An open-closed cobordism category with background space

TL;DR

This paper introduces an open-closed cobordism category with background space maps and identifies its classifying space for specific backgrounds, leveraging homology stability of mapping class groups with twisted coefficients.

Contribution

It defines a new open-closed cobordism category with background space maps and determines its classifying space for certain backgrounds, advancing the understanding of cobordism categories.

Findings

Classifying space identified for specific background spaces

Homology stability of mapping class groups with twisted coefficients established

New framework for open-closed cobordism categories with background maps

Abstract

In this paper we introduce an open-closed cobordism category with maps to a background space. We identify the classifying space of this category for certain classes of background space. The key ingredient is the homology stability of mapping class groups with twisted coefficients.