In the present paper we compute in full mathematical rigor the topological derivative of the elastic homogenized coefficients of periodic microstructures. The expression, here proven for the topological derivative, was successfully used for optimizing homogenized coefficients in an alternate directions optimization algorithm (jointly with the shape derivative) in [C. Barbarosie and A.-M. Toader, Struct. Multidiscip. Optim., 40 (2010), pp. 393-408]. This optimization problem can be viewed as a control problem: the homogenized coefficients are controlled by the shape/topology of the subdomain occupied by material in the cellular problem. The main ingredients for proving the formula of the topological derivative are a generalized adjoint method and a Dirichlet-to-Neumann operator. The techniques employed are general and may be adapted to different functionals depending on elliptic PDEs under periodicity conditions.