Study on the Singer conjecture and the Hopf conjecture

Abstract

The primary aim of this thesis is to survey some interesting results about the $L^2$-Betti numbers and its applications for our later research. Among other things, in this thesis we want to focus our attention to the Singer conjecture and some related topics around the conjecture. The Singer conjecture says that if $it{M}$ is a closed Riemannian manifold with negative sectional curvature, then the $it{p}$-th $L^2$-Betti number vanishes unless $it{p}$ is half the dimension of $it{M}$.