Generator Histories and Parity-Odd Curvature in Lorentzian Topology Change

TL;DR

This paper introduces a generator-history framework for Lorentzian topology change, linking local operations, braid groups, and parity-odd curvature to understand chiral topology transitions in classical Lorentzian geometry.

Contribution

It develops an algebraic and geometric framework representing topology change through elementary local events and identifies parity-odd curvature as a key diagnostic of chiral topology change.

Findings

Braid groups model elementary local exchanges in topology change.

Parity-odd conformal curvature detects chiral generator accumulation.

Generator histories provide a covariant diagnostic independent of endpoint classes.

Abstract

Lorentzian topology change may be resolved into an ordered sequence of localized, orientation-sensitive operations rather than treated solely as a global transition between spatial manifolds. We develop a generator-history framework in which topology-changing spacetimes are represented algebraically as compositions of elementary local events, independent of dynamics, quantization, or anomaly inflow. Braid groups arise as the minimal realization of ordered, invertible pairwise exchanges, while higher-valence generators extend the construction to networked processes. Within this framework we identify parity-odd conformal curvature as the unique nontrivial local curvature pseudoscalar (without derivatives) capable of aggregating oriented generator content in four-dimensional Lorentzian vacuum geometry. The dual Weyl contraction changes sign under orientation reversal and therefore isolates chiral generator accumulation, while parity-even curvature scalars are insensitive to such structure. The associated spacetime integral functions as a covariant geometric diagnostic of chiral topology change that depends on generator histories and does not descend to endpoint-only equivalence classes obtained by Markov-type coarse-graining. The resulting picture isolates a pre-quantum geometric layer beneath spectral asymmetry: oriented generator dynamics induce parity-odd curvature compatible with the Pontryagin density appearing in the Atiyah Patodi Singer index theorem yet remains defined entirely within classical Lorentzian geometry. This framework clarifies the algebraic and geometric substrate underlying chiral topology change without introducing new gravitational dynamics or topological invariants.