Editing Atiyah–Singer index theorem (section)

=== Relation to Grothendieck–Riemann–Roch ===
The [[Grothendieck–Riemann–Roch theorem|Grothendieck–Riemann–Roch]] theorem was one of the main motivations behind the index theorem because the index theorem is the counterpart of this theorem in the setting of real manifolds. Now, if there's a map <math>f:X\to Y</math> of compact stably almost complex manifolds, then there is a commutative diagram<ref>{{Cite web|title=algebraic topology - How to understand the Todd class?|url=https://math.stackexchange.com/questions/41182/how-to-understand-the-todd-class|access-date=2021-02-05|website=Mathematics Stack Exchange}}</ref>
:<math>
\begin{array}{ccc}
 & & & \\
 & K(X) & \xrightarrow[]{\text{Td}(X)\cdot\text{ch}} & H(X;\mathbb{Q}) & \\
 & f_* \Bigg\downarrow && \Bigg\downarrow f_*\\
 & K(Y) & \xrightarrow[\text{Td}(Y)\cdot\text{ch}]{} & H(Y;\mathbb{Q})  & \\
 & & & \\
\end{array}
</math>
if <math>Y = *</math> is a point, then we recover the statement above. Here <math>K(X)</math> is the [[Grothendieck group]] of complex vector bundles. This commutative diagram is formally very similar to the GRR theorem because the cohomology groups on the right are replaced by the [[Chow ring]] of a smooth variety, and the Grothendieck group on the left is given by the Grothendieck group of algebraic vector bundles.