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Register a new userA conjecture on the lengths of filling pairs
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A pair $(alpha, beta)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $alphacupbeta$ in $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In cite{Aou}, Aougab-Huang conjectured that the length of any filling pair on $M$ is at least $frac{m_{g}}{2}$, where $m_{g}$ is the perimeter of the regular right-angled hyperbolic $left(8g-4right)$-gon. In this paper, we prove a generalized isoperimetric inequality for disconnected regions and we prove the Aougab-Huang conjecture as a corollary.
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In our earlier paper (K. Eda, U. Karimov, and D. Repovv{s}, emph{A construction of simply connected noncontractible cell-like two-dimensional Peano continua}, Fund. Math. textbf{195} (2007), 193--203) we introduced a cone-like space $SC(Z)$. In the present note we establish some new algebraic properties of $SC(Z)$.
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 $maxleftlbrace i(gamma_i, gamma_j) | i eq jrightrbrace = 2g-s+1$.
We construct a functor $AC(-,-)$ from the category of path connected spaces $X$ with a base point $x$ to the category of simply connected spaces. The following are the main results of the paper: (i) If $X$ is a Peano continuum then $AC(X,x)$ is a cell-like Peano continuum; (ii) If $X$ is $n-$dimensional then $AC(X, x)$ is $(n+1)-$dimensional; and (iii) For a path connected space $X$, $pi_1(X,x)$ is trivial if and only if $pi_2(AC(X, x))$ is trivial. As a corollary, $AC(S^1, x)$ is a 2-dimensional nonaspherical cell-like Peano continuum.
We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pin(2) symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology.
The classifying space for the framed Haefliger structures of codimension $q$ and class $C^r$ is $(2q-1)$-connected, for $1le rleinfty$. The corollaries deal with the existence of foliations, with the homology and the perfectness of the diffeomorphism groups, with the existence of foliated products, and of foliated bundles.
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