Quasi-holomorphic maps
András Csépai, András Szűcs
TL;DR
This work introduces quasi-holomorphic maps—real smooth maps endowed with complex stratifications that mimic holomorphic singularities on spaces lacking global complex structures. It develops a comprehensive global-singularity framework for these maps using a universal $\tau$-map $f_\tau:Y_\tau\to X_\tau$ and Kazarian-type space $K_\sigma$, enabling a Pontryagin–Thom–style classification of maps with restricted multisingularities and a transversality-based description of singular loci via Thom polynomials. The authors show that Thom polynomials of holomorphic singularities determine the cohomology classes of singular loci for both holomorphic and quasi-holomorphic maps, and they define cobordism groups ${\mathop{\rm Qhol}}_\tau(P)$ with a classifying space $X_\tau$, proving rational decompositions into immersions and analyzing torsion in the fold case. They provide exact computations of the rational parts of these cobordism groups and, in special cases (e.g., 2-codimensional fold maps), obtain detailed torsion information via a key fibration and explicit Thom spaces. The framework has potential applications to representing cohomology classes by quasi-holomorphic subspaces, links with Kazarian’s theory, and broader geometric-topological questions about map cobordisms in settings without global complex structures.
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.