The finer index theorem for the APS e-invariant

Let

be a fiber bundle with fiberwise $K$-theory pushforward

, let

$$denote the Beck\Gottlieb transfer, and let$$

be the APS $e$-invariant map. The notation $[E,F_{t,}]$ and $[B,F_{t,}]$ denotes homotopy classes of maps into the indicated target. Finer APS index conjecture. The following diagram is commutative:

$$\begin{CD} \widetilde K(E,\mathbb C) @>{\bar e_{\operatorname{APS}}}>> [E,F_{t,\mathbb C/\mathbb Z}]\\ @V{\pi^!}VV @VV{\operatorname{tr}_{\operatorname{BG}}^{*}}V\\ \widetilde K(B,\mathbb C) @>{\bar e_{\operatorname{APS}}}>> [B,F_{t,\mathbb C/\mathbb Z}] \end{CD}$$

This is proposed as a finer index theorem refining the corresponding real Chern-character statement; after composing with the indicated Chern character and comparison map, it implies the Bismut\Lott index theorem.