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Sullivan’s theory of minimal models Deschner, Alan Joseph

Abstract

For a simplicial complex K, the de Rham algebra E*(K) is the differential graded algebra (DGA) of Q-coefficient polynomial forms in the barycentric coordinates of the simplices of K which agree as differential forms on common faces. The associated de Rham cohomology algebra is isomorphic to the simplicial cohomology of K with Q-coefficients by integration of forms over simplices. Given a 1-connected DGA, A, the minimal model of A is a DGA, M, which is free as an algebra, has a differential which decomposes the generators, and which computes the cohomology of A. Such minimal models exist and are unique up to isomorphism. The minimal model M(X) of a 1-connected simplicial complex * X is the minimal model of E*(X) . It depends only on the' rational homotopy type of X. For a fibration K(π,n)→ E→ Y, with E and Y 1-connected, we have (under mild hypothesis) M(E) = M(Y)ØH*(K(π,n) ;Q) with a suitably defined differential. This is applied inductively to the Postnikov decomposition of X to show that the free generators of M(X) correspond to the generators of π[sub *](X)ØQ. The number of these generators which are cocycles is the rank of the rational Hurewicz homomorphism.

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