## 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

### Citations:

Zbl 0789.57004; Zbl 0845.57014; Zbl 0830.57011; Zbl 0736.57008
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