# The homology groups

We can now define the homology groups (AKA singular homology groups). In the below definitions, $${\textrm{Ker}}$$ denotes kernel and $${\textrm{Im}}$$ denotes image.

• Cycle: an element of $${\textrm{Ker}\partial_{n}}$$, i.e. an $${n}$$-surface in $${X}$$ that has no boundary
• Boundary: an element of $${\textrm{Im}\partial_{n+1}}$$, i.e. the boundary of an $${(n+1)}$$-volume in $${X}$$
• Homology class: an element of the homology group $${H_{n}(X)\equiv\textrm{Ker}\partial_{n}/\textrm{Im}\partial_{n+1}}$$, i.e. a coset consisting of homologous $${n}$$-cycles that can all be obtained from each other by adding the boundary of some $${(n+1)}$$-volume in $${X}$$
 ◊ We can think of a typical cycle as a loop for $${n=1}$$ or a sphere for $${n=2}$$. Then the typical boundary is the chain of $${n}$$-cycles that form the edge of an arbitrary surface for $${n=1}$$ or the surface of an arbitrary volume for $${n=2}.$$

The cylinder and punctured plane in the figure on triangulations depict examples of homologous loops, two 1-chains that are the boundary of a 2-chain. The abelian group $${H_{n}(X)}$$ is then generated by the cosets of non-homologous $${n}$$-cycles, thus counting the number of “$${n}$$-dimensional holes” in $${X}$$.