be an oriented -simplex, viewed as a basis element of . The boundary operator is the homomorphism defined by: where the oriented simplex is the face of , obtained by deleting its vertex. In , elements of the subgroup are referred to as cycles, and the subgroup is said to consist of boundaries. Boundaries of boundaries Because , where is the second face removed, . In geometric terms, this says that the boundary of anything has no boundary. Equivalently, the abelian groups form a chain complex. Another equivalent statement is that is contained in . As an example, consider a tetrahedron with vertices oriented as . By definition, its boundary is given by: . The boundary of the boundary is given by: . Homology groups The homology group of is defined to be the quotient abelian group It follows that the homology group is nonzero exactly when there are -cycles on which are not boundaries. In a sense, this means that there are -dimensional holes in the complex. For example, consider the complex obtained by gluing two triangles (with no interior) along one edge, shown in the image. The edges of each triangle can be oriented so as to form a cycle. These two cycles are by construction not boundaries (since every 2-chain is zero). One can compute that the homology group is isomorphic to , with a basis given by the two cycles mentioned. This makes precise the informal idea that has two "1-dimensional holes". Holes can be of different dimensions. The rank of the homology group, the number is called the Betti number of . It gives a measure of the number of -dimensional holes in . Example Homology groups of a triangle Let be a triangle (without its interior), viewed as a simplicial complex. Thus has three vertices, which we call , and three edges, which are 1-dimensional simplices. To compute the homology groups of , we start by describing the chain groups : is isomorphic to with basis is isomorphic to with a basis given by the oriented 1-simplices , , and . is the trivial group, since there is no simplex like because the triangle has been supposed without its interior. So are the chain groups in other dimensions. The boundary homomorphism : is given by: Since , every 0-chain is a cycle (i.e. ); moreover, the group of the 0-boundaries is generated by the