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

In this survey paper, we shall derive the following result. Theorem A. Let M denote a closed Riemannian manifold with nonpositive sectional curvature and let M̃ be the universal cover of M with the lifted metric. Suppose that the universal cover M̃ contains no totally geodesic embedded Euclidean plane R (i.e., M is a visibility manifold ). Then Gromov’s simplicial volume ‖M‖ is non-zero. Consequently, M is non-collapsible while keeping Ricci curvature bounded from below. More precisely, if Ricg ≥ −(n− 1), then vol(M , g) ≥ 1 (n−1)nn!‖M ‖ > 0. Among other things, we also outline a proof for the following direct consequence of Perelman’s recent work on 3-manifolds. TheoremB. (Perelman) LetM be a closed a-spherical 3-manifold (K(π, 1)space) with the fundamental group Γ. Suppose that Γ contains no subgroups isomorphic to Z ⊕ Z. Then M is diffeomorphic to a compact quotient of real hyperbolic space H, i.e.,M ≡ H/Γ. Consequently, MinV ol(M) ≥ 1 24‖M ‖ > 0. Minimal volume and simplicial norm of all other compact 3-manifolds without boundary and singular spaces will also be discussed.