Geometriae Dedicata

, Volume 121, Issue 1, pp 89–111

# Stein domains and branched shadows of 4-manifolds

• Francesco Costantino
Original Paper

## Abstract

We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact four-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev’s shadows suitably “smoothed”; the conditions we find are purely algebraic and combinatorial. Applying our results, we provide examples of hyperbolic 3-manifolds admitting “many” positive and negative Stein fillable contact structures, and prove a four-dimensional analog of Oertel’s result on incompressibility of surfaces carried by branched polyhedra.

## Mathematics Subject Classifications (2000)

Primary 57N13 Secondary 32B28

## References

1. 1.
Agol I. (2000) Bounds on exceptional Dehn filling. Geom. Topol. 4, 431–439
2. 2.
Bishop E. (1975) Differentiable manifolds in complex euclidean space. Duke. Math. J. 32, 1–21
3. 3.
Benedetti R., Petronio C. (2000) Branched spines and contact structures on 3-manifolds. Ann. Mat. Pura Appl. 178(4): 81–102
4. 4.
Benedetti R., Petronio C. (1997) Branched standard spines of 3-manifolds. Lecture Notes in Mathematics, Springer-Verlag, Berlin
5. 5.
Costantino, F.: Shadows and branched shadows of 3 and 4-manifolds. Tesi di perfezionamento, Scuola Normale Superiore, Pisa (2004)Google Scholar
6. 6.
Costantino, F.: Branched shadows and complex structures of 4-manifolds. xxx.arXiv.org /math.GT/0502292 (2005)Google Scholar
7. 7.
Costantino F. (2005) A short introduction to shadows of 4-manifolds. Fund. Math. 188, 271–291
8. 8.
Chern S.S., Spanier E. (1951) A theorem on orientable surfaces in four-dimensional space. Comm. Math. Helv. 25, 205–209
9. 9.
Costantino, F., Thurston, D.P.: 3-manifolds efficiently bound 4-manifolds. xxx.arXiv.org/math.GT/ 0506577 (2005)Google Scholar
10. 10.
Eliashberg Y. (1990) Topological characterization of Stein manifolds of dimension ≥  3. Internat. J. Math. 1(1): 119–131
11. 11.
Forstneri&ccedil; F. (2003) Stein domains in complex surfaces. J. Geom. Anal. 13(1): 77–94
12. 12.
Gompf R. (1998) Handlebody construction of Stein surfaces. Ann. of Math. 148, 619–693
13. 13.
Gompf, R., Stipsicz, A.: 4-manifolds and Kirby calculus. Graduate Studies in Mathematics, vol. 20, Amer. Math. Soc., Providence, RI (1999)Google Scholar
14. 14.
Harlamov, V.M., Eliashberg, Y.: On the number of complex points of a real surface in a complex surface. pp. 143–148. Proc. Leningrad Int. Topology Conf. (1982)Google Scholar
15. 15.
Kronheimer P.B., Mrowka T.S. (1997) Monopoles and contact structures. Invent. Math. 130(2): 209–255
16. 16.
Lackenby M. (2000) Word hyperbolic Dehn surgery. Invent. Math. 140, 243–282
17. 17.
Lai H.F. (1972) Characteristic classes of real manifolds immersed in complex manifolds. Trans. Amer. Math. Soc. 172, 1–33
18. 18.
Lisca P. (1998) Symplectic fillings and positive scalar curvature. Geom. Topol. 2, 103–116
19. 19.
Nirenberg R., Wells R.O. (1969) Approximation theorems on differentiable submanifolds of a complex manifold. Trans. Amer. Math. Soc. 142, 15–35
20. 20.
Oertel U. (1984) Incompressible branched surfaces. Invent. Math. 76, 385–410
21. 21.
Polyak, M.: Shadows of Legendrian links and J +-theory of curves. Singularities (Oberwolfach, 1996), pp. 435–458. Progr. Math. 162, Birkhäuser, Basel (1998)Google Scholar
22. 22.
Turaev V.G. (1994) Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics, vol. 18. Walter de Gruyter and Co., BerlinGoogle Scholar
23. 23.
Turaev V.G. (1991) Quantum invariants of 3-manifolds and a glimpse of shadow topology. C. R. Acad. Sci. Paris Sr. I Math. 313(6): 395–398