Nash triviality in families of Nash manifolds

Coste, Michel; Shiota, Masahiro

doi:10.1007/bf02100609

Cited by 39 publications

(50 citation statements)

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“…With respect to (ii), we must wait till [CoSh2], where the existence of global equations is proved: first existence over compact sets, which enables one to, second, compactify the problem and, third, deduce existence in the non-compact case from existence in the compact one. It is well remarkable that the existence over compact sets is based on the semialgebraic (version of) Thom's first isotopy lemma, the main result of [CoSh1]. Also, we recall that separation depends on the finiteness of global sections, as we explained when comparing the usual and the Grothendieck topologies.…”

Section: Solutions In the Non-compact Casementioning

confidence: 88%

“…The usual proofs of Thom's first isotopy lemma use integration of vector fields and cannot produce trivializations of class Nash. So the use of the semialgebraic version of Thom's isotopy lemma of [CoSh1] is indispensable here. Considerations of this nature are present from the very beginning in the study of semialgebraic sets and maps, back in the pioneering papers by Hardt ([Ha]), and remain a matter of high interest.…”

Section: Existence Of Global Equations Over Compact Sets Let I Be a mentioning

confidence: 99%

See 1 more Smart Citation

Global Problems on Nash Functions.

Shiota

Coste²,

Sancho

2004

Rev. Mat. Complut.

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This is a survey on the history of and the solutions to the basic global problems on Nash functions, which have been only recently solved, namely: separation, extension, global equations, Artin-Mazur description and idempotency, also noetherianness. We discuss all of them in the various possible contexts, from manifolds over the reals to real spectra of arbitrary commutative rings.Nash functions appeared for the first time at the early fifties, in a seminal paper by John Nash. Soon other mathematicians were interested on them, and some relevant applications found, but also soon their bad cohomological behaviour was realized. This put an end to a possible systematic development of the theory of Nash functions. Since that moment many specialists have devoted time and efforts to understand the nature of these functions, but only after fifty years we start to understand what is behind their failures, and how the problems generated by those failures should be settled. During this fifty years process the notion of Nash function has evolved in progressively more general settings, with an appealing feedback from concrete to abstract and viceversa. Moreover, a maze of surprising links have been revealed among all the questions involved. These are depicted in the Nash Labyrinth shown below, where each box contains a solved problem, each arrow is an implication, and the doubled boxes are the sources of the flow of arguments. We will try to explain the problems and their solutions in the following sections: §1. Brief history of the topic. §2. The notion of Nash function. §3. Global properties and cohomological failures. §4. Formulation of the problems. §5. Solutions in the compact case. §6. Solutions in the non-compact case. §7. Nash functions over arbitrary real closed fields. §8. Abstract Nash functions.

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Section: Solutions In the Non-compact Casementioning

confidence: 88%

Section: Existence Of Global Equations Over Compact Sets Let I Be a mentioning

confidence: 99%

Global Problems on Nash Functions.

Shiota

Coste²,

Sancho

2004

Rev. Mat. Complut.

Self Cite

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“…Moreover, the integer l is recursive in terms of m, p, q (cf. [CS,Lemma 5.1]). From this we get the following proposition.…”

Section: Nash H-cobordism Theorem Over Any Real Closed Fieldmentioning

confidence: 99%

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

Kartoue¹

2011

Ann. inst. Fourier

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“…This kind of tool was first developed by the third author ( [33], [34]), and M. Coste and the third author have made progress in this direction ( [4], [5]). In this paper, it becomes necessary to show the Nash triviality theorem for a family of pairs of compact Nash manifolds with boundary and compact Nash submanifolds with boundary (Theorem I), in order to prove local modified Nash triviality theorems.…”

Section: Question (03) -Does a Natural And Strong Triviality Which mentioning

confidence: 99%