Geometry & Topology Monographs 2 (1999), Proceedings of the Kirbyfest, paper no. 26, pages 563-569.

Smooth Euclidean 4-spaces with few symmetries

Laurence R Taylor

Abstract. We say that a topologically embedded 3-sphere in a smoothing of Euclidean 4-space is a barrier provided, roughly, no diffeomorphism of the 4-manifold moves the 3-sphere off itself. In this paper we construct infinitely many one parameter families of distinct smoothings of 4-space with barrier 3-spheres. \par The existence of barriers implies, amongst other things, that the isometry group of these manifolds, in any smooth metric, is finite. In particular, S^1 can not act smoothly and effectively on any smoothing of 4-space with barrier 3-spheres.

Keywords. Exotic smoothings, Euclidean spaces, isometries

AMS subject classification. Primary: 57R55.

E-print: arXiv:math.GT/9807143

Submitted: 28 July 1998. Published: 21 November 1999.

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Laurence R Taylor
Department of Mathematics, University of Notre Dame
Notre Dame, IN 46556, USA

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