On Holographic Structures, Traversing Flows, and Exotic Spheres

TL;DR

This paper explores holographic structures on manifolds, demonstrating how boundary causality maps can reconstruct flows and manifolds, and discusses the complexity of such structures on exotic spheres.

Contribution

It generalizes the Holography Theorem to fillable holographic structures on closed manifolds, linking them to traversally generic flows and exotic spheres.

Findings

Holography Theorem is extended to fillable structures.

Boundary causality maps can reconstruct the manifold and flow.

Holographic structures exhibit richness on exotic spheres.

Abstract

Any traversally generic vector flow on a compact manifold $X$ with boundary leaves some residual structure on its boundary $\d X$. A part of this structure is the flow-generated causality map $C_v$, which takes a region of $\d X$ to the complementary region. By the Holography Theorem from \cite{K4}, the map $C_v$ allows to reconstruct $X$ together with the unparametrized flow. The reconstruction is a manifestation of holographic description of the flow. In the paper, we introduce and study the holographic structures on a given closed manifold $Y$, which mimics $\d X$. We generalize the Holography Theorem so that is stated in terms of fillable holographic structures on $Y$. Such structures are intimately linked with traversally generic vector flows on manifolds $X$ whose boundary is $Y$. We conclude with few observations about the richness of holographic structures on smooth exotic spheres.