Proper n-homotopy equivalences of locally compact polyhedra

Abstract

We prove the following theorem which is a locally compact analogue of results of $S$. Ferry and the author. Theorem. Let $f:X\rightarrow Y$ be a proper map between finite dimensional locally compact polyhedra $X$ and Y. Suppose that (1) $\pi_{j}(f):\pi_{i}(X)\rightarrow\pi_{i}(Y)$ is an isomorphism for each $i\leq n$, (2) $f$ induces a surjection between the ends of $X$ and $Y$, and (3) $f$ induces an isomorphism between the $i$-th homotopy groups of ends of $X$ and $Y$ for each $i\leq n$. Then there exist a locally compact polyhedron $Z$ and proper $UV^{n}$-maps $\alpha:Z\rightarrow X$ and $\beta:Z\rightarrow Y$ such that (4) $\dim Z\leq 2\max(\dim X,n)+3$, (5) $f\circ\alpha$ and $\beta$ is properly $n$-homotopic, and (6) $\alpha$ has at most countably many non-contractible fibre all of which have the homotopy type of $S^{n+1}$.