Stability of non-proper functions. (English) Zbl 1503.57030
A smooth map \(f:N\to P\) between manifolds \(N,P\) without boundary is said to be stable if there exists an open neighborhood \(\mathcal{U}\subset C^\infty(N,P)\) of \(f\) in the Whitney \(C^\infty\) topology, and maps \(\Theta:\mathcal{U}\to\mathrm{Diff}(N)\), \(\theta:\mathcal{U}\to\mathrm{Diff}(P)\) such that \(f=\theta(g)\circ g\circ\Theta(g)\) for every \(g\in\mathcal{U}\). It is called strongly stable if \(\Theta\) and \(\theta\) can be chosen to be continuous. J. N. Mather [Ann. Math. (2) 89, 254–291 (1969; Zbl 0177.26002); Adv. Math. 4, 301–336 (1970; Zbl 0207.54303)] proved that proper maps are stable provided they are infinitesimally stable. The main theorem of the present paper yields a criterion for a Morse function \(f:N\to\mathbb{R}\) to be stable. In addition it states that a Morse function is strongly stable iff there exists a neighborhood \(V\subset\mathbb{R}\) of the set of critical values of \(f\) such that the restriction \(f|_{f^{-1}(V)}:f^{-1}(V)\to V\) is proper.
As a consequence it is proved that the map \(\mathbb{R}\to\mathbb{R}\), \(x\mapsto\exp(-x^2)\sin(x)\), is strongly stable but not infinitesimally stable. The paper contains also a criterion for the stability of Nash functions \(f:\mathbb{R}^n\to\mathbb{R}\), and it is proved that Nash functions become stable after a generic linear perturbation.
As a consequence it is proved that the map \(\mathbb{R}\to\mathbb{R}\), \(x\mapsto\exp(-x^2)\sin(x)\), is strongly stable but not infinitesimally stable. The paper contains also a criterion for the stability of Nash functions \(f:\mathbb{R}^n\to\mathbb{R}\), and it is proved that Nash functions become stable after a generic linear perturbation.
Reviewer: Thomas J. Bartsch (Gießen)
MSC:
| 57R35 | Differentiable mappings in differential topology |
| 14P20 | Nash functions and manifolds |
| 26B99 | Functions of several variables |
| 57R70 | Critical points and critical submanifolds in differential topology |
References:
| [1] | Bochnak, J., Coste, M., and Roy, M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 36. Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-3-662-03718-8 D’Acunto, D., and Grandjean, V., On gradient at infinity of semialgebraic functions, Ann. Polon. Math. 87 (2005), 39-49. https://doi.org/10.4064/ap87-0-4 Dias, L. R. G., and Tibăr, M., Detecting bifurcation values at infinity of real polynomials, Math. Z. 279 (2015), no. 1-2, 311-319. https://doi.org/10.1007/s00209-014-1369-4 Dimca, A., Morse functions and stable mappings, Rev. Roumaine Math. Pures Appl. 24 (1979), no. 9, 1293-1297. du Plessis, A., and Vosegaard, H., Characterisation of strong smooth stability, Math. Scand, 88 (2001), no. 2, 193-228. https://doi.org/10.7146/math.scand.a-14323 du Plessis, A., and Wall, T., The geometry of topological stability, London Mathematical Society Monographs. New Series, 9. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. Engelking, R., General topology, Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989. Golubitsky, M., and Guillemin, V., Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14. Springer-Verlag, New York-Heidelberg, 1973. Gay D., and Kirby, R., Trisecting \(4\)-manifolds, Geom. Topol. 20 (2016), no. 6, 3097-3132. https://doi.org/10.2140/gt.2016.20.3097 Ichiki, S., Generic linear perturbations, Proc. Amer. Math. Soc. 146 (2018), no. 11, 4981-4991. https://doi.org/10.1090/proc/14094 Kurdyka, K., Mostowski T., and Parusiński, A., Proof of the gradient conjecture of R. Thom, Ann. of Math. (2), 152 (2000), no. 3, 763-792. https://doi.org/10.2307/2661354 Kurdyka, K., Orro, P., and Simon, S., Semialgebraic Sard theorem for generalized critical values, J. Differential Geom. 56 (2000), no. 1, 67-92. http://projecteuclid.org/euclid.jdg/1090347525 Mather, J. N., Stability of \(C^infty\) mappings. II. Infinitesimal stability implies stability, Ann. of Math. (2) 89 (1969), 254-291. https://doi.org/10.2307/1970668 Mather, J. N., Stability of \(C^infty\) mappings. V. Transversality, Advances in Math. 4 (1970), 301-336. https://doi.org/10.1016/0001-8708(70)90028-9 Mather, J. N., Stratifications and mappings, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 195-232. Academic Press, New York, 1973. Mather, J. N., How to stratify mappings and jet spaces, Singularités d’applications différentiables (Sém., Plans-sur-Bex, 1975), pp. 128-176. Lecture Notes in Math. 535 (1976). Mather, J. N., Notes on topological stability, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 4, 475-506. https://doi.org/10.1090/S0273-0979-2012-01383-6 Murolo, C., du Plessis, A., and Trotman, D., On the smooth Whitney fibering conjecture, HAL-01571382 version 1 (2017), 66 pp. Saeki, O., and Yamamoto, T., Singular fibers of stable maps and signatures of 4-manifolds, Geom. Topol. 10 (2006), 359-399. https://doi.org/10.2140/gt.2006.10.359 Whitney, H., Singularities of mappings of Euclidean spaces, 1958 International symposium on algebraic topology, pp. 285-301. Universidad Nacional Autónoma de México and UNESCO, Mexico City. · doi:10.1007/978-3-662-03718-8{\par}D'Acunto, |
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