Rigid hereditarily indecomposable continua

Abstract

We answer a question of Yohe by showing that there exists a family of continuum many topologically different hereditarily indecomposable Cantor manifolds without any non-trivial weakly infinite-dimensional subcon-tinua. This family may consist either of compacta containing one-dimensional subsets or of compacta containing no weakly infinite-dimensional subsets of positive dimension.

Hereditarily indecomposable continua with exactly n autohomeomorphisms

“…. , the existence of such a family follows, for example, from the following lemma proved by M. Reńska in [18]: for every m-dimensional HI continuum K there exists an HI m-dimensional Cantor manifold M such that K is not embeddable into M . Indeed, let K and M be two mdimensional Cantor manifolds such that K does not embed in M and let a 1 , a 2 , .…”

“…In [5] H. Cook gave an example of a rigid, 1-dimensional, HI continuum. Recently M. Reńska [18] constructed, for every m ∈ N, an HI rigid m-dimensional Cantor manifold. The main goal of this paper is to prove the following theorem.…”

“…[6, Problem 6.1.G]); while the first example of an SID compactum all of whose nontrivial subcontinua are infinite-dimensional was given earlier by Henderson [7]. In [18] M. Reńska constructed a rigid HI hereditarily SID Cantor manifold. We will prove the following theorem.…”

“…In our constructions we apply some ideas of [3], [18] and [16]. It is an interesting question whether the spaces X nm and Y m satisfying the conditions of Theorems 1.1 and 1.2 can be m-dimensional Cantor manifolds (for m > 1) and whether the spaces Z n from Theorem 1.3 can be infinite-dimensional Cantor manifolds.…”

Fully closed maps and non-metrizable higher-dimensional Anderson–Choquet continua

“…A metric Cook continuum must have dimension ≤ 2 (Maćkowiak [38]), and if it is hereditarily indecomposable, then it must be one-dimensional (Krzempek [32]). On the other hand, several authors investigated rigidity properties of higher-dimensional continua (J.J. Charatonik [8], M. Reńska [45], E. Pol [40][41][42][43]34], see [32] for more references). In [32] the present author constructed a metric, n-dimensional (arbitrary n > 1), hereditarily indecomposable continuum no two of whose disjoint n-dimensional subcontinua are homeomorphic, but he was not able to ensure that the continuum be Anderson-Choquet.…”