The topology of the moduli space of arc-length parametrised closed curves in Euclidean space

In this paper we compute the real homotopy type of the space of arc-length parametrised closed curves in Euclidean 3-space modulo orientation-preserving Euclidean motions. We find that this space has the same cohomology ring as the space BS1∨S1∨S3∨S5∨…, although its homotopy type is different. The moduli space we study is a symplectic quotient of the loop space of S2 by SO(3). The zero level set of the moment map in this case is homotopic to the loop space itself.