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 $\Delta_N$?