Geometry & Topology

Exotic open $4$–manifolds which are nonleaves

Carlos Meniño Cotón and Paul A Schweitzer

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Abstract

We study the possibility of realizing exotic smooth structures on finitely punctured simply connected closed 4–manifolds as leaves of a codimension-one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open 4–manifolds which are not diffeomorphic to any leaf of a codimension-one C2 foliation on a compact manifold. These examples include some exotic 4’s and exotic cylinders S3×.

Article information

Source
Geom. Topol., Volume 22, Number 5 (2018), 2791-2816.

Dates
Received: 25 November 2016
Revised: 13 November 2017
Accepted: 25 February 2018
First available in Project Euclid: 26 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.gt/1553565672

Digital Object Identifier
doi:10.2140/gt.2018.22.2791

Mathematical Reviews number (MathSciNet)
MR3811771

Zentralblatt MATH identifier
1397.37024

Subjects
Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx] 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 57R30: Foliations; geometric theory
Secondary: 57R55: Differentiable structures

Keywords
exotic $\mathbb{R}^4$ nonleaves codimension-one foliations

Citation

Meniño Cotón, Carlos; Schweitzer, Paul A. Exotic open $4$–manifolds which are nonleaves. Geom. Topol. 22 (2018), no. 5, 2791--2816. doi:10.2140/gt.2018.22.2791. https://projecteuclid.org/euclid.gt/1553565672


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References

  • T Asselmeyer-Maluga, C H Brans, Exotic smoothness and physics: differential topology and spacetime models, World Sci., Hackensack, NJ (2007)
  • O Attie, S Hurder, Manifolds which cannot be leaves of foliations, Topology 35 (1996) 335–353
  • Ž Bižaca, J Etnyre, Smooth structures on collarable ends of $4$–manifolds, Topology 37 (1998) 461–467
  • A Candel, L Conlon, Foliations, I, Graduate Studies in Mathematics 23, Amer. Math. Soc., Providence, RI (2000)
  • J Cantwell, L Conlon, Poincaré–Bendixson theory for leaves of codimension one, Trans. Amer. Math. Soc. 265 (1981) 181–209
  • J Cantwell, L Conlon, Every surface is a leaf, Topology 26 (1987) 265–285
  • S De Michelis, M H Freedman, Uncountably many exotic ${\bf R}^4$'s in standard $4$–space, J. Differential Geom. 35 (1992) 219–254
  • P R Dippolito, Codimension one foliations of closed manifolds, Ann. of Math. 107 (1978) 403–453
  • S K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279–315
  • M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357–453
  • 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
  • M Furuta, Monopole equation and the $\frac{11}8$–conjecture, Math. Res. Lett. 8 (2001) 279–291
  • S Ganzell, Ends of $4$–manifolds, Topology Proc. 30 (2006) 223–236
  • L J Gerstein, Basic quadratic forms, Graduate Studies in Mathematics 90, Amer. Math. Soc., Providence, RI (2008)
  • E Ghys, Une variété qui n'est pas une feuille, Topology 24 (1985) 67–73
  • R E Gompf, An infinite set of exotic ${\bf R}^4$'s, J. Differential Geom. 21 (1985) 283–300
  • R E Gompf, An exotic menagerie, J. Differential Geom. 37 (1993) 199–223
  • R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc., Providence, RI (1999)
  • R E Greene, Complete metrics of bounded curvature on noncompact manifolds, Arch. Math. $($Basel$)$ 31 (1978) 89–95
  • A Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa 16 (1962) 367–397
  • G Hector, Croissance des feuilletages presque sans holonomie, from “Differential topology, foliations and Gelfand–Fuks cohomology” (P A Schweitzer, editor), Lecture Notes in Math. 652, Springer (1978) 141–182
  • T Inaba, T Nishimori, M Takamura, N Tsuchiya, Open manifolds which are nonrealizable as leaves, Kodai Math. J. 8 (1985) 112–119
  • T Januszkiewicz, Characteristic invariants of noncompact Riemannian manifolds, Topology 23 (1984) 289–301
  • J Król, Exotic smooth $4$–manifolds and gerbes as geometry for quantum gravity, Acta Phys. Polon. B 40 (2009) 3079–3085
  • J A Álvarez López, R Barral Lijó, Bounded geometry and leaves, Math. Nachr. 290 (2017) 1448–1469
  • A Navas, Groups of circle diffeomorphisms, Univ. of Chicago Press (2011)
  • A Phillips, D Sullivan, Geometry of leaves, Topology 20 (1981) 209–218
  • F Quinn, Ends of maps, III: Dimensions $4$ and $5$, J. Differential Geom. 17 (1982) 503–521
  • P A Schweitzer, Riemannian manifolds not quasi-isometric to leaves in codimension one foliations, Ann. Inst. Fourier $($Grenoble$)$ 61 (2011) 1599–1631
  • J-P Serre, A course in arithmetic, Graduate Texts in Mathematics 7, Springer (1973)
  • C H Taubes, Gauge theory on asymptotically periodic $4$–manifolds, J. Differential Geom. 25 (1987) 363–430
  • L R Taylor, An invariant of smooth $4$–manifolds, Geom. Topol. 1 (1997) 71–89
  • L R Taylor, Impossible metric conditions on exotic $\mathbb R^4$'s, Asian J. Math. 12 (2008) 285–287
  • H Whitney, Differentiable manifolds, Ann. of Math. 37 (1936) 645–680
  • A Zeghib, An example of a $2$–dimensional no leaf, from “Geometric study of foliations” (T Mizutani, K Masuda, S Matsumoto, T Inaba, T Tsuboi, Y Mitsumatsu, editors), World Sci., River Edge, NJ (1994) 475–477