Quasi-holomorphic maps

TL;DR

This paper introduces quasi-holomorphic maps, a new class of smooth maps mimicking holomorphic singularities without requiring complex structures, and explores their cobordism groups and associated characteristic classes.

Contribution

It defines quasi-holomorphic maps, establishes a Pontryagin--Thom type construction for them, and extends the applicability of Thom polynomials to these maps.

Findings

Construction of a universal quasi-holomorphic map with prescribed multisingularities

Determination of cohomology classes via Thom polynomials for quasi-holomorphic maps

Complete computation of the free parts of cobordism groups of quasi-holomorphic maps

Abstract

We introduce a new notion, called quasi-holomorphic maps. These are real smooth maps equipped with a structure that imitates the singularities and singularity stratifications of holomorphic maps on the source and target manifolds, although the manifolds themselves carry no global complex structures. Some important examples of quasi-holomorphic maps are branched coverings and links of finitely determined holomorphic map germs. We show a Pontryagin--Thom type construction for a ``universal'' quasi-holomorphic map with prescribed multisingularities, from which all such maps can be induced, and a similar result for maps with prescribed singularities. Applying this, we prove that the Thom polynomials of holomorphic singularities determine the cohomology classes represented by the singular loci of not only holomorphic but quasi-holomorphic maps as well. As another application we define the cobordism groups of quasi-holomorphic maps with restricted multisingularities, whose classifying space was given by the above construction. We completely compute the free parts of these cobordism groups and in some special cases also obtain results on their torsion parts.