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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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