[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:
- 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.
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 .
is called an Elliptic operator if is a bundle isomorphism for all .
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
is the formal adjoint of d.
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
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 .