Some Algebraic Properties of Cerf Diagrams of One-Parameter Function Families

Abstract

We obtain results concerning Arnold's problem about a generalization of the Pontryagin-Thom construction in cobordism theory to real algebraic functions. The Pontryagin-Thom construction in the Wells form is transferred to the space of real functions. The relation of the problem with algebraic K -theory and the h-principle due to Eliashberg and Mishachev is revealed.The Cerf diagram of a family of functions f λ , λ ∈ [0, 1] k , on a manifold is the hypersurface with singularities in [0, 1] k × R consisting of all possible pairs (λ, x), where x is one of the critical values of f λ . Akhmet ev [3] obtained some topological restrictions on the global structure of Cerf diagrams that have proper local singularities and correspond to two-parameter families. In the present paper, we prove a similar result for Cerf diagrams of one-parameter families.