Editing Atiyah–Singer index theorem (section)

===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.