"Euler flag enumeration of Whitney stratified spaces"

Margaret Readdy
University of Kentucky

Monday, October 24, 2011
4:00 pm
845 POT

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 the generalized Dehn-Sommerville relations, that is, all the linear redundancies among the flag vector entries discovered by Bayer and Billera. This result holds for regular CW complexes.

We relax the regularity condition to show the cd-index exists for non-regular CW complexes 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 Eulerian, and obtain a cd-index in the quasi-graded poset arena.

Generally speaking, for an arbitrary quasi-graded poset the weighted zeta function is not unique. However, for a manifold having a Whitney stratification, selecting the weighted zeta function of an interval using the Euler characteristic gives the extended notion of Eulerianess geometric meaning.

This is joint work with Richard Ehrenborg and Mark Goresky.