Abstract
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,…,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 \(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}\), including the moment-angle complex \(\mathcal{Z}_{K}\) and the real moment-angle complex \(\mathbb{R}\mathcal {Z}_{K}\) as special examples. We show that \(H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k})\) has the same graded k-module structure as Tor k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for \(\mathcal{Z}_{K}\) (resp. \(\mathbb{ R}\mathcal{Z}_{K}\)) under the restriction of the natural T m-action on \(\mathcal{Z}_{K}\) (resp. (ℤ2)m-action on \(\mathbb{ R}\mathcal{Z}_{K}\)).
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Supported by grants from FDUROP (No. 080705), NSFC (No. J0730103, No. 10931005) and Shanghai NSF (No. 10ZR1403600).
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Cao, X., Lü, Z. Möbius transform, moment-angle complexes and Halperin–Carlsson conjecture. J Algebr Comb 35, 121–140 (2012). https://doi.org/10.1007/s10801-011-0296-2
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DOI: https://doi.org/10.1007/s10801-011-0296-2