Weakly Confluent Mappings and Finitely-generated Cohomology

In this paper we answer a question of Wayne Lewis by proving that if X is a one-dimensional, hereditarily indecomposable continuum and if HX(X) is finitely generated, then C(X), the hyperspace of subcontinua of X, has dimension 2. Let C(A) be the hyperspace of subcontinua of the continuum X with the topology determined by the Hausdorff metric. A classical theorem of J. L. Kelley [4] asserts that if A is a hereditarily indecomposable continuum and the dimension of X, dim X, exceeds one, then dim C(A") = oo. On the other hand, E. D. Tymchatyn [9] has proved that if A is a nondegenerate hereditarily indecomposable subcontinuum of the plane (hence dim A = 1), then C(A) can be embedded in R3, and consequently dim C(A") = 2. Carl Eberhart and Sam Nadler, Jr. [1], and later Howard Cook, asked if there exists a one-dimensional, hereditarily indecomposable continuum with an infinitedimensional hyperspace. Wayne Lewis [7] has recently given such an example. Lewis' powerful technique yields a continuum with infinitely-generated cohomology, and he has asked (cf. [6]) if a one-dimensional continuum with finitely-generated (co)homology can have an infinite-dimensional hyperspace. In this note, we prove the following theorem. Theorem I. If X is a one-dimensional, hereditarily indecomposablecontinuum and HX(X) is finitely generated, then dim C(X) = 2. Theorem 1 is equivalent, by an observation [1] of Eberhart and Nadler, to the following theorem. Theorem 1. If X is a one-dimensional, hereditarily indecomposable continuum, if HX(X) is finitely generated, and if f: X —» Y is a monotone, open mapping onto a nondegenerate continuum Y, then dim Y = 1. Theorem 2 will follow as a corollary to the following theorem. Received by the editors April 3, 1979; presented to the Society, January 3, 1980. AMS (MOS) subject classifications (1970). Primary 54F45, 54F50; Secondary 54C10.