Euler flag enumeration of Whitney stratified spaces

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.

41 pages, 3 figures. Final version

References in corpus (6)

• Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
• Affine and toric hyperplane arrangements
• The cd-index of Bruhat intervals
• The Tchebyshev transforms of the first and second kind
• Classification of the factorial functions of Eulerian binomial and Sheffer posets
• Bruhat and balanced graphs

Cited by in corpus (3)

• Manifold arrangements
• The $cd$-Index: A Survey
• Bruhat and balanced graphs