Euler characteristic of Fredholm quasicomplexes
The aim of this article is to define the class of so--called Fredholm quasicomplexes, which generalize complexes of Fredholm operators between Hilbert spaces, and to extend the notion of Euler characteristic to this setting. The author defines a quasicomplex to be a sequence of operators \(d_i\) between Hilbert spaces such that \(d_{i+1}\circ d_i\) is a compact operator for all \(i\). Then he defines the appropriate notions of chain mappings and of homotopy of chain mappings in this setting. Generalizing the fact that Fredholm operators are the ones whose image in the Calkin algebra is invertible, the author next introduces a notion of Fredholm quasicomplexes, which is stable under compact perturbations. For bounded complexes, the Fredholm property is proved to be equivalent to the identity mapping being homotopic to zero. The generalization of the Euler characteristic is obtained by proving that there are compact perturbations of the operators in a Fredholm quasicomplex which form a complex, and that any two such perturbations are homotopic.
scientific article; zbMATH DE number 5319693