The causal topology of neutral 4-manifolds with null boundary

TL;DR

This paper explores the topology of 4-manifolds with null boundaries using neutral metrics, constructing examples from conformal compactifications, geodesic spaces, and surface intersections, with potential applications in topology and knot theory.

Contribution

It introduces the concept of neutral causal topology and constructs novel neutral 4-manifolds with null boundaries from classical geometric settings.

Findings

Null hypersurfaces are foliated by their normals and inherit totally null planes.

The conformal compactification yields a boundary with Hopf fibration and integrable null planes.

Boundary tori are totally real and Lorentz if spheres do not intersect.

Abstract

This paper considers aspects of 4-manifold topology from the point of view of the null cone of a neutral metric, a point of view we call neutral causal topology. In particular, we construct and investigate neutral 4-manifolds with null boundaries that arise from canonical 3- and 4-dimensional settings. A null hypersurface is foliated by its normal and, in the neutral case, inherits a pair of totally null planes at each point. This paper focuses on these plane bundles in a number of classical settings The first construction is the conformal compactification of flat neutral 4-space into the 4-ball. The null foliation on the boundary in this case is the Hopf fibration on the 3-sphere and the totally null planes in the boundary are integrable. The metric on the 4-ball is a conformally flat, scalar-flat, positive Ricci curvature neutral metric. The second constructions are subsets of the 4-dimensional space of oriented geodesics in a 3-dimensional space-form, equipped with its canonical neutral metric. We consider all oriented geodesics tangent to a given embedded strictly convex 2-sphere. The third is a neutral geometric model for the intersection of two surfaces in a 4-manifold. The surfaces are the sets of oriented normal lines to two round spheres in Euclidean 3-space, which form Lagrangian surfaces in the 4-dimensional space of all oriented lines. The intersection of the boundaries of their normal neighbourhoods form tori that we prove are totally real and Lorentz if the spheres do not intersect. We conclude with possible topological applications of the three constructions, including neutral Kirby calculus, neutral knot invariants and neutral Casson handles, respectively.