Overlays and group actions (Q284600)

The notion of overlay was introduced by R. H. Fox in 1972 in generalizing the classical classification theorem for covering maps to arbitrary connected metric spaces. Every overlay \(p:X\rightarrow Y\) is a covering map. If \(Y\) is paracompact\(,\) the converse implication holds in the following cases: if \(Y\) is locally connected or if the number of sheets of \(p\) is finite. Fox exhibited an example of a covering map over a metric continuum which is not an overlay. Recently it was shown that a covering map between topological groups which is also a homomorphism of groups is an overlay. NEWLINENEWLINEIn this paper the author gives a new different view on overlays. Firstly\(,\) he introduces the notion of slice of a covering map \(p:X\rightarrow Y\) as an open subset \(U\) of \(X\) having the property that there exists a disjoint union \(\left\{ U_{s}\right\} _{s\in S}\) of open subsets of \(X\) such that \(p(U)\) is evenly covered by \(\left\{ U_{s}\right\} _{s\in S}\) and \(U=U_{s_{0}}\) for some \(s_{0}\in S.\) Given a covering map \(p:X\rightarrow Y,\) a covering structure \(\mathcal{U}\) of \(p\) is an open cover of \(X\) by slices of \(p\) such that for each \(U\in\mathcal{U},\) \(p^{-1}(p(U))\) can be decomposed into a disjoint union \(\left\{ U_{s}\right\} _{s\in S}\) of elements of \(\mathcal{U}\) satisfying \(p(U_{s})=U\) for each \(s\in S.\) Then the author defines an overlay as a covering map \(p:X\rightarrow Y\) provided \(X\) admits a covering structure \(\mathcal{U}\) of \(p\) such that for any \(x\in X\) the star \(st(x,\mathcal{U})=\cup\left\{ U\in\mathcal{U}:x\in U\right\} \) is a slice of \(p.\) In that case \(\mathcal{U}\) is called an overlay structure of \(p.\) Note that this definition of overlays resembles the definition of paracompact spaces via star refinements of open covers. The author shows that his definition of overlays coincides with that of Fox provided the fibers of \(p:X\rightarrow Y\) have the same cardinality. NEWLINENEWLINEUsing his definition he obtains two interesting characterizations of overlays. The first one uses the concept of lifting of chains of points. For a cover \(\mathcal{V}\) of \(Y\), a finite sequence \(\left\{ y_{0} ,y_{1},\ldots,y_{n}\right\} \) of points in \(Y\) is called a \(\mathcal{V} \)-chain if for each \(0\leq i<n\) there is a \(V\in\mathcal{V}\) containing both \(y_{i}\) and \(y_{i+1}.\) It is proved that an open cover \(\mathcal{U}\) of an open surjection \(p:X\rightarrow Y\) is an overlay structure of \(p\) if and only if for a given \(p(\mathcal{U})\)-chain \(\left\{ y_{0},y_{1},\ldots ,y_{n}\right\} \) in \(Y\) and a given \(x_{0}\in p^{-1}(\left\{ y_{0}\right\} )\) there exists a unique \(\mathcal{U}\)-chain \(\left\{ x_{0},x_{1} ,\ldots,x_{n}\right\} \) in \(X\) such that \(p(x_{i})=y_{i}\) for \(0\leq i\leq n.\) Using this characterization the author shows that a surjection \(p:X\rightarrow Y\) between connected metrizable spaces is an overlay if and only if one can metrize \(X\) and \(Y\) in such a way that the restriction \(p\left| B(x,1)\right. :B(x,1)\rightarrow B(p(x),1)\) of \(p\) between related unit balls is an isometry\(,\) for each \(x\in X.\) The second characterization uses nerves of open covers. It is proved that a covering structure \(\mathcal{U}\) of \(p\) is an overlay structure of \(p\) if and only if the induced simplicial map \(\mathcal{N}(p):\mathcal{N}(\mathcal{U})\rightarrow \mathcal{N}(p(\mathcal{U}))\) between the nerves of the covers \(\mathcal{U}\) and \(p(\mathcal{U})\) is a covering map.