Let $M$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:

  1. Rationally, $\Delta_M$ is computed by the Chern character, i.e.
     $$\text{ch}(\Delta_M)=L(M)\cap [M].$$
     Where could I find details of the proof of this formula? 

     Here $\text{ch}$ denotes the Chern character, $L$ denotes the 
     $L$-class and $[M]$ the fundamental class in the ordinary
     homology theory.

  2. Is there a formula for $\Delta_{M\times N}$ in terms of
     $\Delta_M$ and $\Delta_N$?