The Atiyah-Singer Index formula (INCOMPLETE)

[Let , be vector bundles of rank and respectively over a manifold of dimension . A a rankdifferential operator is a map satisfying the following: About each point choose an open set such that and are trivial. and may be expressed locally as , . We say that is a differential operator of order if locally where A is a matrix with coefficients of the form:

and .

Exercise 13.8
  • The Laplace operator. acting on complex functions is a differential operator of degree .
  • The exterior derivative. is a differential operator of degree (exercise).]

Suppose that D is a differential operator of degree . Now for each 1-form we define called the symbol of . This measures the top-order behaviour of the operator and may be described locally:

We form a new matrix from above by forgetting all differentials of order , now write the 1-form in local coordinates . Replace each differential in the matrix with . This matrix gives a linear map and globally we have a bundle homomorphism which is called the symbol.

Exercise 13.9

Consider the cotangent bundle , we may pull back the bundles to obtain . Rearrange the above to show that the symbol can be expressed as a bundle morphism .

Definition 13.4

is called an Elliptic operator if is a bundle isomorphism for all .


Exercise 13.10

For the Exterior derivative we may check that .


By computing the rank of we see that this can't be an isomorphism in all but the most trivial cases, therefore is not an elliptic operator. Define by

. Then is the formal adjoint of d. and , and is an elliptic operator (this is seen by computing the symbol: ). Note that , restricting the operator in this way , .

It turns out that an elliptic complex is always a Fredholm operator ( considered as trivial bundles over ) and we see in this case (exercise).

Elliptic operators are invertible up to lower order operators. Using the compact Rellich lemma (which states that is a compact embedding) shows that

is Fredholm. Index(D) depends only on the homotopy class of the map given and can provide information about the topology of .

Consider . Subtracting to get a class .

We may embed . Suppose that was trivial ().

Now there exists an operator on the trivial bundle over . Whose symbol over , gives the Bott class . We want to form .