Cohomological Index Formula

Part of the Operator Theory: Advances and Applications book series (OT, volume 183)


Apart from the Chern character of the symbol of an elliptic operator, the second main ingredient of our cohomological index formula is the Todd class Td(TM ⊗ ℂ; Г) of the complexified tangent bundle of the manifold M with a Г-action. Let us construct this class. It belongs to the product
$$ \prod\limits_{\left\langle {g_0 } \right\rangle } {H^{ev} \left( {M_{g_0 } ,\mathbb{C}} \right)} $$
of even degree cohomology groups of the fixed point submanifolds Mg0, where g0 runs over representatives of all conjugacy classes in Г. (Recall that the fixed point set of an isometry is a smooth submanifold; see Sec. 1.2.) To define this element, consider the normal bundle NMg0. Since g0 preserves the metric, its differential dg0 preserves the fibers of NMg0 and therefore induces a well-defined automorphism
$$ d_{g_0 } :NM_{g_0 } \to NM_{g_0 } $$
of this bundle. Let
$$ \wedge \left( {NM_{g_0 } } \right) = \wedge ^{even} \left( {NM_{g_0 } } \right) \oplus \wedge ^{odd} \left( {NM_{g_0 } } \right) $$
be the ℤ2-graded exterior algebra bundle of NMg0. The automorphism (9.1) extends to this bundle in a standard way. Let Ω be the curvature form of the connection induced in the bundle (9.2) by the restriction to NMg0 of the Riemannian connection on the tangent bundle TM. Following Atiyah and Singer [11], we define the Chern character localized at g0 ∈ Г,
$$ ch \wedge \left( {NM_{g_0 } \otimes \mathbb{C}} \right)\left( {g_0 } \right) \in H^{ev} \left( {M_{g_0 } ,\mathbb{C}} \right), $$
of the complexification of the ℤ2-graded bundle (9.2) as the cohomology class of the form
$$ ch \wedge \left( {NM_{g_0 } \otimes \mathbb{C}} \right)\left( {g_0 } \right) = str\left( {d_{g_0 } \exp \left( { - \frac{1} {{2\pi i}}} \right)} \right), $$
where str stands for the fiberwise supertrace of endomorphisms of a ℤ2-graded vector bundle.


Vector Bundle Conjugacy Class Tangent Bundle Cohomology Class Topological Index 
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© Birkhäuser Verlag AG 2008

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