Growth of Casson handles and transversality for ASD moduli spaces. (English) Zbl 1148.57022

After proving that various exotic \(4\)-manifolds exist it is natural to measure the complexity of these \(4\)-manifolds. Topologically, any simply-connected \(4\)-manifold with intersection form \(-2k\,E_8\oplus \ell H\) admits an open set homeomorphic to a punctured connected sum of \(\ell\) copies of \(S^2\times S^2\). By the work of Donaldson this homeomorphism cannot be a diffeomorphism and so we see that this open set is obtained by attaching \(2\ell\) Casson handles to the \(4\)-ball. Clearly all of these Casson handles cannot be standard, so one may ask how complicated are the resulting Casson handles. In [Geom. Topol. 8, 779–830 (2004; Zbl 1064.57022)], the present author introduced the notion of a Casson handle of bounded type. These Casson handles arise from sub trees of homogeneous signed trees. Homogeneous signed trees are defined inductively by calling the natural numbers with adjacent edges having a common sign type \((T_1,\pm)\) trees and setting type \((T_{n+1},\pm)\) trees to be obtained by attaching type \((T_n,\pm)\) trees (all with the same sign) to the vertices of a type \((T_1,\pm)\) tree.
In this paper the author describes why Seiberg-Witten theory is insufficient to address the question of the existence bounded type Casson handle decompositions. He then addresses the transversality issues involved in constructing a Donaldson invariant for these open \(4\)-manifolds. (He addressed the index problem in [Contemporary Mathematics 347, 113–129 (2004; Zbl 1072.14052)]). Using his results he is able to prove for example that some of these open sets in single log transforms of a K3 surface cannot be constructed entirely from bounded type Casson handles.


57M30 Wild embeddings
57R57 Applications of global analysis to structures on manifolds
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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