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, p. 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 : https://doi.org/10.5802/aif.2652
Classification:  14P20,  57N70
Keywords: Cobordism, semialgebraic, complexity, effectiveness
     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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {4},
     year = {2011},
     pages = {1573-1597},
     doi = {10.5802/aif.2652},
     mrnumber = {2951505},
     zbl = {1267.57024},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIF_2011__61_4_1573_0/

[1] Acquistapace, F.; Benedetti, R.; Broglia, F. Effectiveness-noneffectiveness in semialgebraic and PL geometry, Invent. Math., Tome 102 (1990) no. 1, pp. 141-156 | Article | MR 1069244 | Zbl 0729.14040

[2] Bochnak, J.; Coste, M.; Roy, M-F Real algebraic geometry, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 36 (1998) (Translated from the 1987 French original, Revised by the authors) | MR 1659509 | Zbl 0633.14016

[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., Tome 312 (1991) no. 5, pp. 395-398 | MR 1096619 | Zbl 0745.14021

[4] Coste, M.; Shiota, M. Nash triviality in families of Nash manifolds, Invent. Math., Tome 108 (1992) no. 2, pp. 349-368 | Article | MR 1161096 | Zbl 0801.14017

[5] Delfs, H.; Knebusch, M. Locally semialgebraic spaces, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1173 (1985) | MR 819737 | Zbl 0582.14006

[6] Fukui, T.; Koike, S.; Shiota, M. Modified Nash triviality of a family of zero-sets of real polynomial mappings, Ann. Inst. Fourier (Grenoble), Tome 48 (1998) no. 5, pp. 1395-1440 | Article | Numdam | MR 1662251 | Zbl 0940.14038

[7] Hudson, J. F. P. Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees (1969) | MR 248844 | Zbl 0189.54507

[8] Kervaire, M. A. Le théorème de Barden-Mazur-Stallings, Comment. Math. Helv., Tome 40 (1965), pp. 31-42 | Article | MR 189048 | Zbl 0135.41503

[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 457126 | Zbl 0383.03002

[10] Ramanakoraisina, R. Complexité des fonctions de Nash, Comm. Algebra, Tome 17 (1989) no. 6, pp. 1395-1406 | Article | MR 997146 | Zbl 0684.14008

[11] Rourke, C. P.; Sanderson, B. J. Introduction to piecewise-linear topology, Springer-Verlag, Berlin, Springer Study Edition (1982) (Reprint) | MR 665919 | Zbl 0254.57010

[12] Shiota, M. Nash manifolds, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1269 (1987) | MR 904479 | Zbl 0629.58002

[13] Shiota, M.; Yokoi, M. Triangulations of subanalytic sets and locally subanalytic manifolds, Trans. Amer. Math. Soc., Tome 286 (1984) no. 2, pp. 727-750 | Article | MR 760983 | Zbl 0527.57014

[14] Smale, S. Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2), Tome 74 (1961), pp. 391-406 | Article | MR 137124 | Zbl 0099.39202

[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, Tome 29 (1974) no. 5(179), pp. 71-168 (Appendix by S. P. Novikov) | MR 405426 | Zbl 0303.57002 | Zbl 0311.57001