Discrete CATS Seminar
U N I V E R S I T Y
K E N T U C K Y
WHERE CATS =
845 PATTERSON OFFICE TOWER
2011 - 2012
"Euler flag enumeration of Whitney stratified spaces"
University of Kentucky
Monday, October 24, 2011
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.