Let $M$ be an oriented Riemannian manifold.Signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.I have 2 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 fin details for the proof of this formula?

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

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