On the genera of moment-angle manifolds associated with dual-neighborly polytopes: combinatorial formulas and sequences (Q6613464)

Toric topology is an emerging field in pure mathematics, whose core object are moment angle complexes and their generalizations. In toric topology, the following notions are widely used to yield moment angle complexes: simplicial complexes, polytopes, polyhedral products and some notations from algebraic geometry (like fans, complements of subspace arrangements). Such research often need tools from different areas in mathematics, like homological algebra, combinatorics, algebraic geometry, etc.\N\NAn interesting object in toric topology is the case that a moment angle complex is the connected sum of sphere products. Research on this topic can be traced back to [\textit{F. Bosio} and \textit{L. Meersseman}, Acta Math. 197, No. 1, 53--127 (2006; Zbl 1157.14313)]. Many researchers have investigated open question in this topic. For example, \textit{F. Fan} et al. [Osaka J. Math. 53, No. 1, 31--45 (2016; Zbl 1335.13018)] gave an example whose cohomology ring is isomorphic to that of a connected sum of sphere products with one product of three spheres. \textit{I. Yu. Limonchenko} [Proc. Steklov Inst. Math. 286, 188--197 (2014; Zbl 1322.13010); translation from Tr. Mat. Inst. Steklova 286, 207--218 (2014)] showed that the nerve complexes of even dimensional dual neighbourly polytopes and certain generalized truncation polytopes will yield a connected sum of sphere products. \N\NIn the present paper, a special case, connected sums of copies of the sphere product \(S^p \times S^p\) is studied. More precisely, the author determines the number of terms in the connected sum for dual neighborly polytopes of even dimension. Using f-vectors and a series of combinatorial identities, the author calculates the Euler characteristic of the moment angle complex in this case. Results in this article also are closely related to the \textit{Sloane Encyclopedia of Sequences}, and some new combinatorial identities emerge as a by-product.\N\NFor the entire collection see [Zbl 1540.57001].