Abstract
We study here the affine space generated by the extendedf-vectors of simplicial homology (d − 1)-spheres which are balanced of a given type. This space is determined, and its dimension is computed, by deriving a balanced version of the Dehn-Sommerville equations and exhibiting a set of balanced polytopes whose extendedf-vectors span it. To this end, a unique minimal complex of a given type is defined, along with a balanced version of stellar subdivision, and such a subdivision of a face in a minimal complex is realized as the boundary complex of a balanced polytope. For these complexes, we obtain an explicit computation of their extendedh-vectors, and, implicitly,f-vectors.
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References
J. W. Alexander, The combinatorial theory of complexes,Ann. of Math. (2)31 (1930), 292–320.
D. Barnette,Map Coloring, Polyhedra, and the Four-Color Problem, Dolciani Mathematical Expositions No. 8, Mathematical Association of America, Washington, DC, 1983.
M. M. Bayer, Facial Enumeration in Polytopes, Spheres and Other Complexes, Ph.D. thesis, Cornell University, Ithaca, NY, 1983.
M. M. Bayer and L. J. Billera, Counting faces and chains in polytopes and posets,Contemp. Math. 34 (1984), 207–252.
M. M. Bayer and L. J. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets,Invent. Math. 79 (1985), 143–157.
L. J. Billera, Polyhedral theory and commutative algebra, inMathematical Programming—Bonn 1982, The State of the Art, (A. Bachem, M. Grötschel, and B. Korte, eds.), 57–77, Springer-Verlag, Berlin, 1983.
H. Bruggesser and P. Mani, Shellable decompositions of cells and spheres,Math. Scand. 29 (1971), 197–205.
R. D. Edwards, An amusing reformulation of the four-color problem,Notices Amer. Math. Soc. 24 (1977), A-257.
G. Ewald and G. C. Shephard, Stellar subdivisions of boundary complexes of convex polytopes,Math. Ann. 210 (1974), 7–16.
J. E. Goodman and H. Onishi, Even triangulations ofS 3 and the coloring of graphs,Trans. Amer. Math. Soc. 246 (1978), 501–510.
B. Grübaum,Convex Polytopes, Wiley-Interscience, New York, 1967.
D. W. Henderson, Simplicial complexes homeomorphic to proper self-subsets have free faces, inContinua, Decompositions, Manifolds, 240–242, University of Texas Press, Austin, TX, 1983.
V. Klee, A combinatorial analogue of Poincaré's duality theorem,Canad. J. Math. 16 (1964), 517–531.
C. C. MacDuffee,The Theory of Matrices, Verlag von Julius Springer, Berlin, 1933.
K. E. Magurn, Subdivision and Enumeration in Balanced Complexes, Ph.D. thesis, Cornell University, 1985.
W. S. Massey,Singular Homology Theory, Springer-Verlag, New York, 1980.
C. R. F. Maunder,Algebraic Topology, Van Nostrand Reinhold, London, 1970.
P. McMullen and G. C. Shephard,Convex Polytopes and the Upper Bound Conjecture, Cambridge University Press, Cambridge, 1971.
D. M. Y. Sommerville, The relations connecting the angle-sums and volume of a polytope in space ofn dimensions,Proc. Roy. Soc. London Ser. A 115 (1927), 103–119.
R. Stanley, Cohen-Macaulay complexes inHigher Combinatorics (M. Aigner, ed.), 51–62, Reidel, Dordrecht and Boston, 1977.
R. Stanley, Balanced Cohen-Macaulay complexes,Trans. Amer. Math. Soc. 249 (1979), 139–157.
R. Stanley, An introduction to combinatorial commutative algebra, inEnumeration and Design (D. A. Jackson and S. A. Vanstone, eds.), 3–18, Academic Press Canada, Toronto, 1984.
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Supported in part by NSF Grant DMS-8403225.
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Billera, L.J., Magurn, K.E. Balanced subdivision and enumeration in balanced spheres. Discrete Comput Geom 2, 297–317 (1987). https://doi.org/10.1007/BF02187885
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DOI: https://doi.org/10.1007/BF02187885