Euler flag enumeration of Whitney stratified spaces

Richard Ehrenborg; Mark Goresky; Margaret Readdy

doi:10.1016/j.aim.2014.09.008

Euler flag enumeration of Whitney stratified spaces

Richard Ehrenborg, Mark Goresky, Margaret Readdy

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

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.

Original language	English
Pages (from-to)	85-128
Number of pages	44
Journal	Advances in Mathematics
Volume	268
DOIs	https://doi.org/10.1016/j.aim.2014.09.008
State	Published - 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.

Keywords

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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10.1016/j.aim.2014.09.008

Cite this

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title = "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.",

keywords = "Eulerian condition, Primary, Quasi-graded poset, Secondary, Semisuspension, Weighted zeta function, Whitney's conditions A and B",

author = "Richard Ehrenborg and Mark Goresky and Margaret Readdy",

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: {\textcopyright} 2014 Elsevier Inc.",

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AU - Goresky, Mark

AU - Readdy, Margaret

N1 - 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.

PY - 2015/1/2

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N2 - 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.

AB - 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.

KW - Eulerian condition

KW - Primary

KW - Quasi-graded poset

KW - Secondary

KW - Semisuspension

KW - Weighted zeta function

KW - Whitney's conditions A and B

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SP - 85

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ER -

Euler flag enumeration of Whitney stratified spaces

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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