$4$-manifolds with boundary and fundamental group $\mathbb{Z}$

Abstract

We classify topological 4-manifolds with boundary and fundamental group Z, under some assumptions on the boundary. We apply this to classify surfaces in simply-connected 4manifolds with S 3 boundary, where the fundamental group of the surface complement is Z. We then compare these homeomorphism classifications with the smooth setting. For manifolds, we show that every Hermitian form over Z[t ±1 ] arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group Z. For surfaces we have a similar result, and in particular we show that every 2-handlebody with S 3 boundary contains a pair of exotic discs.

(Un)knotted nullhomotopic spheres and disks in 4-manifolds produced via knot surgery

“…For smooth knottings of surfaces in the 4-ball the reader is directed towards Hayden [24], Hayden-S. Kim-Miller-Park-Sundberg [25] Hayden-Kjuchukova-Krishna-Miller-Powell-Sunukjian [26], Juhász-Miller-Zemke [23] and Oba [33]. Conway-Piccirillo-Powell [11] showed that every 2-handlebody with a 3-sphere boundary contains a pair of smoothly knotted disks [11].…”