## Examples of Atiyah-Singer Index Theorem

Let’s begin with some notations.

Let be an elliptic differential operator, where and are vector bundles over a closed oriented manifold . Suppose and have smooth inner product structures. Let and be the disk and sphere bundle of the cotangent bundle, respectively. Let be the projection. Let be the associated symbol class. Let be its Chern character. Let denote the pullback of the Todd class . Let be the fundamental class.

**Definition. **The topological index of is defined to be .

**Atiyah-Singer Index Theorem. ***, where is the analytical index.*

**Example. ***[Point Case] *Let be a point. Then and are finite dimensional vector spaces. Any non-trivial differential operator is a linear map between them, and hence of order 0. Thus, . Note that is empty set. Recall the definition of Chern character, then we have and . Since is empty set, then and hence . Recall the definition of Todd class, we then have . Therefore, by the definition of topological index, we have . By Atiyah-Singer, .

**Example** *[ Case] *Let , , Then is a first order elliptic operator. We now claim that . We give four different proofs.

*Proof 1.* We compute directly. Note that and that via .

*Proof 2.* Since , we have

*Proof 3.* , since is self-adjoint.

*Proof 4*. We will show that the topological index vanishes whenever is odd. See next example.

**Example** *[Odd Dimensional Case, Theorem 13.12 in Lawson-Michaelson]*

We will show that the topological index of any elliptic differential operator vanishes whenever is odd.

We want to show that , where is an elliptic differential operator of order . Consider the diffeomorphic involution given by . Since

,

it suffices to show . In fact, , since is homotopic to via .

Next, we need to introduce the Thom Isomorphism to talk about the de Rham operator.

Let be an oriented -vector bundle over , with inner product on each fiber. We now give the notion of Thom class and Thom space.

**Definition. ** is a Thom Class of if it restricts to a generator of on each fiber. The quotient is called the Thom space of , and denoted by .

**Thom Isomorphism Theorem. ***The composition is an isomorphism.*

We denote the composition by and denote its inverse by .

**Remark.** has the following two other interpretations.

(1). *Integration over the fiber.* , where is a bundle over .

Let be given and choose and . Associated to these data is a form , defined as follows. Given and a basis , choose lifts such that , for each , and define

Now is defined by

Integrating over the fibers will give the second formulation of the topological index, which is the next theorem. The factor compensates for the difference between the orientation on induced by the one on , and the canonical orientation on inherited from its almost complex structure.

(2) . Also, we can use the second interpretation of to give that formulation.

is the composition (Poincare duality) (Poincare-Lefschetz duality),

Then, we compute

.

**Theorem.** .

We will then give the third formulation of the topological index. To do this we need the notion of Euler class.

**Definition. **The Euler class of an oriented -bundle over , denoted by $latexe(E)$, is the image of the Thom class under the following isomorphism: . We may denote the composition by .

**Theorem.** *[Gysin Sequence] *To any bundle as above there is associated an exact sequence of the form

**Definition. **The Euler characteristic of is defined to be

* *

From now on, we assume .

We want to analyze to give the third formulation of the topological index. For details please see Lawson-Michaelson, P258, Theorem 13.13.

Since – and are inverse to each other, we have . Applying to both sides, we then get Thus we can write , if .

**Theorem. **, if is not zero.

Now, we are trying to apply this formula to the de Rham operator.

**Example.** *[de Rham operator] *Let Then we already know that is an elliptic complex, that is an elliptic operator and that . We want to use the above theorem to show that the Euler charateristic.

For a complex vector bundle , by the splitting principle, we can write as . Then . It follows that Similarly, we have

Back to our example, applying the real splitting principle to , we compute

since for . Note that , and that , then by the theorem above, we obtain

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