The homology of the stable non-orientable mapping class group

TL;DR

This paper investigates the homology of the stable non-orientable mapping class group, identifying its structure at prime 2 and providing explicit calculations of its integral stable homology up to degree six.

Contribution

It determines the F_2-homology as a Hopf algebra and identifies a polynomial subalgebra of geometric characteristic classes, extending understanding of non-orientable surface symmetries.

Findings

F_2-homology matches that of a known space at odd primes

Integral stable homology tabulated up to degree six

Identified a polynomial subalgebra of geometric characteristic classes

Abstract

Combining results of Wahl, Galatius--Madsen--Tillmann--Weiss and Korkmaz one can identify the homotopy-type of the classifying space of the stable non-orientable mapping class group $N_\infty$ (after plus-construction). At odd primes p, the F_p-homology coincides with that of $Q_0(HP^\infty_+)$, but at the prime 2 the result is less clear. We identify the F_2-homology as a Hopf algebra in terms of the homology of well-known spaces. As an application we tabulate the integral stable homology of $N_\infty$ in degrees up to six. As in the oriented case, not all of these cohomology classes have a geometric interpretation. We determine a polynomial subalgebra of $H^*(N_\infty ; F_2)$ consisting of geometrically-defined characteristic classes.