h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash Categories, Uniform bound and Effectiveness
Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1573-1597.

The h-cobordism theorem is a noted theorem in differential and PL topology. A generalization of the h-cobordism theorem for possibly non simply connected manifolds is the so called s-cobordism theorem. In this paper, we prove semialgebraic and Nash versions of these theorems. That is, starting with semialgebraic or Nash cobordism data, we get a semialgebraic homeomorphism (respectively a Nash diffeomorphism). The main tools used are semialgebraic triangulation and Nash approximation.

One aspect of the algebraic nature of semialgebraic or Nash objects is that one can measure their complexities. We show h and s-cobordism theorems with a uniform bound on the complexity of the semialgebraic homeomorphism (or Nash diffeomorphism) obtained in terms of the complexity of the cobordism data. The uniform bound of semialgebraic h-cobordism cannot be recursive, which gives another example of non effectiveness in real algebraic geometry. Finally we deduce the validity of the semialgebraic and Nash versions of these theorems over any real closed field.

Le théorème de h-cobordisme est bien connu en topologie différentielle et PL. Une généralisation pour les h-cobordismes possiblement non simplement connexe est appelée théorème de s-cobordisme. Dans ce papier, nous démontrons les versions semi-algébrique et Nash de ces théorèmes. C’est-à-dire, avec des données semi-algébriques ou Nash, nous obtenons un homéomophisme semi-algébrique (respectivement un difféomorphisme Nash). Les principaux outils intervenant sont la triangulation semi-algébrique et les approximations Nash.

Un aspect de la nature algébrique des objets semi-algébriques et Nash est qu’on peut mesurer leurs complexités. Nous montrons les théorèmes de h et s-cobordisme avec borne uniforme sur la complexité de l’homéomorphisme semi-algébrique (difféomorphisme Nash) obtenu, en fonction de complexité des données du cobordisme. La borne uniforme pour le h-cobordisme semi-algébrique réelle ne peut être effective. Ce qui donne un autre exemple de non effectivité en géométrie algébrique réelle. Pour finir, nous déduisons la validité de ces théorèmes version semi-algébrique et Nash sur tout corps réel clos.

DOI: 10.5802/aif.2652
Classification: 14P20,  57N70
Keywords: Cobordism, semialgebraic, complexity, effectiveness
Demdah Kartoue , Mady 1

1 Université de Rennes 1 IRMAR (UMR CNRS 6625) Campus de Beaulieu 35042 Rennes cedex (France)
@article{AIF_2011__61_4_1573_0,
     author = {Demdah Kartoue , Mady},
     title = {h-cobordism and s-cobordism {Theorems:} {Transfer} over {Semialgebraic} and {Nash} {Categories,} {Uniform} bound and {Effectiveness}},
     journal = {Annales de l'Institut Fourier},
     pages = {1573--1597},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {4},
     year = {2011},
     doi = {10.5802/aif.2652},
     mrnumber = {2951505},
     zbl = {1267.57024},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2652/}
}
TY  - JOUR
AU  - Demdah Kartoue , Mady
TI  - h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash Categories, Uniform bound and Effectiveness
JO  - Annales de l'Institut Fourier
PY  - 2011
DA  - 2011///
SP  - 1573
EP  - 1597
VL  - 61
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2652/
UR  - https://www.ams.org/mathscinet-getitem?mr=2951505
UR  - https://zbmath.org/?q=an%3A1267.57024
UR  - https://doi.org/10.5802/aif.2652
DO  - 10.5802/aif.2652
LA  - en
ID  - AIF_2011__61_4_1573_0
ER  - 
%0 Journal Article
%A Demdah Kartoue , Mady
%T h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash Categories, Uniform bound and Effectiveness
%J Annales de l'Institut Fourier
%D 2011
%P 1573-1597
%V 61
%N 4
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2652
%R 10.5802/aif.2652
%G en
%F AIF_2011__61_4_1573_0
Demdah Kartoue , Mady. h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash Categories, Uniform bound and Effectiveness. Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1573-1597. doi : 10.5802/aif.2652. http://archive.numdam.org/articles/10.5802/aif.2652/

[1] Acquistapace, F.; Benedetti, R.; Broglia, F. Effectiveness-noneffectiveness in semialgebraic and PL geometry, Invent. Math., Volume 102 (1990) no. 1, pp. 141-156 | DOI | EuDML | MR | Zbl

[2] Bochnak, J.; Coste, M.; Roy, M-F Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 36, Springer-Verlag, Berlin, 1998 (Translated from the 1987 French original, Revised by the authors) | MR | Zbl

[3] Coste, M. Unicité des triangulations semi-algébriques: validité sur un corps réel clos quelconque, et effectivité forte, C. R. Acad. Sci. Paris Sér. I Math., Volume 312 (1991) no. 5, pp. 395-398 | MR | Zbl

[4] Coste, M.; Shiota, M. Nash triviality in families of Nash manifolds, Invent. Math., Volume 108 (1992) no. 2, pp. 349-368 | DOI | EuDML | MR | Zbl

[5] Delfs, H.; Knebusch, M. Locally semialgebraic spaces, Lecture Notes in Mathematics, 1173, Springer-Verlag, Berlin, 1985 | MR | Zbl

[6] Fukui, T.; Koike, S.; Shiota, M. Modified Nash triviality of a family of zero-sets of real polynomial mappings, Ann. Inst. Fourier (Grenoble), Volume 48 (1998) no. 5, pp. 1395-1440 | DOI | EuDML | Numdam | MR | Zbl

[7] Hudson, J. F. P. Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969 | MR | Zbl

[8] Kervaire, M. A. Le théorème de Barden-Mazur-Stallings, Comment. Math. Helv., Volume 40 (1965), pp. 31-42 | DOI | EuDML | MR | Zbl

[9] Manin, Yu. I. A course in mathematical logic, Springer-Verlag, New York, 1977 (Translated from the Russian by Neal Koblitz, Graduate Texts in Mathematics, Vol. 53) | MR | Zbl

[10] Ramanakoraisina, R. Complexité des fonctions de Nash, Comm. Algebra, Volume 17 (1989) no. 6, pp. 1395-1406 | DOI | MR | Zbl

[11] Rourke, C. P.; Sanderson, B. J. Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin, 1982 (Reprint) | MR | Zbl

[12] Shiota, M. Nash manifolds, Lecture Notes in Mathematics, 1269, Springer-Verlag, Berlin, 1987 | MR | Zbl

[13] Shiota, M.; Yokoi, M. Triangulations of subanalytic sets and locally subanalytic manifolds, Trans. Amer. Math. Soc., Volume 286 (1984) no. 2, pp. 727-750 | DOI | MR | Zbl

[14] Smale, S. Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2), Volume 74 (1961), pp. 391-406 | DOI | MR | Zbl

[15] Volodin, I. A.; Kuznecov, V. E.; Fomenko, A. T. The problem of the algorithmic discrimination of the standard three-dimensional sphere, Uspehi Mat. Nauk, Volume 29 (1974) no. 5(179), pp. 71-168 (Appendix by S. P. Novikov) | MR | Zbl

Cited by Sources: