# h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash Categories, Uniform bound and Effectiveness

Mady Demdah Kartoue^{[1]}

- [1] Université de Rennes 1 IRMAR (UMR CNRS 6625) Campus de Beaulieu 35042 Rennes cedex (France)

Annales de l’institut Fourier (2011)

- Volume: 61, Issue: 4, page 1573-1597
- ISSN: 0373-0956

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topDemdah Kartoue, Mady. "h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash Categories, Uniform bound and Effectiveness." Annales de l’institut Fourier 61.4 (2011): 1573-1597. <http://eudml.org/doc/219827>.

@article{DemdahKartoue2011,

abstract = {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.},

affiliation = {Université de Rennes 1 IRMAR (UMR CNRS 6625) Campus de Beaulieu 35042 Rennes cedex (France)},

author = {Demdah Kartoue, Mady},

journal = {Annales de l’institut Fourier},

keywords = {Cobordism; semialgebraic; complexity; effectiveness; cobordism},

language = {eng},

number = {4},

pages = {1573-1597},

publisher = {Association des Annales de l’institut Fourier},

title = {h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash Categories, Uniform bound and Effectiveness},

url = {http://eudml.org/doc/219827},

volume = {61},

year = {2011},

}

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

PB - Association des Annales de l’institut Fourier

VL - 61

IS - 4

SP - 1573

EP - 1597

AB - 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.

LA - eng

KW - Cobordism; semialgebraic; complexity; effectiveness; cobordism

UR - http://eudml.org/doc/219827

ER -

## References

top- F. Acquistapace, R. Benedetti, F. Broglia, Effectiveness-noneffectiveness in semialgebraic and PL geometry, Invent. Math. 102 (1990), 141-156 Zbl0729.14040MR1069244
- J. Bochnak, M. Coste, M-F Roy, Real algebraic geometry, 36 (1998), Springer-Verlag, Berlin Zbl0633.14016MR1659509
- M. Coste, Unicité des triangulations semi-algébriques: validité sur un corps réel clos quelconque, et effectivité forte, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 395-398 Zbl0745.14021MR1096619
- M. Coste, M. Shiota, Nash triviality in families of Nash manifolds, Invent. Math. 108 (1992), 349-368 Zbl0801.14017MR1161096
- H. Delfs, M. Knebusch, Locally semialgebraic spaces, 1173 (1985), Springer-Verlag, Berlin Zbl0582.14006MR819737
- T. Fukui, S. Koike, M. Shiota, Modified Nash triviality of a family of zero-sets of real polynomial mappings, Ann. Inst. Fourier (Grenoble) 48 (1998), 1395-1440 Zbl0940.14038MR1662251
- J. F. P. Hudson, Piecewise linear topology, (1969), W. A. Benjamin, Inc., New York-Amsterdam Zbl0189.54507MR248844
- M. A. Kervaire, Le théorème de Barden-Mazur-Stallings, Comment. Math. Helv. 40 (1965), 31-42 Zbl0135.41503MR189048
- Yu. I. Manin, A course in mathematical logic, (1977), Springer-Verlag, New York Zbl0383.03002MR457126
- R. Ramanakoraisina, Complexité des fonctions de Nash, Comm. Algebra 17 (1989), 1395-1406 Zbl0684.14008MR997146
- C. P. Rourke, B. J. Sanderson, Introduction to piecewise-linear topology, (1982), Springer-Verlag, Berlin Zbl0254.57010MR665919
- M. Shiota, Nash manifolds, 1269 (1987), Springer-Verlag, Berlin Zbl0629.58002MR904479
- M. Shiota, M. Yokoi, Triangulations of subanalytic sets and locally subanalytic manifolds, Trans. Amer. Math. Soc. 286 (1984), 727-750 Zbl0527.57014MR760983
- S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391-406 Zbl0099.39202MR137124
- I. A. Volodin, V. E. Kuznecov, A. T. Fomenko, The problem of the algorithmic discrimination of the standard three-dimensional sphere, Uspehi Mat. Nauk 29 (1974), 71-168 Zbl0311.57001MR405426

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