Abstract
The well known g-conjecture for homology spheres follows from the stronger conjecture that the face ring over the reals of a homology sphere, modulo a linear system of parameters, admits the strong-Lefschetz property. We prove that the strong-Lefschetz property is preserved under the following constructions on homology spheres: join, connected sum, and stellar subdivisions. The last construction is a step towards proving the g-conjecture for piecewise-linear spheres.
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Babson, E., Novik, I., Thomas, R.: Reverse lexicographic and lexicographic shifting. J. Algebr. Comb. 23(2), 107–123 (2006). MR MR2223682 (2007a:13026)
Björner, A., Kalai, G.: An extended Euler-Poincaré theorem. Acta Math. 161(3–4), 279–303 (1988). MR MR971798 (89m:52009)
Björner, A., Kalai, G.: On f-vectors and homology. In: Combinatorial Mathematics: Proceedings of the Third International Conference, New York, 1985. Ann. New York Acad. Sci., vol. 555, pp. 63–80. New York Acad. Sci., New York (1989). MR MR1018610 (90i:52008)
Brenti, F., Welker, V.: f-vectors of barycentric subdivisions. Math. Z. 259(4), 849–865 (2008). MR MR2403744
Harima, T., Migliore, J.C., Nagel, U., Watanabe, J.: The weak and strong Lefschetz properties for Artinian K-algebras. J. Algebra 262(1), 99–126 (2003). MR MR1970804 (2004b:13001)
Harima, T., Watanabe, J.: The finite free extension of Artinian K-algebras with the strong Lefschetz property. Rend. Sem. Mat. Univ. Padova 110, 119–146 (2003). MR MR2033004 (2005a:13002)
Kalai, G.: Characterization of f-vectors of families of convex sets in R d. I. Necessity of Eckhoff’s conditions. Israel J. Math. 48(2–3), 175–195 (1984). MR MR770700 (86c:52003)
Kalai, G.: Hyperconnectivity of graphs. Graphs Comb. 1(1), 65–79 (1985). MR MR796184 (87c:05081)
Kalai, G.: Symmetric matroids. J. Comb. Theory Ser. B 50(1), 54–64 (1990). MR MR1070465 (91m:05053)
Kalai, G.: The diameter of graphs of convex polytopes and f-vector theory. In: Applied Geometry and Discrete Mathematics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 387–411. Amer. Math. Soc., Providence (1991)
Kalai, G.: Algebraic shifting. In: Computational Commutative Algebra and Combinatorics, Osaka, 1999. Adv. Stud. Pure Math., vol. 33, pp. 121–163. Math. Soc. Japan, Tokyo (2002). MR MR1890098 (2003e:52024)
Kubitzke, M., Nevo, E.: The Lefschetz property for barycentric subdivisions of shellable complexes. To appear in Trans. Amer. Math. Soc. Math. arXiv:0712.1560 (2007)
Lee, C.W.: Generalized stress and motions. In: Polytopes: Abstract, Convex and Computational, Scarborough, ON, 1993. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 440, pp. 249–271. Kluwer Acad. Publ., Dordrecht (1994). MR MR1322066 (96c:52039)
Lickorish, W.B.R.: Simplicial moves on complexes and manifolds. In: Proceedings of the Kirbyfest, Berkeley, CA, 1998. Geom. Topol. Monogr., vol. 2, pp. 299–320. Geom. Topol. Publ., Coventry (1999) (electronic)
McMullen, P.: The numbers of faces of simplicial polytopes. Israel J. Math. 9, 559–570 (1971). MR MR0278183 (43 #3914)
McMullen, P.: Weights on polytopes. Discrete Comput. Geom. 15(4), 363–388 (1996). MR MR1384882 (97b:52012)
McMullen, P.: On simple polytopes. Invent. Math. 113(2), 419–444 (1993). MR MR1228132 (94d:52015)
Murai, S.: Algebraic shifting of cyclic polytopes and stacked polytopes. Discrete Math. 307(14), 1707–1721 (2007). MR MR2316811
Murai, S.: Algebraic shifting of strongly edge decomposable spheres. Math. arXiv:0709.4518 (2007)
Nevo, E.: Algebraic shifting and basic constructions on simplicial complexes. J. Algebraic Combin. 22(4), 411–433 (2005). MR MR2191645 (2006h:05239)
Nevo, E.: Higher minors and Van Kampen’s obstruction. Math. Scandi. 101, 161–176 (2007)
Stanley, R.P.: The number of faces of a simplicial convex polytope. Adv. Math. 35(2), 236–238 (1980). MR MR563925 (81f:52014)
Stanley, R.P.: Combinatorics and commutative algebra, 2nd edn. Progress in Mathematics, vol. 41. Birkhäuser Boston, Boston (1996). MR MR1453579 (98h:05001)
Swartz, E.: g-elements, finite buildings and higher Cohen-Macaulay connectivity. J. Comb. Theory Ser. A 113(7), 1305–1320 (2006). MR MR2259062
Swartz, E.: Face enumeration—from spheres to manifolds. Math. arXiv:0709.3998 (2007)
Tay, T.-S., White, N., Whiteley, W.: Skeletal rigidity of simplicial complexes I. Eur. J. Comb. 16(4), 381–403 (1995). MR MR1337142 (96e:52052)
Whiteley, W.: Vertex splitting in isostatic frameworks. Struc. Top. 16, 23–30 (1989)
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Research of E. Nevo was partially supported by an NSF Award DMS-0757828.
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Babson, E., Nevo, E. Lefschetz properties and basic constructions on simplicial spheres. J Algebr Comb 31, 111–129 (2010). https://doi.org/10.1007/s10801-009-0189-9
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DOI: https://doi.org/10.1007/s10801-009-0189-9