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A Note on the Alexander Theorem on the Complex Plane

Volume 60 / 2012

Sylwester Zając Bulletin Polish Acad. Sci. Math. 60 (2012), 249-258 MSC: Primary 32W05; Secondary 58B15. DOI: 10.4064/ba60-3-5

Abstract

We investigate the Banach manifold consisting of complex $\mathcal{C}^r$ functions on the unit disc having boundary values in a given one-dimensional submanifold of the plane. We show that ${\partial}/{\partial\overline{\lambda}}$ restricted to that submanifold is a Fredholm mapping. Moreover, for any such function we obtain a relation between its homotopy class and the Fredholm index.

Authors

  • Sylwester ZającInstitute of Mathematics
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail

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