Generalized topological index
type: Item
instance of: scholarly article
title: Generalized topological index
zbMATH Open document ID: 0889.19004
DOI: 10.1023/A:1007767600747
author: Philip A. Foth
author: D. E. Tamarkin
published in: \(K\)-Theory
publication date: 1998-06-23T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 19L47
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 19K56
zbMATH DE Number: 1121158
zbMATH Keywords: topological equivariant K-theory
zbMATH Keywords: topological index
reviewed by: Cristian N. Costinescu
MaRDI profile type: Publication
full work available at URL: a:1007767600747
OpenAlex ID: W2065365878
Recommended article: Q5427948
Recommended article: Q4734484
Recommended article: On invariant index formulas for spectral boundary value problems
Recommended article: Le théorème de l'indice relatif
Recommended article: Index theory with applications to mathematics and physics
Recommended article: Q3490668
Recommended article: Q3172162
Recommended article: Michael Atiyah’s work in algebraic topology
Recommended article: Q3675210
Recommended article: The index of discontinuous vector fields
description: scientific article; zbMATH DE number 1121158