On the total curvature and Betti numbers of complex projective manifolds

Betti numbers of random manifolds

We study mathematical expectations of Betti numbers of configuration spaces of planar linkages, viewing the lengths of the bars of the linkage as random variables. Our main result gives an explicit asymptotic formulae for these mathematical expectations for two distinct probability measures describing the statistics of the length vectors when the number of links tends to infinity. In the proof ...

-betti Numbers of Aspherical Manifolds

Tight Combinatorial Manifolds and Graded Betti Numbers

In this paper, we study the conjecture of Kühnel and Lutz, who state that a combinatorial triangulation of the product of two spheres S×S with j ≥ i is tight if and only if it has exactly i+2j+4 vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when ...

On the virtual Betti numbers of arithmetic hyperbolic 3--manifolds

An interesting feature of our argument is that although it uses arithmetic in an essential way, it is largely geometric; in particular there is no use of Borel’s theorem [1]. This makes Theorem 1.1 strictly stronger than [1] in this setting, since no congruence assumptions are made. We recall that a group is said to be large if it has a subgroup of finite index which maps onto a free group of r...

Construction of Manifolds of Positive Ricci Curvature with Big Volume and Large Betti Numbers

It is shown that a connected sum of an arbitrary number of complex projective planes carries a metric of positive Ricci curvature with diameter one and, in contrast with the earlier examples of Sha–Yang and Anderson, with volume bounded away from zero. The key step is to construct complete metrics of positive Ricci curvature on the punctured complex projective plane, which have uniform euclidea...