Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds

Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\tilde M^n$ contains no totally geodesic embedded Euclidean plane $\mathbb{R}^2$ (i.e., $M^n$ is a visibility manifold). Then Gromov's simplicial volume $\| M^n \|$ is non-zero. Consequently, $M^n$ is non-collapsible while keeping Ricci curvature bounded from below. More precisely, if $Ric_g \ge -(n-1)$, then $vol(M^n, g) \ge \frac{1}{(n-1)^n n!} \| M^n \| > 0. Theorem B. (Perelman) Let $M^3$ be a closed a-spherical 3-manifold ($K(\pi, 1)$-space) with the fundamental group $\Gamma$. Suppose that $\Gamma$ contains no subgroups isomorphic to $\mathbb{Z}\oplus \mathbb{Z}$. Then $M^3$ is diffeomorphic to a compact quotient of real hyperbolic space $\mathbb{H}^3$, i.e., $M^3 \equiv \mathbb{H}^3/\Gamma$. Consequently, $MinVol(M^3) \ge {1/24}\| M^3 \| > 0$. Minimal volume and simplicial norm of all other compact 3-manifolds without boundary and {\it singular} spaces will also be discussed.