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 to, relationship and/or group:
“Poetry is above all a concentration of the power of language, which is the power of our ultimate relationship to everything in the universe.”
—Adrienne Rich (b. 1929)
“We must introduce a new balance in the relationship between the individual and the government—a balance that favors greater individual freedom and self-reliance.”
—Gerald R. Ford (b. 1913)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)