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K homology and index theory. (English) Zbl 0532.55004
Operator algebras and applications, Proc. Symp. Pure Math. 38, Part 1, Kingston/Ont. 1980, 117-173 (1982).
[For the entire collection see Zbl 0488.00012.]
In this paper the authors present the foundations of their program to refine the Atiyah-Singer index theorem. One may view the index theorem as providing a precise way of matching up analytic and topological data. At the most basic level it gives a topological formula for the analytic index of an elliptic differential operator. The present work intends to make this matching up process more informative and precise.
It was observed by M. F. Atiyah [Proc. int. Conf. funct. Anal. relat. Topics, Tokyo 1969, 21-30 (1970; Zbl 0193.436)] that while K- cohomology played a crucial role in the proof of the index theorem, it was really the dual K-homology theory which was needed for its proper formulation. The present paper provides both topological and analytic definitions of K-homology and defines an explicit natural equivalence between the theories. The groundwork for the definitions was done by Atiyah and Kasparov for \(K_ 0\), while the roots of the \(K_ 1\) definitions are to be found in the earlier work of L. G. Brown, R. G. Douglas and P. A. Fillmore [Ann. Math., II. Ser. 105, 265-324 (1977; Zbl 0376.46036)].
The topological K-homology groups, \(K_*^{top}(X)\), are defined via equivalence classes of ”cycles”. These are triples (M,E,f), M a closed, smooth Spin\({}^ c\) manifold, E a complex vector bundle on M, and f: \(M\to X\) a map. If M is even dimensional one gets an element of \(K_ 0(X)\), and if M is odd dimensional an element of \(K_ 1(X)\). The most important aspects of the equivalence relation between the cycles are a relatively straightforward notion of bordism and ”vector bundle modification”, which has the effect of forcing the Thom isomorphism theorem to hold. The analytic K-homology groups, \(K_*^{an}(X)\), are defined to be classes of generalized elliptic operators for \(K_ 0(X)\) and classes of \(C^*\)-algebra extensions for \(K_ 1(X)\). Let \(D_ E\) denote the Dirac operator on M with coefficients in E. The map between the topological and analytic groups is defined by sending a triple (M,E,f) to \(f_*(D_ E)\) if M is even dimensional. If M is odd- dimensional then the triple is sent to the image of the Toeplitz extension obtained by compressing multiplication operators to the positive eigenspace of \(D_ E\). The key technical fact needed is that the map preserves the equivalence relations. This is discussed here but the complete proofs are to appear in later papers.
This paper is very nicely written and carefully presents several notions which are very useful in index theory, such as \(Spin^ c\) structures and their associated Dirac operators. Possible applications of this work to index theory on singular spaces such as singular algebraic varieties and PL-manifolds, are discussed at length. It is worthwhile reading for the authors’ work itself and also for the insights into index theory in general.
Reviewer: J.Kaminker

55N15 Topological \(K\)-theory
58J20 Index theory and related fixed-point theorems on manifolds
46L05 General theory of \(C^*\)-algebras
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
35J30 Higher-order elliptic equations
14C40 Riemann-Roch theorems
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry