High-Dimensional Topological Field Theory, Positivity, and Exotic Smooth Spheres

TL;DR

This paper develops a positive topological field theory framework applicable to high-dimensional smooth manifolds, demonstrating its ability to detect exotic smooth spheres and extract polynomial invariants in certain dimensions.

Contribution

It introduces a concrete positive TFT for smooth manifolds of dimension greater than 1 that can identify exotic smooth spheres and derive polynomial invariants.

Findings

Positive TFT detects exotic smooth spheres.

Polynomial invariants can be extracted from state sums in dimension ≥ 3.

Framework applies to manifolds of any dimension > 1.

Abstract

In previous work, we proposed a general framework of positive topological field theories (TFTs) based on Eilenberg's notion of summation completeness for semirings. In the present paper, we apply this framework in constructing explicitly a concrete positive TFT defined on smooth manifolds of any dimension greater than 1. We prove that this positive TFT detects exotic smooth spheres. We show further that polynomial invariants (subject to boundary conditions) can be extracted from the state sum if the dimension of the cobordisms is at least 3.