An Additivity theorem for cobordism categories

TL;DR

This paper introduces a new proof of the Genauer fibration sequence for cobordism categories using algebraic K-theory inspired methods, which generalizes to other cobordism-like categories.

Contribution

It provides a novel proof approach that extends the Genauer fibration sequence to broader cobordism-like categories, inspired by Waldhausen's Additivity theorem.

Findings

New proof of the Genauer fibration sequence

Generalization to other cobordism-like categories

Connection to Waldhausen's Additivity theorem

Abstract

Using methods inspired from algebraic $K$-theory, we give a new proof of the Genauer fibration sequence, relating the cobordism categories of closed manifolds with cobordism categories of manifolds with boundaries, and of the B\"okstedt-Madsen delooping of the cobordism category. Unlike the existing proofs, this approach generalizes to other cobordism-like categories of interest. Indeed we argue that the Genauer fibration sequence is an analogue, in the setting of cobordism categories, of Waldhausen's Additivity theorem in algebraic $K$-theory.