The \(E_8\)-manifold, singular fibers and handlebody decompositions. (English) Zbl 0959.57020

Kirby, Robion; Melvin, Paul

Hass, Joel (ed.) et al., Proceedings of the Kirbyfest, Berkeley, CA, USA, June 22-26, 1998. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 2, 233-258 (1999).

Four descriptions, two essentially different, are given for the 4-manifold \(E_8\). One, as a handlebody, is represented by a specific framed link in \(S^3= \partial B^4\). An abbreviated notation, grapes, is introduced for this framed link. \(E_8\) is constructed by adding 2-handles to the framed link. \(E_8\) can also be obtained by taking a \(p\)-fold cover of the 4-ball branched over the standard Seifert surface for the \((q,r)\) torus knot, pushed into the interior of \(B^4\). \((p,q,r)\) can be any cyclic permutation of (2, 3, 5). There result three Seifert surfaces and corresponding branched coverings. These three are shown to be diffeomorphic, and a bunch of grapes are obtained. A move called a slip is introduced on grapes amounting to handle slides and isotopies. These are used to show the covers are diffeomorphic to \(E_8\) as described by adding 2-handles to the framed link in \(S^3=\partial B^4\). The “slippin’ an’ a slidin’ ” method allows for one of the few (the only?) citations of Little Richard in the mathematical literature.
\(E_8\) occurs as a neighborhood of (most of) a singular fiber in an elliptic surface. Section 2 studies regular neighborhoods of singular fibers in minimal elliptic surfaces. These were classified in 1963 by Kodaira. In the current work the regular neighborhoods of each of these are given as handlebodies obtained by attaching specified 2-handles to \(N=T^2 \times B^2\) along \(-1\)-framed meridians and longitudes in successive torus fibers in \(\partial N\).
Gromov’s compactness theorem gives conditions under which a sequence of pseudo-holomorphic curves has a subsequence converging to a cusp curve. The current paper, in its final section, studies examples arising from sequences of regular fibers in an elliptic surface which converge to a singular fiber. These are examined for a number of the singular fibers treated in the second section.
For the entire collection see [Zbl 0939.00055].

Reviewer: G.E.Lang jun.(Fairfield)

MSC:

57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
14J27	Elliptic surfaces, elliptic or Calabi-Yau fibrations
57R65	Surgery and handlebodies

Keywords:

grapes; elliptic surfaces