Algebraic Topology

The field that assigns algebraic invariants (groups, rings, modules) such as homotopy and homology groups to topological spaces to classify and study them up to continuous deformation (homotopy or homeomorphism), capturing global qualitative features.

The branch of topology that studies topological spaces by assigning algebraic invariants—such as fundamental groups, homology and cohomology groups, and homotopy groups—and using algebraic methods to classify, compare, and compute topological phenomena.