# The discrete bordism category in dimension 1

• Jan Steinebrunner (Oxford)
• Friday 18 October 2019, 16:00-17:00
• MR13.

The discrete bordism category hCob_d has as objects closed (d-1)-manifolds and as morphisms diffeomorphism classes of d-dimensional bordisms. This is a simplified version of the topologically enriched bordism category Cob_d whose classifying space B(Cob_d) been completely determined by Galatius-Madsen-Tillmann-Weiss in 2006. In comparison, little is known about the classifying space B(hCob_d).

I will identify B(hCob_1) as a circle bundle over a delooping of BCob_2, showing, in particular, that the rational cohomology ring of hCob_1 is polynomial on classes \kappa_i in degrees 2i+2 for all i>=1. The seemingly simpler category hCob_1 hence has a more complicated classifying space than Cob_1. Moreover, I will give combinatorial formulas for cocycles on hCob_1 that represent \kappa_i.

This talk is part of the Junior Geometry Seminar series.