batu@hound.UUCP (S.PETERSON) (01/20/86)
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______________________________wdh@faron.UUCP (Dale Hall) (01/22/86)
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
weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) (01/23/86)
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