Closed manifolds with transcendentalL2-Betti numbers:

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 ...

Non-formal compact manifolds with small Betti numbers

We show that, for any k ≥ 1, there exist non-formal compact orientable (k−1)-connected n-manifolds with k-th Betti number bk = b ≥ 0 if and only if n ≥ max{4k − 1, 4k + 3− 2b}.

Amenable Covers, Volume and L-betti Numbers of Aspherical Manifolds

We provide a proof for an inequality between volume and LBetti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau’s theory of L-Betti numbers of R-simplicial complexes, and other themes of measurable group theory. Further, we prove new vanishing theorems for ...