Careful said:
Hi, I got to page five and have already loads of technical questions/remarks. The authors start by noticing that a differentiable structure carries lot's of topological information and provides as well the necessary mathematical setting to write out the Einstein Field equations. That is certainly correct, ONE differentiable structure actually determines all Betti numbers (by studying critical points of vectorfields). However, the authors are not pleased with the knowledge of the number of multidimensional handles and want to include exotic differentiable structures associated to a topological manifold. Any good motivation for this is lacking; string theorists would actually jump out of the roof since in ten dimensions, only six inequivalent differential structures exist. It would be instructive to UNDERSTAND why in dim 2 and 3 (one is easy to proof) only one differentiable structure exists and what makes four so special, but no such insight is provided. For example: one should know if an explicit algorithm exists for creating such inequivalent types. The authors do suggest in that respect the use of surjective, smooth (between two inequivalent differentiable structures) but not injective mappings, but this is by far not sufficient.
That is difficult to answer and hopefully the following is not to technical. In dimension 2 and 3 the uniqueness of the differential structure can be shown where the problem is attributed to the 1-dimensional case. In 1982 Freedman classifies all topological, simply-connected manifolds to show that this classification mimics the higher-dimensional case. Thus, it is better to look at the higher-dimensional classification of differential structures by using the h-cobordism theorem. The failure of the smooth h-cobordism (Donaldson, 1987) opens the way to show that there are more than one possible differential structure on simply-connected 4-manifolds. For sufficient complicated 4-manifolds there is an explicite construction by Fintushel and Stern using knots and links (see the pages 9 and 10 of our paper for a description). Now, why is dimension 4 so special? The interior of a h-cobordism between two topologically equivalent 4-manifolds M,N consists of 2-/3-handle pairs. All other handles can be killed by using Morse theory (see Milnor, Lectrures on the h-cobordism theorem). These 2-/3-handle pairs can be killed if and only if there is a special embedded disk (the Whitney Disk). But if the disk has self-intersection then this disk ist not embedded. But that happens in dimension 4 by dimensional reasons. In higher dimensions there is no self-intersections and thus such a Whitney disk always exists. In our paper we use these results to consider a smooth map f:M->N between M,N with different differential structures. The singular set is by definition the set where the map f is not invertable or where df=0. So, we start with a mathematically given situation: two topological equivalent 4-manifolds with different differential structures. In the paper we are not dealing with the question to decide wether two 4-manifolds are diffeomorphic or not. That question has to be addressed later.
Careful said:
Some technical comments regarding section II: the definition of a singular set is very strange, one would expect df_x to have rank < 4 and not df_x=0.
Section III deals with pulling back the tangent structures from a differentiable structure N to a differentiable structure M by a mapping f. The authors define the singular CONNECTION one form G associated to f There is not given any rigorous definition of G = f_{*}^{-1} d f_{*} since this expression is meaningless where df_x has rank < 4 (since
f_{*}^{-1} does not exist there), so at least one should do this in the distributional sense wrt to a volume form determined by an atlas in the differentiable structure.
That is correct. The paper is written for physicists and we are not dealing with the theory of currents which is necessary to understand such singular objects. I know that problem. The whole idea based on the work of Harvey and Lawson about singular bundle maps. The map df:TM->TN is such a singular bundle map and Harvey and Lawson define a current Div(df) which they call a divisor. It is true that the expression in the given form as G = f_{*}^{-1} d f_{*} is mathematically not rigorous but it can be defined by the theory of Harvey and Lawson (see HL, A theory of characteristic currents associated with a singular connection, Asterisque 213, 1993). In the following we consider the cohomology class associated to the current (in the sense of Federer, Geometric measure theory). Thus, a diffeomorphism of M AND/OR N does not change this class. Secondly, by a result of Freedman, two homotopy-equivalent 4-manifolds are homeomorphic. Thus, the cohomology classes are connected to the differential strcuture. That agrees also with the results of Seiberg-Witten theory where special cohomology classes (called basic classes) determine the differential structure. What I want to say is that one can give the experssion G a sense by using the theory of Harvey and Lawson. I see that we forgot to write it.
Careful said:
A second comment is that G is not anything intrinsic - it is just a (distributional) gauge term and NOT a one form. Therefore, it is an uninteresting object related to a specific mapping f and to a choice of coordinate systems on M AND N (and especially this last property is very bad) - admittedly, it depends slightly upon the change of differentiable structure (through f) and does give rise to a distributional source in the energy momentum tensor. Nevertheless, the authors want to do something with it and give two inequivalent definitions for G; one based on nontrivial connections and one on the flat connection.
Yes it is right that the G in that form depends on the differential structure. That is the reason why we take the trace of the connection or curvature to exclude the dependence of the diffeomorpism. We always use in the paper the fact that a cohomology class can be associated to a current and vice versa.
