Elliptic Theory and Noncommutative Geometry pp 99-114 | Cite as

# Cohomological Index Formula

Chapter

## Abstract

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( of even degree cohomology groups of the fixed point submanifolds of this bundle. Let be the ℤ of the complexification of the ℤ where str stands for the fiberwise supertrace of endomorphisms of a ℤ

*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$$
\prod\limits_{\left\langle {g_0 } \right\rangle } {H^{ev} \left( {M_{g_0 } ,\mathbb{C}} \right)}
$$

*M*_{g0}, where*g*0 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*NM*_{g0}. Since*g*0 preserves the metric, its differential*dg*0 preserves the fibers of*NM*_{g0}and therefore induces a well-defined automorphism$$
d_{g_0 } :NM_{g_0 } \to NM_{g_0 }
$$

(9.1)

$$
\wedge \left( {NM_{g_0 } } \right) = \wedge ^{even} \left( {NM_{g_0 } } \right) \oplus \wedge ^{odd} \left( {NM_{g_0 } } \right)
$$

(9.2)

_{2}-graded exterior algebra bundle of*NM*_{g0}. 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*NM*_{g0}of the Riemannian connection on the tangent bundle*TM*. Following Atiyah and Singer [11], we define the*Chern character localized at**g*0 ∈ Г,$$
ch \wedge \left( {NM_{g_0 } \otimes \mathbb{C}} \right)\left( {g_0 } \right) \in H^{ev} \left( {M_{g_0 } ,\mathbb{C}} \right),
$$

(9.3)

_{2}-graded bundle (9.2) as the cohomology class of the form$$
ch \wedge \left( {NM_{g_0 } \otimes \mathbb{C}} \right)\left( {g_0 } \right) = str\left( {d_{g_0 } \exp \left( { - \frac{1}
{{2\pi i}}} \right)} \right),
$$

(9.4)

_{2}-graded vector bundle.## Keywords

Vector Bundle Conjugacy Class Tangent Bundle Cohomology Class Topological Index
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Birkhäuser Verlag AG 2008