×

An extended Euler-Poincaré theorem. (English) Zbl 0667.52008

The Euler-Poincaré theorem is a linear relation between the face-vector f and the Betti sequence b of a finite simplicial d-dimensional complex. In the paper a set s of d nonlinear relations is stated which are satisfied by f and b. It is shown that the Euler-Poincaré relation and s completely characterize pairs (f,b) which arise as f-vectors and Betti sequences of finite simplicial complexes. Related results are discussed, too. [Cf. the first author, Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 2, 1408-1418 (1987), and R. Stanley, Discrete geometry and convexity. Proc. Conf., New York 1982, Ann. N. Y. Acad. Sci. 440, 212-223 (1985; Zbl 0573.52008)].
Reviewer: E.Jucovič

MSC:

52Bxx Polytopes and polyhedra

Citations:

Zbl 0573.52008
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] [Ba]Bayer, M., Barycentric subdivisions.Pacific J. Math., 134 (1988), in press. · Zbl 0627.57020
[2] [Bj]Björner, A., Face numbers of complexes and polytopes, inProc. Intern. Congress of Math. Berkeley 1986, 1408–1418.
[3] [BK]Björner, A. & Kalai, G., Onf-vectors and homology. Preprint, 1985. To appear inProc. 3rd Intern. Conf. on Combinat. Math., New York, 1985.Ann. N.Y. Acad. Sci.
[4] [BW]Björner, A. &Walker, J., A homotopy complementation formula for partially ordered sets.European J. Combin., 4 (1983), 11–19. · Zbl 0508.06005
[5] [Bo]Bourbaki, N.,Algebra 1. Chapter 1–3. Addison-Wesley, Reading, Ma., 1974. · Zbl 0281.00006
[6] [C]Clements, G., A minimization problem concerning subsets.Discrete Math., 4 (1973), 123–128. · Zbl 0257.05003 · doi:10.1016/0012-365X(73)90074-5
[7] [DGH]Daykin, D. E., Godfrey, J. &Hilton, A. J. W., Existence theorems for Sperner families.J. Combin. Theory Ser. A, 17 (1974), 245–251. · Zbl 0287.05003 · doi:10.1016/0097-3165(74)90011-9
[8] [EKR]Erdös, P., Ko, C. &Rado, R., Intersection theorems for systems of finite sets.Quart. J. Math. Oxford, 12 (1961), 313–318. · Zbl 0100.01902 · doi:10.1093/qmath/12.1.313
[9] [F1]Frankl, P., A new short proof for the Kruskal-Katona theorem.Discrete Math., 48 (1984), 327–329. · Zbl 0539.05006 · doi:10.1016/0012-365X(84)90193-6
[10] [F2]–, The shifting technique in extremal set theory, inProc. British Combinat. Coll. London 1987. Cambridge Univ. Press, Cambridge, 1987.
[11] [GK]Greene, C. &Kleitman, D. J., Proof techniques in the theory of finite sets, inStudies in Combinatorics (G. C. Rota, ed.), The Mathematical Association of America, Washington, D.C., 1978, 22–79.
[12] [K1]Kalai, G., Characterization off-vectors of families of convex sets inR d , Part I: Necessity of Eckhoff’s conditions.Israel J. Math., 48 (1984), 175–195. · Zbl 0572.52006 · doi:10.1007/BF02761163
[13] [K2]–,f-vectors of acyclic complexes,Discrete Math., 55 (1984), 97–99. · Zbl 0579.57015 · doi:10.1016/S0012-365X(85)80024-8
[14] [K3]Kalai, G., Algebraic shifting methods and iterated homology groups. To appear.
[15] [Ka]Katona, G. O. H., A theorem of finite sets, inTheory of graphs (Proc. Tihany Conf., 1966, P. Erdös and G. Katona, eds.), Academic Press, New York, and Akadémia Kiadó, Budapest, 1968, 187–207.
[16] [Kr]Kruskal, J. B., The number of simplices in a complex, inMathematical Optimization Techniques (R. Bellman, ed.). Univ. of California Press, Berkeley-Los Angeles, 1963, 251–278.
[17] [LW]Lundell, A. T. &Weingram, S.,The topology of CW complexes. Van Nostrand, New York, 1969. · Zbl 0207.21704
[18] [M]Mayer, W., A new homology theory II.Ann. of Math. (2), 43 (1942), 594–605. · Zbl 0061.40313 · doi:10.2307/1968815
[19] [P1]Poincaré, H., Sur la généralisation d’un théorème d’Euler relatif aux polyèdres.C.R. Acad. Sci. Paris, 117 (1893), 144–145. · JFM 25.1031.03
[20] [P2]–, Complément à l’Analysis situs.Rend. Circ. Mat. Palermo, 13 (1899), 285–343. · JFM 30.0435.02 · doi:10.1007/BF03024461
[21] [Sa]Sarkaria, K. S., Heawood inequalities.J. Combin. Theory Ser. A, 46 (1987), 50–78. · Zbl 0628.05029 · doi:10.1016/0097-3165(87)90076-8
[22] [Sp]Spanier, E. H.,Algebraic Topology. McGraw Hill, New York, 1966.
[23] [Spe]Sperner, E., Ein Satz über Untermenge einer endlichen Menge.Math. Z., 27 (1928), 544–548. · JFM 54.0090.06 · doi:10.1007/BF01171114
[24] [St]Stanley, R., The number of faces of simplicial polytopes and spheres, inDiscrete Geometry and Convexity (J. E. Goodman et al., eds.).Ann. New York Acad. Sci., 440 (1985), 212–223.
[25] [W]Wegner, G., Kruskal-Katona’s theorem in generalized complexes, inFinite and infinite sets, Vol. 2, Coll. Math. Soc. János Bolyai, Vol. 37. North-Holland, Amsterdam, 1984, 821–827.
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.