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 $\operatorname{tr} : H_\ast (\Lambda M / G) \to H_\ast (\Lambda M)$ 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 $\vartheta : \Lambda M \to \Lambda M$ which reverses orientation, $\vartheta ( \gamma (t) := \gamma (1-t)$, is of particular interest to us. We compute $H_\ast (\Lambda S^n / \vartheta ; \mathbb{Q}), n \gt 2$, and the product\[P_\vartheta : H_i (\Lambda S^n / \vartheta ; \mathbb{Q}) \times H_j (\Lambda S^n / \vartheta ; \mathbb{Q)} \to H_{i+j-n} (\Lambda Sn/\vartheta ; \mathbb{Q})\]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 geodesics between non-reversible and reversible Finsler metrics on $S^n$, the latter might always be infinite.