Editing Atiyah–Singer index theorem (section)

==Topological index==
The '''topological index''' of an elliptic differential operator <math>D</math> between smooth vector bundles <math>E</math> and <math>F</math> on an <math>n</math>-dimensional compact manifold <math>X</math> is given by

:<math>(-1)^n\operatorname{ch}(D)\operatorname{Td}(X)[X] = (-1)^n\int_X \operatorname{ch}(D)\operatorname{Td}(X)</math>

in other words the value of the top dimensional component of the mixed [[cohomology class]] <math>\operatorname{ch}(D) \operatorname{Td}(X)</math> on the [[fundamental homology class]] of the manifold <math>X</math> up to a difference of sign.
Here,

*<math>\operatorname{Td}(X)</math> is the [[Todd class]] of the complexified tangent bundle of <math>X</math>.
*<math>\operatorname{ch}(D)</math> is equal to <math>\varphi^{-1}(\operatorname{ch}(d(p^*E,p^*F, \sigma(D)))) </math>, where
**<math>\varphi: H^k(X;\mathbb{Q}) \to H^{n+k}(B(X)/S(X);\mathbb{Q})</math> is the [[Thom isomorphism]] for the sphere bundle <math>p:B(X)/S(X) \to X</math>
**<math>\operatorname{ch}:K(X)\otimes\mathbb{Q} \to H^*(X;\mathbb{Q})</math> is the [[Chern character]]
**<math>d(p^*E,p^*F,\sigma(D))</math> is the "difference element" in <math>K(B(X)/S(X))</math> associated to two vector bundles <math>p^*E</math> and <math>p^*F</math> on <math>B(X)</math> and an isomorphism <math>\sigma(D)</math> between them on the subspace <math>S(X)</math>.
**<math>\sigma(D)</math> is the symbol of <math>D</math>

In some situations, it is possible to simplify the above formula for computational purposes. In particular, if <math>X</math> is a <math>2m</math>-dimensional orientable (compact) manifold with non-zero [[Euler class]] <math>e(TX)</math>, then applying the [[Thom isomorphism]] and dividing by the Euler class,<ref>{{citation|last=Shanahan|first= P.|title=The Atiyah-Singer Index Theorem|isbn=978-0-387-08660-6 |series=Lecture Notes in Mathematics |volume=638|publisher= Springer|year= 1978|doi=10.1007/BFb0068264|citeseerx= 10.1.1.193.9222}}</ref><ref>{{citation|first1=H. Blane|last1=Lawson|author1-link=H. Blaine Lawson|first2=Marie-Louise|last2=Michelsohn|author2-link=Marie-Louise Michelsohn|title=Spin Geometry|year=1989|isbn=0-691-08542-0|publisher=Princeton University Press}}</ref> the topological index may be expressed as 
:<math>(-1)^m\int_X \frac{\operatorname{ch}(E)-\operatorname{ch}(F)}{e(TX)}\operatorname{Td}(X)</math>
where division makes sense by pulling <math>e(TX)^{-1}</math> back from the cohomology ring of the [[classifying space]] <math>BSO</math>.

One can also define the topological index using only K-theory (and this alternative definition is compatible in a certain sense with the Chern-character construction above). If ''X'' is a compact submanifold of a manifold ''Y'' then there is a pushforward (or "shriek") map from K(''TX'') to K(''TY''). The topological index of an element of 
K(''TX'') is defined to be the image of this operation with ''Y'' some Euclidean space, for which K(''TY'') can be naturally identified with the integers '''Z''' (as a consequence of Bott-periodicity). This map is independent of the embedding of ''X'' in Euclidean space. Now a differential operator as above naturally defines an element of K(''TX''), and the image in '''Z''' under this map "is" the topological index.

As usual, ''D'' is an elliptic differential operator between vector bundles ''E'' and ''F'' over a compact manifold ''X''.

The ''index problem'' is the following: compute the (analytical) index of ''D'' using only the symbol ''s'' and ''topological'' data derived from the manifold and the vector bundle. The Atiyah–Singer index theorem solves this problem, and states:

:'''The analytical index of ''D'' is equal to its topological index.'''

In spite of its formidable definition, the topological index is usually straightforward to evaluate explicitly. So this makes it possible to evaluate the analytical index. (The cokernel and kernel of an elliptic operator are in general extremely hard to evaluate individually; the index theorem shows that we can usually at least evaluate their '''difference'''.) Many important invariants of a manifold (such as the signature) can be given as the index of suitable differential operators, so the index theorem allows us to evaluate these invariants in terms of topological data.

Although the analytical index is usually hard to evaluate directly, it is at least obviously an integer. The topological index is by definition a rational number, but it is usually not at all obvious from the definition that it is also integral. So the Atiyah–Singer index theorem implies some deep integrality properties, as it implies that the topological index is integral.

The index of an elliptic differential operator obviously vanishes if the operator is self adjoint. It also vanishes if the manifold ''X'' has odd dimension, though there are '''pseudodifferential''' elliptic operators whose index does not vanish in odd dimensions.

=== 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.