The causal topology of neutral 4-manifolds with null boundary (1605.09576v2)

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.

• The paper introduces a conformal compactification of flat neutral 4-space into a 4-ball with a null hypersurface, setting up a framework for neutral Kirby calculus.
• The paper characterizes tangent hypersurfaces by identifying α-planes and β-planes as contact structures that support the development of neutral knot invariants.
• The paper provides a geometric model for intersecting Lagrangian surfaces that manifest as disjoint or linked tori, offering new insights into 4-manifold topology.

An Examination of the Causal Topology of Neutral 4-Manifolds with Null Boundary

The research conducted by Nikos Georgiou and Brendan Guilfoyle provides an in-depth exploration of neutral 4-manifolds with null boundaries, focusing on topological and geometric properties arising from the presence of neutral metrics (significantly, pseudo-Riemannian metrics with signature (2,2)). This paper is positioned within the nascent framework of what the authors term "neutral causal topology," aiming to leverage the distinct characteristics of neutral metrics to gain insights into the manifold structures.

Key Contributions and Results

1. Conformal Compactification: The paper delineates the conformal compactification of flat neutral 4-space into a 4-ball. Notably, the boundary of this compactification inherits a null hypersurface structure, characterized by a degenerate Lorentz metric whose null cone comprises transverse α-planes and β-planes. This compactification sets the stage for neutral Kirby calculus, providing a canonical class for surgery via handle attachments, guided by the null foliation structure.
2. Analysis of Tangent Hypersurfaces: Exploring sets of oriented geodesics, the authors explore the neutral metric geometry of tangent hypersurfaces. These hypersurfaces, including circle bundles over strictly convex surfaces, inherently exhibit nullity and are foliated by null geodesic circles. A significant result is the characterization of both α-planes and β-planes as contact structures, laying the groundwork for developing neutral knot invariants.
3. Geometric Models of Intersection Tori: Extending the above analysis, the authors provide a geometric model for intersections of two Lagrangian surfaces within the space of oriented lines in Euclidean space. Formalized through precise inequalities, these intersections manifest variously as disjoint or linked tori, contingent on the inter-sphere distance, furnishing topological insights pertinent to Lagrangian embeddings and neutral surgery techniques.
4. Neutral Causal Topology Implications: The paper contemplates the broader implications for 4-manifold topology. It extends classical understandings of open 4-manifolds by questioning how null boundaries reflect interior manifold properties—mirroring principles inherent in the Radon transform and linking to physical interpretations akin to the holographic principle.

Implications and Future Directions

Practically, this research could inform the construction of neutral metrics suited for explorations into topological structures of 4-manifolds with boundary components. The neutral Kirby calculus introduced offers a pathway for an array of applications, including investigating neutral Casson handles and neutral knot invariants. In theoretical terms, the exploration of contact structures in α-planes and β-planes suggests potential applications in diffeomorphism groups and related symmetry considerations in higher-dimensional topology.

Looking forward, potential avenues of development may include an elaboration on neutral surgery techniques, expanding on the neutral metric's ability to bridge topology and geometry, and applying the findings to complex link structures within 4-manifolds exhibiting more exotic topologies. There is also scope for refining the methods to include computations of further knot invariants and more thorough classifications of Lagrangian surface neighborhoods concerning various neutral Kähler metrics.

In summation, this paper provides a meticulous framework and compendium of techniques for engaging with the topology of 4-manifolds through a neutral metric lens. It proposes methodologies and augments existing tools, creating avenues for profound insights into both known and novel topological manifolds.