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Normal triangulations in o-minimal structures. (English) Zbl 1207.03048

The paper under review concerns improvements of the triangulation theorem for definable sets in o-minimal expansions of fields. This theorem states that if \(M\) is an o-minimal expansion of a field and \(X\subseteq M^n\) is a definable set, then there is a simplicial complex \(K\) a definable homeomorphism \(\phi\) from the realization \(|K|\) of \(K\) to \(X\). In fact, the triangulation can be chosen to be compatible with given definable subsets \(A_1,\ldots,A_k\) of \(X\). (See Chapter 8 of L. van den Dries’ book [Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series. 248. Cambridge: Cambridge University Press (1998; Zbl 0953.03045)] for the precise statement and for the proof). The author proves several refinements of this result. Some of the statements are a little too technical to repeat here, but the following (taken from the introduction to the paper) is representative. Suppose that we are given a definable triangulation \((K,\phi)\) of a definable set \(X\) compatible with some definable subsets \(A_1,\ldots,A_k\) of \(X\). Then given further definable subsets \(B_1,\ldots,B_l\) of \(X\), there is a subdivision \(K'\) of \(K\) and a definable triangulation \((K',\phi)\) of \(X\) compatible with \(A_1,\ldots,A_k,B_1,\ldots,B_l\) such that \(\phi'\) is definably homotopic to \(\phi\).
The paper also contains several applications of these results. For example, the author extends the semialgebraic Hauptvermutung from the real field to arbitrary real closed fields. See the paper for the precise result. (In a recent preprint, Shiota has proved a version of the Hauptvermutung for arbitrary o-minimal expansions of fields). As a second application, the author shows that his results can be used to give an alternative proof of Woerheide’s theorem on the existence of an o-minimal simplicial homology theory.

MSC:

03C64 Model theory of ordered structures; o-minimality
14P10 Semialgebraic sets and related spaces
57Q25 Comparison of PL-structures: classification, Hauptvermutung

Citations:

Zbl 0953.03045
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References:

[1] DOI: 10.1090/S0002-9947-1984-0760983-2
[2] DOI: 10.1112/S0024609306018832 · Zbl 1116.03032
[3] An open mapping theorem for o-minimal structures 66 pp 1817– (2001) · Zbl 0993.03052
[4] Tame topology and o-minimal structures 248 (1998)
[5] Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 312 pp 395– (1991)
[6] The Quarterly Journal of Mathematics pp 15– (2009)
[7] Locally semialgebraic spaces 1173 (1985)
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