Topology

Geometric Sub-Bundles

Authors: Ryan J. Buchanan
Comments: 8 Pages.

Let $mathfrak{X}$ be a topological stack, and $LocSys(mathfrak{X})$ a local system taking varieties $v in mathfrak{X}$ to their projective resolutions over an affine coordinate system. Let $alpha$ and $beta$ be smooth charts encompassing non-degenerate loci of the upper-half plane, and let $varphi$ be the map $beta circ alpha^{-1}$. Our goal is to describe a class of vector bundles, called $emph{geometric sub-bundles}$, which provide holonomic transport for n-cells (for small values of n) over a $G_delta$-space which models the passage $mathfrak{X} ightrightarrows LocSys(mathfrak{X})$. We will first establish the preliminary definitions before advancing our core idea, which succinctly states that for a pointed, stratified space $Strat_M^ast$, there is a canonical selection of transition maps $[varphi]$ which preserves the intersection of a countable number of fibers in some sub-bundle of the bundle $Bun_V$ over $LocSys(mathfrak{X})$
Category: Topology