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In a volume entitled "Real Analytic and Algebraic Geometry" (ed. F. Broglia, M. Galbiati, and A. Tognoli), there is a paper by Coste and Shiota which proves that if the data in Thom's first isotopy lemma are semi-algebraic, then there exists a semi-algebraic trivialization (in the classical proofs involving integration of controlled vector fields, the flows will not in general be semi-algebraic even if the vector fields are, so this Coste-Shiota result is rather refined).

I'm wondering whether anyone has proved an o-minimal generalization of this result: that if the data of the first isotopy lemma are definable, then there will be a definable trivialization? Guides to relevant literature are highly appreciated!

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Update: I went ahead and contacted Michel Coste about this, and the answer is apparently 'yes' according to some work by Jesús Escribano which I now have. I don't know if this means the discussion should now be closed, but I'm happy to keep it open in case anyone else has something illuminating to say. –  Todd Trimble May 6 '10 at 17:06
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3 Answers

Shiota sent this preprint on the arXiv not long ago : http://arxiv.org/abs/1002.1508

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Thanks, Serge. I did see this and skimmed through it, but wasn't able to extract what I need from it, unfortunately. –  Todd Trimble May 2 '10 at 16:57
It was indeed not a very sharp reference. More relevant could be the appendix (Theorem A.5.) of ### Computing o-minimal topological invariants using differential topology Author(s): Ya'acov Peterzil; Sergei Starchenko Journal: Trans. Amer. Math. Soc. 359 (2007), 1375-1401. ### (it states the result for a proper function, not a general proper mapping.) –  Serge R. May 2 '10 at 17:21
Thanks for the reference, Serge. It's definitely a step in the right direction, and there are some nice technical results in there. It's not exactly what I need, though, as this is ordinary Morse theory done in the o-minimal context, whereas what I need is stratified Morse theory done in the o-minimal context. (NB: even without the o-minimal constraint, stratified Morse theory is much trickier than ordinary Morse theory -- the Harvard notes by Mather make that plain!) Thanks again, though. –  Todd Trimble May 2 '10 at 19:40
(Also a limitation of the results in Shiota's 97 book is that they "only" deals with expansions of the field of reals and not with general o-minimal expansions of a real closed fields.) –  Serge R. May 2 '10 at 22:56
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This Coste-Shiota result contradicts what I thought I knew about Thom's career. Isn't the point that there are examples where you just can't improve "topological stability" to "smooth stability"? (see Todd's comment for a simple explanation.)

Here is a famous example of Whitney's. Take in R^3 the four hypersurfaces x = 0, y = 0, x = y, and x = zy. Let's restrict our attention to where 0 < z < 1. Stratify so that the union of these hypersurfaces is the 2-skeleton and the line x = y = 0 is the 1-skeleton. Consider the map $(x,y,z) \mapsto z$. (Maybe we'd prefer a proper map, so require $x^2 + y^2 \leq 1$ as well and stratify the boundary in the obvious way.) The fibers of this map are R^2's (or disks) stratified by four lines through the origin. One of those lines moves around as z changes.

This map is a submersion on each stratum, so by Thom's isotopy lemma the fibers are all homeomorphic in a stratum-preserving way. But any such trivialization must fail to be C^1 along z = 0, since any C^1 map from one fiber to another has to preserve the cross ratio of those four lines.

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"Semi-algebraic implies smooth, right?" No. Recall that a semi-algebraic set is just a union of intersections of sets defined by polynomial inequalities. By definition, a map is semi-algebraic if its graph is; for example, a PL map from the interval to itself is semi-algebraic. It need not be $C^1$. –  Todd Trimble May 2 '10 at 17:27
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Have you checked Shiota's 1997 book from Birkhauser:

Geometry of Subanalytic and Semialgebraic Sets (Progress in Mathematics)

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Thanks, Carl. Yes, I have the book in my library, and spurred by your comment I'm looking at it again. The material is surely closely related to the theory of o-minimal structures, but I find the axioms of X-sets not at all easy to work with. For me, a huge advantage of working with o-minimal structures is their closure under first-order logical operations and the order relation. By contrast, axiom 3 in the definition of X-sets is for my purposes awkward. Anyway, I really do want to work in the o-minimal context. (Disclaimer: I'm a category theorist, not a hard-core geometer!) –  Todd Trimble May 2 '10 at 22:53
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