Newsgroups: sci.math From: rusin@mp.cs.niu.edu (David Rusin) Subject: Re: exotic R^4 Date: Mon, 14 Sep 1992 17:47:01 GMT In article ara@zurich.ai.mit.edu (Allan Adler) asks if it is possible to find exotic R^4's as hypersurfaces in R^5. I am not a person who would know but I can point to a related example. The first exotic manifolds were the 27 non-standard structures on S^7 deduced by Milnor. (1957 I think). It turns out that they can be described easily in the way proposed. Indeed, let V be the hypersurface in C^5 described by the equation 4k+1 3 2 2 2 z + z + z + z + z = 0 1 2 3 4 5 Then V has one singular point at the origin, so that V - {0} is a complex 4-manifold (real 8-manifold). You can show it meets the unit sphere S^9 in R^10 transversely, so that the intersection is a real 7-manifold M. That M is a topological 7-sphere is not hard (see Milnor's Singular Points of Complex Hypersurfaces). However, the manifolds M are only diffeomorphic if the k's are comgruent mod 28. (I think this is due to Brieskorn. maybe it was Hirzebruch.) I love to show this example to beginning topology grad students. If a similar example is available for R^4 I would like to see it. dave rusin@math.niu.edu