Homology of a complex

The starting point for various homological constructions. Let $A$ be an Abelian category. A chain complex in $A$ is a family $K_\bullet = (K_n,d_n)$ of objects $(K_n)_{n \in \mathbf{Z}}$ in $A$ and morphisms $d_n : K_n \rightarrow K_{n-1}$ such that $d_{n-1} \circ d_n = 0$ for all $n$. The quotient object $\ker d_n / \text{im} d_{n+1}$ is called the $n$-th homology of the complex $K_\bullet$ and is denoted by $H_n(K_\bullet)$. The family $(H_n(K_\bullet))_{n \in \mathbf{Z}}$ is also denoted by $H_\bullet(K_\bullet)$. 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 $X$ defines a chain complex in the category $\textsf{Ab}$ of Abelian groups: $(C_n(X),\partial_n)$. Here $C_n(X)$ is the group of $n$-dimensional singular chains of $X$, while $\partial_n$ is the boundary homomorphism. The $n$-th homology of this complex is said to be the $n$-th singular homology group of $X$ and is denoted by $H_n(X)$.

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