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simplicial complex; algebraic de Rham complex; Sullivan’s de Rham complex
This paper shows that the simplicial type of a finite simplicial complex $K$ is determined by its algebra $A$ of polynomial functions on the baricentric coordinates with coefficients in any integral domain. The link between $K$ and $A$ is done through certain admissible matrix associated to $K$ in a natural way. This result was obtained for the real numbers by I. V. Savel’ev [5], using methods of real algebraic geometry. D. Kan and E. Miller had shown in [2] that $A$ determines the homotopy type of the polyhedron associated to $K$ and not only its rational homotopy type as it was previously proved by D. Sullivan in [6].
[1] Halperin, S.: Lectures on minimal models. Memoires SMF, Nouvelle Série (1983), 9–10. MR 0736299 | Zbl 0536.55003
[2] Kan, D. M. and Miller, E. Y.: Homotopy types and Sullivan’s algebras of 0-forms. Topology 16 (1977), 193–197. MR 0440539
[3] Kan, D. M. and Miller, E. Y.: Sullivan’s de Rham complex is definable in terms of its 0-forms. Proc. A.M.S. 57 2 (1976), 337–339. MR 0410737
[4] Matsumura, H.: Commutative Algebra. Benjamin, 1980. MR 0266911 | Zbl 0655.00011
[5] Savel’ev, I. V.: Simplicial complexes and ruled manifolds. Math. Zam. 50 1 (1991), 92–97. MR 1140356
[6] Sullivan, D.: Infinitesimal computations in topology. Publ. I.H.E.S. 47 (1977), 269–331. MR 0646078 | Zbl 0374.57002
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