Far Eastern Mathematical Journal

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Families of minimally non-Golod simplicial complexes and polyhedral products


Limonchenko I.Yu.

2015, issue 2, . 222-237


Abstract
We consider families of simple polytopes P and simplicial complexes K well-known in polytope theory and convex geometry, and show that their moment-angle complexes have some remarkable homotopy properties which depend on combinatorics of the underlying complexes and algebraic properties of their Stanley-Reisner rings. We introduce infinite series of Golod and minimally non-Golod simplicial complexes K with moment-angle complexes k having free integral cohomology but not homotopy equivalent to a wedge of spheres or a connected sum of products of spheres respectively. We then prove a criterion for a simplicial multiwedge and composition of complexes to be Golod and minimally non-Golod and present a class of minimally non-Golod polytopal spheres.

Keywords:
simple polytopes, Golod rings, moment-angle complexes, Stanley-Reisner rings

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References

[1] A.A. Ajzenberg, "Podstanovki mnogogrannikov, simplicial'nyh kompleksov i mul'tigraduirovannye chisla Betti", Tr. MMO, 74:2 (2013), 211-245.
[2] A. Bahri, M. Bendersky, F.R. Cohen, S. Gitler, Operations on polyhedral products and a new topological construction of infinite families oftoric manifolds, Preprint, 2010, arXiv: 1011.0094v5.
[3] A. Bahri, M. Bendersky, F.R. Cohen, S. Gitler, "The polyhedral product functor: A method of decomposition for moment-angle complexes, arrangements and related spaces", Advances in Mathematics, 225:3 (2010), 1634-1668.
[4] Piotr Beben and Jelena Grbic, Configuration spaces and polyhedral products, Preprint, 2014, arXiv: 1409.4462v11.
[5] Alexander Berglund and Michael Jollenbeck, "On the Golod property of Stanley-Reisner rings", J. Algebra, 315:1 (2007), 249-273.
[6] Victor M. Buchstaber and Taras E. Panov, "Toric Topology", Mathematical Surveys and Monographs. V. 204, American Mathematical Society, RI, Providence, 2015.
[7] N.Ju. Erohovec, "Teorija invarianta Buhshtabera simplicial'nyh kompleksov i vypuklyh mnogogrannikov", Tr. MIAN, 286 (2014), 144-206.
[8] E.S. Golod, "O gomologijah nekotoryh lokal'nyh kolec", DAN SSSR, 144:3 (1962), 479-482.
[9] Jelena Grbic, Taras Panov, Stephen Theriault and Jie Wu, "Homotopy types of moment-angle complexes for flag complexes", Transactions of the AMS, 2015 (to appear), arXiv: 1211.0873.
[10] T.H. Gulliksen and G. Levin, "Homology of local rings", Queen's Papers in Pure and Applied Mathematics. V. 20, Queen's University, Kingston, Ontario, 1969.
[11] M. Hochster, "Cohen-Macaulay rings, combinatorics, and simplicial complexes, in Ring theory, II", Lecture Notes in Pure and Appl. Math. V.26, Proc. Second Conf. (Univ. Oklahoma, Norman, Okla., 1975), Dekker, New York, 1977, 171-223.
[12] Kouyemon Iriye and Daisuke Kishimoto, Fat wedge filtrations and decomposition of polyhedral products, Preprint, 2014, arXiv: 1412.4866v3.
[13] W. Kuhnel, T.F. Banchoff, "The 9-Vertex Complex Projective Plane", Math. Int., 5:3 (1983), 11-22.
[14] W. Kuhnel, G. Lassmann, "The Unique 3-Neighbourly 4-Manifold with Few Vertices", Series A, 35, 1983, 173-184.
[15] I.Ju. Limonchenko, "Kol'ca Stenli-Rajsnera obobshhennyh mnogogrannikov usechenija i ih moment-ugol mnogoobrazija", Tr. MIAN, 286 (2014), 207-218.
[16] Macaulay 2, A software system devoted to supporting research in algebraic geometry and commutative algebra, Available at http://www.math.uiuc.edu/Macaulay2/.
[17] S. Lopez de Medrano, "Topology of the intersection of quadrics in Rn", Lecture Notes in Mathematics, 1370 (1989), 280-292.
[18] Taras E. Panov, "Cohomology of face rings, and torus actions", Surveys in Contemporary Mathematics. V. 347, London Math. Soc. Lecture Note Series, Cambridge, U.K., 2008, 165-201, arXiv:math.AT/0506526.
[19] Gunter M. Ziegler, Lectures on Polytopes, Springer-Verlag, New York, 2007.

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