Torus Topology/Fundamental Group

Fundamental Group of the Torus The fundamental group is one of the most important algebraic invariants in topology, providing information about the "holes" in a topological space. For the torus, this concept reveals its distinctive topological structure. Definition of the Fundamental Group The fundamental group of a topological space with basepoint consists of equivalence classes of loops based at , where loops are considered equivalent if they can be continuously deformed into one another while keeping the basepoint fixed. The Fundamental Group of the Torus The fundamental group of the torus is: This can be understood as follows: 1. The torus has two basic types of non-contractible loops: * Meridian loops (going around the "tube") * Longitudinal loops (going through the "hole") 2. Any loop on the torus can be expressed as a combination of these two basic loops. 3. The two generators and correspond to these basic loops, and they commute: Calculation Methods Using Van Kampen's Theorem We can calculate using the Seifert-van Kampen theorem: 1. Represent the torus as a square with opposite edges identified. 2. Remove a small open disk to get a space that deformation retracts to a wedge of two circles. 3. Let be a neighborhood of the removed disk. 4. Apply van Kampen's theorem to the covering . (free group on two generators) (trivial group) The theorem yields: Using the Universal Cover Another approach uses the universal cover of the torus: 1. The universal cover of is . 2. The covering map is given by . 3. The deck transformation group is , acting by translations. 4. The fundamental group is isomorphic to the deck transformation group, hence . Homomorphisms and Representations The fundamental group of the torus admits various homomorphisms and representations: Relation to Covering Spaces The subgroups of correspond to covering spaces of the torus: 1. For any subgroup , there exists a covering space with . 2. Finite-index subgroups correspond to finite-sheeted covering spaces, which are themselves tori. 3. The universal cover corresponds to the trivial subgroup . Applications in Physics and Geometry The fundamental group of the torus has applications in: 1. Crystallography - describing symmetry groups of crystal structures 2. String theory - compactification on tori 3. Dynamical systems - flows on the torus 4. Electrical circuit theory - analyzing circuits with toroidal topology Exercises 1. Prove that any two loops representing the same element in are homotopic. 2. Calculate the fundamental group of the n-dimensional torus . 3. Find all the subgroups of of index 2 and describe the corresponding double covers of the torus. 4. Show that the commutator subgroup of is trivial. 5. Prove that if is continuous, then cannot be injective.