×

Euler flag enumeration of Whitney stratified spaces. (English) Zbl 1302.05215

Summary: 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 \(\mathbf{cd}\)-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 \(\mathbf{cd}\)-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 \(\mathbf{cd}\)-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.

MSC:

05E45 Combinatorial aspects of simplicial complexes
06A07 Combinatorics of partially ordered sets
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B22 Shellability for polytopes and polyhedra
57N80 Stratifications in topological manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aguiar, M.; Bergeron, N.; Sottile, F., Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math., 142, 1-30 (2006) · Zbl 1092.05070
[2] Bayer, M.; Billera, L., Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math., 79, 143-157 (1985) · Zbl 0543.52007
[3] Bayer, M.; Klapper, A., A new index for polytopes, Discrete Comput. Geom., 6, 33-47 (1991) · Zbl 0761.52009
[4] Billera, L. J.; Brenti, F., Quasisymmetric functions and Kazhdan-Lusztig polynomials, Israel J. Math., 184, 317-348 (2011) · Zbl 1269.20030
[5] Billera, L. J.; Ehrenborg, R., Monotonicity properties of the cd-index for polytopes, Math. Z., 233, 421-441 (2000) · Zbl 0966.52014
[6] Billera, L. J.; Ehrenborg, R.; Readdy, M., The \(c - 2 d\)-index of oriented matroids, J. Combin. Theory Ser. A, 80, 79-105 (1997) · Zbl 0886.05043
[7] Billera, L. J.; Lee, C. W., A proof of the sufficiency of McMullen’s conditions for \(f\)-vectors of simplicial polytopes, J. Combin. Theory Ser. A, 31, 237-255 (1981) · Zbl 0479.52006
[8] Björner, A., Topological Methods. Handbook of Combinatorics, vols. 1, 2, 1819-1872 (1995), Elsevier: Elsevier Amsterdam · Zbl 0851.52016
[9] Björner, A.; Korte, B.; Lovász, L., Homotopy properties of greedoids, Adv. in Appl. Math., 6, 447-494 (1985) · Zbl 0642.05014
[10] Björner, A.; Wachs, M., Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., 348, 1299-1327 (1996) · Zbl 0857.05102
[11] Borsuk, K., On the imbedding of systems of compacta in simplicial complexes, Fund. Math., 35, 217-234 (1948) · Zbl 0032.12303
[12] Dehn, M., Die Eulersche Formel in Zusammenhang mit dem Inhalt in der Nicht-Euklidischen Geometrie, Math. Ann., 61, 561-586 (1906) · JFM 37.0492.01
[13] Denkowska, Z.; Wachta, K.; Stasica, J., Stratifications des ensembles sous-analytiques avec les propriétés (A) et (B) de Whitney, Univ. Iagel. Acta Math., 25, 183-188 (1985) · Zbl 0584.32013
[14] du Plessis, A.; Wall, T., The Geometry of Topological Stability, London Mathematical Society Monographs, New Series, Oxford Science Publications, vol. 9 (1995), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York · Zbl 0870.57001
[15] Ehrenborg, R., Lifting inequalities for polytopes, Adv. Math., 193, 205-222 (2005) · Zbl 1079.52007
[16] Ehrenborg, R.; Johnston, D.; Rajagopalan, R.; Readdy, M., Cutting polytopes and flag \(f\)-vectors, Discrete Comput. Geom., 23, 261-271 (2000) · Zbl 0956.52012
[17] Ehrenborg, R.; Karu, K., Decomposition theorem for the \(c d\)-index of Gorenstein posets, J. Algebraic Combin., 26, 225-251 (2007) · Zbl 1134.52012
[18] Ehrenborg, R.; Readdy, M., Sheffer posets and \(r\)-signed permutations, Ann. Sci. Math. Qué., 19, 173-196 (1995) · Zbl 0843.06003
[19] Ehrenborg, R.; Readdy, M., Coproducts and the \(c d\)-index, J. Algebraic Combin., 8, 273-299 (1998) · Zbl 0917.06001
[20] Ehrenborg, R.; Readdy, M., Classification of the factorial functions of Eulerian binomial and Sheffer posets, J. Combin. Theory Ser. A, 114, 339-359 (2007) · Zbl 1109.06003
[21] Ehrenborg, R.; Readdy, M., The Tchebyshev transforms of the first and second kind, Ann. Comb., 14, 211-244 (2010) · Zbl 1230.06002
[22] Ehrenborg, R.; Readdy, M., The \(c d\)-index of Bruhat and balanced graphs (2013), preprint
[23] Ehrenborg, R.; Readdy, M.; Slone, M., Affine and toric hyperplane arrangements, Discrete Comput. Geom., 41, 481-512 (2009) · Zbl 1168.52018
[24] Hetyei, G., On the cd-variation polynomials of André and simsun permutations, Discrete Comput. Geom., 16, 259-276 (1996) · Zbl 0862.05002
