1. Atiyah-Singer index theorem
Write $ M $ for a closed oriented Riemannian manifold of dimension $ n $. Let $ E_1, E_2 $ be complex vector bundles over $ M $ of rank $ r_1, r_2 $, respectively. A map of sections
\[F \colon \Gamma(E_1) \to \Gamma(E_2)\]which can locally be written as a sum of partial derivatives
\[F \phi(x) = \sum_{|\alpha| \leq m} f^{\alpha}(x) (\partial^\alpha \phi)(x),\]where
\[\partial^\alpha = \frac{\partial^{|\alpha|}} {\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}}\]is called a differential operator of order at most $ m $. Here, $ \alpha = (\alpha_1, …, \alpha_n) $ is a multi-index in $ \mathbb{N}_0^n $ with $ \lvert\alpha\rvert = \sum \alpha_i $, we let $ U \subseteq M $ be any open subset where $ E_1, E_2 $ are trivial, and
\[f^\alpha \in C^{\infty}(U, \text{Hom}(\mathbb{C}^{r_1}, \mathbb{C}^{r_2})).\]The highest-order part of $ F $ defines its principal symbol. If the symbol is invertible away from the zero section of the cotangent bundle, then $ F $ is called elliptic. On a closed manifold, an elliptic operator extends to a Fredholm operator between suitable Sobolev spaces, so its index
\[\text{Ind}(F) = \dim \ker F - \dim \text{coker} F\]is finite. Its symbol defines a compactly supported K-theory class
\[[\sigma(F)]\in K_c^0(T^\ast M).\]The Atiyah-Singer index theorem states
\[\text{Ind}(F)=\operatorname{ind}_{\mathrm{top}}([\sigma(F)]),\]where the topological index is constructed from the symbol class by the Thom isomorphism and K-theoretic pushforward. Under the Chern character, this topological index can be written using $ \operatorname{ch}([\sigma(F)]) $ and the Todd class of the complexified tangent bundle. The essential point is that the topological side depends on the symbol of $ F $, not merely on the bundles $ E_1 $ and $ E_2 $.
The following Theorems are corollaries of the Atiyah-Singer index theorem:
- Chern-Gauss-Bonnet theorem;
- Hirzebruch–Riemann–Roch theorem;
- Hirzebruch signature theorem.
We focus on the first, which states that the Euler characteristic of $ M $ equals the Euler class of the tangent bundle $ e(TM) $ integrated over $ M $:
\[\chi(M) = \int_M e(TM)\]In particular, we will show that the Euler characteristic is the index of a Dirac operator. We will complexify the de Rham complex and check ellipticity by computing the operator's symbol.
2. The de Rham complex
Write $ \Omega^k(M) = \Gamma(\bigwedge^k T^{\ast}M) $ for the (real) vector space of differential $ k $-forms of $ M $. In what follows, we omit $ M $ simply writing $ \Omega^k $. We further set $ \Omega^{\ast} = \bigoplus_k \Omega^k $ to be the associated graded vector space. It has a decomposition into components with $ k $ even or odd:
\[\Omega^{\ast} = \Omega^{\text{ev}} \oplus \Omega^{\text{od}}.\]Recall that there is a natural map $ d \colon \Omega^k \to \Omega^{k+1} $ such that $ d \circ d = 0 $ called the differential. We see that the image of $ d $ is a subobject of its kernel, and we define
\[H_{\text{dR}}^k(M) = \frac{\ker d \colon \Omega^k \to \Omega^{k+1}}{\text{im} d \colon \Omega^{k-1} \to \Omega^k}\]to be the $ k $th de Rham cohomology group of $ M $.
We can dually define the codifferential $ \delta \colon \Omega^{k} \to \Omega^{k-1} $ using the Hodge star operator $ \star \colon \Omega^{k} \to \Omega^{n-k} $. The Riemannian metric and orientation uniquely characterize this operator by
\[\omega\wedge\star\eta=\langle\omega,\eta\rangle\,d\operatorname{vol}_M.\]The codifferential on $ \Omega^k $ is the formal adjoint of $ d $ and, with our sign convention, is given by $ \delta = (-1)^k \star^{-1} d \star $.
The Hodge star operator yields a natural inner product structure on $ \Omega^{\ast} $. For $ \omega, \eta \in \Omega^k $, we set
\[\left< \omega, \eta \right> = \int_M \omega \wedge \star \eta,\]and then we define differential forms of unequal degrees to be orthogonal. For a differential operator $ T \colon \Omega^{\ast} \to \Omega^{\ast} $, its formal adjoint is an operator $ T^{\ast} $ such that $ \left< T \omega, \eta \right> = \left< \omega, T^{\ast} \eta \right> $. Analytic adjoints will be taken after completing the spaces of forms in suitable Sobolev norms.
