Modifying surfaces in 4–manifolds by twist spinning

Kim, Hee Jung¹

¹ Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada

Geometry & topology, Tome 10 (2006) no. 1, pp. 27-56.

Résumé

In this paper, given a knot $K$, for any integer $m$ we construct a new surface ${\Sigma }_K\left(m\right)$ from a smoothly embedded surface $\Sigma$ in a smooth 4–manifold $X$ by performing a surgery on $\Sigma$. This surgery is based on a modification of the ‘rim surgery’ which was introduced by Fintushel and Stern, by doing additional twist spinning. We investigate the diffeomorphism type and the homeomorphism type of $\left(X,\Sigma \right)$ after the surgery. One of the main results is that for certain pairs $\left(X,\Sigma \right)$, the smooth type of ${\Sigma }_K\left(m\right)$ can be easily distinguished by the Alexander polynomial of the knot $K$ and the homeomorphism type depends on the number of twist and the knot. In particular, we get new examples of knotted surfaces in $ℂ{\mathbf{P}}^2$, not isotopic to complex curves, but which are topologically unknotted.

Détail

DOI : 10.2140/gt.2006.10.27

Keywords: Twist spinning, Seiberg–Witten invariants, branched covers, ribbon knots

Bibliographie

[1] S Finashin, Knotting of algebraic curves in CP2, Topology 41 (2002) 47 | DOI

[2] R Fintushel, R J Stern, Surfaces in 4–manifolds : Addendum

[3] R Fintushel, R J Stern, Surfaces in 4–manifolds, Math. Res. Lett. 4 (1997) 907

[4] R Fintushel, R J Stern, Knots, links, and 4–manifolds, Invent. Math. 134 (1998) 363 | DOI

[5] R H Fox, Free differential calculus III: Subgroups, Ann. of Math. 64 (1956) 407

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

[7] R E Gompf, A new construction of symplectic manifolds, Ann. of Math. 142 (1995) 527

[8] C M Gordon, Ribbon concordance of knots in the 3–sphere, Math. Ann. 257 (1981) 157 | DOI

[9] R Kirby, Problems in low-dimensional topology, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35

[10] P Kronheimer, T Mrowka, Floer homology for Seiberg–Witten monopoles, in preparation

[11] J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358

[12] J Morgan, H Bass, The Smith conjecture, 112, Academic Press (1984)

[13] C H Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809

[14] C H Taubes, The Seiberg–Witten invariants and 4–manifolds with essential tori, Geom. Topol. 5 (2001) 441 | DOI

[15] V Turaev, Introduction to combinatorial torsions, , Birkhäuser Verlag (2001)

[16] S Vidussi, Seiberg–Witten invariants for manifolds diffeomorphic outside a circle, Proc. Amer. Math. Soc. 129 (2001) 2489

[17] E C Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965) 471