The Disk-Based Origami Theorem and a Glimpse of Holography for Traversing Flows

TL;DR

This paper introduces a novel method to encode the topology of a manifold with boundary using a disk origami map derived from a traversally generic flow, revealing residual smooth structure information and enabling topological reconstruction.

Contribution

It presents a new construction of a CW-complex from a traversally generic flow, capturing residual smooth structure and enabling topological reconstruction via an origami map and Lyapunov function.

Findings

The CW-complex $\,\mathcal T(v)$ is homotopy equivalent to the manifold $X$.

The origami map $O$ has fibers of at most $(n+1)$ points.

Knowledge of $O$ and a Lyapunov function allows topological reconstruction of $(X, \mathcal F(v))$.

Abstract

This paper describes a mechanism by which a traversally generic flow $v$ on a smooth connected manifold $X$ with boundary produces a compact $CW$-complex $\mathcal T(v)$, which is homotopy equivalent to $X$ and such that $X$ embeds in $\mathcal T(v)\times \mathbf R$. The $CW$-complex $\mathcal T(v)$ captures some residual information about the smooth structure on $X$ (such as the stable tangent bundle of $X$). Moreover, $\mathcal T(v)$ is obtained from a simplicial \emph{origami map} $O: D^n \to \mathcal T(v)$, whose source space is a disk $D^n \subset \partial X$ of dimension $n = \dim(X) -1$. The fibers of $O$ have the cardinality $(n+1)$ at most. The knowledge of the map $O$, together with the restriction to $D^n$ of a Lyapunov function $f:X \to \mathbf R$ for $v$, make it possible to reconstruct the topological type of the pair $(X, \mathcal F(v))$, were $\mathcal F(v)$ is the $1$-foliation, generated by $v$. This fact motivates the use of "holography" in the title.