5 Coda: The ABS Orientation
Suppose is a rank vector bundle with a Euclidean metric. Then taking the Clifford algebra with respect to (the negation of) the metric fiberwise gives us a -bundle over . Let be the Grothendieck group of -modules. If is a graded -module, then the pullbacks and are isomorphic over , the complement of the zero section, where the isomorphism at is given by multiplication by (with inverse a multiple of , since ). This thus gives us an element of , where is the Thom space. In other words, we have a canonical map
Suppose came from a -module, then by a linear deformation, we can use a vector in the trivial factor as the isomorphism between and instead, and this extends to an isomorphism defined on all of . So the resulting class is trivial in . Moreover, this map is natural with respect to products, namely if and are two bundles, then the following diagram commutes:
In particular, taking , we obtain ring homomorphisms
In fact, this is the same map as the one we constructed above, which is just a matter of tracing through all the constructions we have done.
Let be the even invertible elements of that preserve under the conjugation action, and let consist of those elements of ``spinor norm'' . The natural action on begets a map that is a non-trivial double cover. Crucially, acts as automorphisms of the whole of .
Thus, has the structure of a -bundle, so that there is a principal -bundle with , then and there is a natural map that sends an module to . This gives a long composite
In particular, if , and is the generator of , then by naturality, the image of restricts to a generator of . So this is a Thom class of the bundle . Stabilizing, we get
Every spin bundle is naturally -oriented.