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

Abstract

This paper describes a mechanism by which a traversally generic flow v on a smooth connected $(n+1)$-dimensional manifold X with boundary produces a compact n-dimensional CW-complex ${\mathcal {T}}(v)$, which is homotopy equivalent to X and such that X embeds in ${\mathcal {T}}(v)\times \mathbb R$. 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 origami map$O: D^n \rightarrow {\mathcal {T}}(v)$, whose source space is a ball $D^n \subset \partial X$. 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 $f: X \rightarrow \mathbb R$ 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 the word “holography” in the title. In a qualitative formulation of the holography principle, for a massive class of ODE’s on a given compact manifold X, the solutions of the appropriately staged boundary value problems are topologically rigid.