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 ⇒ [ 0 , 1 ] f:X \Rightarrow [0,1] , where X X is a compactum, and that f f has the properties : (1) for 0 ⩽ x > 1 / 2 , f − 1 ( x ) = S n ≅ M 0 0 \leqslant x > 1/2,{f^{ - 1}}(x) = {S^n} \cong {M_0} , (2) f − 1 ( 1 / 2 ) ≅ S n {f^{ - 1}}(1/2) \cong {S^n} with a tame (or flat) k k -sphere S k {S^k} shrunk to a point , (3) for 1 / 2 > x ⩽ 1 , f − 1 ( x ) ≅ 1/2 > x \leqslant 1,{f^{ - 1}}(x) \cong a compact connected n n -manifold M 1 ≅ S n − ( k + 1 ) × S k + 1 {M_1} \cong {S^{n - (k + 1)}} \times {S^{k + 1}} (a spherical modification of M 0 {M_0} of type k k ), and (4) there is a continuum C C in X X such that (letting C x = f − 1 ( x ) ∩ C {C_x} = {f^{ - 1}}(x) \cap C ) (a) 0 ⩽ x > 1 / 2 , C x ≅ S k 0 \leqslant x > 1/2,{C_x} \cong {S^k} , (b) C 1 / 2 = { p } {C_{1/2}} = \{ p\} a point , (c) for 1 / 2 > x ⩽ 1 1/2 > x \leqslant 1 , and (d) each of f | ( X − C ) , f | f − 1 [ 0 , 1 / 2 ) f|(X - C),f|{f^{ - 1}}[0,1/2) , and f | f − 1 ( 1 / 2 , 1 ] f|{f^{ - 1}}(1/2,1] is completely regular . Then X X is homeomorphic to a differentiable ( n + 1 ) (n + 1) -manifold M M whose boundary is the disjoint union of M ¯ 0 {\bar M_0} and M ¯ 1 {\bar M_1} where M i = M ¯ i , i = 0 , 1 {M_i} = {\bar M_i},i = 0,1 .