Definable continuous mappings and Whyburn’s conjecture
HTML articles powered by AMS MathViewer
- by Sĩ Tiệp Dinh and Tien-Son PhẠm
- Proc. Amer. Math. Soc. 151 (2023), 2081-2095
- DOI: https://doi.org/10.1090/proc/16272
- Published electronically: February 10, 2023
- HTML | PDF | Request permission
Abstract:
For a definable continuous mapping $f$ from a definable connected open subset $\Omega$ of $\mathbb R^n$ into $\mathbb R^n$, we show that the following statements are equivalent:
- The mapping $f$ is open.
- The fibers of $f$ are finite and the Jacobian of $f$ does not change sign on the set of points at which $f$ is differentiable.
- The fibers of ${f}$ are finite and the set of points at which $f$ is not a local homeomorphism has dimension at most $n - 2$.
As an application, we prove that Whyburn’s conjecture is true for definable mappings: A definable surjective open continuous mapping of one closed ball into another which maps boundary homeomorphically onto boundary is necessarily a homeomorphism.
References
- Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. MR 1659509, DOI 10.1007/978-3-662-03718-8
- J. M. Borwein and D. M. Zhuang, Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps, J. Math. Anal. Appl. 134 (1988), no. 2, 441–459. MR 961349, DOI 10.1016/0022-247X(88)90034-0
- A. V. Černavskiĭ, Finite-to-one open mappings of manifolds, Mat. Sb. (N.S.) 65(107) (1964), 357–369 (Russian). MR 172256
- A. V. Černavskiĭ, Addendum to the paper “Finite-to-one open mappings of manifolds”, Mat. Sb. (N.S.) 66(108) (1965), 471–472 (Russian). MR 220254
- P. T. Church, Differentiable open maps on manifolds, Trans. Amer. Math. Soc. 109 (1963), 87–100. MR 154296, DOI 10.1090/S0002-9947-1963-0154296-6
- P. T. Church, Differentiable maps with non-negative Jacobian, J. Math. Mech. 16 (1967), 703–708. MR 205263
- P. T. Church, Discrete maps on manifolds, Michigan Math. J. 25 (1978), no. 3, 351–357. MR 512905
- Philip T. Church and Erik Hemmingsen, Light open maps on $n$-manifolds, Duke Math. J. 27 (1960), 527–536. MR 116315
- M. Coste, An Introduction to O-Minimal Geometry. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica. Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000.
- Jane Cronin and L. F. McAuley, Whyburn’s conjecture for some differentiable maps, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 405–412. MR 202116, DOI 10.1073/pnas.56.2.405
- Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
- Maciej P. Denkowski and Jean-Jacques Loeb, On open analytic and subanalytic mappings, Complex Var. Elliptic Equ. 62 (2017), no. 1, 27–46. MR 3575850, DOI 10.1080/17476933.2016.1200035
- A. Dold, Lectures on algebraic topology, Die Grundlehren der mathematischen Wissenschaften, Band 200, Springer-Verlag, New York-Berlin, 1972 (German). MR 415602
- J. M. Gamboa and F. Ronga, On open real polynomial maps, J. Pure Appl. Algebra 110 (1996), no. 3, 297–304. MR 1393117, DOI 10.1016/0022-4049(95)00106-9
- M. Seetharama Gowda and Roman Sznajder, On the Lipschitzian properties of polyhedral multifunctions, Math. Programming 74 (1996), no. 3, Ser. A, 267–278. MR 1407688, DOI 10.1016/0025-5610(96)00006-8
- H. V. Hà and T. S. Phạm, Genericity in Polynomial Optimization, volume 3 of Series on Optimization and Its Applications. World Scientific, Singapore, 2017.
