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From: wdh@faron.UUCP (Dale Hall)
Newsgroups: net.math
Subject: Re: Poincare conjecture
Message-ID: <441@faron.UUCP>
Date: Tue, 21-Jan-86 19:44:15 EST
Article-I.D.: faron.441
Posted: Tue Jan 21 19:44:15 1986
Date-Received: Thu, 23-Jan-86 21:22:02 EST
References: <1613@hound.UUCP>
Reply-To: wdh@faron.UUCP (Dale Hall)
Organization: The MITRE Coporation, Bedford, MA
Lines: 111
Keywords: Poincare' Conjecture, low-dimensional topology

In article <1613@hound.UUCP> batu@hound.UUCP (S.PETERSON) 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, ...

	Any simply-connected (i.e., fundamental group trivial) closed three-
	manifold is homeomorphic to the standard three-sphere.

	This is the original Poincare' Conjecture. That this is equivalent
	to the usual formulation:

      "	a homotopy three-sphere is homeomorphic to the standard three-sphere "

	follows from Poincare' duality (to get homology groups right) and
	the Whitehead Theorem (to produce a homotopy equivalence between the
	candidate space and the standard three-sphere. (there may be much 
	more elementary methods to show equivalence, I'm just lazy today)

	As far as references, I think any text on geometric topology
	should suffice:

	C.Rourke, D.Sanderson: Introduction to Piecewise-Linear Topology,
	Springer-Verlag (Ergebnisse der Mathematik ..., band 69, 1972)

	J.Milnor: Lectures on the h-Cobordism Theorem, (notes by L.Siebenmann
	and J.Sondow), Princeton University 1965

	This list is to be taken with a grain of salt, as I have been unable
	to locate a reference myself. The above list amounts to a guess as to
	where I would look next. Another line of attack might be Smale's paper
	on the Generalized Poincare' Conjecture (dimensions > 4), which 
	appeared in the Annals of Mathematics, 1961.

>ii) the current state of affairs, viz. the "fake" R**4 of Freidman, Donaldson,
>    et. al.

	According to D.Freed, K.Uhlenbeck: Instantons and Four-Manifolds, MSRI
	Publication 1 (Springer-Verlag, 1984):

	Freedman proved that compact 4-manifolds M with trivial fundamental
	group are in 1-1 correspondence with pairs <w,a>, where 

		w = unimodular symmetric bilinear form over Z

		  = m x m symmetric integer matrix, with det(w) = +/- 1.

		  = intersection pairing on two-dimensional homology of M.

	and 	a = Kirby-Siebenmann invariant of M

		  = obstruction to lifting the TOP tangent microbundle to
						      1
		    BPL (alternatively, a = 0 if M x S  can be smoothed,
		    a = 1 if it can't)

	there is in addition a restriction if w is even: s(w)/8 = a (mod 2)
	where s(*) is the signature (= (number of +'s) - (number of -'s)
	when * is in diagonal form).

	This relates to earlier known results of C.T.C.Wall and S.P.Novikov,
	which showed the same result stably: i.e., up to connected sum
			    2	2
	with finitely many S x S  's, in the smooth category, so "a" above
	isn't necessary.
			       4
	To continue, the fake R  emerges by taking the Kummer surface:

		 4     4     4     4             3
		w   + x   + y   + z    = 0  in CP

			      2   2
	removing 3 copies of S x S  , and applying existing results on
	what can be done smoothly (due to Rokhlin) to obtain a contradiction
	to smoothness for this (now non-compact) manifold.

	The 4-dimensional Poincare' conjecture follows from extensive further 
	analysis, making use of Atiyah - Hitchin - Singer work on Yang-Mills
	equations. Again, this is done in the topological category, and so
	does not relate directly to the structure of smooth 4-manifolds.

>iii) any information of the conjecture when n is 3 (Milgram once mentioned a 
>     possible counterexample) ?

	This sounds like an oblique reference to the known existence of
	problems with either the 4-dimensional or 5-dimensional s-cobordism
	theorem (it's known that one or the other of these versions breaks
	down, but which one was not known, as of 1980 [Matumoto and 
	Siebenmann, Math.Proc.Camb.Phil.Soc 1978]).

	On the other hand, it could be a reference to the fact that
	the existing classification of homotopy spheres up to CAT (= DIFF
	or PL) equivalence agrees with the corresponding homotopy theory
	of TOP/CAT, except in unknown dimensions (3,4 for DIFF, 3 for PL).
	Since the third homotopy group of TOP/PL or TOP/DIFF is known
	to be Z  (Kirby-Siebenman), this would contradict the Poincare'
	       2
	conjecture.

	OK, there may be a final alternative: E. Brieskorn has produced 
	some (weighted homogeneous) ploynomial singularities which, in
	higher dimensions, yield exotic spheres -- topological spheres
	which are not DIFF equivalent to the standard "round" sphere.
	My recollection from graduate school is that there may have been 
	some candidates for counterexamples to the Poincare' conjecture
	in that collection.

	Sorry for the length. I must have gotten carried away.

						Dale Hall