Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture

Cao, Xiangyu ; Lü, Zhi

Journal of Algebraic Combinatorics, Tome 35 (2012) no. 1, pp. 121-140.

Résumé

Summary: The motivation for this paper comes from the Halperin-Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the Möbius transform of an abstract simplicial complex $K$ on $[ m]={1,\cdots , m}$ in terms of the Betti numbers of the Stanley-Reisner face ring $k( K)$ of $K$ over a field k. We then employ a way of compressing $K$ to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes $Z _{ K} ^{(\mathbb D, \mathbb S)}$ mathcalZ_K^(underlinemathbb D, underlinemathbb S), including the moment-angle complex $Z _{ K}$ mathcalZ_K and the real moment-angle complex $\mathbb R Z _{ K}$ mathbbR$\mathcal $Z_K as special examples. We show that $H ^{*}( Z _{ K} ^{(\mathbb D, \mathbb S)}$; k) H^*(mathcalZ_K^(underlinemathbb D, underlinemathbb S);mathbfk) has the same graded k-module structure as Tor $^{ k[ v]}( k( K), k)$. Finally we show that the Halperin-Carlsson conjecture holds for $Z _{ K}$ mathcalZ_K (resp. $\mathbb R Z _{ K}$ mathbb RmathcalZ_K) under the restriction of the natural $T ^{ m }$-action on $Z _{ K}$ mathcalZ_K (resp. $(\Bbb Z _{2}) ^{ m }$-action on $\mathbb R Z _{ K}$ mathbb RmathcalZ_K).

Détail

Keywords: keywords Möbius transform, moment-angle complex, Halperin-Carlsson conjecture