Geometric torsions and invariants of manifolds with triangulated boundary

TL;DR

This paper introduces invariants for manifolds with triangulated boundaries using geometric torsions, which behave consistently under boundary changes and manifold gluings, aligning with topological quantum field theory principles.

Contribution

It develops a new method to construct boundary invariants from geometric torsions that transform linearly under boundary triangulation changes and satisfy TQFT axioms.

Findings

Invariants form vectors that change linearly with boundary triangulation modifications.

Gluing manifolds results in scalar multiplication of the invariants vectors.

The approach aligns with Atiyah's axioms for topological quantum field theory.

Abstract

Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a manifold with a triangulated boundary. These invariants can be naturally united in a vector, and a change of the boundary triangulation corresponds to a linear transformation of this vector. Moreover, when two manifolds are glued by their common boundary, these vectors undergo scalar multiplication, i.e., they work according to M. Atiyah's axioms for a topological quantum field theory.