Subdivisions and transgressive chains

TL;DR

This paper explores combinatorial transgressions arising from subdivisions in triangulated spaces, characterizing when they are path-independent and providing a local formula for a specific case related to the Euler class.

Contribution

It introduces a cohomological framework for understanding subdivision transgressions and derives a canonical local formula for a key combinatorial transgression.

Findings

Characterization of path-independent transgressions using cohomology on posets

Equivalence of this cohomology to higher derived functors of inverse limits

A local formula for the transgression related to the Euler class

Abstract

Combinatorial transgressions are secondary invariants of a space admitting triangulations. They arise from subdivisions and are analogous to transgressive forms such as those arising in Chern-Weil theory. Unlike combinatorial characteristic classes, combinatorial transgressions have not been previously studied. First, this article characterizes transgressions that are path-independent of subdivision sequence. The result is obtained by using a cohomology on posets that is shown to be equivalent to higher derived functors of the inverse (or projective) limit over the opposite poset. Second, a canonical local formula is demonstrated for a particular combinatorial transgression: namely, that relative the difference of Poincar\'{e} duals to the Euler class.