  We consider the energy critical nonlinear Schr\"odinger equation on periodic
domains of the form R^m x T^{4-m} with m=0,1,2,3. Assuming that a certain L^4
Strichartz estimate holds for solutions to the corresponding linear
Schr\"odinger equation, we prove that the nonlinear problem is locally
well-posed in the energy space. Then we verify that the L^4 estimate holds if
m=2,3, leaving open the cases m=0,1.
