Volume 19, issue 3 (2019)

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

Jack S Calcut and Robert E Gompf

Algebraic & Geometric Topology 19 (2019) 1299–1339
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.

Keywords
end sum, connected sum at infinity, Mittag-Leffler, semistable end, exotic smoothing
Mathematical Subject Classification 2010
Primary: 57N99, 57Q99, 57R99
Publication
Received: 8 January 2018
Revised: 3 October 2018
Accepted: 2 December 2018
Published: 21 May 2019
Authors
 Jack S Calcut Department of Mathematics Oberlin College Oberlin, OH United States http://www.oberlin.edu/faculty/jcalcut/ Robert E Gompf Department of Mathematics The University of Texas at Austin Austin, TX United States https://www.ma.utexas.edu/users/gompf/