Clifford Algebras and Bott Periodicity — Coda: The ABS Orientation

5 Coda: The ABS Orientation

Suppose π:VX\pi : V \to X is a rank kk vector bundle with a Euclidean metric. Then taking the Clifford algebra with respect to (the negation of) the metric fiberwise gives us a CkC_ k-bundle C(V)C(V) over XX. Let M(V)M(V) be the Grothendieck group of C(V)C(V)-modules. If E=E0E1E = E^0 \oplus E^1 is a graded C(V)C(V)-module, then the pullbacks πE1\pi ^* E_1 and πE0\pi ^* E_0 are isomorphic over V#V^\# , the complement of the zero section, where the isomorphism at vV#v \in V^\# is given by multiplication by vv (with inverse a multiple of vv, since v2=vv^2 = -\| v\| ). This thus gives us an element of KO~(V,V#)KO~(XV)\widetilde{KO}(V, V^\# ) \cong \widetilde{KO}(X^ V), where XVX^ V is the Thom space. In other words, we have a canonical map

M(V)KO~(XV). M(V) \to \widetilde{KO}(X^ V).

Suppose EE came from a C(V1)C(V \oplus 1)-module, then by a linear deformation, we can use a vector in the trivial factor as the isomorphism between πE1\pi ^*E_1 and πE2\pi ^*E_2 instead, and this extends to an isomorphism defined on all of VV. So the resulting class is trivial in KO~(XV)\widetilde{KO}(X^ V). Moreover, this map is natural with respect to products, namely if VXV \to X and WYW \to Y are two bundles, then the following diagram commutes:

\begin{useimager} 
  \[
    \begin{tikzcd}
      A(V) \otimes A(W) \ar[d] \ar[r] & A (V \oplus W) \ar[d]\\
      \widetilde{KO}(X^V) \otimes \widetilde{KO}(Y^W) \ar[r] & \widetilde{KO}(X \times Y^{V \oplus W}).
    \end{tikzcd}
  \]
\end{useimager}

In particular, taking X=X = *, we obtain ring homomorphisms

Ak0KOk() A_* \to \sum _{k \geq 0} KO^{-k}(*)

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 GSpin(k)\mathrm{GSpin}(k) be the even invertible elements of CkC_ k that preserve RkCk\mathbb {R}^ k \subseteq C_ k under the conjugation action, and let Spin(k)\mathrm{Spin}(k) consist of those elements of ``spinor norm'' 11. The natural action on Rk\mathbb {R}^ k begets a map Spin(k)SO(k)\mathrm{Spin}(k) \to \mathrm{SO}(k) that is a non-trivial double cover. Crucially, Spin(k)\mathrm{Spin}(k) acts as automorphisms of the whole of CkC_ k.

Thus, VV has the structure of a Spin(k)\mathrm{Spin}(k)-bundle, so that there is a principal Spin(k)\mathrm{Spin}(k)-bundle PXP \to X with V=P×Spin(k)RkV = P \times _{\mathrm{Spin}(k)} \mathbb {R}^ k, then C(V)=P×Spin(k)CkC(V) = P \times _{\mathrm{Spin}(k)} C_ k and there is a natural map AkA(V)A_ k \to A(V) that sends an CkC_ k module MM to P×Spin(k)MP \times _{\mathrm{Spin}(k)} M. This gives a long composite

AkKO~(XV). A_ k \to \widetilde{KO}(X^ V).

In particular, if k=8mk = 8m, and λ\lambda is the generator of A8mA_{8m}, then by naturality, the image of λm\lambda ^ m restricts to a generator of KO~(S8m)\widetilde{KO}(S^{8m}). So this is a Thom class of the bundle VV. Stabilizing, we get

Theorem

Every spin bundle is naturally KOKO-oriented.