Cyclic homology model for the fiberwise negative TC transfer

Establish that for fibrations f: X → Y and p: Y → B between nilpotent spaces of finite rational type, modeled by cofibrations φ: R → S and ι: k → R of commutative differential graded algebras over Q, with the fiber of f simply connected and finitely dominated and the fiber of p simply connected, the fiberwise transfer f^*: TC^-_B(Y) → TC^-_B(X) in negative topological cyclic homology is modeled by the cyclic homology transfer φ^*: HC_k(S) → HC_k(R).

Background

The paper proves that, under suitable hypotheses on fibrations of nilpotent spaces modeled by Sullivan algebras, the fiberwise THH transfer is rationally modeled by the Hochschild homology transfer. Motivated by analogous structures for K-theory and by Keller's construction of a transfer for cyclic homology, the authors propose an extension from THH to negative topological cyclic homology.

Goodwillie's work relates negative topological cyclic homology rationally to algebraic K-theory, so verifying this conjecture would yield a rational model for a close approximation of the fiberwise A-theory transfer. The conjecture posits that the cyclic homology transfer on the Sullivan model computes, rationally, the fiberwise transfer on TC^-.