Geometric finiteness theorems via controlled topology (Q1812780)

For fixed numbers \(n\in\mathbb{N}\), \(k\in\mathbb{R}\), \(D>0\), \(v>0\), \(i>0\) let \({\mathcal M}_ 1={\mathcal M}^ n_ 1(k,v,D)\) (resp. \({\mathcal M}_ 2={\mathcal M}^ n_ 2(k,i)\)) be the class of \(n\)-dimensional closed Riemannian manifolds \(M\) with sectional curvature \(K_ M\geq k\), volume \(v_ M\geq v\), diameter \(d_ M\leq D\) (resp. sectional curvature \(K_ M\geq k\), injectivity radius \(i_ M\geq i\)). The authors show that both classes contain only finitely many homeomorphism types if \(n\neq 3\). As a consequence, for \(n\geq 5\) these classes only contain finitely many diffeomorphism types as well. This result is an immediate consequence of a more general theorem stating, that for every concave function \(\rho: [0,R]\to[0,\infty)\) with \(\rho\geq Id\) and \(\rho(0)=0\) a Gromov-Hausdorff- precompact class of closed \(n\)-dimensional manifolds with the property, that every concentric \(\varepsilon\)-ball is contractible within the concentric \(\rho(\varepsilon)\)-ball with the same center \((\varepsilon\leq R)\), contains only finitely many homeomorphism types when \(n\neq 3\).