zbMATH — the first resource for mathematics

Buchstaber invariants of skeleta of a simplex. (English) Zbl 1238.57031
This paper provides some computations of the real Buchstaber invariant of the skeleta of a simplex.
The Buchstaber invariant of a finite simplicial complex $$K$$ is defined in the following way: M. W. Davis and T. Januszkiewicz [Duke Math. J. 62, No. 2, 417–451 (1991; Zbl 0733.52006)] associated to $$K$$ a space $${\mathcal Z}_K$$, referred to as the moment-angle complex of $$K$$, and [V. M. Buchstaber and T. E. Panov, Torus actions and their applications in topology and combinatorics. University Lecture Series. 24. Providence, RI:American Mathematical Society (AMS). (2002; Zbl 1012.52021)] realized it as an $$(S^1)^m$$-invariant subspace of the polydisc $$(D^2)^m\subset {\mathbb C}^m$$, where $$m$$ is the number of vertices of $$K$$. Then the Buchstaber invariant of $$K$$ is the largest dimension of a subtorus of $$(S^1)^m$$ acting freely on $${\mathbb Z}_K$$.
It is bounded from above by the real Buchstaber invariant, which is defined analogously by replacing the action of $$(S^1)^m$$ on the moment-angle complex $${\mathcal Z}_K$$ by an action of the two-torus $$(S^0)^m$$ on the real moment-angle complex $${\mathbb R}{\mathcal Z}_K$$.
The authors denote the real Buchstaber invariant of the $$(m-p-1)$$-skeleton of the $$(m-1)$$-simplex by $$s_{\mathbb R}(m,p)$$ and prove several results about these numbers; for instance, they calculate $$s_{\mathbb R}(m,p)$$ for $$m-3 \leq p\leq m$$. Moreover, they investigate the values of $$m$$ and $$p$$ for which $$s_{\mathbb R}(m,p)$$ equals a fixed number $$k$$, and formulate the problem of finding the range of $$m$$ for which this equality holds (depending on fixed $$p$$ and $$k$$) as a problem of integer linear programming. For various choices of $$p$$ and $$k$$ they are able to solve it, and they formulate a conjecture for the general case.

MSC:
 57S17 Finite transformation groups 90C10 Integer programming 52B99 Polytopes and polyhedra 57Q99 PL-topology
Full Text:
References:
 [1] A. Aizenberg: Graduate thesis , Moscow State University (2009). [2] V.M. Buchstaber: Lectures on toric topology , Trends in Mathematics, Information Center for Mathematical Sciences 11 (2008), 1-55. [3] V.M. Buchstaber and T.E. Panov: Torus Actions and Their Applications in Topology and Combinatorics, University Lecture Series 24 , Amer. Math. Soc., Providence, RI, 2002. · Zbl 1012.52021 [4] X. Cao and Z. Lü: Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture , preprint (2009), [5] M.W. Davis and T. Januszkiewicz: Convex polytopes, Coxeter orbifolds and torus actions , Duke Math. J. 62 (1991), 417-451. · Zbl 0733.52006 [6] N. Erokhovets: Buchstaber invariant of simple polytopes , · Zbl 1184.52013 [7] J. Grbić and S. Theriault: The homotopy type of the complement of the codimension-two coordinate subspace arrangement , Uspekhi Mat. Nauk 59 (2004), 203-204, English translation in Russian Math. Surveys 59 (2004), 1207-1209. · Zbl 1072.55014 [8] J. Grbić and S. Theriault: The homotopy type of the complement of a coordinate subspace arrangement , Topology 46 (2007), 357-396, · Zbl 1118.55006 [9] M. Harada, Y. Karshon, M. Masuda and T. Panov (eds), Toric Topology, Proc. of the International Conference held at Osaka City University in 2006, Contemporary Mathematics 460 , Amer. Math. Soc., Providence, RI, 2008. [10] Y. Ustinovsky: Toral rank conjecture for moment-angle complexes , preprint (2009),
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.