An Explicit Family of Exotic Casson Handles

Abstract

This paper contains a proof that Casson handle that contains only one, positive, self-intersection on each level, CH+ , is exotic in sense that attaching circle of this Casson handle is not smoothly slice in its interior. The proof is an easy consequence of L. Rudolph's result (Bull. Amer. Math. Soc. (N.S.) 29 (1993), 51-59) that no iterated positive untwisted doubles of positive trefoil knot is smoothly slice. An explicit infinite family of Casson handles is constructed by using non-product h-cobordism from Z. Bilaca (A handle decomposition of an exotic R4, J. Differential Geom. (to appear)), CH,,, n > 0, such that CHO is above-described CH+ and each CH,+, is obtained by reimbedding algorithm (2. Bilaca, A reimbedding algorithm for Casson handles, Trans. Amer. Math. Soc. 345 (1994), 435-510) in first six levels of CH, . An argument that no two of those exotic Casson handles are diffeomorphic is outlined, and it mimics one from S. DeMichelis and M. Freedman (J. Differential Geom. 17 (1982), 357-453). The purpose of this paper is to present explicit examples of exotic Casson handles. For a description of Casson handles reader is referred to [C], [F], or [K2]. The notion of 'exoticness' is defined below. A way to describe any handle, and so a Casson handle in particular, is to think of it as a pair consisting of a manifold and a selected piece of its boundary, the attaching area. One of biggest break-throughs in four-dimensional topology was M. Freedman's proof [F] that every Casson handle is homeomorphic to standard open 2-handle, (D2 x R2, Si x R2). Andrew Casson constructed these manifold pairs in an attempt to mimic proof of h-cobordism theorem for higher-dimensional manifolds in setting of four-dimensional manifolds [C]. The piece of higher-dimensional proof that does not apply to dimension four is trick (see [RS, ?6]). Casson handles were constructed in places where embedded 2-handles, or Whitney discs, are needed in order to perform trick [C]. Therefore, a consequence of Freedman's result is topological hcobordism theorem for four-dimensional manifolds [F]; that is, an h-cobordism between two simply-connected closed four-dimensional manifolds, say MO and M, , is necessarily homeomorphic to product cobordism, MO x [0, 1]. If all Casson handles were diffeomorphic to standard open 2-handle, Received by editors July 28, 1993. 1991 Mathematics Subject Classification. Primary 57M99, 57N 13, 57R65.