COLLOQUIUM MATHEMATICUM VOL. 103 2005 NO. 2 THE GENERALIZED SCHOENFLIES THEOREM FOR ABSOLUTE SUSPENSIONS BY DAVID P. BELLAMY (Newark, DE) and JANUSZ M. LYSKO (Chester, PA) Abstract. The aim of this paper is to prove the generalized Schoenflies theorem for the class of absolute suspensions. The question whether the finite-dimensional absolute suspensions are homeomorphic to spheres remains open. Partial solution to this question was obtained in [Sz] and [Mi]. Morton Brown gave in [Br] an ingenious proof of the generalized Schoenflies theorem. Careful analysis of his proof reveals that modulo some technical adjustments a similar argument gives an analogous result for the class of absolute suspensions. 1. A brief history of the problem. The original question whether finite-dimensional absolute suspensions are topological spheres was asked by de Groot in [Gr]. It was also observed by de Groot that in order to provide the positive answer to the above question one needs “only” prove that such spaces are topological manifolds. Szyma´ nski proved in [Sz] that the answer to de Groot’s question was positive for spaces of dimension not exceeding three. Finite-dimensional absolute suspensions are homogeneous ANR’s (see e.g. [Mi], [Sz]). We see therefore that the positive answer to de Groot’s question would provide a partial answer to the long standing question of R. H. Bing and K. Borsuk who asked in [BB] whether finite-dimensional, homogeneous ANR’s are topological manifolds. The answer to the Bing–Borsuk question is known only for homogeneous ANR’s of dimension not exceeding two. In a very interesting paper [Ja] W. Jakobsche indicates the level of difficulty of the Bing–Borsuk question by showing that the positive answer to their question would provide the solution to the Poincar´ e conjecture. 2. Terminology, definitions and the statements of results used in the proof. All spaces considered in this paper will be finite-dimensional and metric. If Y is a compact space then the suspension over Y , denoted by S(Y ), is the quotient of Y × [−1, 1] with two nondegenerate equivalence classes: Y ×{1} and Y × {−1}. The two points obtained by the identifica- tion of these sets are called the vertices of the suspension. The quotient of 2000 Mathematics Subject Classification : 54F15, 54C55, 57P05. Key words and phrases : ANR, absolute suspension. [241]