Online Talk

   
 If a finite CW-complex $X$ admits a free  $(\mathbb{Z}_2)^r$-action, a conjecture by G.~Carlsson says that the sum of mod-$2$ Betti numbers $X$ should be at least $2^r$. In this talk, we will discuss this conjecture in the cases when $\dim(X)\leq 3$ or the orbit space is a small cover. In particular, in these two cases we can determine all possible patterns of the homology groups of $X$ when the lower bound $2^r$ is reached.  These results motivate us to study the sum of bigraded Betti numbers of a simplicial complex $K$ with $m$ vertices, which is known to have a universal lower bound that depends only on $m$ and the dimension of $K$. We will show how to classify all the simplicial complexes with $m$ vertices that have the minimal or the maximal sum of bigraded Betti numbers, respectively.