This paper shows that real K-theory can be obtained from tensoring down
from Spin bordism. That is, there is an isomorphism KO(X) = MSpin(X)
tensor over MSpin of a point with KO of a point. The map from MSpin to
a point is the Atiyah-Bott-Shapiro orientation, also known as the index
of the Dirac operator. In some sense this logically completed that old
book of Conner and Floyd, where they showed symplectic bordism
determines KO. (Ochanine showed in the 80s that SU bordism does not
determine KO in this sense).
This was my first real foray into stable homotopy theory. The problem
was suggested to me by Haynes Miller, and he was something of a second
thesis advisor for me, for which I am eternally grateful.