Date: 2024-05-19 | Article ID: 28539 |

Homology groups are examples of topological invariants: topologically equivalent spaces have the same homology groups. The idea behind homology groups is to consider a special family of topological spaces C for which the concept of a *boundary* makes sense, namely spaces made of simple polyhedra, and to use maps from those spaces into another topological space X as a way of exploring the topology of X. Roughly, the nth **homology group** of X describes continuous maps into X from those special n-dimensional spaces C that cannot be extended to a continuous map into X from any of the special (n+1)-dimensional spaces whose boundary is C. This article introduces homology groups. A brief overview of related topological invariants called **cohomology groups** and **cohomology rings** is also included.

**2024-05-19**(adjusted two section titles, moved one statement to a footnote, added a derivation of the homology groups of an n-dimensional torus)**2024-04-20**(added references to a book by Whitehead, added a result about the nth homology group of an n-dimensional manifold, added results related to Hurewicz isomorphism, added definition of homology equivalence)**2024-04-16**(expanded the section about notation, separated the section about products and sums into two sections and expanded the tensor product section, reworded the comparison between bordism homology and singular homology, separated the section about torsion into a section about abelian groups and a section about compact manifolds and slightly expanded both of them, added a section reviewing the definitions of ring and principal ideal domain and field, added sections about cohomology groups and cohomology rings, added sections about (co)homology with coefficients in a field, added a section about cartesian products of spheres)**2024-02-25**(first version)

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