arXiv · math/0607768
This paper has been withdrawn by the author. Much simpler proof of the main result was obtained which led to major changes in the presentation.
Explore connections, maps & timelines
A. Chigogidze. 2009-11-09. $Z$-set unknotting in large cubes. https://arxiv.org/abs/math/0607768
Cite the original work for its findings. Save a collection to share your selection of sources.
KEEP EXPLORING
For infinite cardinals $λ\leq κ$, we calculate the calibers of topological spaces $X$ of cardinality $κ$ whose open sets are the subsets $A$ satisfying $|X \setminus A| < λ$. We also prove that the set of calibers of $X$ coincides with the set of calibers of the Vietoris hyperspace $\mathcal{H}(X)$, where the space $\mathcal{H}(X)$ has as its underlying set a collection of subsets of $X$ that includes all its finite subsets and is contained within the collection of its closed subsets.
We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every $\varepsilon>0$ there are two continuous self-maps of the circle, depending on $\varepsilon$, whose images cover the circle and an integer $N$ such that every composition of $N$ of them has image of diameter less than $\varepsilon$. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.
Let $A$ be an abelian group and $E=\Endo_{\Z}(A)$. We give a module-theoretic description of the singular $E$-submodules asked for in Fuchs' Problem~1.2. For a unital ring $R$ and a left $R$-module $M$, put $W=I_R({}_RR)\oplus I_R(M)$ and $J=\Jaco(\Endo_R(W))$, and let $u$ be the image of $1_R$. The classical essential-kernel criterion yields \[ Z_R(M)=M\cap Ju. \] For $R=E$ and $M=A$, all singular submodules are therefore the $E$-submodules of $A\cap Ju$. Writing $T=t(A)$ and $B=A/T$, we prove a torsion-transfer formula, identify the torsion part as $\bigoplus_p pT_p$, and describe simultaneous prime lifting by a canonical obstruction. The resulting extension gives an $\Extt/\Homm$ parametrization of all singular submodules. We obtain explicit formulas for torsion groups and for $\Z(p^\infty)\oplus B$ with $B$ torsion-free; in the latter case the fully invariant subgroup lattice of $B$ occurs as an interval. The general description retains the induced endomorphism action and extension data, rather than giving a classification by classical group invariants.