Relating Cut and Paste Invariants and TQFTs

TL;DR

This paper explores the relationship between topological quantum field theories (TQFTs) and cut-and-paste invariants of smooth manifolds, revealing a natural group homomorphism and characterizing the structure of these invariants.

Contribution

It establishes a group homomorphism linking invertible TQFTs and SKK invariants, and shows all positive real-valued SKK invariants derive from invertible TQFTs.

Findings

A natural group homomorphism between invertible TQFTs and SKK invariants.

An exact sequence describing the relationship among these groups.

All positive real-valued SKK invariants can be realized as restrictions of invertible TQFTs.

Abstract

In this paper, we shall be concerned with a relation between TQFTs and cut and paste invariants introduced by Karras, Kreck, Neumann and Ossa. Cut and paste invariants, or SK invariants, are functions on the set of smooth manifolds that are invariant under the cutting and pasting operation. Central to the work in this paper are also SKK invariants, whose values on cut and paste equivalent manifolds differ by an error term depending only on the glueing diffeomorphism. Here we investigate a surprisingly natural group homomorphism between the group of invertible TQFTs and the group of SKK invariants and describe how these groups fit into an exact sequence. We conclude in particular that all positive real-valued SKK invariants can be realized as restrictions of invertible TQFTs. All manifolds are smooth and oriented throughout unless stated otherwise.