Homology, Homotopy and Applications

Volume 24 (2022)

Number 2

On the string topology coproduct for Lie groups

Pages: 327 – 345

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n2.a17


Maximilian Stegemeyer (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany; and Universität Leipzig, Germany)


The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space.We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into the diagonal map on the group and a based coproduct on the homology of the based loop space. This result implies that the coproduct is trivial for even-dimensional Lie groups. Using results by Bott and Samelson, we show that the coproduct is trivial as well for a large family of simply connected Lie groups.


string topology, Lie group

2010 Mathematics Subject Classification

55P50, 57T10

Received 22 September 2021

Received revised 10 December 2021

Accepted 10 December 2021

Published 14 September 2022