## Algebraic & Geometric Topology

### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 19, Number 3 (2019), 1299-1339.

Dates
Revised: 3 October 2018
Accepted: 2 December 2018
First available in Project Euclid: 29 May 2019

https://projecteuclid.org/euclid.agt/1559095427

Digital Object Identifier
doi:10.2140/agt.2019.19.1299

Mathematical Reviews number (MathSciNet)
MR3954283

Zentralblatt MATH identifier
07078605

#### Citation

Calcut, Jack S; Gompf, Robert E. On uniqueness of end sums and $1$–handles at infinity. Algebr. Geom. Topol. 19 (2019), no. 3, 1299--1339. doi:10.2140/agt.2019.19.1299. https://projecteuclid.org/euclid.agt/1559095427

#### References

• D R Anderson, Homotopy type and obstructions to smoothing, Topology 7 (1968) 333–348
• I Belegradek, Obstructions to nonpositive curvature for open manifolds, Proc. Lond. Math. Soc. 109 (2014) 763–784
• J Bennett, Exotic smoothings via large $\mathbb{R}^4$'s in Stein surfaces, PhD thesis, The University of Texas at Austin (2015) Available at \setbox0\makeatletter\@url https://tinyurl.com/Bennett-UT {\unhbox0
• J Bennett, Exotic smoothings via large $\mathbb{R}^4$'s in Stein surfaces, Algebr. Geom. Topol. 16 (2016) 1637–1681
• Ž Bižaca, R E Gompf, Elliptic surfaces and some simple exotic ${\mathbb R}^4$'s, J. Differential Geom. 43 (1996) 458–504
• M Brown, Locally flat imbeddings of topological manifolds, Ann. of Math. 75 (1962) 331–341
• J S Calcut, P V Haggerty, Connected sum at infinity and $4$–manifolds, Algebr. Geom. Topol. 14 (2014) 3281–3303
• J S Calcut, H C King, L C Siebenmann, Connected sum at infinity and Cantrell–Stallings hyperplane unknotting, Rocky Mountain J. Math. 42 (2012) 1803–1862
• J C Cantrell, Separation of the $n$–sphere by an $(n{-}1)$–sphere, Trans. Amer. Math. Soc. 108 (1963) 185–194
• K Cieliebak, Y Eliashberg, From Stein to Weinstein and back: symplectic geometry of affine complex manifolds, American Mathematical Society Colloquium Publications 59, Amer. Math. Soc., Providence, RI (2012)
• R Connelly, A new proof of Brown's collaring theorem, Proc. Amer. Math. Soc. 27 (1971) 180–182
• J Dancis, General position maps for topological manifolds in the ${2\over 3}$rds range, Trans. Amer. Math. Soc. 216 (1976) 249–266
• D A Edwards, H M Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics 542, Springer (1976)
• M H Freedman, F Quinn, Topology of $4$–manifolds, Princeton Mathematical Series 39, Princeton Univ. Press (1990)
• M H Freedman, L R Taylor, A universal smoothing of four-space, J. Differential Geom. 24 (1986) 69–78
• H Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z. 33 (1931) 692–713
• R Geoghegan, Topological methods in group theory, Graduate Texts in Mathematics 243, Springer (2008)
• R E Gompf, Three exotic ${\mathbb R}\sp{4}$'s and other anomalies, J. Differential Geom. 18 (1983) 317–328
• R E Gompf, An infinite set of exotic ${\mathbb R}^4$'s, J. Differential Geom. 21 (1985) 283–300
• R E Gompf, A moduli space of exotic ${\mathbb R}^4$'s, Proc. Edinburgh Math. Soc. 32 (1989) 285–289
• R E Gompf, Constructing Stein manifolds after Eliashberg, from “New perspectives and challenges in symplectic field theory” (M Abreu, F Lalonde, L Polterovich, editors), CRM Proc. Lecture Notes 49, Amer. Math. Soc., Providence, RI (2009) 229–249
• R E Gompf, Minimal genera of open $4$–manifolds (2013)
• R E Gompf, Minimal genera of open $4$–manifolds, Geom. Topol. 21 (2017) 107–155
• R E Gompf, Quotient manifolds of flows, J. Knot Theory Ramifications 26 (2017) art. id. 1740005
• R E Gompf, Group actions, corks and exotic smoothings of $\mathbb{R}^4$, Invent. Math. 214 (2018) 1131–1168
• R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc., Providence, RI (1999)
• M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer (1994)
• T Homma, On the imbedding of polyhedra in manifolds, Yokohama Math. J. 10 (1962) 5–10
• B Hughes, A Ranicki, Ends of complexes, Cambridge Tracts in Mathematics 123, Cambridge Univ. Press (1996)
• M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504–537
• R C Kirby, L C Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton Univ. Press (1977)
• K W Kwun, F Raymond, Mapping cylinder neighborhoods, Michigan Math. J. 10 (1963) 353–357
• B Mazur, On embeddings of spheres, Bull. Amer. Math. Soc. 65 (1959) 59–65
• M L Mihalik, Semistability at the end of a group extension, Trans. Amer. Math. Soc. 277 (1983) 307–321
• R Myers, End sums of irreducible open $3$–manifolds, Quart. J. Math. Oxford Ser. 50 (1999) 49–70
• F Quinn, Ends of maps, III: Dimensions $4$ and $5$, J. Differential Geom. 17 (1982) 503–521
• P Sparks, The double $n$–space property for contractible $n$–manifolds, Algebr. Geom. Topol. 18 (2018) 2131–2149
• J R Stallings, On infinite processes leading to differentiability in the complement of a point, from “Differential and combinatorial topology” (S S Cairns, editor), Princeton Univ. Press (1965) 245–254
• L R Taylor, An invariant of smooth $4$–manifolds, Geom. Topol. 1 (1997) 71–89
• F C Tinsley, D G Wright, Some contractible open manifolds and coverings of manifolds in dimension three, Topology Appl. 77 (1997) 291–301