Finally, we define the Euler characteristic of $ M $ as the alternating sum
\[\chi(M) = \sum_{k=0}^n (-1)^k \dim H_{\text{dR}}^k(M).\]It is a topological invariant of our manifold.
3. Main Proof
We call $ \Delta = (d + \delta)^2 $ the Laplace operator. It abstracts the traditional definition of the divergence of the gradient. We set
\[\mathcal{H}^k(M) = \ker \{\Delta \colon \Omega^k \to \Omega^k\}\]and call elements of $ \mathcal{H}^k(M) $ harmonic $ k $-forms. They allow us to compute the de Rham cohomology groups.
Hodge isomorphism. Let $ M $ be a closed oriented Riemannian manifold. Then there exists a canonical isomorphism $ \mathcal{H}^k(M) \cong H_{\text{dR}}^k(M) $.
Associated to the Laplace operator is the de Rham-Dirac operator $ D = d + \delta $. It is formally self-adjoint and satisfies $ D^2 = \Delta $. From now on we complexify the spaces of differential forms, but suppress this from the notation. Complexification does not change the dimensions of the harmonic spaces.
Lemma. We have $ \ker D\cap\Omega^k = \mathcal{H}^k(M) $.
Proof. Since $ D $ is formally self-adjoint,
\[\|D\omega\|^2 = \left< D^2\omega,\omega \right> = \left< \Delta\omega,\omega \right>.\]Thus $ D\omega=0 $ implies $ \Delta\omega=0 $, and if $ \Delta\omega=0 $ then the displayed norm vanishes, so $ D\omega=0 $. $ \blacksquare $
Using decomposition of $ \Omega^{\ast} $ into even and odd components, we can restrict $ D $ to get adjoint operators
\[\begin{aligned} D^{\text{ev}} & \colon \Omega^{\text{ev}} \to \Omega^{\text{od}}, \\ D^{\text{od}} & \colon \Omega^{\text{od}} \to \Omega^{\text{ev}}. \end{aligned}\]We verify that these operators are elliptic. For a nonzero covector $ \xi\in T_x^\ast M $, the principal symbol of $ D $, up to a harmless nonzero scalar depending on convention, is Clifford multiplication
\[c(\xi)\omega=\xi\wedge\omega-\iota_{\xi^\sharp}\omega.\]The identity
\[c(\xi)^2=-\|\xi\|^2\operatorname{Id}\]shows that the symbol is invertible for $ \xi\neq0 $. Since Clifford multiplication reverses parity, the symbol of $ D^{\text{ev}} $ is an isomorphism from even to odd forms away from the zero section. Hence $ D^{\text{ev}} $ is elliptic.
After Sobolev completion, for example
\[D^{\text{ev}}\colon H^1(\Omega^{\text{ev}})\longrightarrow L^2(\Omega^{\text{od}}),\]elliptic regularity makes $ D^{\text{ev}} $ Fredholm with closed range. Its formal adjoint differential operator is $D^{\text{od}}$. The $L^2$-orthogonal complement of the range consists of weak solutions of $D^{\text{od}}\eta=0$, and elliptic regularity identifies these with the smooth kernel of $D^{\text{od}}$ (equivalently, with the kernel of its $H^1\to L^2$ realization). Therefore,
\[\text{coker}(D^{\text{ev}}) \cong \ker(D^{\text{od}})\]and all of these spaces are finite dimensional. Combining this with the Hodge isomorphism gives
\[\begin{aligned} \text{Ind}(D^{\text{ev}}) & = \dim \ker D^{\text{ev}} - \dim \ker D^{\text{od}} \\ & = \sum_{k \text{ even}} \dim \mathcal{H}^k(M) - \sum_{k \text{ odd}} \dim \mathcal{H}^k(M) \\ & = \sum_{k=0}^{n} (-1)^k \dim H_{\text{dR}}^k(M) \\ & = \chi(M). \end{aligned}\]Theorem. We have the equality $ \chi(M) = \text{Ind}(D^{\text{ev}}) $.
4. References
- Michael Atiyah and Isadore Singer, The index of elliptic operators on compact manifolds, Bulletin of The American Mathematical Society 69 (1963), no. 3, 422–433.
- Daniel Freed, The Atiyah-Singer index theorem, Bulletin of The American Mathematical Society 58 (2021), no. 4, 517–566.
- Frank Warner, Foundations of differentiable manifolds and Lie groups, Springer, 1983.