Numerical calculation of three-point branched covers of the projective line

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.Comment: 58 pages, 24 figures; referee's comments incorporate

Counting homomorphisms from surface groups to finite groups

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group $G$, where conjugacy classes of the boundary components of the surface must map to prescribed conjugacy classes in $G$, to a sum over values of irreducible characters of $G$ weighted by Frobenius-Schur multipliers. The proof is structured so that the corresponding results for closed and possibly orientable surfaces, as well as some generalizations, are derived using the same methods. We then apply these results to the specific case of the symmetric group.Comment: Comments welcome

A relative version of Rochlin's theorem

Rochlin proved \cite{VR} that a closed 4-dimensional connected smooth oriented manifold $X^4$ with vanishing second Stiefel-Whitney class has signature $\sigma(X)$ divisible by 16. This was generalized by Kervaire and Milnor \cite{kervaire_milnor_spheres} to the statement that if $\xi \in H_2(X;\mathbb{Z})$ is an integral lift of an element in $H_2(X; \mathbb{Z}/2\mathbb{Z})$ that is dual to $w_2(X)$, and if $\xi$ can be represented by an embedded sphere in $X$, then the self-intersection number $\xi^2$ is divisible by 16. This was subsequently generalized further by Rochlin (see Theorem \ref{matsumoto} below) and various alternative proofs of this result where given by Freedman and Kirby \cite{freedman1978geometric}, Matsumoto \cite{matsumoto}, and Kirby \cite{kirbybook}. We give further generalizations of this result concerning 4-manifolds with boundary. Given a smooth compact orientable four manifold $X^4$ with integral homology sphere boundary and a connected orientable characteristic surface with connected boundary $F^2$ properly embedded in $X$, we prove a theorem relating the Arf invariant of $\partial F$, and the Arf invariant of $F$, and the Rochlin invariant of $\partial X$. We then proceed to generalize this result to the case where $X$ is a topological compact orientable 4-manifold (which brings in the Kirby-Siebenmann invariant), $\partial F$ is not connected (which brings in the condition of being proper as a link), $F$ is not orientable (which brings in Brown invariants), and finally where $\partial X$ is an arbitrary 3-manifold (which brings in pin structures). The final result gives a ``combinatorial'' description of the Kirby-Siebenmann invariant of a compact orientable 4-manifold with nonempty boundary.Comment: Added several generalizations of the main result. Comments welcome

Deep and shallow slice knots in 4-manifolds

We consider slice disks for knots in the boundary of a smooth compact 4-manifold $X^{4}$. We call a knot $K \subset \partial X$ deep slice in $X$ if there is a smooth properly embedded 2-disk in $X$ with boundary $K$, but $K$ is not concordant to the unknot in a collar neighborhood $\partial X \times I$ of the boundary. We point out how this concept relates to various well-known conjectures and give some criteria for the nonexistence of such deep slice knots. Then we show, using the Wall self-intersection invariant and a result of Rohlin, that every 4-manifold consisting of just one 0- and a nonzero number of 2-handles always has a deep slice knot in the boundary. We end by considering 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. A generalization of the Murasugi-Tristram inequality is used to show that there does not exist a compact, oriented 4-manifold $V$ with spherical boundary such that every knot $K \subset S^3 = \partial V$ is slice in $V$ via a null-homologous disk.Comment: 14 pages, 5 figures; v3 is the final draft which has been accepted for publication in Proceedings of the AMS, Series B; v3 includes improvements to the exposition thanks to the anonymous refere

Desenvolvimento de uma Aeronave VTOL de Baixo Custo do Tipo Quadrirotor

Resumo- Neste trabalho aborda-se o desenvolvimento e construção de uma aeronave de decolagem e pouso vertical (VTOL, do inglês Vertical Take-off and Landing) do tipo quadrirotor. Esta topologia se destaca pelas propriedades de manobrabilidade, e é comumente utilizada em aplicações como monitoramento, inspeção remota, mapeamento, entre outras. Para a construção do frame optou-se pela utilização de materiais de baixo custo, objetivando a popularização deste tipo de tecnologia. Resultados são apresentados com intuito de demonstrar o funcionamento do dispositivo construído