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 $\mathrm{\parallel }{M}^n\mathrm{\parallel }$ is non-zero. Consequently, $M^n$ is non-collapsible while keeping Ricci curvature bounded from below. More precisely, if $Ri{c}_g\ge -(n-1)$, 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)withthefundamentalgroup$\Gamma$\mathrm{.}Supposethat$\Gamma$ 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., $M^3\equiv {\mathbb{H}}^3\mathrm{/}\mathrm{\Gamma }$. 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.