H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, *Spin Geometry*, (1989), p. xi:



> 
> ...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic operator canonically associated to the underlying riemannian metric. ...Twisting the Dirac-type operator...led...to a general formula for the index of any elliptic operator. [Then,] Atiyah and Singer went on to understand the index in the more proper setting of K-theory. This led in particular to the formulation of certain $KO$-invariants which have had profound applications in geometry and topology. These invariants touch questions unapproachable by other means.
> 
> 
> 


I'm curious--at the time, what were these unapproachable questions? (Perhaps this is standard fare when one properly learns this area. And I'm not even sure what the '$KO$-invariants' are. But my ignorance is probably a less interesting point for discussion.)


Just for context: I am a student, right now trying to understand the context in which Quillen first studied $MU$ and its formal group law, and more generally I'm trying to 'connect the dots' of the development of elliptic cohomology. I came across this text while trying to understand the relationship with genera (specifically the Chern character). So any answer or commentary shedding light on this would be especially appreciated.



What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?
Taken in the context of the introduction, I would guess that at least in part they are referring to applications of the index theorem to questions about positive scalar curvature. Before Lichnerowicz's application (1963) of the index theorem for the spin Dirac operator, it might have been conceivable that every manifold of dimension greater than 3 has a metric with positive scalar curvature. Lichnerowicz showed that such a metric implies vanishing of the (numerical) index of the Dirac operator in dimensions 4k; this was extended by Hitchin to various KO-valued index theoretic invariants. Those invariants also apply to families of such metrics and to diffeomorphism groups.


But I'm just speculating about which of the many applications of the index theorem in geometry & topology the authors had in mind. They do list a few other directions a couple of paragraphs up (in discussing the spin representations).


