Overview
This research introduces a framework termed 'cotangent buildings' for the analysis of Weinstein manifolds. This framework is presented as a refinement and complement to existing methods, specifically Weinstein handlebodies. The authors indicate that while Weinstein handlebodies are suited for a “bottom-up” analytical approach to Weinstein structures, cotangent buildings facilitate a “top-down” analysis. A key aspect of cotangent buildings is the precise control they offer over the interactions among a subcollection of their constituent building blocks. Each of these building blocks is modeled on the cotangent bundle of a manifold with corners.
Approach
The work establishes cotangent buildings as a framework by defining their structure and properties relative to Weinstein manifolds. It articulates how this framework extends or refines the existing understanding derived from Weinstein handlebodies. The approach centers on demonstrating a specific relationship between Weinstein manifolds and cotangent buildings. This involves showing that a Weinstein manifold can be related to a cotangent building through a particular mathematical transformation or equivalence.
Findings
The principal finding is that every Weinstein manifold is Weinstein homotopic to a manifold that exhibits the structure of a cotangent building. This establishes a fundamental connection between these two mathematical concepts. The framework of cotangent buildings provides precise control over the interaction of any subcollection of its building blocks. Each of these blocks is modeled on the cotangent bundle of a manifold with corners. The distinction in analytical utility between cotangent buildings and Weinstein handlebodies is also a finding, with the former enabling “top-down” analysis alongside the latter's “bottom-up” focus.