The combinatorial model for the Sullivan functor on simplicial sets
It is well known that Sullivan's rational de Rham algebra \(A_{\mathrm{PL}}(\_\_ )\) leads to a purely algebraic model of rational homotopy type for nilpotent spaces [\textit{D. Sullivan}, Differential forms and the topology of manifolds. Manifolds, Proc. int. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 37--49 (1975; Zbl 0319.58005)]. In Appendix G of this paper, Sullivan asserts -- without proof -- that \(A_{\mathrm{PL}}( K )\) admits a purely combinatorial description for \(K\) a finite simplicial complex. Specifically, Sullivan asserts \(A_{\mathrm{PL}}(K)\) is isomorphic to the DG algebra \(A(K) = \Lambda(t_1, \dots, t_{n+1}, dt_1, \dots, dt_{n+1})/J\) where the \(t_i\) are the vertices of \(K\) (of degree \(0\)), the \(dt_i\) are their derivatives (of degree \(1\)) and where \(J\) is the smallest differential ideal generated by \(\sum_{i=1}^{n+1}t_i -1\) and the monomials \(t_{i+1}\cdots t_{i_r} dt_{t_{i_{r+1}}}\cdots dt_{i_l}\) for \(\{ t_{i_1}, \dots, t_{i_l} \}\) not a simplex of \(K.\) The authors prove Sullivan's assertion constructing an isomorphism \(\psi_K \colon A(K) \to A_{\mathrm{PL}}(K)\) which is natural with respect to simplicial maps. Applying this result and the Whitney map \(C^*(K; \mathbb{Q}) \to A(K)\), the authors obtain some explicit descriptions of products in \(H^*(K; \mathbb{Q})\) and of the Malcev completion of \(\pi_1(K).\)
scientific article; zbMATH DE number 5376983
scientific article