Abstract

The simplicial volume is a non-negative real valued homotopy invariant of closed connected
manifolds measuring how efficient the fundamental class can be represented by real singular
cycles. The problem of determining whether the simplicial volume of a given manifold is
non-zero has been a challenge. It is known that the simplicial volume of negatively curved
manifolds is positive [15]. Losing the negative bound on the sectional curvature, it has
been shown that locally symmetric spaces of non-compact type have positive simplicial
volume [21]. In their 2018 paper, C.Connell and S.Wang showed that the simplicial volume
Xn \
of n-manifolds with non-positive sectional curvature and negative 4 ` 1 -Ricci curvature
have positive simplicial volume, which confirms the Gromov’s conjecture in special cases. The conjecture states that the simplicial volume of manifolds with non-positive sectional curvature and negative Ricci-curvature is positive. In this master thesis we will introduce required notions and preliminaries and present detailed proofs of the results mentioned above.