# The principal symbol as an element in the K-theory

This line

The symbol may naturally be thought of as an element in the K-theory of X

appears in the nLab page on principal symbols for differential operators. What does this mean? Are they talking about K-theory or K-homology? How does one produce a class from the symbol of the operator?

• As with every statement on nLab, my first guess would be either a) article author have silently redefined any number of words involved in it; or b) statement is simply false. Aug 14 at 6:09

It's a bit easier to see this using a slightly non-standard definition of topological K-theory. Given a locally compact Hausdorff space $$X$$, let $$\bf{E}$$ be a complex of vector bundles, i.e. a sequence

$$0 \to E_0 \xrightarrow{\alpha_0} E_1 \xrightarrow{\alpha_1} \ldots \xrightarrow{\alpha_{n-1}} E_n \to 0$$

where the $$\alpha_i$$'s are bundle maps and $$\alpha_{i+1} \circ \alpha_i = 0$$. The support of $$\bf{E}$$ is by definition the set of all $$x \in X$$ such that the fiber of $$\bf{E}$$ over $$x$$ is not exact. A homotopy between complexes $$\bf{E}$$ and $$\bf{F}$$ is a complex over $$X \times [0,1]$$ whose restriction to $$X \times 0$$ is isomorphic to $$\bf{E}$$ and whose restriction to $$X \times 1$$ is isomorphic to $$\bf{F}$$. Finally, declare that two compactly supported complexes of vector bundles over $$X$$ are equivalent if there is a compactly supported homotopy between them, and let $$C(X)$$ denote the set of all equivalence classes.

$$C(X)$$ is an abelian group under Whitney sum of complexes, and it has a subgroup $$C_0(X)$$ consisting of complexes with empty support, i.e. the complex is exact over every point in $$X$$.

Proposition: $$K(X) \cong C(X) / C_0(X)$$

There is a proof in Atiyah's book on K-theory, for example.

With that in hand, let $$D$$ be an elliptic operator mapping smooth sections of a vector bundle $$E$$ to smooth sections of a vector bundle $$F$$ over the same compact base manifold $$M$$. Let $$\pi \colon TM \to M$$ denote the tangent bundle of $$M$$. For each $$x \in M$$ and $$V \in T_x M$$, the symbol of $$D$$ is a linear map

$$\sigma(x, V) \colon E_x \to F_x$$

which varies smoothly in $$x$$ and $$V$$. Thus it defines a complex of vector bundles over TM:

$$0 \to \pi^* E \xrightarrow{\sigma} \pi^* F \to 0$$

The condition that $$D$$ is elliptic means that $$\sigma$$ is invertible - and hence the complex above is exact - outside of the zero section of $$TM$$, which is compact since $$M$$ is compact. So by the definition of K-theory above we get a class $$[\sigma] \in K(TM)$$.

• The ncatlab page says $[\sigma]\in K(M)$ (not $TM$). To get this, one has to apply the Thom isomorphism in $K$-theory. But this only works if $TM$ is $K$-orientable (Spin$^c$ in the complex case) and even then, it is only well-defined if there is just one $K$-orientation. Aug 13 at 8:15
• @SebastianGoette Good catch - I missed that point. The nLab page points to Freed's notes on Dirac operators, and so in that context there is probably a fixed choice of Spin or Spin$^c$ structure lying around - but the nLab page itself should probably be corrected. Aug 13 at 12:09