Clifford Algebras and Bott PeriodicityCoda: The ABS Orientation

5 Coda: The ABS Orientation

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

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

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

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

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

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 $\mathrm{GSpin}(k)$ be the even invertible elements of $C_k$ that preserve $\mathbb {R}^k \subseteq C_k$ under the conjugation action, and let $\mathrm{Spin}(k)$ consist of those elements of “spinor norm” $1$. The natural action on $\mathbb {R}^k$ begets a map $\mathrm{Spin}(k) \to \mathrm{SO}(k)$ that is a non-trivial double cover. Crucially, $\mathrm{Spin}(k)$ acts as automorphisms of the whole of $C_k$.

Thus, $V$ has the structure of a $\mathrm{Spin}(k)$-bundle, so that there is a principal $\mathrm{Spin}(k)$-bundle $P \to X$ with $V = P \times _{\mathrm{Spin}(k)} \mathbb {R}^k$, then $C(V) = P \times _{\mathrm{Spin}(k)} C_k$ and there is a natural map $A_k \to A(V)$ that sends an $C_k$ module $M$ to $P \times _{\mathrm{Spin}(k)} M$. This gives a long composite

$A_k \to \widetilde{KO}(X^V).$

In particular, if $k = 8m$, and $\lambda$ is the generator of $A_{8m}$, then by naturality, the image of $\lambda ^m$ restricts to a generator of $\widetilde{KO}(S^{8m})$. So this is a Thom class of the bundle $V$. Stabilizing, we get

Theorem

Every spin bundle is naturally $KO$-oriented.