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Register a new userOn 2-dimensional nonaspherical cell-like Peano continua: A simplified approach
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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.
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Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts by a noncontractible n-dimensional Peano continuum for any n>0, then our construction yields a simply connected noncontractible (n + 1)-dimensional cell-like Peano continuum. In particular, starting with the circle $mathbb{S}^1$, one gets a noncontractible simply connected cell-like 2-dimensional Peano continuum.
We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths space from the 1950s.
Let $X$ be a Peano continuum having a free arc and let $C^0(X)$ be the semigroup of continuous self-maps of $X$. A subsemigroup $Fsubset C^0(X)$ is said to be sensitive, if there is some constant $c>0$ such that for any nonempty open set $Usubset X$, there is some $fin F$ such that the diameter ${rm diam}(f(U))>c$. We show that if $X$ admits a sensitive commutative subsemigroup $F$ of $C^0(X)$ consisting of continuous open maps, then either $X$ is an arc, or $X$ is a circle.
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.
We prove a homological characterization of $Q$-manifolds bundles over $C$-spaces. This provides a partial answer to Question QM22 from cite{w}.
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