Fundamental Group - Relationship To First Homology Group

Relationship To First Homology Group

The fundamental groups of a topological space X are related to its first singular homology group, because a loop is also a singular 1-cycle. Mapping the homotopy class of each loop at a base point x₀ to the homology class of the loop gives a homomorphism from the fundamental group π₁(X, x₀) to the homology group H₁(X). If X is path-connected, then this homomorphism is surjective and its kernel is the commutator subgroup of π₁(X, x₀), and H₁(X) is therefore isomorphic to the abelianization of π₁(X, x₀). This is a special case of the Hurewicz theorem of algebraic topology.