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 Trudy Mat. Inst. Steklova, 2011, Volume 275, Pages 22–54 (Mi tm3337)

Nerve complexes and moment–angle spaces of convex polytopes

A. A. Aizenberga, V. M. Buchstaberb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, Russia
b Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

Abstract: 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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 275, 15–46

Bibliographic databases:

UDC: 515.164.8

Citation: A. A. Aizenberg, V. M. Buchstaber, “Nerve complexes and moment–angle spaces of convex polytopes”, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Collected papers. In commemoration of the 120th anniversary of Boris Nikolaevich Delone's birth, Trudy Mat. Inst. Steklova, 275, MAIK Nauka/Interperiodica, Moscow, 2011, 22–54; Proc. Steklov Inst. Math., 275 (2011), 15–46

Citation in format AMSBIB
\Bibitem{AyzBuc11} \by A.~A.~Aizenberg, V.~M.~Buchstaber \paper Nerve complexes and moment--angle spaces of convex polytopes \inbook Classical and modern mathematics in the wake of Boris Nikolaevich Delone \bookinfo Collected papers. In commemoration of the 120th anniversary of Boris Nikolaevich Delone's birth \serial Trudy Mat. Inst. Steklova \yr 2011 \vol 275 \pages 22--54 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3337} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2962969} \elib{https://elibrary.ru/item.asp?id=17238814} \transl \jour Proc. Steklov Inst. Math. \yr 2011 \vol 275 \pages 15--46 \crossref{https://doi.org/10.1134/S0081543811080025} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000305482400002} \elib{https://elibrary.ru/item.asp?id=18034701} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84856477713} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Aizenberg, “Topological applications of Stanley-Reisner rings of simplicial complexes”, Trans. Moscow Math. Soc., 73 (2012), 37–65
2. A. A. Aizenberg, “Substitutions of polytopes and of simplicial complexes, and multigraded Betti numbers”, Trans. Moscow Math. Soc., 74 (2013), 175–202
3. N. Yu. Erokhovets, “Buchstaber invariant theory of simplicial complexes and convex polytopes”, Proc. Steklov Inst. Math., 286 (2014), 128–187
4. Ayzenberg A., “Locally standard torus actions and $h'$-numbers of simplicial posets”, J. Math. Soc. Jpn., 68:4 (2016), 1725–1745
5. Ayzenberg A., “Buchstaber Invariant, Minimal Non-Simplices and Related”, Osaka J. Math., 53:2 (2016), 377–395
6. Lopez de Medrano S., “Singular Intersections of Quadrics i”, Singularities in Geometry, Topology, Foliations and Dynamics: a Celebration of the 60Th Birthday of Jose Seade, Trends in Mathematics, eds. CisnerosMolina J., Le D., Oka M., Snoussi J., Springer International Publishing Ag, 2017, 155–170
7. Lopez de Medrano S., “Samuel Gitler and the topology of intersections of quadrics”, Bol. Soc. Mat. Mex., 23:1, SI (2017), 5–21
8. A. A. Ayzenberg, V. M. Buchstaber, “Manifolds of isospectral arrow matrices”, Sb. Math., 212:5 (2021), 605–635
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