Abstract
We consider an operation K ↦ L(K) on the set of simplicial complexes, which we call the “doubling operation.” This combinatorial operation was recently introduced in toric topology in an unpublished paper of Bahri, Bendersky, Cohen and Gitler on generalized moment-angle complexes (also known as K-powers). The main property of the doubling operation is that the moment-angle complex can be identified with the real moment-angle complex for the double L(K). By way of application, we prove the toral rank conjecture for the spaces by providing a lower bound for the rank of the cohomology ring of the real moment-angle complexes . This paper can be viewed as a continuation of the author’s previous paper, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.
Similar content being viewed by others
References
X. Cao and Z. Lu, Möbius Transform, Moment-Angle Complexes, and Halperin-Carlsson Conjecture, arXiv: math. CO/0908.3174 (2009).
A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, “On an infinite family of toric manifolds associated to a given one” (to appear).
Yu.M. Ustinovskii, “Doubling operation for polytopes and torus actions,” Uspekhi Mat. Nauk 64(5), 181–182 (2009) [Russian Math. Surveys 64 (5), 952–954 (2009)].
S. Gitler and S. Lopez de Medrano, Intersections of Quadrics, Moment-Angle Manifolds, and Connected Sums, arXiv: math. GT/0901.2580 (2009).
S. Halperin, “Rational homotopy and torus actions,” in Aspects of Topology, London Math. Soc. Lecture Notes Ser. (Cambridge Univ. Press, Cambridge, 1985), Vol. 93, pp. 293–306.
M. W. Davis and T. Januszkiewicz, “Convex polytopes, Coxeter orbifolds, and torus actions,” Duke Math. J. 62(2), 417–451 (1991).
V. M. Bukhshtaber and T. E. Panov, Toric Actions in Topology and Combinatorics (MTsNMO, Moscow, 2004) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu. M. Ustinovskii, 2011, published in Matematicheskie Zametki, 2011, Vol. 90, No. 2, pp. 300–305.
Rights and permissions
About this article
Cite this article
Ustinovskii, Y.M. Toral rank conjecture for moment-angle complexes. Math Notes 90, 279 (2011). https://doi.org/10.1134/S0001434611070273
Received:
Accepted:
Published:
DOI: https://doi.org/10.1134/S0001434611070273