An explicit family of exotic Casson handles

Abstract

This paper contains a proof that the Casson handle that contains only one, positive, self-intersection on each level, CH+ , is exotic in the sense that the 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 the positive trefoil knot is smoothly slice. An explicit infinite family of Casson handles is constructed by using the 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 the above-described CH+ and each CH,+, is obtained by the reimbedding algorithm (2. Bilaca, A reimbedding algorithm for Casson handles, Trans. Amer. Math. Soc. 345 (1994), 435-510) in the first six levels of CH, . An argument that no two of those exotic Casson handles are diffeomorphic is outlined, and it mimics the 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 the 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 the biggest break-throughs in four-dimensional topology was M. Freedman's proof [F] that every Casson handle is homeomorphic to the standard open 2-handle, (D2 x R2, Si x R2). Andrew Casson constructed these manifold pairs in an attempt to mimic the proof of the h-cobordism theorem for higher-dimensional manifolds in the setting of four-dimensional manifolds [C]. The piece of the higher-dimensional proof that does not apply to dimension four is the Whitney trick (see [RS, ?6]). Casson handles were constructed in places where embedded 2-handles, or "Whitney discs", are needed in order to perform the Whitney trick [C]. Therefore, a consequence of Freedman's result is the 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 the product cobordism, MO x [0, 1]. If all Casson handles were diffeomorphic to the standard open 2-handle, the Received by the editors July 28, 1993. 1991 Mathematics Subject Classification. Primary 57M99, 57N 13, 57R65.