Manifold properties of planar polygon spaces

TL;DR

This paper investigates the topological properties of spaces formed by planar polygons with fixed side lengths, revealing their tangent bundle structure and implications for orientability and other manifold characteristics.

Contribution

It establishes a canonical isomorphism for the tangent bundle of generic planar polygon spaces, extending understanding of their topological and geometric properties.

Findings

Tangent bundle plus trivial line bundle is isomorphic to (n-2) times a canonical line bundle.

Results have implications for orientability, cobordism, immersions, and parallelizability of these spaces.

Provides a foundational topological analysis of planar polygon spaces.

Abstract

We prove that the tangent bundle of a generic space of planar n-gons with specified side lengths, identified under isometry, plus a trivial line bundle is isomorphic to (n-2) times a canonical line bundle. We then discuss consequences for orientability, cobordism class, immersions, and parallelizability.