An explicit family of exotic Casson handles

Bižaca, Žarko

doi:10.1090/S0002-9939-1995-1246517-X

An explicit family of exotic Casson handles
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by Žarko Bižaca

Proc. Amer. Math. Soc. 123 (1995), 1297-1302

DOI: https://doi.org/10.1090/S0002-9939-1995-1246517-X

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Abstract:

This paper contains a proof that the Casson handle that contains only one, positive, self-intersection on each level, $C{H^ + }$, 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 Ž. Bižaca (A handle decomposition of an exotic ${\mathbb {R}^4}$, J. Differential Geom. (to appear)), $C{H_n},n \geq 0$, such that $C{H_0}$ is the above-described $C{H^ + }$ and each $C{H_{n + 1}}$ is obtained by the reimbedding algorithm (Ž. Bižaca, A reimbedding algorithm for Casson handles, Trans. Amer. Math. Soc. 345 (1994), 435-510) in the first six levels of $C{H_n}$. 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. Differential Geom. 17 (1982), 357-453).

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