Mathematics - Topology - Algebraic topology - Counting Holes with Algebra

10 steps: (1) Algebraic topology: assign algebraic objects (groups, rings) to topological spaces to distinguish them. (2) FUNDAMENTAL GROUP π₁(X): the group of loops in X (up to continuous deformation). (3) π₁(circle) = ℤ. A loop can wind around 0, 1, 2, ... times (or negative = clockwise). (4) π₁(s