On face numbers of manifolds with symmetry
The author considers symmetric simplicial complexes~\(\Gamma\), specifically those that admit a proper action of the cyclic group~\(\mathbb Z_p\) of prime order~\(p\). Throughout, we follow the author and write the number of vertices of~\(\Gamma\) as \(n=l+pm\), where \(l\)~is the number of \(\mathbb Z_p\)-invariant vertices and \(m\)~the number of \(\mathbb Z_p\)-orbits of non-invariant vertices. The paper under review is roughly divided into two parts. The first part is dedicated to the case that \(\Gamma\)~is Cohen-Macaulay (e.g.,~a simplicial sphere) or Buchsbaum (e.g.,~a simplicial manifold); in these situations, the author gives necessary conditions on the \(f\)-vector of~\(\Gamma\). The second part treats the Upper Bound Theorem for numbers of faces, as well as bounds on the reduced Euler characteristic of a \(\mathbb Z_p\)-symmetric simplicial manifold, and on a certain weighted sum of its Betti numbers. The special case of \(p=2\), i.e., centrally symmetric manifolds, also receives attention. For each class of complexes considered in the first part of this paper, i.e., Cohen-Macaulay and Buchsbaum complexes, the necessary conditions on the \(f\)-vector just alluded to take two forms. The first one (treated in Theorems~1.1 and~3.3) says that the \(h\)-vector \(h(\Gamma)=(h_0(\Gamma),\dots, h_d(\Gamma))\) of any simplicial \((d-1)\)-dimensional Cohen-Macaulay complex~\(\Gamma\) must be the face vector~\(F(\Gamma)=(F_0(\Gamma),F_1(\Gamma),\dots)\) of a suitable multicomplex \[ M \;\subset \;S\big((p-1)^m, \infty^{n-d-m}\big); \] here \(S(a^r, b^s)\)~denotes the multicomplex of all monomials \(x_1^{a_1}\cdots x_{r+s}^{a_{r+s}}\) with \(0\leq a_i\leq a\) for \(1\leq i\leq r\) and \(0\leq a_{r+i}\leq b\) for \(1\leq i\leq s\), ordered by divisibility, and \(F_i(\Gamma)=| \{\mu\in\Gamma:\deg\mu=i\}| \). This strengthens \textit{R. P. Stanley}'s [Higher Comb., Proc. NATO Adv. Study Inst., Berlin (West) 1976, 51--62 (1977; Zbl 0376.55007)] earlier characterization of the face numbers of \(\mathbb Z_p\)-symmetric Cohen-Macaulay complexes. The principal proof technique here involves a variant of algebraic shifting, more precisely the study of the combinatorics of initial ideals that are not quite generic, but retain enough genericity to interact well with Buchsbaum complexes [cf. \textit{R. P. Schenzel}, Math. Z. 178, 125--142 (1981; Zbl 0472.13012)]. This notion is made precise by the Kind-Kleinschmidt condition, which by another result of \textit{R. P. Stanley} [Studies Appl. Math. 54, 135--142 (1975; Zbl 0308.52009) and Combinatorics and commutative algebra. 2nd ed. Progress in Mathematics (Boston, Mass.). 41. Basel: Birkhäuser. (1996; Zbl 0838.13008)] permits one to construct a multicomplex whose \(F\)-vector is the \(h\)-vector of~\(\Gamma\). To extend this result to the case where \(\Gamma\)~is \(k\)-Buchsbaum, the author considers the modified \(h\)-vector \[ h^\prime_j(\Gamma) \;= \;h_j(\Gamma) + \binom{d}{j} \sum_{i=0}^{j-1} (-1)^{j-i-1} \beta_{i-1}(\Gamma) \qquad\text{for } j=0,1,\dots,d, \] where \(\beta_j=\dim_k \widetilde H_i(\Gamma;k)\) are the reduced Betti numbers of~\(\Gamma\). (From now on, the dependence on the ground field~\(k\) will be suppressed; also note that \(h^\prime_j(\Gamma)=h_j(\Gamma)\) if \(\Gamma\)~is Cohen-Macaulay.) The author proves in Theorems~1.3 and~3.9 the upper bounds \(h_0^\prime(\Gamma)=1\), \(h_1^\prime(\Gamma)=n-d\), and \[ \partial_{S,j} \big( h^\prime_{j+1}(\Gamma) \big ) \;\leq \;h^\prime_j(\Gamma) - \binom{d-1}{j}\beta_{j-1}(\Gamma) \qquad\text{for } j=1,2,\dots, d-1, \] where \(\partial_{S,j}\) is the Clements-Lindström function associated to \(S=S((p-1)^m, \infty^{n-d-m})\) [\textit{G. F. Clements} and \textit{B. Lindström}, J. Comb. Theory 7, 230--238 (1969; Zbl 0186.01704)]. The author points out that a special case of this result has already been stated in one of her previous papers [\textit{I. Novik}, Isr. J. Math. 108, 45--82 (1998; Zbl 0922.52004)], but that the proof given there (to date) only applies to ground fields~\(k\) of characteristic zero. The problem is that it relies on the fact that the