We define the crosscap number of a 2-component link as the minimum of the first Betti numbers of connected, nonorientable surfaces bounding the link. We discuss some properties of the crosscap numbers of 2-component links.

A bst r ac t This paper is focused on the study of perfect discrete Morse functions on a 2-simplicial complex. These are those discrete Morse functions such that the number of critical i-simplices coincides with the i-th Betti number of the complex. In particular, we establish conditions under which a 2-complex admits a perfect discrete Morse function and conversely, we get topological properti...

We generalise to the setting of 3-graphs a combinatorial analogue of the Jordan curve theorem due to Stahl [9, 10]. More specifically, using combinatorial techniques only, we give a graph theoretic version of the theorem that the first Betti number of a surface is the largest number of closed curves that can be drawn on the surface without dividing it into two or more regions.

We study the structure of manifolds with almost nonnegative Ricci curvature. We prove a compact Riemannian manifold with bounded curvature, diameter bounded from above, and Ricci curvature bounded from below by an almost nonnegative real number such that the first Betti number having codimension two is an infranilmanifold or a finite cover is a sphere bundle over a torus. Furthermore, if we ass...

The maximum gen'us of connected of orient able surface on which G has 2-ce11 C:U.UYC;UUULl.".. to 2,M(G) where (3(G) the Betti n'umber of G.

The Betti numbers of moduli spaces of representations of a universal central extension of a surface group in the groups U(2, 1) and SU(2, 1) are calculated. The results are obtained using the identification of these moduli spaces with moduli spaces of Higgs bundles, and Morse theory, following Hitchin’s programme [14]. This requires a careful analysis of critical submanifolds which turn out to ...

We calculate the Betti numbers of moduli spaces of representations of a universal central extension of a surface group in the groups U (2; 1) and S U (2; 1). In order to obtain our results we use the identiication of this space with an appropriate mod-uli space of Higgs bundles and Morse theory, following Hitchin's programme 11]. This requires a careful analysis of critical subman-ifolds which ...

In this paper we study the monodromy action on the first Betti and de Rham nonabelian cohomology arising from a family of smooth curves. We describe sufficient conditions for the existence of a Zariski dense monodromy orbit. In particular we show that for a Lefschetz pencil of sufficiently high degree the monodromy action is dense.