Chains on suspension spectra (Q731421)

The polynomial differential forms \(A(S)\) on a simplicial set \(S\), as introduced by Sullivan, are a classical functorial bridge between the rational homotopy type of finite type complexes and commutative differential graded algebras over the rationals. Moreover, integration on \(A(S)\) satisfies a de Rham theorem and thus, the rational cohomology algebra of \(S\) is that of its differential forms. One may think that, dualizing the forms, one obtains in a functorial way a chain complex whose homology produces the rational homology of the given complex. However \(A(S)\) is not of finite type, even when \(S\) is finite. In this paper, the author builds a monoidal symmetric functor \(\Phi\) from simplicial sets to rational chain complexes such that, for a given simplicial set \(S\), the homology \(H_*(\Phi(S))\) is the rational homology of \(S\). The chain complex \(\Phi(S)\) is built from a subtle and smaller subcomplex of the dual of \(A(S)\) perturbed with a new differential. Moreover, the author gives also an interpretation of \(\Phi(S)\) as a certain colimit of groups which are not related, as it strongly was in the original definition, to the differential forms on \(S\).