Topological Methods in Nonlinear Analysis

Removing isolated zeroes by homotopy

Adam Coffman and Jiří Lebl

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Suppose that the inverse image of the zero vector by a continuous map $f\colon {\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. The existence of a continuous map $g$ which approximates $f$ but is nonvanishing near $P$ is equivalent to a topological property we call ``local inessentiality of zeros'', generalizing the notion of index zero for vector fields, the $q=n$ case. We consider the problem of constructing such an approximation $g$ and a continuous homotopy $F(x,t)$ from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic map, then there exists $F$ also semialgebraic. If $q=2$ and $f$ is real analytic with a locally inessential zero, then there exists a Hölder continuous homotopy $F(x,t)$ which, for $(x,t)\ne(P,0)$, is real analytic and nonvanishing. The existence of a smooth homotopy, given a smooth map $f$, is stated as an open question.

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Topol. Methods Nonlinear Anal., Volume 54, Number 1 (2019), 275-296.

First available in Project Euclid: 16 July 2019

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Coffman, Adam; Lebl, Jiří. Removing isolated zeroes by homotopy. Topol. Methods Nonlinear Anal. 54 (2019), no. 1, 275--296. doi:10.12775/TMNA.2019.042.

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  • H. Aikawa, Hölder continuity of the Dirichlet solution for a general domain, Bull. London Math. Soc. 34 (2002), no. 6, 691–702.
  • D. Anker, On Removing Isolated Zeroes of Vector Fields by Perturbation, Ph.D. Thesis, University of Michigan, 1981.
  • D. Anker, On removing isolated zeroes of vector fields by perturbation, Nonlinear Anal. 8 (1984), no. 9, 1005–1112.
  • P. Baum, Quadratic maps and stable homotopy groups of spheres, Illinois J. Math. 11 (1967), 586–595.
  • J. Bochnak, M. Coste and M. Roy, Real Algebraic Geometry, MSM, vol. 36, Springer, 1998.
  • R. Brown, M. Furi, L. Górniewicz and B. Jiang (eds.), Handbook of Topological Fixed Point Theory, Springer, 2005.
  • A. Coffman, CR singular immersions of complex projective spaces, Beiträge zur Algebra und Geometrie \bf43 (2002), no. 2, 451–477.
  • A. Coffman, Real congruence of complex matrix pencils and complex projections of real Veronese varieties, Linear Algebra Appl. 370 (2003), 41–83.
  • E.N. Dancer, On the existence of zeros of perturbed operators, Nonlinear Anal. 7 (1983), no. 7, 717–727.
  • E.N. Dancer, Bifurcation under continuous groups of symmetries, Systems of Nonlinear Partial Differential Equations (Oxford, 1982), 343–350; NATO ASI Ser. C \bf111, Reidel, Dordrecht, 1983.
  • E.N. Dancer, Perturbation of zeros in the presence of symmetries, J. Austral. Math. Soc. Ser. A \bf36 (1984), no. 1, 106–125.
  • F. Deloup, The fundamental group of the circle is trivial, Amer. Math. Monthly \bf112 (2005), no. 5, 417–425.
  • A. Elgindi, A topological obstruction to the removal of a degenerate complex tangent and some related homotopy and homology groups, Internat. J. Math. 26 (2015), no. 5, 1550025, 16 pp.
  • M. Fenille, Epsilon Nielsen coincidence theory, Cent. Eur. J. Math. 12 (2014), no. 9, 1337–1348.
  • P. Garabedian, Partial Differential Equations, Wiley, 1964.
  • D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, CIM, 2001.
  • O.A. Ladyzhenskaya and N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations, 1964 (in Russian); English transl.: Academic Press, 1968.
  • J. Lee, Introduction to Smooth Manifolds, second ed., GTM, vol. 218, Springer, 2013.
  • V.G. Maz'ya, Notes on Hölder regularity of a boundary point with respect to an elliptic operator of second order, Problemy Matematicheskogo Analiza 74 (2013), 117–121; English transl.: J. Math. Sci. (N.Y.) 196 (2014), no. 4, 572–577.
  • M. Nestler, I. Nitschke, S. Praetorius and A. Voigt, Orientational order on surfaces\rom: the coupling of topology, geometry, and dynamics, J. Nonlinear Sci. \bf28 (2018), no. 1, 147–191.
  • J. Palis and C. Pugh, Fifty problems in dynamical systems, Dynamical Systems, Warwick 1974, 345–353; LNM 468, Springer, 1975.
  • C. Simon and C. Titus, Removing index-zero singularities with C$^1$-small perturbations, Dynamical Systems, Warwick 1974, 278–286; LNM \bf468, Springer, 1975.
  • E. Spanier, Algebraic Topology, McGraw-Hill, 1966.
  • R. Wood, Polynomial maps from spheres to spheres, Invent. Math. 5 (1968), 163–168.