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

Abstract

We show that no exotic ${\mathbb{R}}^4$ admits a complete Riemannian metric with uniformly positive isotropic curvature and with bounded geometry. This is essentially 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 diffeomorphism type of an infinite connected sum of some connected smooth $n$-manifolds ($n\ge 2$) according to a locally finite graph does not depend on the gluing maps used.

arXiv comment5 pages