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.