Diagonal Chain Approximation for the torus

We write down the chain complex C∗ of the universal cover of the torus, and indicate how to do likewise for any 2-dimensional CW complex arising as the 2-skeleton of a K(G, 1). We exhibit a contracting chain homotopy for this chain complex, and again explain how to do so for the chain complex of the universal cover of our more general CW complex. This yields a contracting homotopy for the tensor product C∗ ⊗ C∗, and formulae of Davis ([2]) allow us to construct a diagonal chain approximation ∆0 : C∗ → C∗ ⊗C∗, close to the diagonal map x 7→ (x, x). Formulae of Trotter ([3]) exist for this, but include the possibility of a 3-cell. In the case of a 2-dimensional group such as Z⊕ Z, the fundamental group of the torus, Trotter’s formulae are unnecessarily long. He remarks that the solution of the word problem is pre-requisite to using the method of contracting chain homotopies; in this case, however, a canonical form for a word is readily available, and so we follow the method through.