On the Reeb Spaces of Definable Maps

被引：0
作者：
Saugata Basu
Nathanael Cox
Sarah Percival
机构：
[1] Purdue University,Department of Mathematics
来源：
Discrete & Computational Geometry | 2022年 / 68卷
关键词：
Reeb spaces; O-minimal structures; Betti numbers; Semi-algebraic maps; 03C64; 14P10; 62R40;
D O I：
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摘要：
We prove that the Reeb space of a proper definable map f:X→Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:X \rightarrow Y$$\end{document} in an arbitrary o-minimal expansion of a real closed field is realizable as a proper definable quotient. This result can be seen as an o-minimal analog of Stein factorization of proper morphisms in algebraic geometry. We also show that the Betti numbers of the Reeb space of f can be arbitrarily large compared to those of X, unlike in the special case of Reeb graphs of manifolds. Nevertheless, in the special case when f:X→Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:X \rightarrow Y$$\end{document} is a semi-algebraic map and X is closed and bounded, we prove a singly exponential upper bound on the Betti numbers of the Reeb space of f in terms of the number and degrees of the polynomials defining X, Y, and f.
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页码：372 / 405
页数：33
相关论文
共 26 条
[1]  
Basu S(1999)On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets Discrete Comput. Geom. 22 1-18
[2]  
Basu S(2015)A complexity theory of constructible functions and sheaves Found. Comput. Math. 15 199-279
[3]  
Basu S(2005)On the Betti numbers of sign conditions Proc. Am. Math. Soc. 133 965-974
[4]  
Pollack R(1974)Sur certaines applications génériques d’une variété close à Enseign. Math. 20 275-292
[5]  
Roy M-F(2005) dimensions dans le plan Discrete Comput. Geom. 33 395-401
[6]  
Burlet O(2009)Betti numbers of semialgebraic sets defined by quantifier-free formulae J. Lond. Math. Soc. 80 35-54
[7]  
de Rham G(2004)Approximation of definable sets by compact families, and upper bounds on homotopy and homology J. Lond. Math. Soc. 69 27-43
[8]  
Gabrielov A(1964)Betti numbers of semialgebraic and sub-Pfaffian sets Proc. Am. Math. Soc. 15 275-280
[9]  
Vorobjov N(1986)On the Betti numbers of real varieties Trans. Am. Math. Soc. 295 565-592
[10]  
Gabrielov A(1988)Definable sets in ordered structures. I Trans. Am. Math. Soc. 309 469-476
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