I am trying to understand the family version of the Atiyah-Singer Index Theorem as described in the book "Spin Geometry" by Lawson/Michelsohn. In Part III.§8, they define the operator bundle $$ Op^m(\mathscr{E}, \mathscr{F}) \to A $$ as the ''bundle associated to the principal $\mathscr{D}$-bundle of $\mathscr{E} \oplus \mathscr{F} \to A$ by the action of $\mathscr{D}$ on $Op^m(E,F)$''. Here, $\mathscr{E} \oplus \mathscr{F} \to A$ is a continuous pair of vector bundles over some compact manifold $X$ and $\mathscr{D} = \operatorname{Diff}(E,F;X)$. The space $A$ is a compact Hausdorff parameter space.
- What is the ''principal bundle of $\mathscr{E} \oplus \mathscr{F}$''?
- How is $Op^m(E,F)$ topologized?
- What is a good reference for the theory of continuous fibre bundles (which are not necessarily principal $G$-bundles for some Lie group $G$), which is implicitely used here?
I guess $\mathscr{D}$ is topologized with the $\mathscr{C}^{\infty}$-topology and the fibre over some $a$ of $Op^m(\mathscr{E}, \mathscr{F})$ is (isomorphic to) $Op^m(\mathscr{E}_a, \mathscr{F}_a)$.