# Homology

In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes.

##### Relations

In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is conce...

Edit details Edit relations Attach new author Attach new topic Attach new resource

##### Resources

treated in Computing homology groups | Algebraic Topology | NJ Wildberger

9.0 rating 3.0 level 9.0 clarity 3.0 background – 1 rating

The definition of the homology groups H_n(X) of a space X, say a simplicial complex, is quite abstrac...

treated in Homology Theory — A Primer

10.0 rating 6.0 level 9.0 clarity 4.0 background – 1 rating

This series on topology has been long and hard, but we’re are quickly approaching the topics where we...