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

In this paper, we present an elementary proof of the following result. Theorem A. Let Mn denote a closed Riemannian manifold with nonpositive sectional curvature and let M̃n be the universal cover of Mn with the lifted metric. Suppose that the universal cover M̃n contains no totally geodesic embedded Euclidean plane R2 (i.e., Mn is a visibility manifold ). Then Gromov’s simplicial volume ∥Mn∥ is non-zero. Consequently, Mn is non-collapsible while keeping Ricci curvature bounded from below. More precisely, if Ricg ≥ −(n − 1), then vol(Mn, g) ≥ 1 (n−1)nn!∥M n∥ > 0. Among other things, we also outline a proof for the following direct consequence of Perelman’s recent work on 3-manifolds. Theorem B. (Perelman) Let M3 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 M3 is diffeomorphic to a compact quotient of real hyperbolic space H3, i.e., M3 ≡ H3/Γ. Consequently, MinV ol(M3) ≥ 1 24 ∥M3∥ > 0. Minimal volume and simplicial norm of all other compact 3-manifolds without boundary and singular spaces will also be discussed. 2000 Mathematics Subject Classification: 53C99, 58C99