An additivity theorem for cobordism categories

Steimle, Wolfgang

doi:10.2140/agt.2021.21.601

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An additivity theorem for cobordism categories

Wolfgang Steimle

Algebraic & Geometric Topology 21 (2021) 601–646

DOI: 10.2140/agt.2021.21.601

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

References

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

Volume 21, issue 2 (2021)