Homology of Stein manifolds and affine hypersurfaces

Abstract

This paper has two parts. In the first part we investigate the independence of homology groups of Stein manifolds. We apply tools from Morse theory and other topological aspects to address this problem. As smooth affine varieties are a sub-class of Stein manifolds the similar problem leads us to several results about independence of homology groups of smooth affine varieties. The second part consists of some results related to affine fibrations on an affine space. Mainly, we analyze the topology of the general fiber of a given polynomial map under certain hypothesis such as triviality of rational homology, Zarisky locally triviality, isolated singularity etc.