Александровские чтения-2016 | 22-26 мая 2016
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 Участник: Pavel Gevorgyan
 Имя: Pavel GevorgyanНазвание: Movability of shape morphisms and The Whitehead theorem in shape theory Город, страна: Moscow Организация: Moscow State Pedagogical University Абстракт: The classical theorem of J. H. C. Whitehead has played an important role in homotopy theory. In shape theory there are several results analogous to the Whitehead theorem, which are proved for movable spaces. In this paper we introduce the notions of movable and co-movable shape morphisms and state the Whitehead theorem in shape theory for thees morphisms. \begin{definition} Let $\mathbf{X}=(X_\lambda,p_{\lambda\lambda'},\Lambda)$ and $\mathbf{Y}=(Y_\mu,q_{\mu\mu'},M)$ inverse systems in a category $\mathcal{C}$ and $(f_\mu,\phi):\mathbf{X}\rightarrow \mathbf{Y}$ a morphism of inverse systems. We say that the morphism $(f_\mu,\phi)$ is \textsl{movable} if every $\mu\in M$ admits $\lambda\in \Lambda$, $\lambda\geq \phi(\mu)$, such that each $\mu'\in M$, $\mu'\geq\mu$, admits a morphism $u:X_{\lambda}\rightarrow Y_{\mu'}$, in the category $\mathcal{C}$, which satisfies \begin{equation*}\label{2.2} f_\mu\circ p_{\phi(\mu)\lambda}=q_{\mu\mu'}\circ u \end{equation*} \end{definition} \begin{definition} Let $\mathbf{X}=(X_\lambda,p_{\lambda\lambda'},\Lambda)$ and $\mathbf{Y}=(Y_\mu,q_{\mu\mu'},M)$ be inverse systems in a category $\mathcal{C}$ and $(f_\mu,\phi):\mathbf{X}\rightarrow \mathbf{Y}$ a morphism of inverse systems. We say that the $(f_\mu,\phi)$ is \textsl{co-movable morphism} provided every $\mu\in M$ admits $\lambda\in \Lambda$, $\lambda\geq \phi(\mu)$ (called a \textsl{co-movability index} of $\mu$ relative to $(f_\mu,\phi)$) such that each $\lambda'\geq \phi(\mu)$ admits a morphism $r:X_{\lambda}\rightarrow X_{\lambda'}$ of $\mathcal{C}$ which satisfies \begin{equation*} f_{\mu\lambda}=f_{\mu\lambda'}\circ r. \end{equation*} \end{definition} \begin{definition} A morphism in a pro-category pro-$\mathcal{C}$, $\mathbf{f}:\mathbf{X}\rightarrow \mathbf{Y}$, is called \textsl{movable} (\textsl{co-movable}) if $\mathbf{f}$ admits a representation $(f_\mu,\phi):\mathbf{X}\rightarrow \mathbf{Y}$ which is movable (co-movable). \end{definition} \begin{definition} A shape morphism $F:X\rightarrow Y$ is movable (co-movable) if it can be represented by a movable (co-movable) pro-morphism $\mathbf{f:X\rightarrow Y}$. \end{definition} \begin{theorem} An inverse system $\mathbf{X}=(X_\lambda,p_{\lambda\lambda'},\Lambda)$ is movable if and only if the identity morphism $1_{\mathbf{X}}$ is movable (co-movable). \end{theorem} Now we can formulate two variants of Dydak's infinite-dimensional Whitehead theorem in shape theory \cite{Dydak}. \begin{theorem} Let $F:(X,\ast)\rightarrow (Y,\ast)$ be a weak shape equivalence of pointed connected topological spaces. Suppose that $X$ is of finite shape dimension, $sh X<\infty$, and that $F$ is a movable pointed shape domination. Then $F$ is a pointed shape equivalence. \end{theorem} \begin{theorem} Let $F:(X,\ast)\rightarrow (Y,\ast)$ be a weak shape equivalence of pointed connected topological space. Suppose that $Y$ is of finite shape dimension, $sh Y<\infty$, and that $F$ is a co-movable pointed shape morphism which has a left inverse. Then $F$ is a pointed shape equivalence. \end{theorem} \begin{thebibliography}{99} \bibitem{Dydak} J. Dydak, \textsl{The Whitehead and Smale Theorems in Shape Theory}, Dissertations Math. \textbf{156} (1979), 1-55. \bibitem{Gev} P.S. Gevorgyan, \textsl{Movable categories}, Glas. Mat. Ser. III, \textbf{38} (2003),177-183. \bibitem{Gev-Pop}P.S.Gevorgyan and I.Pop, \textsl{Uniformly movable categories and uniformly movability of toplogical sapces}, Bull. Pol. Acad. Sci. Math., \textbf{55} (2007), 229-242. \bibitem{Pop2} I. Pop, \textsl{On movability of pro-morphisms}, Analele Universitatii de Vest, Timisoara. Seria Matematica-Informatica , Vol. \textbf{XLVIII}, \textbf{1,2} (2010), 223-238. \end{thebibliography}

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