A model for cyclic homology and algebraic K-theory of 1-connected topological spaces (Q1077756)

Let X be a 1-connected space of finite type, with Sullivan minimal model (\(\bigwedge Z,d)\). Define a degree -1 derivation \(\beta\) in \(\bigwedge Z\otimes \bigwedge \bar Z\) \((\bar Z=\Sigma^{-1}Z)\) by \(\beta(z)=\bar z\) and \(\beta(\bar z)=0\). Construct degree \(+1\) derivations \(\delta\) in \(\bigwedge Z\otimes \bigwedge \bar Z\) and D in \(\bigwedge \alpha \otimes \bigwedge Z\otimes \bigwedge \bar Z\) \((| \alpha | =2)\) as follows: \(\delta (z)=dz\), \(\delta(\bar z)=-\beta(dz)\); \(D(\alpha)=0\) and \(D(u)=\delta(u)+ \alpha \cdot \beta (u)\) for \(u\in \wedge Z\otimes \bigwedge \bar Z\). Main result: the algebraic fibration \[ (\bigwedge \alpha,0)\to^{incl}(\bigwedge \alpha \otimes \bigwedge Z\otimes \bigwedge \bar Z,D)\to^{proj}(\bigwedge Z\otimes \bigwedge \bar Z,\delta) \] represents a model for the fibration defining equivariant cohomology of the free loop space: \(X^{S^ 1}\to ES^ 1\times_{S^ 1} X^{S^ 1}\to BS^ 1\). The usefulness of this is twofold: firstly, the Gysin sequence of this fibration was identified by \textit{T. G. Goodwillie} [Topology 24, 187-215 (1985; Zbl 0569.16021)] and the second author and \textit{Z. Fiedorowicz} [ibid. 25, 303-317 (1986)] with the Connes sequence relating Hochschild and cyclic cohomology of X; second, one has: \(\tilde K_{*+1}(X)\otimes Q=^{\sim}_*(X;Q)\), as shown by the second author [Contemp. Math. 55, Part I, 89-115 (1986)]. The model (of cyclic homology and algebraic K-theory) under review is shown to be quite manageable, by performing an explicit computation of \(HC^*(X;Q)\), including the extra structure coming from Connes' periodicity \(HC^*\to^{S}HC^{*+2}\), in the case \(H^*(X;Q)\) is a truncated algebra on one generator, and by showing that \(H_{*+1}(X;Q)\) is a direct summand in \(HC_*(X;Q)\), for a general X.