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An explicit family of exotic Casson handles. (English) Zbl 0845.57015

Summary: 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. Am. Math. Soc., New Ser. 29, 51-59 (1993; Zbl 0789.57004)] 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 [the author, J. Differ. Geom. 39, No. 3, 491-508 (1994; Zbl 0845.57014), see the review above] \(CH_n\), \(n \geq 0\), such that \(CH_0\) is the above-described \(CH^+\) and each \(CH_{n+1}\) is obtained by the reimbedding algorithm [the author, Trans. Am. Math. Soc. 345, 435-510 (1994; Zbl 0830.57011)] in the first six levels of \(CH_n\). 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. Differ. Geom. 35, 219-254 (1992; Zbl 0736.57008)].

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M99 General low-dimensional topology
57R65 Surgery and handlebodies
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