Generalized topological index
The topological index was defined by Atiyah and Singer [see \textit{M. F. Atiyah} and \textit{I. M. Singer}, ``The index of elliptic operators. I'', Ann. Math., II. Ser. 87, 484-530 (1968; Zbl 0164.24001)] and they showed that this index satisfies three axioms: excision, multiplicativity and normalization. The normalization property introduced by Aityah and Singer is the most natural one for the purposes of the index theorem and all possible changes of this axiom do not yield substantially new results. In this paper, the authors define an index which is a generalization of the topological index of Atiyah and Singer, the new index having a different normalization property: \[ I^{\mathbb{R}^n}_{O(n)}(j!(1))= b_n,\quad\text{where } b_n\in RO(n), \] \(j!: RO(n)\to K_{O(n)}(T\mathbb{R}^n)\) is the Gysin homomorphism induced by the inclusion \(j\) of the origin in \(\mathbb{R}^n\), and we denote by \(RO(n)\) the ring characters for \(O(n)\). The main purpose of the authors is to introduce an operation in K-theory and to find a formula for the new index in terms of this operation. The existence and uniqueness of the generalized topological index is also proved.
scientific article; zbMATH DE number 1121158
scientific article