Interpreting the number of ramified covering of the sphere by Riemann surface as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of the sphere by Riemann surface with any genus, with elementary branch points and prescribed ramification type over infinity.

We show that any homologically non-trivial Dehn twist of a compact surface F with boundary is the lifting of a half-twist in the braid group Bn, with respect to a suitable branched covering p : F → B. In particular, we allow the surface to have disconnected boundary. As a consequence, any allowable Lefschetz fibration on B is a branched covering of B × B.

For a finite connected graph let be the spectral radius of its universal cover. We prove that for any graph of average degree and derive from it the following generalization of the Alon Boppana bound. If the average degree of the graph after deleting any radius ball is at least , then its second largest eigenvalue in absolute value is at least for some absolute constant . This result is tight i...

The main theorem states that if K is a finite CW-complex with finite fundamental group G and universal cover homotopy equivalent to a product of spheres X, then G acts smoothly and freely on X×Sn for any n greater than or equal to the dimension of X. If the G-action on the universal cover of K is homologically trivial, then so is the action on X × Sn. Ünlü and Yalçın recently showed that any fi...

A closed Riemann surface X which can be realised as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. A p-gonal Riemann surface is called real p-gonal if there is an anticonformal involution (symmetry) σ of X commuting with the p-gonal morphism. If the p-gonal morphism is a cyclic regular covering the Riemann surface is called real c...