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

Funding Information:
The authors thank the Institute for Advanced Study where this research was done and the two referees for their careful comments. The first author is partially supported by National Science Foundation grant 0902063 . The second author is grateful to the Defense Advanced Research Projects Agency for its support under DARPA grant number HR0011-09-1-0010 . This work was partially supported by a grant from the Simons Foundation (# 206001 to Margaret Readdy). The first and third authors were also partially supported by National Science Foundation grants DMS-0835373 and CCF-0832797 and the second author by a Bell Companies Fellowship for 2010.

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

  • Mathematics (all)


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