A Note on the Alexander Theorem on the Complex Plane
Volume 60 / 2012
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.
