  We consider the periodic cubic-quintic nonlinear Schr\"odinger equation
\begin{align}\label{cqnls_abstract} (i\partial_t +\Delta )u=\mu_1 |u|^2 u+\mu_2
|u|^4 u\tag{CQNLS} \end{align} on the three-dimensional torus $\mathbb{T}^3$
with $\mu_1,\mu_2\in \mathbb{R} \setminus\{0\}$. As a first result, we
establish the small data well-posedness of \eqref{cqnls_abstract} for
arbitrarily given $\mu_1$ and $\mu_2$. By adapting the crucial perturbation
arguments in \cite{zhang2006cauchy} to the periodic setting, we also prove that
\eqref{cqnls_abstract} is always globally well-posed in $H^1(\mathbb{T}^3)$ in
the case $\mu_2>0$.
