Exotic smoothings via large ℝ4’s in Stein surfaces

Bennett, Julia¹

¹Department of Mathematics, University of Texas at Austin, Austin, TX 78712, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1637-1681

Résumé

We study the relationship between exotic ℝ4’s and Stein surfaces as it applies to smoothing theory on more general open 4–manifolds. In particular, we construct the first known examples of large exotic ℝ4’s that embed in Stein surfaces. This relies on an extension of Casson’s embedding theorem for locating Casson handles in closed 4–manifolds. Under sufficiently nice conditions, we show that using these ℝ4’s as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant.

Détail

DOI : 10.2140/agt.2016.16.1637

Classification : 57N13, 57R55
Keywords: exotic smooth structures, open $4$–manifolds, Stein surfaces

Bibliographie

[1] S Akbulut, R Matveyev, Exotic structures and adjunction inequality, Turkish J. Math. 21 (1997) 47

[2] Ž Bižaca, J Etnyre, Smooth structures on collarable ends of 4–manifolds, Topology 37 (1998) 461

[3] J S Calcut, R E Gompf, On uniqueness of end-sums and 1–handles at infinity,

[4] A J Casson, Three lectures on new infinite constructions in 4–dimensional manifolds, from: "À la recherche de la topologie perdue" (editors L Guillou, A Marin), Progr. Math. 62, Birkhäuser (1986) 201

[5] F Ding, Smooth structures on some open 4–manifolds, Topology 36 (1997) 203

[6] S K Donaldson, Connections, cohomology and the intersection forms of 4–manifolds, J. Differential Geom. 24 (1986) 275

[7] F Fang, Smooth structures on Σ × R, Topology Appl. 99 (1999) 123

[8] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357

[9] M Freedman, R Kirby, A geometric proof of Rochlin’s theorem, from: "Algebraic and geometric topology" (editor R J Milgram), Proc. Sympos. Pure Math. XXXII Part 2, Amer. Math. Soc. (1978) 85

[10] M H Freedman, F Quinn, Topology of 4–manifolds, 39, Princeton Univ. Press (1990)

[11] M H Freedman, L R Taylor, A universal smoothing of four-space, J. Differential Geom. 24 (1986) 69

[12] M Furuta, Monopole equation and the –conjecture, Math. Res. Lett. 8 (2001) 279

[13] R E Gompf, Three exotic R4’s and other anomalies, J. Differential Geom. 18 (1983) 317

[14] R E Gompf, An infinite set of exotic R4’s, J. Differential Geom. 21 (1985) 283

[15] R E Gompf, An exotic menagerie, J. Differential Geom. 37 (1993) 199

[16] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998) 619

[17] R E Gompf, Stein surfaces as open subsets of C2, J. Symplectic Geom. 3 (2005) 565

[18] R E Gompf, Constructing Stein manifolds after Eliashberg, from: "New perspectives and challenges in symplectic field theory" (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 229

[19] R E Gompf, Minimal genera of open 4–manifolds, preprint (2013)

[20] R E Gompf, Smooth embeddings with Stein surface images, J. Topol. 6 (2013) 915

[21] R E Gompf, T S Mrowka, Irreducible 4–manifolds need not be complex, Ann. of Math. 138 (1993) 61

[22] R E Gompf, A I Stipsicz, 4–manifolds and Kirby calculus, 20, Amer. Math. Soc. (1999)

[23] R C Kirby, The topology of 4–manifolds, 1374, Springer (1989)

[24] E Laitinen, End homology and duality, Forum Math. 8 (1996) 121

[25] F Quinn, Ends of maps, III : Dimensions 4 and 5, J. Differential Geom. 17 (1982) 503

[26] C P Rourke, D P Sullivan, On the Kervaire obstruction, Ann. of Math. 94 (1971) 397

[27] C H Taubes, Gauge theory on asymptotically periodic 4–manifolds, J. Differential Geom. 25 (1987) 363

[28] L R Taylor, An invariant of smooth 4–manifolds, Geom. Topol. 1 (1997) 71