An Additivity theorem for cobordism categories

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.

Final version. To appear in Algebraic and Geometric Topology

References in corpus (7)

• Higher Topos Theory
• The positive scalar curvature cobordism category
• The cobordism category and Waldhausen's K-theory
• A cobordism model for Waldhausen $K$-theory
• Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans
• Stable moduli spaces of hermitian forms
• The homotopy type of the topological cobordism category

Cited by in corpus (5)

• The positive scalar curvature cobordism category
• Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans
• A cobordism model for Waldhausen $K$-theory
• A K-theory spectrum for cobordism cut and paste groups
• The Disc-structure space