# zbMATH — the first resource for mathematics

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)].

##### MSC:
 55U10 Simplicial sets and complexes in algebraic topology 55P62 Rational homotopy theory 58A10 Differential forms in global analysis
Full Text: