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We show that no exotic $\mathbb{R}^4$ admits a complete Riemannian metricwith uniformly positive isotropic curvature and with bounded geometry. This isessentially a corollary of the main result in [Hu1], and was stated in [Hu2]without proof. In the process of the proof we also show that the diffeomorphismtype of an infinite connected sum of some connected smooth $n$-manifolds($n\geq 2$) according to a locally finite graph does not depend on the gluingmaps used.

Exotic $\mathbb{R}^4$'s and positive isotropic curvature