zbMATH — the first resource for mathematics

Wedge operations and doubling operations of real toric manifolds. (English) Zbl 1387.57040
Summary: This paper deals with two things. First, the cohomology of canonical extensions of real topological toric manifolds is computed when the coefficient ring \(G\) is a commutative ring in which 2 is a unit. Second, the author focuses on specific canonical extensions called doublings and presents their various properties. They include existence of infinitely many real topological toric manifolds admitting complex structures, and a way to construct infinitely many real toric manifolds which have an odd torsion in their cohomology groups. Moreover, some questions about real topological toric manifolds related to Halperin’s toral rank conjecture are presented.
57N65 Algebraic topology of manifolds
57S17 Finite transformation groups
05E45 Combinatorial aspects of simplicial complexes
Full Text: DOI
[1] Bahri, A.; Bendersky, M.; Cohen, F. R.; Gitler, S., Operations on polyhedral products and a new topological construction of infinite families of toric manifolds, Homology Homotopy Appl., 17, 137-160, (2015) · Zbl 1342.13029
[2] Cai, L., On products in a real moment-angle manifold, J. Math. Soc. Japan., 69, 503-528, (2017) · Zbl 1421.55004
[3] Choi, S.; Park, H., Wedge operations and torus symmetries, Tohoku Math. J. (2), 68, 91-138, (2016) · Zbl 1362.14052
[4] Choi, S.; Park, H., Wedge operations and torus symmetries II, Canad. J. Math., 69, 767-789, (2017) · Zbl 1388.14139
[5] Choi, S.; Park, H., On the cohomology and their torsion of real toric objects, Forum Math., 29, 543-554, (2017) · Zbl 1377.57022
[6] Ewald, G., Spherical complexes and nonprojective toric varieties, Discrete Comput. Geom., 1, 115-122, (1986) · Zbl 0597.52009
[7] Ishida, H.; Fukukawa, Y.; Masuda, M., Topological toric manifolds, Mosc. Math. J., 13, 57-98, (2013) · Zbl 1302.53091
[8] Panov, T.; Ustinovsky, Y., Complex-analytic structures on moment-angle manifolds, Mosc. Math. J., 12, 149-172, (2012) · Zbl 1257.32019
[9] Suciu, A. and Trevisan, A., Real toric varieties and abelian covers of generalized Davis-Januszkiewicz spaces, 2012, unpublished.
[10] Trevisan, A., Generalized Davis-Januszkiewicz spaces and their applications in algebra and topology, Ph D thesis, Vrije University Amsterdam, Amsterdam, 2012, http://dspace.ubvu.vu.nl/handle/1871/32835.
[11] Ustinovsky, Y., Doubling operation for polytopes and torus actions, Uspekhi Mat. Nauk, 64, 181-182, (2009)
[12] Ustinovsky, Y., The toral rank conjecture for moment-angle complexes, Mat. Zametki, 90, 300-305, (2011)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.