Abstract

In this paper, we exhibit an infinite loop space structure on the nerve of certain spin bordism $2$-categories and compare it to the classifying space of suitably stabilised spin mapping class groups. We show that the stable spin mapping class group has the homology of an infinite loop space. In order to do this, we adapt Harer's homology stabilisation results for spin mapping class groups to a setting compatible with the methods Tillmann used to prove that the classifying space of (non-spin) mapping class groups has the homology of an infinite loop space. We also study a variant of the spin mapping class groups, due to Masbaum, and show that its homology also stabilises as the genus tends to infinity.