Extension dimension of inverse limits (Q2709931)

The author improves on the results of two preceding articles [\textit{L. R. Rubin} and \textit{P. J. Schapiro}, Pac. J. Math. 187, No. 1, 177-186 (1999; Zbl 0951.55002); \textit{L. R. Rubin}, A stronger limit theorem in extension theory, Glas. Mat., III. Ser. 36(56), No. 1, 95-103 (2001)]. These works involve limit theorems in extension theory for metrizable spaces. The paper under review adds to the theory by showing that the main results apply to a larger class of spaces, those called stratifiable, and also by providing a somewhat different approach to the proof techniques. NEWLINENEWLINENEWLINEThe starting point in this area is the notion of extension theory. For a space \(X\) and a CW-complex \(K\), one writes \(X\tau K\) to mean that for every closed subset \(A\) of \(X\) and map \(f:A\rightarrow K\) there exists a map \(F:X\rightarrow K\) which is an extension of \(f\). If we restrict our attention say to stratifiable spaces \(X\), then \(X\tau S^n\) is equivalent to the statement that the covering dimension of \(X\) is \(\leq n\). A similar statement can be made about cohomological dimension modulo an abelian group \(G\) if one replaces \(S^n\) by an Eilenberg-MacLane complex \(K(G,n)\). The first theorem proved is: NEWLINENEWLINENEWLINETheorem. Let \((X_i,p_i^{i+1})\) be an inverse sequence of stratifiable spaces, \(X\) be its limit, \(K\) be a CW-complex, and suppose that \(X_i\tau K\) for each \(i\). Then \(X\tau K\).NEWLINENEWLINENEWLINEThe second theorem improves this one by replacing the requirement \(X_i\tau K\) by the stipulation that for each \(i\), closed set \(A\) of \(X_i\), and map \(f:A\rightarrow K\), there exists \(j\geq i\) and a map \(F:X_j\rightarrow K\) such that \(F(x)=f\circ p_i^j(x)\) for each \(x\in(p_i^j)^{-1}(A)\). NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINEWe remark that since the class of stratifiable spaces contains that of polyhedra (weak or metric topology), then as a corollary, the limit theorems apply to inverse sequences of polyhedra.