Characterizing k-dimensional universal Menger compacta.

*(English)*Zbl 0645.54029
Mem. Am. Math. Soc. 71, No. 380, 110 p. (1988).

Over 30 years ago R. D. Anderson [Ann. Math., II. Ser. 67, 313-324 (1958; Zbl 0083.176); Ann. Math., II. Ser. 68, 1-16 (1958; Zbl 0083.176)] provided a characterization of the Menger universal (1-dimensional) curve as the locally connected continuum having no local cut points and having no nonvoid open subset that embeds in the plane. He also showed it to be homogeneous. Studying here the k-dimensional \((k>1)\) universal Menger continuum, \(\mu\) k, the author derives analogous results and much, much more. The main theorem is the striking characterization of Menger manifolds: a locally compact metric space X is covered by open sets, each homeomorphic to an open subset of \(\mu^ k\), if and only if X is k- dimensional, locally \((k-1)\)-connected, and satisfies the disjoint k- cells property (i.e., any two maps of the standard k-cell to X can be approximated by maps having disjoint images). Moreover, a k-dimensional compact Menger manifold is homeomorphic to \(\mu\;k\) if and only if it is k-connected. A satisfying consequence is that varius natural constructions appearing in the literature, all previously suspected of producing \(\mu^ k\), now are known to do so. At a broad level the manuscript possesses similarities with two powerful characterizations of other kinds of manifolds, one of infinite dimensional Q-manifolds by H. Torunczyk [Fundam. Math. 106, 31-40 (1980; Zbl 0432.57004)] and another of (resolvable) Euclidean n-manifolds (n\(\geq 5)\) by R. D. Edwards [Proc. int. Congr. Math., Helsinki 1978, Vol. 1, 111-127 (1980; Zbl 0428.57004)]. Philosophically all strive to veryfy that spaces having certain local homotopy theoretic properties plus some general position property are manifolds of the desired sort. Methodologically the influences share a less common basis, but there are places where the strategies prevalent in either Torunczyk or Edwards transparently affect the manuscript at hand.

Another significant antecedent involves the following universality property of the Menger curve derived by D. C. Wilson [Trans. Am. Math. Soc. 168, 497-515 (1972; Zbl 0239.54007)]: every locally path- connected metric continuum is the continuous image of the Menger universal curve under a UV 0 map. Bestvina develops an extremely useful generalization, a fundamental link between locally (k-1)-connected spaces and \(\mu\) k-manifolds, which he calls the resolution theorem: every locally (k-1)-connected metric continuum Y is the image of \(\mu^ k\) under a \(UV^{k-1}\) mapping f (meaning that for each neighborhood U of\(f^{-1}(y)\), \(y\in Y\), there exists a smaller neighborhood V such that every map of \(\partial B^ i\) into V extends to a map of \(B^ i\) into U, \(i\in \{0,1,...,k\}).\)

An inspection of the individual chapters will reveal the scope of this work. Chapter 1 introduces the notion of a k-partition, the starting point for the basic combinatorial gadgetry employed throughout as leverage. Here \(\mu\) k-manifolds are described as nested intersections of compact Euclidean manifolds (of dimension at least \(2k+1)\), each equipped with a k-partition. The point is that, under appropriate and expected restrictions on features such as compatibility, connectivity, and mesh size, two \(\mu^ k\)-manifolds are homeomorphic if they have combinatorially identical nests, namely, nests in which there are bijections \(T_ i\) between the k-partitions at stage i such that both \(T_ i\) and its inverse preserve intersections. The lengthy Chapter 2 sets forth the technical heart of the manuscript, describing moves on partitions and adjustments to maps aimed at improving connectivity properties. Bestvina correctly describes it as a kind of PL-theory for \(\mu^ k\)-manifolds. Here is where it is verified that \(\mu^ k\)- manifolds fulfill the various conditions appearing in the characterization. Also included here is a low dimensional version of the Z-set approximation theorem from Q-manifold theory: every map of a k- dimensional compactum C into X, a continuum with the local properties of a \(\mu^ k\)-manifold and satisfying the disjoint k-cells property, can be approximated by a nice embeding \(\lambda\), called a Z-embedding, where maps of \(B^ k\) into X can be adjusted to avoid \(\lambda(C)\). Chapter 3 presents the important Z-set unknotting theorem, asserting that any controlled homeomorphism between Z-sets in a comact \(\mu^ k\)-manifold M can be extended to a controlled homeomorphism of M onto itself. As an immediate corollary, any connected \(\mu^ k\)-manifold, of course including \(\mu^ k\) itself, is homogeneous. Treating the decomposition theory of \(\mu^ k\)-manifolds, Chapter 4 shows that here UV\({}^{k-1}\)- maps play the same role as cell-like maps do for Euclidean and Q- manifolds. This unit builds through assorted cases to the main shrinking theorem, which attests that every \(UV^{k-1}\) mapping from a compact \(\mu^ k\)-manifold M onto a k-dimensional space X satisfying the disjoint k-cells property can be approximated by homeomorphisms. Thus, a map f between compact \(\mu^ k\)-manifolds can be approximated by homeomorphisms if and only if f is \(UV^{k-1}\). Chapter 5 devotes the bulk of its attention to the aforementioned resolution theorem. With all the other results already in place, the characterization theorem (compact case) follows quickly. Chapter 6 exposes the modifications needed to extend the preceeding to the realm of noncompact \(\mu^ k\)-manifolds.

