[Q] Were you already thinking about indices?

No, this is before that. The real theory in the context of vector bundles and periodicity theorem, because in the complex case everything can be done, there's no torsion, so everything can be done by using cohomology calculation, rational cohomology, which gives you a machinery. In the real case there are all these… mod 2 problems, which means that… to do with signs and so on… which you cannot do by cohomology calculations, so you had to do those by going back to basic homotopy theory. And in some senses, in the real case, Bott's theorems about homotopy are that more refined and there's no alternative way except going back to the fundamental principles.

And so I had to learn that.  But for example, Hirzebruch and I had made an application of the real theory to the questions about vector fields on spheres, and parallelisable spheres, so there are some nice applications you can make quite early on with the real theory, which are purely topological. They've got nothing to do with the index, that's a much later story. And at the beginning we did everything with the complex case, but the real case was one you brought along when you could, and got refinements and improvements of various kinds.