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](https://ncatlab.org/nlab/show/symbol+of+a+differential+operator) 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?



The principal symbol as an element in the K-theory
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 $[\sigma] \in K(TM)$.


