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On Davis-Januszkiewicz homotopy types II: Completion and globalisation. (English) Zbl 1198.55005
For any finite simplicial complex \(K\), the Stanley-Reisner algebra \(\mathbb Z[K]\) is an important combinatorial invariant that may be graded by assigning degree 2 to each of its generators. In their fundamental work on toric topology, Davis and Januszkiewicz constructed for any \(K\) a space \(hc(K)\) with \(H^*(hc(K);\mathbb Z)\cong \mathbb Z[K]\). The space \(hc(K)\) is the homotopy colimit of the \(BT^\sigma\) where \(\sigma\) runs over all faces \(\sigma\) of \(K\), and \(T^\sigma\) means \(T^{\dim\, \sigma}\).
In this paper the authors investigate the number of possible homotopy types of spaces that realize \(\mathbb Z[K]\). They use Sullivan’s arithmetic square to decompose the problem into a rational and a \(p\)-adic problem. The rational case was the subject of the first paper of the authors on the subject [cf. Algebr. Geom. Topol. 5, 31–51 (2005; Zbl 1065.55006)]. Their purpose here is to address the \(p\)-adic version. They prove the following theorems.
Theorem 1 : If \(X\) is a \(p\)-complete CW-complex, if \(K\) is an iterated join \(\Delta^{(r_1)}(U_1) * \cdots * \Delta^{(r_q)}(U_q)\) of skeleta of simplices and if \(H^*(X;\mathbb Z_p^\land ) \cong H^*(hc(K);\mathbb Z_p^\land)= \mathbb Z_p^\land[K]\), then \(X\) and \(hc(K)\) are homotopy equivalent.
Theorem 2 : If \(X\) is a nilpotent CW-complex and \(\mathbb Q[K]\) is a complete intersection with \(H^*(X;\mathbb Z) \cong \mathbb Z[K]\), then \(X\) is homotopy equivalent to \(hc(K)\).

MSC:
55P15 Classification of homotopy type
55P60 Localization and completion in homotopy theory
05E99 Algebraic combinatorics
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[1] A Bahri, M Bendersky, F R Cohen, S Gitler, The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces · Zbl 1197.13021
[2] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer (1972) · Zbl 0259.55004
[3] W Bruns, J Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Math. 39, Cambridge Univ. Press (1998) · Zbl 0909.13005
[4] V M Buchstaber, T E Panov, Torus actions and their applications in topology and combinatorics, Univ. Lecture Series 24, Amer. Math. Soc. (2002) · Zbl 1012.52021
[5] M W Davis, T Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417 · Zbl 0733.52006
[6] F X Dehon, J Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un groupe de Lie compact commutatif, Inst. Hautes Études Sci. Publ. Math. (1999) · Zbl 0967.55013
[7] J Grbić, S Theriault, The homotopy type of the complement of a coordinate subspace arrangement, Topology 46 (2007) 357 · Zbl 1118.55006
[8] J Hollender, R M Vogt, Modules of topological spaces, applications to homotopy limits and \(E_\infty\) structures, Arch. Math. \((\)Basel\()\) 59 (1992) 115 · Zbl 0766.55006
[9] N J Kuhn, M Winstead, On the torsion in the cohomology of certain mapping spaces, Topology 35 (1996) 875 · Zbl 0861.55019
[10] J Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un \(p\)-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. (1992) 135 · Zbl 0857.55011
[11] D Notbohm, N Ray, On Davis-Januszkiewicz homotopy types. I. Formality and rationalisation, Algebr. Geom. Topol. 5 (2005) 31 · Zbl 1065.55006
[12] B Oliver, Higher limits via Steinberg representations, Comm. Algebra 22 (1994) 1381 · Zbl 0815.55003
[13] L Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Math., Univ. of Chicago Press (1994) · Zbl 0871.55001
[14] R P Stanley, Combinatorics and commutative algebra, Progress in Math. 41, Birkhäuser (1996) · Zbl 0838.13008
[15] R Thom, L’homologie des espaces fonctionnels, Georges Thone (1957) 29 · Zbl 0077.36301
[16] R M Vogt, Convenient categories of topological spaces for homotopy theory, Arch. Math. \((\)Basel\()\) 22 (1971) 545 · Zbl 0237.54001
[17] Z Wojtkowiak, On maps from \(\mathrm{ho}\varinjlim F\) to \(\mathbfZ\) (editors J Aguadé, R Kane), Lecture Notes in Math. 1298, Springer (1987) 227
[18] A Zabrodsky, On phantom maps and a theorem of H Miller, Israel J. Math. 58 (1987) 129 · Zbl 0638.55020
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