On the Betti and Tachibana Numbers of Compact Einstein Manifolds

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}.

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

On the Betti Numbers of Irreducible Compact Hyperkähler Manifolds of Complex Dimension Four

The study of higher dimensional hyperkähler manifolds has attracted much attention: we have [Wk], [Bg1,2,3,4], [Fj1,2], [Bv1], [Vb1,2], [Sl1,2], [HS], [Huy], [Gu3,4,5] etc. It is evident that there are only a few known examples of these manifolds and the obvious question is: can we classify them as in the case of complex dimension 4? The Riemann-Roch formula plays an important role in the surfa...

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 ...