Explore Homology on MathWorld

Homology is a mathematical concept used in many branches of algebra and topology that involves a topological invariant known as a homology group.

Homology is a graduate-level concept that would be first encountered in a topology course.


Algebra: (1) Algebra is a subject taught in grade school and high school, sometimes referred to as "arithmetic", that includes the solution of polynomial equations in one or more variables and basic properties of functions and graphs. (2) In higher mathematics, the term algebra generally refers to abstract algebra, which involves advanced topics that deal with abstract algebraic structures rather than the usual number systems. (3) In topology, an algebra is a vector space that also possesses a vector multiplication.
Group: A mathematical group is a set of elements and a binary operation that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property.
Topology: (1) As a branch of mathematics, topology is the mathematical study of object's properties that are preserved through deformations, twistings, and stretchings. (2) As a set, a topology is a set along with a collection of subsets that satisfy several defining properties.

Classroom Articles on Topology (Up to Graduate Level)

  • Closed Set
  • Neighborhood
  • Differential Topology
  • Open Set
  • Dimension
  • Point-Set Topology
  • Homeomorphism
  • Projective Plane
  • Homotopy
  • Projective Space
  • Knot
  • Subspace
  • Link
  • Tangent Space
  • Manifold
  • Topological Space
  • Metric
  • Torus
  • Metric Space
  • Vector Bundle
  • Möbius Strip