Euler flag enumeration of Whitney stratified spaces

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6 Scopus citations


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.

Original languageEnglish
Pages (from-to)85-128
Number of pages44
JournalAdvances in Mathematics
StatePublished - Jan 2 2015

Bibliographical note

Publisher Copyright:
© 2014 Elsevier Inc.


  • Eulerian condition
  • Primary
  • Quasi-graded poset
  • Secondary
  • Semisuspension
  • Weighted zeta function
  • Whitney's conditions A and B

ASJC Scopus subject areas

  • General Mathematics


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