Editing Atiyah–Singer index theorem (section)

==Examples==

===Chern-Gauss-Bonnet theorem===
Suppose that <math>M</math> is a compact oriented manifold of dimension <math>n = 2r</math>. If we take <math>\Lambda^\text{even}</math> to be the sum of the even exterior powers of the cotangent bundle, and <math>\Lambda^\text{odd}</math> to be the sum of the odd powers, define <math>D = d + d^*</math>, considered as a map from <math>\Lambda^\text{even}</math> to <math>\Lambda^\text{odd}</math>.  Then the analytical index of <math>D</math> is the [[Euler characteristic]] <math>\chi (M)</math> of the [[Hodge cohomology]] of <math>M</math>, and the topological index is the integral of the [[Euler class]] over the manifold. The index formula for this operator yields the [[Chern–Gauss–Bonnet theorem]].

The concrete computation goes as follows: according to one variation of the [[splitting principle]], if <math>E</math> is a real vector bundle of dimension <math>n = 2r</math>, in order to prove assertions involving characteristic classes, we may suppose that there are complex line bundles <math>l_1,\, \ldots,\, l_r</math> such that <math>E \otimes \mathbb{C} = l_1 \oplus \overline{l_1} \oplus \dotsm l_r \oplus \overline{l_r}</math>. Therefore, we can consider the Chern roots <math>x_i (E \otimes \mathbb{C}) = c_1(l_i)</math>, <math>x_{r+i} (E \otimes \mathbb{C}) = c_1\mathord\left(\overline{l_i}\right) = -x_i(E \otimes \mathbb{C})</math>, <math>i = 1,\, \ldots,\, r</math>.

Using Chern roots as above and the standard properties of the Euler class, we have that <math display="inline">e(TM) = \prod^r_i x_i(TM \otimes \mathbb{C})</math>. As for the Chern character and the Todd class,<ref>{{citation|first= Mikio|last=Nakahara|title=Geometry, topology and physics|year=2003|isbn=0-7503-0606-8|publisher=Institute of Physics Publishing}}</ref>
:<math>\begin{align}
  \operatorname{ch}\mathord\left(\Lambda^\text{even} - \Lambda^\text{odd}\right)
    &= 1 - \operatorname{ch}(T^* M \otimes \mathbb{C}) + \operatorname{ch}\mathord\left(\Lambda^2 T^* M \otimes \mathbb{C}\right) - \ldots + (-1)^n \operatorname{ch}\mathord\left(\Lambda^n T^* M \otimes \mathbb{C}\right) \\
    &= 1 - \sum_i^n e^{-x_i}(TM \otimes \mathbb{C}) + \sum_{i<j} e^{-x_i}e^{-x_j}(TM \otimes \mathbb{C}) + \ldots + (-1)^n e^{-x_1} \dotsm e^{-x_n}(TM \otimes \mathbb{C}) \\
    &= \prod_i^n \left(1 - e^{-x_i}\right)(TM \otimes \mathbb{C}) \\[3pt]
  \operatorname{Td}(TM \otimes \mathbb{C})
    &= \prod_i^n \frac{x_i}{1 - e^{-x_i}} (TM \otimes \mathbb{C})
\end{align}</math>

Applying the index theorem, 
:<math>\chi(M) = (-1)^r \int_M \frac{\prod_{i}^n \left(1 - e^{-x_i}\right)}{\prod_i^r x_i} \prod_i^n \frac{x_i}{1 - e^{-x_i}}(TM \otimes \mathbb{C}) = (-1)^r \int_{M}(-1)^r\prod_i^r x_i(TM \otimes \mathbb{C}) = \int_M e(TM)</math>
which is the "topological" version of the Chern-Gauss-Bonnet theorem (the geometric one being obtained by applying the [[Chern-Weil homomorphism]]).

