Nerve complexes and moment--angle spaces of convex polytopes

A. A. Aizenberg ; V. M. Buchstaber

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 22-54

Résumé

We introduce spherical nerve complexes that are a far-reaching generalization of simplicial spheres, and consider the differential ring of simplicial complexes. We show that spherical nerve complexes form a subring of this ring, and define a homomorphism from the ring of polytopes to this subring that maps each polytope $P$ to the nerve $K_P$ of the cover of the boundary $\partial P$ by facets. We develop a theory of nerve complexes and apply it to the moment–angle spaces $\mathcal Z_P$ of convex polytopes $P$. In the case of a polytope $P$ with $m$ facets, its moment–angle space $\mathcal Z_P$ is defined by the canonical embedding in the cone $\mathbb R_\geq^m$. It is well-known that the space $\mathcal Z_P$ is homeomorphic to the polyhedral product $(D^2,S^1)^{\partial P^*}$ if the polytope $P$ is simple. We show that the homotopy equivalence $\mathcal Z_P\simeq(D^2,S^1)^{K_P}$ holds in the general case. On the basis of bigraded Betti numbers of simplicial complexes, we construct a new class of combinatorial invariants of convex polytopes. These invariants take values in the ring of polynomials in two variables and are multiplicative with respect to the direct product or join of polytopes. We describe the relation between these invariants and the well-known $f$-polynomials of polytopes. We also present examples of convex polytopes whose flag numbers (in particular, $f$-polynomials) coincide, while the new invariants are different.