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arXiv · 1201.3377

Euler flag enumeration of Whitney stratified spaces

Richard Ehrenborg·Mark Goresky·Margaret Readdy
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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.

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Richard Ehrenborg, Mark Goresky, Margaret Readdy. 2014-09-29. Euler flag enumeration of Whitney stratified spaces. https://doi.org/10.1016/j.aim.2014.09.008

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