Geometry and topology of proper polynomial mappings

Abstract

We give a review of recent results on the fibers of proper polynomial mappings obtained by A. Agrachev, R. Gamkrelidze, Z. Jelonek, V. Tretyakov, H. Żoląndek, and the present author and apply them to a number of concrete problems. In particular, we indicate effectively verifiable sufficient conditions of properness and list general geometric properties of the nonproperness set. We also establish basic geometric properties of proper homogeneous and quadratic mappings. Special attention is given to stable quadratic mappings. Namely, we give estimates for various topological invariants of their fibers and give a complete description of the possible topological structure of fibers in a number of cases. A few applications of these results are also indicated.