Stability of non-proper functions


  • Kenta Hayano



The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney topology). We show that a Morse function is stable if it is end-trivial at any point in its discriminant, where end-triviality (which is also called local triviality at infinity) is a property concerning behavior of functions around the ends of the source manifolds. We further show that a Morse function is strongly stable if (and only if) it is quasi-proper. This result yields existence of a strongly stable but not infinitesimally stable function. Applying our result on stability, we give a sufficient condition for stability of Nash functions, and show that any Nash function becomes stable after a generic linear perturbation.


Bochnak, J., Coste, M., and Roy, M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 36. Springer-Verlag, Berlin, 1998.

D'Acunto, D., and Grandjean, V., On gradient at infinity of semialgebraic functions, Ann. Polon. Math. 87 (2005), 39–49.

Dias, L. R. G., and Tibăr, M., Detecting bifurcation values at infinity of real polynomials, Math. Z. 279 (2015), no. 1–2, 311–319.

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.

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.

Ichiki, S., Generic linear perturbations, Proc. Amer. Math. Soc. 146 (2018), no. 11, 4981–4991.

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.

Kurdyka, K., Orro, P., and Simon, S., Semialgebraic Sard theorem for generalized critical values, J. Differential Geom. 56 (2000), no. 1, 67–92.

Mather, J. N., Stability of $C^infty $ mappings. II. Infinitesimal stability implies stability, Ann. of Math. (2) 89 (1969), 254–291.

Mather, J. N., Stability of $C^infty $ mappings. V. Transversality, Advances in Math. 4 (1970), 301–336.

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.

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.

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.



How to Cite

Hayano, K. (2022). Stability of non-proper functions. MATHEMATICA SCANDINAVICA, 128(2).