Swallowtails and Whitney umbrellas are homeomorphic

Abstract

. We show that the surface of all degree 2n real polynomials with multiple real zeros is homeomorphic to the surface of all pairs of degree n real polynomials with common real zeros. This proves the simplest case of the Szucs' conjecture suggesting a homeomorphism of generalized swallowtails and Whitney umbrellas. x1. Introduction Swallowtails and Whitney umbrellas are naturally arising as the simplest singularities of smooth maps and functions. Increasing interest in the geometrical properties of these surfaces is connected with the study of the topology of various functional spaces (see [A],[V]). Here we deal exclusively with the real situation. It should be mentioned that the correspondent complex objects are nonhomeomorphic: the "handles of Whitney umbrella and the swallow tail lie in different connected components. However it was proved recently that the homology of the complements to these complex objects are isomorphic (see [V1], [CCMM]). Our main objects can be also defined as...