Homology group

Homology groups

Given a sufficiently-nice topological space ${\displaystyle X}$, a choice of appropriate homology theory, and a chain complex ${\displaystyle (C_{\bullet },d_{\bullet })}$ associated to ${\displaystyle X}$ that is compatible with that homology theory, the ${\displaystyle n}$th homology group${\displaystyle H_{n}(X)}$ is then given by the quotient group${\displaystyle H_{n}(X)=Z_{n}/B_{n}}$ of ${\displaystyle n}$-cycles (${\displaystyle n}$-dimensional cycles) modulo ${\displaystyle n}$-dimensional boundaries. In other words, the elements of ${\displaystyle H_{n}(X)}$, called homology classes, are equivalence classes whose representatives are ${\displaystyle n}$-cycles, and any two cycles are regarded as equal in ${\displaystyle H_{n}(X)}$ if and only if they differ by the addition of a boundary. This also implies that the "zero" element of ${\displaystyle H_{n}(X)}$ is given by the group of ${\displaystyle n}$-dimensional boundaries, which also includes formal sums of such boundaries.