CUTTING AND PASTING OF MORSE FUNCTIONS (Research on topology and differential geometry using singularity theory of differentiable maps)

Abstract

Cobordism groups of various types of Morse functions have been studied separately by several authors including Ikegami, Kalmar, Saeki, Yamamoto, and the author. In this article, we propose a conceptually new approach for studying cobordism groups of several types of Morse functions within a single unifying framework. Our method is crucially based on certain cutting and pasting relations for manifolds that have been used before to define SKK-groups of manifolds. We provide an explicit isomorphism between the cobordism group of Morse functions and SKK-groups. Moreover, we sketch an application of our framework to cobordism theory for Morse functions with boundary, and raise some problems for future study concerning Morse functions with index constraints and circle-valued Morse functions.