Homology transfer products on free loop spaces: orientation reversal on spheres

Abstract

We consider the space $\Lambda M:=H^1(S^1,M)$ of loops of Sobolev class $H^1$ of a compact smooth manifold $M$, the so-called free loop space of $M$. We take quotients $\Lambda M/G$ where $G$ is a finite subgroup of $O(2)$ acting by linear reparametrization of $S^1$. We use the existence of transfer maps $tr:H_*(\Lambda M/G)\rightarrow H_*(\Lambda M)$ to define a homology product on $\Lambda M/G$ via the Chas-Sullivan loop product. We call this product $P_G$ the transfer product. The involution $\vartheta:\Lambda M\rightarrow \Lambda M$ which reverses orientation, $ \vartheta\big(\gamma(t)\big):=\gamma(1-t)$, is of particular interest to us. We compute $H_*(\Lambda S^n/\vartheta;\mathbb{Q})$, $n>2$, and the product $P_\vartheta:H_i(\Lambda S^n/\vartheta;\mathbb{Q})\times H_j(\Lambda S^n/\vartheta;\mathbb{Q})\rightarrow H_{i+j-n}(\Lambda S^n/\vartheta;\mathbb{Q})$ associated to orientation reversal. Rationally $P_\vartheta$ can be realized "geometrically" using the concatenation of equivalence classes of loops. There is a qualitative difference between the homology of $\Lambda S^n/\vartheta$ and the homology of $\Lambda S^n/G$ when $G\subset S^1\subset O(2)$ does not "contain" the orientation reversal. This might be interesting with respect to possible differences in the number of closed geodesic between non-reversible and reversible Finsler metrics on $S^n$, the latter might always be infinite.