[25] Hetyei, G.; Reiner, E., Permutation trees and variation statistics, European J. Combin., 19, 847-866 (1998) · Zbl 0929.05001
[26] Gibson, C. G.; Wirthmüller, K.; du Plessis, A. A.; Looijenga, E. J.N., Topological Stability of Smooth Mappings, Lecture Notes in Mathematics, vol. 552 (1976), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0377.58006
[27] Goresky, M.; MacPherson, R., Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 14 (1988), Springer-Verlag: Springer-Verlag Berlin · Zbl 0639.14012
[28] Hardt, R. M., Stratifications of real analytic mappings and images, Invent. Math., 28, 193-208 (1975) · Zbl 0298.32003
[29] Hironaka, H., Subanalytic sets, (Number Theory, Algebraic Geometry and Commutative Algebra (1973), Kinokuniya: Kinokuniya Tokyo), 453-493, (volume in honor of Y. Akizuki) · Zbl 0297.32008
[30] Hironaka, H., Stratification and flatness, (Real and Complex Singularities. Real and Complex Singularities, Nordic Summer School (Oslo, 1976) (1977), Sijthoff-Noordhoff: Sijthoff-Noordhoff Groningen), 199-265
[31] Kalai, G., A new basis of polytopes, J. Combin. Theory Ser. A, 49, 191-209 (1988) · Zbl 0691.05006
[32] Karu, K., The cd-index of fans and posets, Compos. Math., 142, 701-718 (2006) · Zbl 1103.14029
[33] Klain, D.; Rota, G.-C., Introduction to Geometric Probability, Lezioni Lincee (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0896.60004
[34] Lojasiewicz, S., Triangulation of semi-analytic sets, Ann. Sc. Norm. Super. Pisa (3), 18, 449-474 (1964) · Zbl 0128.17101
[35] Mather, J., Notes on topological stability, Harvard University, 1970, Bull. Amer. Math. Soc. (N.S.), 49, 475-506 (2012) · Zbl 1260.57049
[36] Mather, J., Stratifications and mappings, (Dynamical Systems, Proc. Sympos.. Dynamical Systems, Proc. Sympos., Univ. Bahia, Salvador, 1971 (1973), Academic Press: Academic Press New York), 195-232
[37] McMullen, P., The maximum numbers of faces of a convex polytope, Mathematika, 17, 179-184 (1970) · Zbl 0217.46703
[38] Mulmuley, K., A generalization of Dehn-Sommerville relations to simple stratified spaces, Discrete Comput. Geom., 9, 47-55 (1993) · Zbl 0768.52008
[39] Reading, N., The cd-index of Bruhat intervals, Electron. J. Combin., 11, R74 (2004), 25 p · Zbl 1067.20050
[40] Rota, G.-C., On the combinatorics of the Euler characteristic, (Studies in Pure Mathematics (1971), Academic Press: Academic Press London), 221-233, (presented to Richard Rado)
[41] Rota, G.-C., On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 2, 340-368 (1964) · Zbl 0121.02406
[42] Schanuel, S., Negative sets have Euler characteristic and dimension, (Category Theory. Category Theory, Como, 1990. Category Theory. Category Theory, Como, 1990, Lecture Notes in Mathematics, vol. 1488 (1991), Springer: Springer Berlin), 379-385 · Zbl 0748.18005
[43] Sommerville, D. M.Y., The relations connecting the angle-sums and volume of a polytope of a polytope in space of \(n\) dimensions, Proc. R. Soc. Lond. Ser A, 115, 103-119 (1927) · JFM 53.0578.03
[44] Stanley, R. P., The number of faces of a simplicial convex polytope, Adv. Math., 35, 236-238 (1980) · Zbl 0427.52006
[45] Stanley, R. P., Subdivisions and local \(h\)-vectors, J. Amer. Math. Soc., 5, 805-851 (1992) · Zbl 0768.05100
[46] Stanley, R. P., Flag \(f\)-vectors and the cd-index, Math. Z., 216, 483-499 (1994) · Zbl 0805.06003
[47] Stanley, R. P., Combinatorics and Commutative Algebra, Progress in Mathematics, vol. 41 (1996), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0838.13008
[48] Stanley, R. P., Enumerative Combinatorics, vol. 1 (2012), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1247.05003
[49] Steinitz, E., Über die Eulerischen Polyderrelationen, Rev. Math. Phys., 11, 86-88 (1906) · JFM 37.0500.01
[50] Thom, R., Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc., 75, 240-284 (1969) · Zbl 0197.20502
[51] Verdier, J. L., Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math., 36, 295-312 (1976) · Zbl 0333.32010
[52] Whitney, H., Local properties of analytic varieties, (Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (1965), Princeton University Press: Princeton University Press Princeton, NJ), 205-244 · Zbl 0129.39402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.