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)
“... the Wall became a magnet for citizens of every generation, class, race, and relationship to the war perhaps because it is the only great public monument that allows the anesthetized holes in the heart to fill with a truly national grief.”
—Adrienne Rich (b. 1929)
“I can’t think of a single supposedly Black issue that hasn’t wasted the original Black target group and then spread like the measles to outlying white experience.”
—June Jordan (b. 1936)