The Disk-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 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.

12 pages, 2 figures. arXiv admin note: text overlap with arXiv:1409.0588

References in corpus (2)

• Convexity of Morse Stratifications and Gradient Spines of 3-Manifolds
• Traversally Generic & Versal Vector Flows: Semi-Algebraic Models of Tangency to the Boundary