Products of elements of cobordism-like modules induced from generic maps

TL;DR

This paper extends cobordism-like modules induced from generic maps with negative codimension by defining a product operation, generalizing the classical cobordism product and enhancing tools in differential topology and singularity theory.

Contribution

It introduces a product structure for cobordism-like modules from generic maps, broadening the algebraic framework beyond classical cobordism classes.

Findings

Defined a product operation for cobordism-like modules

Extended classical cobordism product to new modules

Provided tools for applications in topology and singularity theory

Abstract

Recently the author has introduced cobordism-like modules induced from generic maps whose codimensions are negative. They are generalizations of cobordism modules of manifolds. They have been introduced in generalizing the following theorem shown by Hiratuka and Saeki in 2013--14; for a generic map whose codimension is negative including a connected component of an inverse image of a regular value being not null-cobordant and for a space defined as all connected components of inverse images, which is a polyhedron of dimension equal to that of the target space, the top-dimensional homology group does not vanish. Note that such spaces are fundamental and important tools in general, in the differential topological theory of Morse functions and their higher dimensional versions and application to algebraic and differential topology of manifolds, or the global singularity theory. In this paper, the author succeeded in defining suitable elements as the products for pairs of elements in cobordism modules which may be distinct, as in the case of the ordinary cobordism modules. This is an extension of the product of two ordinary cobordism classes of manifolds.