On uniqueness of end sums and 1-handles at infinity

TL;DR

This paper investigates the conditions under which the operation of end summing in noncompact manifolds is unique, revealing cases of nonuniqueness and establishing criteria for when uniqueness holds, with applications to 4-manifold smoothings.

Contribution

It identifies when end sum uniqueness fails and proves conditions for its validity, extending to handle slides and cancellations at infinity.

Findings

Nonuniqueness occurs with complicated fundamental group behavior at infinity.

Uniqueness holds for Mittag-Leffler ends.

Applications include analyzing smoothings of noncompact 4-manifolds.

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. The present paper examines 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.