Monotone Maps of Hereditarily Indecomposable Continua

Lewis, W. S.

doi:10.2307/2042773

Cited by 4 publications

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“…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.…”

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confidence: 99%

Weakly confluent mappings and finitely-generated cohomology

Rogers¹

1980

Proc. Amer. Math. Soc.

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Abstract. 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 2 will follow as a corollary to the following theorem.

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confidence: 99%

Weakly confluent mappings and finitely-generated cohomology

Rogers¹

1980

Proc. Amer. Math. Soc.

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“…In particular, the hereditarily indecomposable, one-dimensional continua constructed by Lewis [5] have uncountably many orbits under the action of their homeomorphism groups.…”

Section: Old Results From Polandmentioning

confidence: 99%

Orbits of Higher-Dimensional Hereditarily Indecomposable Continua

Rogers¹

1985

Proceedings of the American Mathematical Society

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ABSTRACT. Let X be a continuum.The following theorems are proved.THEOREM. i/dimX > 1, tkenX contains uncountably many nonhomeomorphic continua.THEOREM. If dim X > 1 and X is hereditarily indécomposable, then X has uncountably many orbits under the action of its hcrmex/rnorphisrn group.In 1970, Howard Cook [3] gave a beautiful proof that hereditarily equivalent continua are tree-like. In 1982, the author [9] proved that homogeneous, hereditarily indecomposable continua are tree-like. A step in both these proofs was to show that such continua must be one-dimensional.In this paper, we give a simple and independent proof that these continua must be one-dimensional.Moreover, the results are even stronger: namely, if X is a continuum and dim AT > 1, then X contains uncountably many nonhomeomorphic subcontinua. If, in addition, X is hereditarily indecomposable, then X has uncountably many orbits under the action of its homeomorphism group.A continuum is a compact, connected, nonvoid metric space. A continuum is hereditarily equivalent if it is homeomorphic to each of its proper nondegenerate subcontinua.A continuum is indecomposable if it is not the union of two of its proper subcontinua.A continuum is hereditarily indecomposable if each of its subcontinua is indecomposable.R. H. Bing [2] constructed examples of hereditarily indecomposable continua of dimension n, 1 < n < oo. Bing [2] showed that, for 1 < n < oo, no such continuum is homogeneous.

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“…Clearly the same result can also be obtained from Theorem 1.1. The first examples of such continua were given by Lewis [8]. (This should be compared with Theorem 1.9.…”

Section: Introductionmentioning

confidence: 99%