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In this paper, we first prove global well-posedness for the defocusing cubicnonlinear Schr\"odinger equation (NLS) on 4-dimensional tori - either rationalor irrational - and with initial data in $H^1$. Furthermore, we prove that if amaximal-lifespan solution of the focusing cubic NLS $u: I\times\mathbb{T}^4\to\mathbb{C}$ satisfies $\sup_{t\inI}\|u(t)\|_{\dot{H}^1(\mathbb{T}^4)}<\|W\|_{\dot{H}^1(\mathbb{R}^4)}$, then itis a global solution. $W$ denotes the ground state on Euclidean space, which isa stationary solution of the corresponding focusing equation in $\mathbb{R}^4$.

Global Well-Posedness of the Energy-Critical Nonlinear Schr\"odinger Equation on $\mathbb{T}^4$