CE equivalence and shape equivalence of LC\(^ n\) compacta
The author continues to investigate the relationship of shape equivalence and \(UV^ m\) equivalence of path connected compacta. It is known that \(UV^ m\)-equivalent \(m\)-dimensional compacta are shape equivalent, while shape equivalent \(m\)-dimensional compacta need not be \(UV^ m\) equivalent. S. Ferry has proved the following: If \(X\) is a continuum with pro-\(\pi_ 1X\) profinite and \(Y\) shape equivalent to \(X\), then \(Y\) is \(UV^ m\) equivalent to \(X\) for each \(m \geq 2\). [\textit{S. C. Ferry}, `Geometric topology and shape theory', Lect. Notes Math. 1283, 88-114 (1987; Zbl 0634.55008)]. The main result of reviewed paper is: If \(X\) is an \(LC^{n+1}\) continuum, \(n\geq 1\), with \(\pi_ 1(X)\) infinite, then there exists an \(LC^ n\) continuum \(X'\), \(\dim X'\geq \max\{\dim X,n+2\}\) such that \(X\) and \(X'\) are shape equivalent but not \(UV^{n+1}\) equivalent. In particular, \(X\) and \(X'\) are not \(CE\) equivalent. Ferry's first constructed example of shape equivalent compacta which are not \(CE\) equivalent (they are not even \(UV^ 1\) equivalent) falls under this theorem. The core of the paper occupies the founding of \(UV^ m\) and \(CE\) invariants that are later used to detect \(UV^ m\) and \(CE\) inequivalence, respectively. For a pointed space \((X,x_ 0)\), these invariants are named \(UV^ m\) groups and \(CE\) groups, denoted \(\pi^{(m)}_ k(X,x_ 0)\) and \(\pi^{CE}_ k(X,x_ 0)\), \(k = 1,2,\dots\). The \(UV^ m\) groups turned out to be trivial for \(k > m\) and any space \(X\), and isomorphic to the \(k\)-th shape group \(\check\pi_ k(X,x_ 0)\) when \(X\) is an \(LC^ n\) continuum and \(k\leq n,m\). They are formally introduced as certain equivalence classes of triples \((C,\alpha,\beta)\) where \(C\) is \(UV^ m\) or \(CE\) compactum, respectively, \(\alpha: S^{k-1}\to C\) and \(\beta: C\to X\) maps \(S^{k- 1}\) is the \((k-1)\)-sphere and \(\alpha\beta(S^{k-1}) = \{x_ 0\}\). Intuitively, it is a generalisation of elements of \(\pi_ k(X,x_ 0)\) interpreted as the homotopy classes of the composition of inclusion of \(S^{k-1}\) into \(B^ k\) and a map \(\beta: B^ k\to X\) with the property that \(\beta(\partial B^ k = S^{k-1}) = \{x_ 0\}\). The proof of the quoted main result is by a certain construction (developed in the 3rd paragraph of the paper) of an \(LC^ n\) continuum \(X'\) shape equivalent to \(X\) (the construction uses \((n+1)\)-dimensional Hawaiian earring) such that its group \(\pi^{(n+1)}_{n+1}\) is not isomorphic to the analogues of \(X\), therefore \(X\) and \(X'\) are not \(UV^{n+1}\) equivalent.
scientific article; zbMATH DE number 416864
scientific article