Careful said:
The definition of the support is fine (since one wants to single out the singular part). With the definition of the product, something strange happens: the authors seem to consider G as a ONE FORM (which it isn't) and POSTULATE that the singular support of G is a three manifold and want to associate a specific generator of the first fundamental group to it. Poincare duality as far as I know is a duality between cell complexes of dimension k and n-k or homology classes of dimensions k and n-k, and this is clearly not the case. What the authors seem to allude to is the duality between the first homotopy class and the first homology class, which is the de Rahm duality and this could be only appropriate in case the singular support of G is a three manifold but still there is NO CANONICAL ONE FORM given, which is the other essential part of de Rahm theory. The same comment applies to the use Seifert theory; this COULD be only meaningful when the singular support of G is a three manifold, which is NOT necessarily the case (for a generic surjective, non injective, smooth f, the singular support could not even be a manifold) - the authors should provide a theorem that this is so. The latter is necessary since the theory of knots makes only sense in three dimensions (and M is a four dimensional manifold).
I think these issues need clarification otherwhise it seems to go wrong from the beginning...
It is not necessary to consider G as a one form. You can also consider G as a current with support \Sigma. Let f:M->N be a singular map. Now I will say some words about the structure of \Sigma. For that purpose, we have to say some words about the theory of singular maps. In topology, we are only interested in such topological characteristics like intersection points which are coupled to the question when two sub-manifolds intersect transversal. In most cases that happens and
we are done, but according to Sard's theorem a smooth mapping f:X\to Y between smooth manifolds X,Y has a set of critical
values of f of measure zero. That means, there are some (countable
many) cases where we don't get a transversal intersection between
sub-manifolds represented by some map f:X\to Y. The question is
now: which deformation of the smooth map f to \tilde{f} given by
a deformation of the smooth manifolds X,Y eliminates the critical
(or singular) values of f. Such a procedure is called unfolding of f and Hironaka proves the general theorem that for every singular map f there is a
sequence of operations which unfolds f. These operations are
usually called blow-up and blow-down. In our case f:M\to M' a
blow-up leads to a map \tilde{f}:M\#{\mathbb C}P^2\to M' and a
blow-down to \tilde{f}:M\#\overline{{\mathbb C}P}^2\to M'.
Hironakas theorem means that the unfolding of f leads to a
diffeomorphism
<br />
\tilde{f}:M\underbrace{\#{\mathbb C}P^2\#\cdots\#{\mathbb C}P^2}_n\#<br />
\underbrace{\overline{{\mathbb C}P}^2\cdots\#\overline{{\mathbb<br />
C}P}^2}_m \to M' <br />
and by using the diffeomorphism (see Kirby, Topology of 4-manifolds)
<br />
(S^2\times S^2)\#{\mathbb C}P^2={\mathbb C}P^2\#\overline{{\mathbb<br />
C}P}^2\#{\mathbb C}P^2<br />
we obtain a diffeomorphism
<br />
\tilde{f}:M\underbrace{\#S^2\times S^2\cdots\#S^2\times S^2}_m\#<br />
\underbrace{\#{\mathbb C}P^2\#\cdots\#{\mathbb C}P^2}_{n-m}\to M'<br />
where we assume w.l.o.g. m<n. But that is nothing than a weaker
version of the famous theorem of Wall about diffeomorphisms between
4-manifolds (see Kirby). A very important concept is the
stable mapping. Let f\in C^\infty(X,Y) be a smooth mapping f:X\to<br />
Y. Then f is stable if there is a neighborhood W_f of f in
C^\infty(X,Y) (we use the compact-open topology for that space)
such that each f' in W_f is equivalent to f. According to
Mather stable smooth mappings between 4-manifolds are
dense in the set of smooth mappings. Thus according to Stingley
(see the phd thesis under supervisition of Lawson) one has to focus on that particular subset to study
maps between homeomorphic but non-diffeomorphic 4-manifolds.
Locally such maps are given by stable maps between {\mathbb<br />
R}^4\to{\mathbb R}^4, where there is two types: 2 maps (rank 2
singularities) with a 2-dimensional singular subset and 5 maps
(Morin singularities or rank 3 singularities) with a 3-dimensional
singular subset. Stingley extends this result to
smooth 4-manifolds and shows that the rank 2 singularities can be
killed by an isotopy for maps f:M\to M' between two homeomorphic
but non-diffeomorphic 4-manifolds. Thus we are left with the rank 3 singularities. Furthermore the corresponding manifold is closed. That supports the use of Seifert theory.
That agrees with a result of Freedman, Hsiang and Stong. They analyse the failure of the smooth h-cobordism and prove a structure theorem. Then the h-cobordism can be divided into two parts: a trivial h-cobordism inducing the homeomorphism between the two manifolds and a subcobordism between two contractable submanifolds A1, A2 of M and N, respectively. The boundary of this submanifolds A1,A2 are homology 3-spheres (see Freedman, 1982). By the usual association between critical points of Morse functions and cobordism, it was shown (I forgot the reference, maybe Milnor) that a singular map and the cobordism are associated to each other.
Some words about the Poincare duality. Yes you are right. I use a combination of the Poincare duality to relate the k form to an n-k cycle. Then I use the duality of an k cycle and an n-k cycle for a compact manifold. The element of the fundamental group is related to homology class by using the Hurewicz isomorphism, i.e. I can only relate the elements of the fundamental group which are not belong to the commutator subgroup.
Hopefully you are satisfied with that explanation. Otherwise please write.
