Homology theory formulas for generalized Riemann-Hurwitz and generalized monoidal transformations

TL;DR

This paper develops generalized homology formulas extending classical Riemann-Hurwitz and monoidal transformation results, applicable to singular spaces and involving characteristic classes and cap product pairings.

Contribution

It introduces a homology-theoretic framework for generalized formulas related to branched coverings and monoidal transformations, including applications to singular varieties and characteristic classes.

Findings

Formulas expressed as cap product pairings in homology theory.

Applications to branched coverings and singular varieties.

Connections to characteristic classes like Chern-Schwartz-MacPherson and L-classes.

Abstract

In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann-Hurwitz formula for regulating the topology of branched covering maps and that for monoidal transformations which include the standard blowing-up process. Here the results are presented as cap product pairings, which will be elements of a suitable homology theory, rather than characteristic numbers as would be the case when taking Kronecker products once Poincar\'e duality is defined. We further consider possible applications and examples including branched covering maps, singular varieties involving virtual tangent bundles, the Chern-Schwartz-MacPherson class, the homology L-class, generalized signature, and the cohomology signature class.