The flag $f$-vectors of Gorenstein* order complexes of dimension $3$
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- by Satoshi Murai and Eran Nevo PDF
- Proc. Amer. Math. Soc. 142 (2014), 1527-1538 Request permission
Abstract:
We characterize the $\mathbf {cd}$-indices of Gorenstein* posets of rank $5$, equivalently the flag $f$-vectors of order complexes triangulating rational homology $3$-spheres, and show they are also the characterization of the flag $f$-vectors of the subfamily of regular CW-complexes homeomorphic to the $3$-sphere. As a corollary, we characterize the $f$-vectors of Gorenstein* order complexes in dimensions $3$ and $4$. This characterization gives rise to a speculated intimate connection between the $f$-vectors of flag homology spheres and the $f$-vectors of Gorenstein* order complexes.References
- Michał Adamaszek and Jan Hladký. Dense flag triangulations of $3$-manifolds via extremal graph theory. To appear in Trans. Amer. Math. Soc.
- Margaret M. Bayer and Louis J. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math. 79 (1985), no. 1, 143–157. MR 774533, DOI 10.1007/BF01388660
- Margaret M. Bayer and Andrew Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33–47. MR 1073071, DOI 10.1007/BF02574672
- A. Björner, Posets, regular CW complexes and Bruhat order, European J. Combin. 5 (1984), no. 1, 7–16. MR 746039, DOI 10.1016/S0195-6698(84)80012-8
- Światosław R. Gal, Real root conjecture fails for five- and higher-dimensional spheres, Discrete Comput. Geom. 34 (2005), no. 2, 269–284. MR 2155722, DOI 10.1007/s00454-005-1171-5
- Kalle Karu, The $cd$-index of fans and posets, Compos. Math. 142 (2006), no. 3, 701–718. MR 2231198, DOI 10.1112/S0010437X06001928
- Satoshi Murai and Eran Nevo, On the cd-index and $\gamma$-vector of $\textrm {S}^*$-shellable CW-spheres, Math. Z. 271 (2012), no. 3-4, 1309–1319. MR 2945609, DOI 10.1007/s00209-011-0917-4
- Eran Nevo and Eyal Novinsky, A characterization of simplicial polytopes with $g_2=1$, J. Combin. Theory Ser. A 118 (2011), no. 2, 387–395. MR 2739491, DOI 10.1016/j.jcta.2009.12.011
- Nathan Reading, The cd-index of Bruhat intervals, Electron. J. Combin. 11 (2004), no. 1, Research Paper 74, 25. MR 2097340
- Richard P. Stanley, Flag $f$-vectors and the $cd$-index, Math. Z. 216 (1994), no. 3, 483–499. MR 1283084, DOI 10.1007/BF02572336
- Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
- Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1
Additional Information
- Satoshi Murai
- Affiliation: Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan
- MR Author ID: 800440
- Email: murai@yamaguchi-u.ac.jp
- Eran Nevo
- Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva 84105, Israel
- MR Author ID: 762118
- Email: nevoe@math.bgu.ac.il
- Received by editor(s): September 30, 2011
- Received by editor(s) in revised form: June 12, 2012
- Published electronically: February 17, 2014
- Additional Notes: The research of the first author was partially supported by KAKENHI 22740018
The research of the second author was partially supported by Marie Curie grant IRG-270923 and by an ISF grant - Communicated by: Jim Haglund
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 1527-1538
- MSC (2010): Primary 05E45; Secondary 13F55, 06Axx
- DOI: https://doi.org/10.1090/S0002-9939-2014-11897-1
- MathSciNet review: 3168460