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Simplicial types and polynomial algebras. (English) Zbl 1088.55014
Let \(R\) be an integral domain. Consider a finite simplicial complex \(K\). If \(\sigma \) is a simplex of \(K\), denote its \(j\)-th face as \(\partial ^j\sigma \). Associated to each simplex \(\sigma \in K\) one can define the ring \[ R_{\sigma }=R[x_i]_{i\in \sigma }\left /\left (\sum _{i\in \sigma }x_i-1\right )\right . \] and face maps \(\partial ^j:R_{\sigma }\to R_{\partial ^j\sigma }\) given by sending \(x_j\) to zero. The algebra \(A_R^0(K)\) of polynomial functions with coefficients in \(R\) on the barycentric coordinates of \(K\) is defined as follows: an element \(f\) of \(A_R^0(K)\) associates to each simplex \(\sigma \) of \(K\) an element \(f(\sigma )\in R_{\sigma }\) such that \(f(\partial ^j\sigma )=\partial ^j(f(\sigma ))\). This algebra coincides with Sullivan’s algebra of \(0\)-forms.
The main result of the paper says that a finite simplicial complex is uniquely (up to simplicial equivalence) determined by its algebra \(A_R^0(K)\).
This generalizes and completes the results obtained by D. Sullivan [Publ. Math., Inst. Hautes Étud. Sci. 47, 269–331 (1977; Zbl 0374.57002)], D. M. Kan and E. Y. Miller [Topology 16, 193–197 (1977; Zbl 0367.55007)] and I. V. Savel’ev [Math. Notes 50, 719–722 (1991); translation from Mat. Zametki 50, No. 1, 92–97 (1991; Zbl 0739.57014)].

55U10 Simplicial sets and complexes in algebraic topology
55P62 Rational homotopy theory
58A10 Differential forms in global analysis
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