On uniqueness of end sums and $1$–handles at infinity

Abstract

For oriented manifolds of dimension at least $4$ that are simply connected at infinity, it is known that end summing is a uniquely defined operation. Calcut and Haggerty showed that more complicated fundamental group behavior at infinity can lead to nonuniqueness. We examine how and when uniqueness fails. Examples are given, in the categories top, pl and diff, of nonuniqueness that cannot be detected in a weaker category (including the homotopy category). In contrast, uniqueness is proved for Mittag-Leffler ends, and generalized to allow slides and cancellation of (possibly infinite) collections of $0$– and $1$–handles at infinity. Various applications are presented, including an analysis of how the monoid of smooth manifolds homeomorphic to ${ℝ}^4$ acts on the smoothings of any noncompact $4$–manifold.