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

TL;DR

This paper proves that exotic alf4^4f4s cannot admit complete Riemannian metrics with uniformly positive isotropic curvature and bounded geometry, and explores the diffeomorphism types of infinite connected sums of smooth manifolds.

Contribution

It establishes a non-existence result for certain metrics on exotic alf4^4f4s and shows the invariance of diffeomorphism types of infinite connected sums under different gluing maps.

Findings

Exotic alf4^4f4s do not admit complete metrics with positive isotropic curvature.

The diffeomorphism type of infinite connected sums is independent of the gluing maps.

The result follows from and extends previous work by Hu.

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 ($n\geq 2$) according to a locally finite graph does not depend on the gluing maps used.