Another significant antecedent involves the following universality property of the Menger curve derived by D. C. Wilson [Trans. Am. Math. Soc. 168, 497-515 (1972; Zbl 0239.54007)]: every locally path- connected metric continuum is the continuous image of the Menger universal curve under a UV 0 map. Bestvina develops an extremely useful generalization, a fundamental link between locally (k-1)-connected spaces and \(\mu\) k-manifolds, which he calls the resolution theorem: every locally (k-1)-connected metric continuum Y is the image of \(\mu^ k\) under a \(UV^{k-1}\) mapping f (meaning that for each neighborhood U of\(f^{-1}(y)\), \(y\in Y\), there exists a smaller neighborhood V such that every map of \(\partial B^ i\) into V extends to a map of \(B^ i\) into U, \(i\in \{0,1,...,k\}).\)

An inspection of the individual chapters will reveal the scope of this work. Chapter 1 introduces the notion of a k-partition, the starting point for the basic combinatorial gadgetry employed throughout as leverage. Here \(\mu\) k-manifolds are described as nested intersections of compact Euclidean manifolds (of dimension at least \(2k+1)\), each equipped with a k-partition. The point is that, under appropriate and expected restrictions on features such as compatibility, connectivity, and mesh size, two \(\mu^ k\)-manifolds are homeomorphic if they have combinatorially identical nests, namely, nests in which there are bijections \(T_ i\) between the k-partitions at stage i such that both \(T_ i\) and its inverse preserve intersections. The lengthy Chapter 2 sets forth the technical heart of the manuscript, describing moves on partitions and adjustments to maps aimed at improving connectivity properties. Bestvina correctly describes it as a kind of PL-theory for \(\mu^ k\)-manifolds. Here is where it is verified that \(\mu^ k\)- manifolds fulfill the various conditions appearing in the characterization. Also included here is a low dimensional version of the Z-set approximation theorem from Q-manifold theory: every map of a k- dimensional compactum C into X, a continuum with the local properties of a \(\mu^ k\)-manifold and satisfying the disjoint k-cells property, can be approximated by a nice embeding \(\lambda\), called a Z-embedding, where maps of \(B^ k\) into X can be adjusted to avoid \(\lambda(C)\). Chapter 3 presents the important Z-set unknotting theorem, asserting that any controlled homeomorphism between Z-sets in a comact \(\mu^ k\)-manifold M can be extended to a controlled homeomorphism of M onto itself. As an immediate corollary, any connected \(\mu^ k\)-manifold, of course including \(\mu^ k\) itself, is homogeneous. Treating the decomposition theory of \(\mu^ k\)-manifolds, Chapter 4 shows that here UV\({}^{k-1}\)- maps play the same role as cell-like maps do for Euclidean and Q- manifolds. This unit builds through assorted cases to the main shrinking theorem, which attests that every \(UV^{k-1}\) mapping from a compact \(\mu^ k\)-manifold M onto a k-dimensional space X satisfying the disjoint k-cells property can be approximated by homeomorphisms. Thus, a map f between compact \(\mu^ k\)-manifolds can be approximated by homeomorphisms if and only if f is \(UV^{k-1}\). Chapter 5 devotes the bulk of its attention to the aforementioned resolution theorem. With all the other results already in place, the characterization theorem (compact case) follows quickly. Chapter 6 exposes the modifications needed to extend the preceeding to the realm of noncompact \(\mu^ k\)-manifolds.

Reviewer: R.J.Daverman

##### MSC:

54F65 | Topological characterizations of particular spaces |

55M10 | Dimension theory in algebraic topology |

57N15 | Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010) |

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |