Stiefel-Whitney Classes and the Conormal Cycle of a Singular Variety

TL;DR

This paper introduces a geometric method to construct Stiefel-Whitney classes for real analytic varieties using conormal cycles, linking them to Chern-MacPherson classes in complex cases.

Contribution

It provides a new geometric construction of Stiefel-Whitney classes via conormal cycles and establishes their relation to Chern-MacPherson classes for complex varieties.

Findings

Stiefel-Whitney classes are constructed as natural transformations from constructible functions to homology.

The classes are shown to be mod 2 reductions of Chern-MacPherson classes for complex varieties.

A geometric interpretation of Stiefel-Whitney classes for singular varieties is provided.

Abstract

A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety $X$ is given by means of the conormal cycle of an embedding of $X$ in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. We also show that, for a complex analytic variety, these classes are the mod 2 reductions of the Chern-MacPherson classes.