Generalized Topological Index

Abstract

This paper deals with an index which is different in general from the topological index defined by Atiyah and Singer because we loosen the normalization property. The intrinsic relation of this new index with operations in K-theory is explained. It is also shown that if we change the normalization axiom, the corresponding index is well-defined and may be expressed in terms of the topological index. Mathematics Subject Classifications (1991): Primary 19L47; Secondary 19K56. The purpose of this paper is to define an index which is a generalization of the topological equivariant index introduced by Atiyah and Singer. The new index is allowed to have a different normalization condition. The main results of the paper are presented in Theorems 4.2 and 4.3. It turns out that our modification of the normalization axiom cannot be completely arbitrary. There are strong relations imposed on the set of parameters involved. In the third section of the paper we find all those relations. We prove that under those restrictions the generalized topological index exists and, moreover, it is unique. The final formula for the new index is written in terms of an operation in K-theory. The normalization proposed by Atiyah and Singer is the most natural one for the index theorem purposes and our results imply the following statements. First, the normalization axiom for the index is not redundant and it does not follow from the other axioms. Second, all possible changes of the normalization property do not have substantially new features because of the close relations to the classical theory which we exhibit. The authors want to thank Victor Nistor for the statement of the problem. 2. Equivariant K-Theory and Topological Index