On The Topology of Polygonal Meshes

This paper is an introductory and informal exposition on the topology of polygonal meshes. We begin with a broad overview of topological notions and discuss how homeomorphisms, homotopy, and homology can be used to characterise topology. We move on to define polygonal meshes and make a distinction between intrinsic topology and extrinsic topology which depends on the space in which the mesh is immersed. A distinction is also made between quantitative topological properties and qualitative properties. Next, we outline proofs of the Euler and the Euler-Poincar\'e formulas. The Betti numbers are then defined in terms of the Euler-Poincar\'e formula and other mesh statistics rather than as cardinalities of the homology groups which allows us to avoid abstract algebra. Finally, we discuss how it is possible to cut a polygonal mesh such that it becomes a topological disc.