  The M-theory fieldstrength and its dual, given by the integral lift of the
left hand side of the equation of motion, both satisfy certain cohomological
properties. We study the combined fields and observe that the multiplicative
structure on the product of the corresponding degree four and degree eight
cohomology fits into that given by Spin K-theory. This explains some earlier
results and leads naturally to the use of Spin characteristic classes. We
reinterpret the one-loop term in terms of such classes and we show that it is
homotopy invariant. We argue that the various anomalies have natural
interpretations within Spin K-theory. In the process, mod 3 reductions play a
special role.
