Euler flag enumeration of Whitney stratified spaces

TL;DR

This paper extends the concept of the cd-index from Eulerian posets to Whitney stratified spaces by introducing quasi-graded posets, enabling a broader combinatorial analysis of stratified manifolds.

Contribution

It generalizes the cd-index to Whitney stratified manifolds through quasi-graded posets, relaxing regularity conditions and extending classical combinatorial tools.

Findings

The cd-index exists for Whitney stratified manifolds.

Extension of semi-suspension to boundary embeddings.

Analysis of simplicial shelling components.

Abstract

The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi-graded poset. This is a poset endowed with an order-preserving rank function and a weighted zeta function. This allows us to generalize the classical notion of Eulerianness, and obtain a cd-index in the quasi-graded poset arena. We also extend the semi-suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the simplicial shelling components.