Geometric finiteness in large families in dimension \(3\). (Q5950610)

The author provides explicit bounds for the number of distinct diffeomeorphism types in families of closed Riemannian 3-manifolds satisfying some geometric conditions. More precisely, let \({\mathcal C}_0\) be the family of all the closed Riemannian 3-manifolds which admit a maximal \(\varepsilon\)-net \(Q\) (\(\varepsilon = i(M)/12\), \(i(M)\) being the injectivity radius) of cardinality at most \(A\) and twist (defined explicitly in the paper in terms of the maximal number of points equidistant from any 4 distinct points of the net) at most \(B\). The author shows that the number of distinct diffeomeorphism types of members of \({\mathcal C}_0\) is at most \(\exp (n\log n)\), where \(n = 4A^4B\). The same bound of the number of distinct diffeomorphism types is obtained for any family \(\mathcal C\) of closed 3-manifolds which can be covered minimally by at most \(A\) distance-like functions in tight and general position (all the terms being defined by a number of explicit technical conditions in the paper) with twist at most \(B\).