#### Volume 21, issue 2 (2021)

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An additivity theorem for cobordism categories

### Wolfgang Steimle

Algebraic & Geometric Topology 21 (2021) 601–646
##### 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 Bökstedt–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.

##### Keywords
cobordism category, algebraic $K$–theory
##### Mathematical Subject Classification 2010
Primary: 57R90
Secondary: 19D06, 57R15, 57R56
##### Publication
Received: 20 February 2019
Revised: 20 April 2020
Accepted: 22 June 2020
Published: 25 April 2021
##### Authors
 Wolfgang Steimle Institut für Mathematik Universität Augsburg Augsburg Germany