  We prove the following result: Let $(X,g_0)$ be a complete, connected
4-manifold with uniformly positive isotropic curvature and with bounded
geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the
form $\mathbb{S}^3 \times \mathbb{R} /G$, where $G$ is a fixed point free
discrete subgroup of the isometry group of the standard metric on
$\mathbb{S}^3\times \mathbb{R}$, such that $X$ is diffeomorphic to a (possibly
infinite) connected sum of copies of $\mathbb{S}^4,\mathbb{RP}^4$ and/or
members of $\mathcal{F}$. This extends recent work of Chen-Tang-Zhu and Huang.
We also extend the above result to the case of orbifolds. The proof uses Ricci
flow with surgery on complete orbifolds.
