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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 \(hc(K)\).

55P15 Classification of homotopy type
55P60 Localization and completion in homotopy theory
05E99 Algebraic combinatorics
Full Text: DOI
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