three elements on the right of these equations, creating a two-dimensional subgroup of . So the 0th homology group is isomorphic to , with a basis given (for example) by the image of the 0-cycle (). Indeed, all three vertices become equal in the quotient group; this expresses the fact that is connected. Next, the group of 1-cycles is the kernel of the homomorphism ∂ above, which is isomorphic to , with a basis given (for example) by . (A picture reveals that this 1-cycle goes around the triangle in one of the two possible directions.) Since , the group of 1-boundaries is zero, and so the 1st homology group is isomorphic to . This makes precise the idea that the triangle has one 1-dimensional hole. Next, since by definition there are no 2-cycles, (the trivial group). Therefore the 2nd homology group is zero. The same is true for for all not equal to 0 or 1. Therefore, the homological connectivity of the triangle is 0 (it is the largest for which the reduced homology groups up to are trivial). Homology groups of higher-dimensional simplices Let be a tetrahedron (without its interior), viewed as a simplicial complex. Thus has four 0-dimensional vertices, six 1-dimensional edges, and four 2-dimensional faces. The construction of the homology groups of a tetrahedron is described in detail here. It turns out that is isomorphic to , is isomorphic to too, and all other groups are trivial.Therefore, the homological connectivity of the tetrahedron is 0. If the tetrahedron contains its interior, then is trivial too. In general, if is a -dimensional simplex, the following holds: If is considered without its interior, then and and all other homologies are trivial; If is considered with its interior, then and all other homologies are trivial. Simplicial maps Let S and T be simplicial complexes. A simplicial map f from S to T is a function from the vertex set of S to the vertex set of T such that the image of each simplex in S (viewed as a set of vertices) is a simplex in T. A simplicial map f: determines a homomorphism of homology groups for each integer k. This is the homomorphism associated to a chain map from the chain complex of S to the chain complex of T. Explicitly, this chain map is given on k-chains by if are all distinct, and otherwise . This