section_path: Topics/Algebraic topology
section_name: Algebraic topology
paragraph_text: Algebraic topology
  
  Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
  
  The most important of these invariants are homotopy groups, homology, and cohomology.
  
  Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.