Quillen notebook

January 8, 1984

Problem: Odd $K$-elements

The question is how one should represent odd $K$-elements. Originally I believed that this was the point of introducing $C_n$-bundles, where $C_n$ is the Clifford algebra. Somehow $K^{-n}$ is the $K$-theory of bundles of graded $C_n$-modules.

But the picture is more subtle. What happens is that $K^{-n}(X)$ is the relative $K$-theory of graded $C_n$-bundles modulo graded $C_{n-1}$-bundles. One can see this as follows. First of all, a graded $C_{n-1}$-bundle is the same as an ungraded $C_n$-bundle. For $n$ even $C_n$ is simple, and for $n$ odd $C_n$ is the product of two simple algebras. Thus, for $n$ even, there is one irreducible $C_n$-module which is ungraded, and two irreducible graded $C_n$-modules. So \begin{gather} K(X, \text{graded $C_{n+1}$}) \to K(X, \text{graded $C_n$})
\begin{cases} K(X) \stackrel{\Delta}{\longrightarrow} K(X)^2 & \text{$n$ even}
K(X)^2 \stackrel{+}{\longrightarrow} K(X) & \text{$n$ odd} \end{cases} \end{gather} and we see the relative or cokernel theory coincides with $K^{-n}(X)$.

However, notice that $+ \colon K(X)^2 \to K(X)$ is always onto so that an element of $K^{-1}(X)$ is not realized by a $C_1$-bundle. Now somehow this state of affairs improves in the infinite dimensional theory.

In the infinite dimensional version of $K^0(X)$, a $K$-element is represented by a graded Hilbert bundle $\mathcal{H} = \mathcal{H}^+ \oplus \mathcal{H}^-$ together with a bundle map $F$ of degree $1$ such that $F^2 - I$ is compact. This amounts to giving a Fredholm operator $\mathcal{H}^+ \to \mathcal{H}^-$ and an inverse modulo compacts. If we use the Kuiper theorem to trivialize the Hilbert bundle, then we get a map from $X$ to Fredholm operators.

For $K^{\text{odd}}$ one requires in addition a $C_1$-structure, more precisely a graded $C_1$-module structure, on $\mathcal{H}$ such that $F$ anti-commutes with the generator $\gamma$ of $C_1$. Thus, $\mathcal{H}^+ = \mathcal{H}^-$ and we have $$\epsilon = \begin{pmatrix} 1 & \ & -1 \end{pmatrix} \quad \gamma^1 = \begin{pmatrix} & 1 \ 1 & \end{pmatrix} \quad F = \begin{pmatrix} & iQ \ iP & \end{pmatrix},$$ where $\gamma^1 F \gamma^1 = -F \iff P = -Q$. Thus, \begin{gather} F = i \begin{pmatrix} 0 & -P \ P & 0 \end{pmatrix} \text{ where $P^2 - I$ is compact}
F = F^* \quad \iff \quad P = P^* \end{gather} Thus we have the self-adjoint Fredholm operators with essential spectrum $\subset {\pm 1}$.

Note that if $F$ can be homotoped to an $F$ of square $I$, then we obtain a graded $C_2$-structure. Thus the obstruction of lifting $F$ from $F^2-I \in \text{Compact}$ to $F^2 = I$ is the important thing, if we think of the relative theory of graded $C_n$-modules modulo graded $C_{n+1}$-modules.

So we seem to be able to conclude that there are interesting (topologically) $F$ which do not come from a finite dimensional graded $C_1$-bundle.

The relative theory of graded $C_n$-bundles modulo graded $C_{n+1}$-bundles over $X$ seems to be described by maps of $X$ to the space $\mathcal{F}_n$ consisting of odd endomorphisms $F$ of $C_n \otimes \mathcal{H}$ such that $F^2-I$ is compact.

The only interesting question I see is whether one could obtain something for the differential forms. We can do something interesting for $n=0$, whence a graded $C_0$-bundle is a pair $E^+, E^-$ and our $F$ is an odd endomorphism $L$ of $E$.