Analytic \(K\)-theory on manifolds with corners. (English) Zbl 0761.55002

Melrose, Richard B.; Piazza, Paolo

Adv. Math. 92, No. 1, 1-26 (1992).

M. F. Atiyah introduced [“Global theory of elliptic operators”, Proc. Int. Conf. Funct. Anal. Rel. Topics, Tokyo 1969, 21–30 (1970; Zbl 0193.43601)] an analytic definition of the \(K\)-homology groups \(K_ *(X)\), where \(X\) denotes a compact manifold; cycles are “abstract elliptic operators” — i.e.: bounded linear operators between Hilbert spaces satisfying additional conditions derived from the properties of elliptic (pseudo-) differential operators. The homology class depends only on the principal symbol and only on the \(K\)-cohomology class it represents on the cotangent bundle; the resulting map: \(K^ i(T^*X)\to K_ i(X)\) is an isomorphism, realizing Poincaré duality. The analogue of the previous map for the absolute \(K\)-homology is the isomorphism: \[ K^ i(T^*X,T^*_{\partial X}X)\to K_i(X), \tag{1} \] which is considered as a “quantization map” through pseudodifferential operators which are trivial.
In this paper the authors prove that the quantization map for totally characteristic pseudodifferential operators (which they call more succinctly \(b\)-pseudodifferential operators) gives an explicit realization of Poincaré duality for (1) and, using Kasparov’s results on the \(K\)-spaces, for relative \(K\)-homology of a compact manifold with boundary (or corners): \[ K^ i(T^*X)\to K_ i(X,\partial X). \tag{2} \] This is an important paper, but unfortunately it is not easy to read. From the identification of (2) with Poincaré duality, it follows in particular that any elliptic differential operator on \(X\) defines a relative \(K\)-class. This fact generalizes the results of P. Baum, R. G. Douglas and M. E. Taylor, who showed [J. Differ. Geom. 30, No. 3, 761–804 (1989; Zbl 0697.58050)] that an elliptic boundary problem defines a relative \(K\)-homology class independent of the boundary conditions.

Reviewer: Cristian Costinescu (Bucureşti)

MSC:

55N15

Topological \(K\)-theory

Keywords:

\(K\)-homology groups; elliptic operators; principal symbol; Poincaré duality

Citations:

Zbl 0193.43601; Zbl 0697.58050

References:

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