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:
“Artists have a double relationship towards nature: they are her master and her slave at the same time. They are her slave in so far as they must work with means of this world so as to be understood; her master in so far as they subject these means to their higher goals and make them subservient to them.”
—Johann Wolfgang Von Goethe (1749–1832)
“Living in cities is an art, and we need the vocabulary of art, of style, to describe the peculiar relationship between man and material that exists in the continual creative play of urban living. The city as we imagine it, then, soft city of illusion, myth, aspiration, and nightmare, is as real, maybe more real, than the hard city one can locate on maps in statistics, in monographs on urban sociology and demography and architecture.”
—Jonathan Raban (b. 1942)
“...Women’s Studies can amount simply to compensatory history; too often they fail to challenge the intellectual and political structures that must be challenged if women as a group are ever to come into collective, nonexclusionary freedom.”
—Adrienne Rich (b. 1929)