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'''Algebraic topology''' is a branch of mathematics that uses [[abstract algebra]] to understand [[topological space]]s, and [[topology]] to understand [[abstract algebra]].   

'''Algebraic topology''' is a branch of mathematics that uses [[abstract algebra]] to understand [[topological space]]s, and [[topology]] to understand [[abstract algebra]].   

The primary method of algebraic topology is to identify some kind of relation between a [[group|Group (mathematics)]] (or a [[ring|Ring (mathematics)]] or other structure) and a [[topological space]].  Once such a relation is established, known results in one field can yield results in the other field.

The primary method of algebraic topology is to identify some kind of relation between a [[group (mathematics)|group]] (or a [[ring (mathematics)|ring]] or other structure) and a [[topological space]].  Once such a relation is established, known results in one field can yield results in the other field.

Important results in this field include the [[Poincare conjecture]], the unclassifiable nature of four-dimensional manifolds, and the proof that subgroups of free groups are necessarily free themselves.

Important results in this field include the [[Poincare conjecture]], the unclassifiable nature of four-dimensional manifolds, and the proof that subgroups of free groups are necessarily free themselves.