|analytic integration||cohomological integration|
|measurable space||Poincaré duality|
|measure||orientation in generalized cohomology|
|volume form||(virtual) fundamental class|
|Riemann/Lebesgue integration of differential forms||push-forward in generalized cohomology/in differential cohomology|
The topological index of topological K-theory on is the composite
One can prove that this is independent of all the occurring choices. In particular it does not depend on the specific choice of embedding of the manifold into to the Euclidean space. The topological index function is uniquely fixed by two properties (this is the content of the Atiyah-Singer index theorem):
For a point we have .
Index functions commute with the maps .
From this one defines the topological index of an elliptic operator . The principal symbol of the operator defines a homogenous length-one chain complex of bundles on exact outside the null section. Elements of this kind are precisely cycles for the compactly supported K-theory of hence an elliptic operator has a topological index only depending on its principal symbol.
On the other hand, analysis associates to its analytical index that is
on one (hence all) Sobolev space that is defined on.
Notice here the reverse functoriality: for the base space is contravariant while for the total spaces of the tangent bundles it is covariant. This uses the Thom mapping: if is a compact manifold and a real vector bundle over there is a natural map
One of the most important results of K-theory, namely Bott periodicity, can be seen as the statement of the fact that this map is an isomorphism. Now apply this construction to the normal bundle of in to get
and (looking at as a tubular neighbourhood of in ) compose it with the natural map
to get .
The topological index is defined to be