generic initial ideal of~\(I_\Gamma\subset k[x]\) is strongly stable; but this is currently only known for fields~\(k\) of characteristic zero. The proof of the more general result presented in the paper under review avoids this restriction and is somewhat simpler conceptually (it relies directly on algebraic properties of Buchsbaum complexes). The second form of the necessary conditions (Corollary~1.2 and Theorem~1.5) is derived from the first one, and comes in the guise of an upper bound for the \(h\)-vector of any \((d-1)\)-dimensional \(\mathbb Z_p\)-symmetric Cohen-Macaulay or Buchsbaum complex~\(\Gamma\) as above, namely \[ h_j(\Gamma)\leq F_j(S((p-1)^m, \infty^{n-d-m})) \qquad\text{for } j=0,1,\dots, d. \] In the Cohen-Macaulay case, this bound is originally due to \textit{R. M. Adin} [Contemp. Math. 178, 1--20 (1994; Zbl 0820.52006)]; in the Buchsbaum case, it is only proved for complexes on at least \(n\geq3d-2\)~vertices. A crucial ingredient in the proof of the Buchsbaum case is the use of the normalized matching property for ranked posets, thus sidestepping the unwieldy Clements-Lindström function. This property is also responsible for the restriction \(n\geq 3d-2\), which may therefore well be an artefact of the proof. As for the second part of the paper, Kühnel's conjecture [\textit{W. Kühnel}, Tight polyhedral submanifolds and tight triangulations. Lecture Notes in Mathematics. 1612. Berlin: Springer-Verlag (1995; Zbl 0834.53004)] asserts that the reduced Euler characteristic \(\widetilde\chi(\Gamma)\) of every \(2k\)-dimensional simplicial manifold~\(\Gamma\) with \(n\)~vertices satisfies the inequality \[ (-1)^k \binom{2k+1}{k} \big(\widetilde\chi(\Gamma)-1\big) \;\leq \;\binom{n-k-2}{k+1}. \] An analogous conjecture of \textit{E. Sparla} [Discrete Comput. Geom. 19, No. 4, 575--593 (1998; Zbl 0960.57019)] states that for \(\Gamma\)~a centrally symmetric manifold on \(n=2m\) vertices, \[ (-1)^k \binom{2k+1}{k} \big(\widetilde\chi(\Gamma)-1\big) \;\leq \;4^{k+1}\binom{\frac12(m-1)}{k+1}. \] Noting that \[ \binom{n-k-2}{k+1} \;= \;F_{k+1}\big(S(\infty^{n-2k-1})\big) - F_k\big(S(\infty^{n-2k-1})\big) \] and \[ 4^{k+1}\binom{\frac12(m-1)}{k+1} \;= \;F_{k+1}\big(S(1^m, \infty^{m-2k-1})\big) - F_k\big(S(1^m, \infty^{m-2k-1})\big), \] the author proves the following common generalization of these conjectures for \(\mathbb Z_p\)-symmetric \(2k\)-dimensional manifolds with \(n\geq 6k+3\)~vertices (Theorem~1.4): \[ (-1)^k \binom{2k+1}{k} \big(\widetilde\chi(\Gamma)-1\big) \;\leq \;F_{k+1}(S)-F_k(S), \] where \(S=S\big((p-1)^m, \infty^{n-2k-1-m}\big)\). Moreover, as Conjecture~6.1 she asserts that the restriction \(n\geq 6k+3\) is superfluous. A final objective of the paper under review is to establish a version of the Upper Bound Theorem (UBT) for centrally symmetric (cs) manifolds. The paper contains a quite informative review of the history and state of the art of the Upper Bound Theorem, from its inception regarding simplicial polytopes to its validity for odd-dimensional simplicial manifolds and even-dimensional simplicial manifolds of reduced Euler characteristic~\(1\), and some generalizations. Some new results proved here are as follows: \(\bullet\) If \(\Gamma\) is a \((d-1)\)-dimensional cs~Buchsbaum complex on \(n=2m\geq 2d+r-2\) vertices, then \(h_j(\Gamma)\leq h_j(\text{skel}_{d-1} C^\Delta_m)\) for \(j=0,1,\dots, \min\{r,d\}\), where \(C^\Delta_m\)~denotes the boundary complex of the \(m\)-dimensional cross-polytope (Corollary~4.5). \(\bullet\) If \(\Gamma\)~is a \((d-1)\)-dimensional cs~simplicial sphere or odd-dimensional cs~manifold on \(2m\)~vertices, then \(f_i(\Gamma)\leq CS_i(d,m)\) for \(i=0,1,\dots, d-1\), where \(CS_i(d,m)\) is the number of \(i\)-dimensional faces that a nearly neighborly cs~\((d-1)\)-sphere would have (Theorem 5.6). \(\bullet\) Similarly, if \(\Gamma\)~is a \(2k\)-dimensional cs~homology manifold on \(n=2m\) vertices with reduced Euler characteristic~\(1\) and \(n\geq 5k\), then \(f_i(\Gamma)\leq CS_i(2k+1,m)\) for \(i=0,1,\dots,2k\) (Theorem~5.7).
scientific article; zbMATH DE number 2153794
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