Zero-and One-Dimensional Cases via Multivalued Perturbation

Abstract

In Chaps. 7, 8, 9 and 10, we discussed smooth correspondence and defined virtual fundamental chains based on de Rham theory and CF-perturbations. In this chapter, we discuss another method based on multivalued perturbations. Here we restrict ourselves to the case when the dimension of K-spaces of our interest is 1, 0 or negative, and define a virtual fundamental chain over ${\mathbb Q}$ in the 0-dimensional case. In spite of this restriction, the argument of this chapter is enough for the purpose, for example, to prove all the results stated in [FOn2]. We recall that in [FOn2] we originally used a triangulation of the zero set of a multisection to define a virtual fundamental chain. In this chapter we present a different way from [FOn2].