Nuclei of elliptic surfaces
Elliptic surfaces are smooth closed 4-manifolds with significant properties. The simply connected, minimal ones fall into infinite families, one or two such families corresponding to each positive integer, in which all members of a given family have identical intersection forms, so they are smoothly h-cobordant and therefore homeomorphic, by Freedman's topological h-cobordism theorem. Fundamental distinctions arise in the smooth category, however; Donaldson has shown that one family contains more than one diffeomorphism type and, using Donaldson's invariants, Friedman and Morgan improved on this by showing that each family in fact contains infinitely many diffeomorphism types. Seeking to circumvent or simplify what he views as the topologically cumbersome nature of elliptic surfaces, here the author extracts a comparatively small nucleus, aptly termed, which embodies most of the topology. Specifically, the nucleus is simply connected when the manifold is, and the homeomorphism and diffeomorphism classifications of nuclei are closely related to those of the associated surfaces (in the simply connected case, two nuclei are the same, topologically and/or diffeomorphically, precisely when the associated surfaces are). As the boundaries of nuclei examined here are Brieskorn homology 3-spheres, different diffeomorphism types of nuclei can be distinguished by their Donaldson-Floer invariants, which in turn are distinguished by the Donaldson invariants of the corresponding surfaces. The author gives elementary examples of topologically identical but smoothly distinct nuclei, each a handlebody comprised of one 0-handle and exactly two 2-handles. Among other useful analyses, he provides a Kirby- calculus description of nuclei and shows which automorphisms of intersection form are realized by diffeomorphisms.
scientific article; zbMATH DE number 4211084
scientific article