Torus Topology/Definition and Properties

Definition and Properties A torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle but does not intersect it. More formally, a torus can be defined as the product of two circles: . Topological Definition From a topological perspective, a torus is a closed surface that is homeomorphic to the product of two circles. This means that a torus has the same topological properties as the product space , where represents a circle. Key Properties The torus possesses several distinctive properties that make it significant in topology: Genus: The torus has genus 1, meaning it has exactly one "hole". This distinguishes it from a sphere (genus 0) and higher-genus surfaces. Euler Characteristic: The Euler characteristic of a torus is , which can be calculated using the formula , where is the genus. Orientability: The torus is an orientable surface, meaning it has two distinct sides. Compactness: The torus is a compact surface without boundary. Parallelizability: Unlike the sphere, the torus is parallelizable, meaning its tangent bundle is trivial. Flat Structure: The torus can be given a flat geometry, which means it can be equipped with a Riemannian metric of zero Gaussian curvature. Quotient Space Representation The torus can be represented as a quotient space of the Euclidean plane. Specifically, , where acts on by translation. This representation is often visualized as identifying opposite edges of a square, creating the familiar "video game screen" topology where exiting one edge causes reentry from the opposite edge. Classification The torus plays a crucial role in the classification of compact surfaces. According to the classification theorem, every compact, connected, orientable surface is homeomorphic to either a sphere or a connected sum of tori. Exercises 1. Prove that the Euler characteristic of the torus is zero by triangulating the surface and counting vertices, edges, and faces. 2. Show that the fundamental group of the torus is isomorphic to . 3. Demonstrate that any continuous map from a sphere to a torus is homotopic to a constant map. 4. Prove that the torus cannot be embedded in without self-intersection. 5. Calculate the homology groups of the torus directly from its CW-complex structure.