesics is finite.

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