Any simplicial map ϕ: K → L between simplicial complexes K and L induces well-defined homomorphisms ϕq: Cq(K) → Cq(L) of chain groups, where
ϕq(〈v0, v1, . . . , vq〉) = 〈ϕ(v0), ϕ(v1), . . . , ϕ(vq)〉
whenever v0, v1, . . . , vq span a simplex of K. Note that ϕq (〈v0, v1, . . . , vq〉) = 0 unless ϕ(v0), ϕ(v1), . . . , ϕ(vq) are all distinct.
Now ϕq−1 ◦ ∂q = ∂q ◦ ϕq for each integer q. Therefore ϕq(Zq(K)) ⊂ Zq(L) and ϕq(Bq(K)) ⊂ Bq(L) for all integers q. It follows that any simplicial map ϕ: K → L induces well-defined homomorphisms ϕ∗: Hq(K) → Hq(L) of homology groups, where ϕ∗([z]) = [ϕq(z)] for all q-cycles z ∈ Zq(K). It is a trivial exercise to verify that if K, L and M are simplicial complexes and if ϕ: K → L and ψ: L → M are simplicial maps then the induced homomorphisms of homology groups satisfy (ψ ◦ ϕ)∗ = ψ∗ ◦ ϕ∗.
