Geodesic
Families of minimally non-Golod simplicial complexes and polyhedral products
Dalʹnevostočnyj matematičeskij žurnal, Tome 15 (2015) no. 2, pp. 222-237.

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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 $\mathcal Z_K$ 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. 
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I. Yu. Limonchenko. Families of minimally non-Golod simplicial complexes and polyhedral products. Dalʹnevostočnyj matematičeskij žurnal, Tome 15 (2015) no. 2, pp. 222-237. http://geodesic.mathdoc.fr/item/DVMG_2015_15_2_a6/

[1] A. A. Aizenberg, “Podstanovki mnogogrannikov, simplitsialnykh kompleksov i multigraduirovannye 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 of toric 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 | DOI | MR | Zbl

[4] Piotr Beben and Jelena Grbić, Configuration spaces and polyhedral products, Preprint, 2014, arXiv: 1409.4462v11

[5] Alexander Berglund and Michael Jöllenbeck, “On the Golod property of Stanley–Reisner rings”, J. Algebra, 315:1 (2007), 249–273 | DOI | MR | Zbl

[6] Victor M. Buchstaber and Taras E. Panov, Toric Topology, Mathematical Surveys and Monographs, 204, American Mathematical Society, Providence, RI, 2015 | MR | Zbl

[7] N. Yu. Erokhovets, “Teoriya invarianta Bukhshtabera simplitsialnykh kompleksov i vypuklykh mnogogrannikov”, Tr. MIAN, 286 (2014), 144–206 | Zbl

[8] E. S. Golod, “O gomologiyakh nekotorykh lokalnykh kolets”, DAN SSSR, 144:3 (1962), 479–482 | MR | Zbl

[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 | MR | Zbl

[11] M. Hochster, “Cohen-Macaulay rings, combinatorics, and simplicial complexes”, Ring theory, II, Proc. Second Conf. (Univ. Oklahoma, Norman, Okla., 1975), Lecture Notes in Pure and Appl. Math., 26, Dekker, New York, 1977, 171–223 | MR

[12] Kouyemon Iriye and Daisuke Kishimoto, Fat wedge filtrations and decomposition of polyhedral products, Preprint, 2014, arXiv: 1412.4866v3 | MR

[13] W. Kühnel, T. F. Banchoff, “The 9-Vertex Complex Projective Plane”, Math. Int., 5:3 (1983), 11–22 | DOI | MR | Zbl

[14] W. Kühnel, G. Lassmann, “The Unique 3-Neighbourly 4-Manifold with Few Vertices”, J. of Comb. Theory, Series A, 35, 1983, 173–184 | DOI | MR

[15] I. Yu. Limonchenko, “Koltsa Stenli–Raisnera obobschennykh mnogogrannikov usecheniya i ikh moment-ugol mnogoobraziya”, Tr. MIAN, 286 (2014), 207–218 | Zbl

[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 $\mathbb{R}^n$”, Lecture Notes in Mathematics, 1370 (1989), 280–292 | DOI | MR | Zbl

[18] Taras E. Panov, “Cohomology of face rings, and torus actions”, Surveys in Contemporary Mathematics, London Math. Soc. Lecture Note Series, 347, Cambridge, U.K., 2008, 165–201, arXiv: math.AT/0506526 | MR | Zbl

[19] Günter M. Ziegler, Lectures on Polytopes, Springer-Verlag, New York, 2007