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From: weemba@brahms.BERKELEY.EDU (Matthew P. Wiener)
Newsgroups: net.math
Subject: Re: Poincare conjecture
Message-ID: <11506@ucbvax.BERKELEY.EDU>
Date: Thu, 23-Jan-86 01:05:24 EST
Article-I.D.: ucbvax.11506
Posted: Thu Jan 23 01:05:24 1986
Date-Received: Fri, 24-Jan-86 21:26:34 EST
References: <1613@hound.UUCP>
Sender: usenet@ucbvax.BERKELEY.EDU
Reply-To: weemba@brahms.UUCP (Matthew P. Wiener)
Organization: University of California, Berkeley
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I am posting this because the mailer truncated the e-mail address.  If
anyone could tell me (via e-mail) how to get around that, I'd be obliged.

In article <1613@hound.UUCP> batu@hound.UUCP writes:
>
>Can someone give an exact statement and reference for 
>
>i) the "original" Poincare conjecture, i.e. on when homotopy spheres are 
>   diffeomorphic or topological spheres, and
>ii) the current state of affairs, viz. the "fake" R**4 of Freidman, Donaldson, 
>    et. al.
>iii) any information of the conjecture when n is 3 (Milgram once mentioned a 
>     possible counterexample) ?
>______________________________
>----------------------------------- SPP in Holmdel_____________________________

ad i)	In dimensions 5 and up, look at C Rourke and B Sanderson, _Piecewise
	Linear Topology_.  Start near the end and work your way backwards.
	An essentially similar proof, using critical points of Morse
	functions instead of Smale's handlebodies, is in J Milnor, _Lectures
	on the h-cobordism Theorem_.  Both are rated R.  In dimension 4,
	only M H Freedman, "The topology of 4-dimensional manifolds", J of
	Diff Geom 17 (1982) pp 357-453, is available.  Rated X.  No one knows
	if the Poincare conjecture holds in 4 dimensions in the differentiable
	category.  There are candidate fake S^4s.
ad ii)	For starts, there are two good books, D S Freed and K K Uhlenbeck,
	_Instantons and Four-Manifolds_, and H Blaine Lawson Jr, _The Theory
	of Gauge Fields in Four Dimensions_.  Both are rated XX, but the
	introduction to the latter is PG-13.  Then there is the paper of
	R Gompf, "Three exotic R^4s and other anomalies", J Diff Geom (1983)
	pp 317-328.  Rated PG.
	Latest results, not yet published, are:
	There are infinitely many fake R^4s.  (Gompf)
	There is a natural semigroup operation on R^4s, gotten from gluing
	along (???), with neutral element std R^4 and having an absorbing
	element (the universal fake R^4).   (Freedman)
	There are continuum many fake R^4s.  (Taubes)
ad iii)	R Kirby says there are no putative counterexamples.  Lots of
	scattered work, plus, of course, the works of Thurston.  Have fun.
	However, these are rated X^n, for some n>3, so the public decency is
	protected by not letting anyone see them.  (There is an XXX rated
	ultrasimplified introduction in J Morgan and H Bass, _The Smith
	Conjecture_).

ucbvax!brahms!weemba	Matthew P Wiener/UCB Math Dept/Berkeley CA 94720