What is the exact statement of the Poincare conjecture, and where does it stand (fake R^4s, the case n = 3)?

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_).

-- Matthew P Wiener <ucbvax!brahms!weemba>

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 Freedman
>    [Editor: "Freidman" in the original], 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-
                                                     2
Siebenman), this would contradict the Poincare' conjecture.

OK, there may be a final alternative: E. Brieskorn has produced some
(weighted homogeneous) polynomial 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