Introduction to Algebraic Topology Summary

Topology · Axiom Academy

SUMMARY Introduction to Algebraic Topology Unit 8 - Complete Review of Key Concepts Algebraic topology bridges geometry and algebra, allowing us to study topological spaces through algebraic invariants. This unit introduced fundamental concepts that reveal the deep structure of continuous transformations and topological properties. A homotopy is a continuous deformation between two continuous functions. Two functions and are homotopic if there exists a continuous map: The fundamental group is the group of homotopy classes of loops based at a point . The group operation is concatenation of loops: traverse the first loop, then traverse the second. The identity element is the constant loop at . A space is simply connected if it is path-connected and every loop can be continuously contracted to a point. Examples: is simply connected. The circle and torus are not simply connected. A covering space is a space with a map such that each point in has a neighborhood evenly covered by . Example: The real line covers the circle via . Van Kampen's Theorem allows us to compute the fundamental group of a space by decomposing it into simpler pieces. If and , then is the free product with amalgamation . The Euler characteristic is a topological invariant defined for polyhedra and simplicial complexes: where is the number of vertices, is edges, and is faces. Examples: for sphere, for torus, for Klein bottle.

This is the written version of the interactive lesson above. See the full Topology course.