NetNews Usenet Archive 1992 #20/NN_1992_20.iso/spool/sci/math/11422

Text File | 1992-09-14 | 1.6 KB | 37 lines

Newsgroups: sci.math
Path: sparky!uunet!europa.asd.contel.com!darwin.sura.net!zaphod.mps.ohio-state.edu!sdd.hp.com!ux1.cso.uiuc.edu!mp.cs.niu.edu!rusin
From: [e-mail] (David Rusin)
Subject: Re: exotic R^4
Message-ID: <[e-mail]>
Organization: Northern Illinois University
References: <[e-mail]>
Distribution: sci
Date: Mon, 14 Sep 1992 17:47:01 GMT
Lines: 25

In article <[e-mail]> [e-mail] (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 [e-mail]