Homology of a complex

The starting point for various homological constructions. Let be an Abelian category and let be a chain complex in , i.e. a family of objects in and morphisms such that for all . The quotient object is called the -th homology of the complex and is denoted by . The family is also denoted by . The concept of the homology of a complex serves as the base for a number of important constructions in homological algebra, commutative algebra, algebraic geometry, and topology. Thus, in topology, each topological space defines a chain complex in the category of Abelian groups: . Here is the group of -dimensional singular chains of , while is the boundary homomorphism. The -th homology of this complex is said to be the -th singular homology group of and is denoted by . The concept of the cohomology of a cochain complex is defined in a dual manner.

References

[1]	S. MacLane, "Homology" , Springer (1963) Zbl 0818.18001 Zbl 0328.18009

References

[a1]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303