Decompositions of suspensions of spaces involving polyhedral products

Iriye, Kouyemon¹ ; Kishimoto, Daisuke²

¹Department of Mathematics and Information Sciences, Osaka Prefecture University, Sakai 599-8531, Japan
²Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

Algebraic and Geometric Topology, Tome 16 (2016) no. 2, pp. 825-841

Résumé

Two homotopy decompositions of suspensions of spaces involving polyhedral products are given. The first decomposition is motivated by the decomposition of suspensions of polyhedral products by Bahri, Bendersky, Cohen and Gitler, and is a generalization of a retractile argument of James. The second decomposition is on the union of an arrangement of subspaces called diagonal subspaces, and generalizes a result of  Labassi.

Détail

DOI : 10.2140/agt.2016.16.825

Classification : 55P15, 55U10, 52C35
Keywords: polyhedral products, retractile spaces over posets, diagonal arrangements

Bibliographie

[1] A Adem, A Bahri, M Bendersky, F R Cohen, S Gitler, On decomposing suspensions of simplicial spaces, Bol. Soc. Mat. Mexicana 15 (2009) 91

[2] A Adem, F R Cohen, Commuting elements and spaces of homomorphisms, Math. Ann. 338 (2007) 587

[3] A Adem, F R Cohen, J M Gómez, Stable splittings, spaces of representations and almost commuting elements in Lie groups, Math. Proc. Cambridge Philos. Soc. 149 (2010) 455

[4] A Adem, F R Cohen, E Torres Giese, Commuting elements, simplicial spaces and filtrations of classifying spaces, Math. Proc. Cambridge Philos. Soc. 152 (2012) 91

[5] 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, Adv. Math. 225 (2010) 1634

[6] E D Farjoun, Cellular spaces, null spaces and homotopy localization, 1622, Springer (1996)

[7] M Goresky, R Macpherson, Stratified Morse theory, 14, Springer (1988)

[8] J Grbić, S Theriault, The homotopy type of the complement of a coordinate subspace arrangement, Topology 46 (2007) 357

[9] A Hatcher, Algebraic topology, Cambridge University Press (2002)

[10] P Hilton, G Mislin, J Roitberg, On co-H–spaces, Comment. Math. Helv. 53 (1978) 1

[11] K Iriye, D Kishimoto, Decompositions of polyhedral products for shifted complexes, Adv. Math. 245 (2013) 716

[12] K Iriye, D Kishimoto, Fat wedge filtrations and decomposition of polyhedral products, preprint (2015)

[13] I M James, On H–spaces and their homotopy groups, Quart. J. Math. Oxford Ser. 11 (1960) 161

[14] S Kallel, I Saihi, Homotopy groups of diagonal complements, preprint (2013)

[15] Y Kamiyama, S Tsukuda, The configuration space of the n–arms machine in the Euclidean space, Topology Appl. 154 (2007) 1447

[16] S Kim, Shellable complexes and topology of diagonal arrangements, Discrete Comput. Geom. 40 (2008) 190

[17] D N Kozlov, A class of hypergraph arrangements with shellable intersection lattice, J. Combin. Theory Ser. A 86 (1999) 169

[18] F Labassi, Sur les diagonales épaisses et leurs complémentaires, J. Homotopy Relat. Struct. 10 (2015) 375

[19] J Lillig, A union theorem for cofibrations, Arch. Math. (Basel) 24 (1973) 410

[20] M S Miller, Massey products and k–equal manifolds, Int. Math. Res. Not. 2012 (2012) 1805

[21] M Miller, M Wakefield, Formality of Pascal arrangements, Proc. Amer. Math. Soc. 139 (2011) 4461

[22] I Peeva, V Reiner, V Welker, Cohomology of real diagonal subspace arrangements via resolutions, Compositio Math. 117 (1999) 99

[23] G M Ziegler, R T Živaljević, Homotopy types of subspace arrangements via diagrams of spaces, Math. Ann. 295 (1993) 527