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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
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References:
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