A topological version of the argument principle and Rouche's theorem

In this paper, the topological nature of the well-known in multidimensional complex analysis generalization of the classic argument principle is discussed. The topological approach offered here, ensures some topological results on the structure of pole and zero sets of holomorphic maps of bounded domains in complex manifolds. Some connections with integral representations of holomorphic functions are studied and a geometric interpretation of the Martinelli–Bochner complex-valued differential-form realization is given.