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

Abstract

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 $\Vert M^n\Vert$ 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 $vol(M^n,g)\ge \frac{1}{(n-1{)}^{n}n!}\Vert M^n\Vert >0.TheoremB.(Perelman)Let$M^3$beacloseda-spherical3-manifold($K(π, 1)$-space)withthefundamentalgroup$Γ$.Supposethat$Γ$containsnosubgroupsisomorphicto$\mathbb{Z}\oplus \mathbb{Z}$.Then$M^3$isdiffeomorphictoacompactquotientofrealhyperbolicspace$\mathbb{H}^3$,i.e.,$M^3 \equiv \mathbb{H}^3/Γ$.Consequently,$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.

arXiv commentIn this updated version, we were able to extend our results on smooth visibility manifolds to possibly singular visibility spaces by using a result of Martin Bridson

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