A characterization of manifolds

Abstract:

The purpose of this paper is (1) to give a proof of one general theorem characterizing certain manifolds and (2) to illustrate a technique which should be useful in proving various theorems analogous to the one proved here. Theorem. Suppose that $f:X \Rightarrow [0,1]$, where $X$ is a compactum, and that $f$ has the properties: (1) for $0 \leqslant x < 1/2,{f^{ - 1}}(x) = {S^n} \cong {M_0}$, (2) ${f^{ - 1}}(1/2) \cong {S^n}$ with a tame (or flat) $k$-sphere ${S^k}$ shrunk to a point, (3) for $1/2 < x \leqslant 1,{f^{ - 1}}(x) \cong$ a compact connected $n$-manifold ${M_1} \cong {S^{n - (k + 1)}} \times {S^{k + 1}}$ (a spherical modification of ${M_0}$ of type $k$), and (4) there is a continuum $C$ in $X$ such that (letting ${C_x} = {f^{ - 1}}(x) \cap C$) (a) $0 \leqslant x < 1/2,{C_x} \cong {S^k}$, (b) ${C_{1/2}} = \{ p\}$ a point, (c) for $1/2 < x \leqslant 1$, and (d) each of $f|(X - C),f|{f^{ - 1}}[0,1/2)$, and $f|{f^{ - 1}}(1/2,1]$ is completely regular. Then $X$ is homeomorphic to a differentiable $(n + 1)$-manifold $M$ whose boundary is the disjoint union of ${\bar M_0}$ and ${\bar M_1}$ where ${M_i} = {\bar M_i},i = 0,1$.