Exotic \(\mathbb{R}^4\)'s and positive isotropic curvature (Q2325734)

It is well known that \(\mathbb{R}^4\) admits uncountably many ``exotic'' structures, i.e., uncountably many non-diffeomorphic smooth structures. The author shows (Theorem 1.2) that no exotic \(\mathbb{R}^4\) admits a complete Riemannian metric with uniformly positive isotropic curvature and bounded geometry. Section 1 contains a statement of the problem. Section 2 treats infinite connected sums and shows that the diffeomorphism type of an infinite connected sum of some connected \(n\)-dimensional manifolds according to a locally finite graph is independent of the gluing maps used. Section 3 uses this observation to establish Theorem 1.2. The author also shows that no exotic \(\mathbb{S}^3\times\mathbb{R}\) (if one exists) can admit a complete Riemannian metric with uniformly positive isotropic curvature and bounded geometry.