===Hirzebruch–Riemann–Roch theorem===
Take ''X'' to be a [[complex manifold]] of (complex) dimension ''n'' with a holomorphic vector bundle ''V''. We let the vector bundles ''E'' and ''F'' be the sums of the bundles of differential forms with coefficients in ''V'' of type (0, ''i'') with ''i'' even or odd, and we let the differential operator ''D'' be the sum

:<math>\overline\partial + \overline\partial^*</math>

restricted to ''E''. 

This derivation of the Hirzebruch–Riemann–Roch theorem is more natural if we use the index theorem for elliptic complexes rather than elliptic operators. We can take the complex to be

:<math>0 \rightarrow V \rightarrow V \otimes \Lambda^{0,1}T^*(X) \rightarrow V \otimes \Lambda^{0,2}T^*(X) \rightarrow \dotsm</math>

with the differential given by <math>\overline\partial</math>. Then the ''i'''th cohomology group is just the coherent cohomology group H<sup>''i''</sup>(''X'', ''V''), so the analytical index of this complex is the [[holomorphic Euler characteristic]] of ''V'':

:<math>\operatorname{index}(D) = \sum_p (-1)^p \dim H^p(X, V) = \chi(X, V)</math>

Since we are dealing with complex bundles, the computation of the topological index is simpler. Using Chern roots and doing similar computations as in the previous example, the Euler class is given by <math display="inline">e(TX) = \prod_{i}^{n}x_i(TX)</math> and

:<math>\begin{align}
  \operatorname{ch}\left(\sum_{j}^{n} (-1)^j V \otimes \Lambda^{j}\overline{T^*X}\right) &= \operatorname{ch}(V)\prod_{j}^{n}\left(1 - e^{x_j}\right)(TX) \\
  \operatorname{Td}(TX \otimes \mathbb{C}) = \operatorname{Td}(TX)\operatorname{Td}\left(\overline{TX}\right) &= \prod_i^n\frac{x_i}{1 - e^{-x_i}} \prod_j^n\frac{-x_j}{1 - e^{x_j}}(TX)
\end{align}</math>

Applying the index theorem, we obtain the [[Hirzebruch-Riemann-Roch theorem]]:
:<math>\chi(X, V)=\int _X \operatorname{ch}(V)\operatorname{Td}(TX)</math>

In fact we get a generalization of it to all complex manifolds: Hirzebruch's proof only worked for '''projective''' complex manifolds ''X''.

===Hirzebruch signature theorem===
The [[Hirzebruch signature theorem]] states that the signature of a compact oriented manifold ''X'' of dimension 4''k'' is given by the [[L genus]] of the manifold. This follows from the Atiyah–Singer index theorem applied to the following [[signature operator]].

The bundles ''E'' and ''F'' are given by the +1 and −1 eigenspaces of the operator on the bundle of differential forms of ''X'', that acts on ''k''-forms as <math>i^{k(k - 1)}</math> times the [[Hodge dual|Hodge star operator]]. The operator ''D'' is  the [[Hodge Laplacian]]

:<math>D \equiv \Delta \mathrel{:=} \left(\mathbf{d} + \mathbf{d^*}\right)^2</math>

restricted to ''E'', where '''d''' is the Cartan [[exterior derivative]] and '''d'''* is its adjoint.

The analytic index of ''D'' is the signature of the manifold ''X'', and its topological index is the L genus of ''X'', so these are equal.

===Â genus and Rochlin's theorem===
The [[Â genus]] is a rational number defined for any manifold, but is in general not an integer. Borel and Hirzebruch showed that it is integral for spin manifolds,  and an even integer if in addition the dimension is 4 mod 8. This can be deduced from the index theorem, which implies that the Â genus for spin manifolds is the index of a Dirac operator. The extra factor of 2 in dimensions 4 mod 8 comes from the fact that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so the index is even.

In dimension 4 this result implies [[Rochlin's theorem]] that the signature of a 4-dimensional spin manifold is divisible by 16: this follows because in dimension 4 the Â genus is minus one eighth of the signature.