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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|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.

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.

== The Homotopy Groups ==

== The Homotopy Groups ==