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.
Read more about this topic: Fundamental Group
Famous quotes containing the words relationship and/or group:
“If one could be friendly with women, what a pleasure—the relationship so secret and private compared with relations with men. Why not write about it truthfully?”
—Virginia Woolf (1882–1941)
“The trouble with tea is that originally it was quite a good drink. So a group of the most eminent British scientists put their heads together, and made complicated biological experiments to find a way of spoiling it. To the eternal glory of British science their labour bore fruit.”
—George Mikes (b. 1912)