In 1957, Grothendieck introduces the K-theory in algebraic geometry. Later, Atiyah and Singer apply its real counterpart, the topological K-theory, to give a proof of the famous index theorem. In 1990's, in Arakelov geometry and arithmetic algebraic geometry, the K-theory is extended to the arithmetic K-theory. In this century, motivated by the study of the superstring theory and the quantum field theory, people extend the topological K-theory to the differential K-theory as the real analogue of the arithmetic K-theory. Naturally, people expect that a property holds in one K-theory may also holds in the other three and imply the nontrivial consequences in their respective fields.
In this talk, we will compare the lambda-ring property, Riemann-Roch property and the Lefschetz formula in four K-theories. As a consequence of the Lefschetz formula in differential K-theory, we obtain a localization formula of eta invariants, which is a purely geometric formula but cannot be proved geometrically until now. This is a joint work with Xiaonan Ma recently.