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تسجيل مستخدم جديدFilling systems on surfaces
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Let $F_g$ be a closed orientable surface of genus $g$. A set $Omega = { gamma_1, dots, gamma_s}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a emph{filling system} or simply a emph{filling} of $F_g$, if $F_gsetminus Omega$ is a union of $b$ topological discs for some $bgeq 1$. A filling system is called emph{minimal}, if $b=1$. The emph{size} of a filling is defined as the number of its elements. We prove that the maximum size of a filling of $F_g$ with $b$ complementary discs is $2g+b-1$. Next, we show that for $ggeq 2, bgeq 1text{ with }(g,b) eq (2,1)$ (resp. $(g,b)=(2,1)$) and for each $2leq sleq 2g+b-1$ (resp. $3leq sleq 2g+b-1$), there exists a filling of $F_g$ of size $s$ with $b$ complementary discs. Furthermore, we study geometric intersection number of curves in a minimal filling. For $ggeq 2$, we show that for a minimal filling $Omega$ of size $s$, the emph{geometric intersection numbers} satisfy $max leftlbrace i(gamma_i, gamma_j)| i eq jrightrbraceleq 2g-s+1$, and for each such $s$ there exists a minimal filling $Omega=leftlbrace gamma_1, dots, gamma_s rightrbrace$ such that $maxleftlbrace i(gamma_i, gamma_j) | i eq jrightrbrace = 2g-s+1$.
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