- Truong Xuan Duc Ha and Tiến-Sơn Phạm, Some classical analysis results for continuous definable mappings, J. Math. Anal. Appl. 515 (2022), no. 1, Paper No. 126380, 19. MR 4435919, DOI 10.1016/j.jmaa.2022.126380
- Robert M. Hardt, Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math. 102 (1980), no. 2, 291–302. MR 564475, DOI 10.2307/2374240
- Morris W. Hirsch, Jacobians and branch points of real analytic open maps, Aequationes Math. 63 (2002), no. 1-2, 76–80. MR 1891276, DOI 10.1007/s00010-002-8006-8
- A. Ioffe, A Sard theorem for tame set-valued mappings, J. Math. Anal. Appl. 335 (2007), no. 2, 882–901. MR 2345506, DOI 10.1016/j.jmaa.2007.01.104
- A. D. Ioffe, An invitation to tame optimization, SIAM J. Optim. 19 (2008), no. 4, 1894–1917. MR 2486055, DOI 10.1137/080722059
- Joseph Johns, An open mapping theorem for o-minimal structures, J. Symbolic Logic 66 (2001), no. 4, 1817–1820. MR 1877024, DOI 10.2307/2694977
- Jae Hyoung Lee and Tien-Son Pham, Openness, Hölder metric regularity, and Hölder continuity properties of semialgebraic set-valued maps, SIAM J. Optim. 32 (2022), no. 1, 56–74. MR 4358476, DOI 10.1137/20M1331901
- N. G. Lloyd, Degree theory, Cambridge Tracts in Mathematics, No. 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978. MR 493564
- Stanisław Łojasiewicz, Introduction to complex analytic geometry, Birkhäuser Verlag, Basel, 1991. Translated from the Polish by Maciej Klimek. MR 1131081, DOI 10.1007/978-3-0348-7617-9
- Morris L. Marx, Whyburn’s conjecture for $C^{2}$ maps, Proc. Amer. Math. Soc. 19 (1968), 660–661. MR 226604, DOI 10.1090/S0002-9939-1968-0226604-9
- Louis F. McAuley, Concerning a conjecture of Whyburn on light open mappings, Bull. Amer. Math. Soc. 71 (1965), 671–674. MR 176448, DOI 10.1090/S0002-9904-1965-11392-1
- B. S. Mordukhovich, Variational Analysis and Applications. Springer, New York, 2018.
- Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR 217337
- Jean-Paul Penot, Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal. 13 (1989), no. 6, 629–643. MR 998509, DOI 10.1016/0362-546X(89)90083-7
- Ya’acov Peterzil and Sergei Starchenko, Computing o-minimal topological invariants using differential topology, Trans. Amer. Math. Soc. 359 (2007), no. 3, 1375–1401. MR 2262855, DOI 10.1090/S0002-9947-06-04220-6
- B. H. Pourciau, Univalence and degree for Lipschitz continuous maps, Arch. Rational Mech. Anal. 81 (1983), no. 3, 289–299. MR 683357, DOI 10.1007/BF00250804
- Stefan Scholtes, Introduction to piecewise differentiable equations, SpringerBriefs in Optimization, Springer, New York, 2012. MR 2953259, DOI 10.1007/978-1-4614-4340-7
- Chris Shannon, Regular nonsmooth equations, J. Math. Econom. 23 (1994), no. 2, 147–165. MR 1266510, DOI 10.1016/0304-4068(94)90003-5
- C. J. Titus and G. S. Young, A Jacobian condition for interiority, Michigan Math. J. 1 (1952), 89–94. MR 49548
- Jussi Väisälä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I 392 (1966), 10. MR 200928
- Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
- Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337, DOI 10.1215/S0012-7094-96-08416-1
- G. T. Whyburn, An open mapping approach to Hurwitz’s theorem, Trans. Amer. Math. Soc. 71 (1951), 113–119. MR 42693, DOI 10.1090/S0002-9947-1951-0042693-0
- A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4, 1051–1094. MR 1398816, DOI 10.1090/S0894-0347-96-00216-0
- David Wilson, Open mappings on manifolds and a counterexample to the Whyburn conjecture, Duke Math. J. 40 (1973), 705–716. MR 320989
- N. D. Yen, J.-C. Yao, and B. T. Kien, Covering properties at positive-order rates of multifunctions and some related topics, J. Math. Anal. Appl. 338 (2008), no. 1, 467–478. MR 2386430, DOI 10.1016/j.jmaa.2007.05.041
Bibliographic Information
- Sĩ Tiệp Dinh
- Affiliation: Institute of Mathematics, VAST, 18, Hoang Quoc Viet Road, Cau Giay District 10307, Hanoi, Vietnam
- MR Author ID: 885696
- ORCID: 0000-0001-9116-4534
- Email: dstiep@math.ac.vn
- Tien-Son PhẠm
- Affiliation: Department of Mathematics, Dalat University, 1 Phu Dong Thien Vuong, Dalat, Vietnam
- MR Author ID: 632781
- Email: sonpt@dlu.edu.vn
- Received by editor(s): May 24, 2022
- Received by editor(s) in revised form: September 6, 2022, September 7, 2022, and September 11, 2022
- Published electronically: February 10, 2023
- Additional Notes: This work was supported by the International Centre of Research and Postgraduate Training in Mathematics (ICRTM) under grant number ICRTM01\rule{0.1cm}0.15mm2022.01
The first author is the corresponding author. - Communicated by: Nageswari Shanmugalingam
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 2081-2095
- MSC (2020): Primary 26B10, 54C10, 03C64
- DOI: https://doi.org/10.1090/proc/16272
- MathSciNet review: 4556202