For oriented manifolds of dimension at least 4 that are simply connected at infinity, it is known that end summing (the noncompact analogue of boundary summing) is a uniquely defined operation. Calcut and Haggerty showed that more complicated fundamental group behavior at infinity can lead to nonuniqueness. We will examine how and when uniqueness fails. There are examples in various categories (homotopy, TOP, PL and DIFF) of nonuniqueness that cannot be detected in a weaker category. In contrast, we will present a group-theoretic condition that guarantees uniqueness. As an application, the monoid of smooth manifolds homeomorphic to $\mathbb{R}^4$ acts on the set of smoothings of any noncompact 4-manifold.

This work is joint with Jack Calcut.