##
**General position properties in fiberwise geometric topology.**
*(English)*
Zbl 1301.57019

This volume deals with general position properties, the type that historically have played an important role in geometric and infinite-dimensional topology. Among these are \(\mathrm{DD}^n\), the property of disjoint \(n\)-cells. For example, each Polish \(\mathrm{LC}^{n-1}\)-space enjoying the \(\mathrm{DD}^n\)-property contains a topological copy of every compact \(n\)-dimensional metrizable space. The authors introduce the 3-parameter property \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\) which branches out from some of the other \(\mathrm{DD}^n\)-like properties. They give “arithmetic” tools for revealing such a property in a given space.

The work is written in two parts, the first being a survey of certain results that appear in the second part. The authors of course give in the first part various definitions; they discuss some applications and interplay of their theory with existing knowledge in this area. The second part is dedicated to providing proofs of what was given in the first part.

A few definitions are in order. We presume in the following that “approximation” means approximation up to some open cover (of the given space). A space \(X\) has the \(\mathrm{DD}^n\)-property if any two maps \(f\), \(g\) of \([0,1]^n\) to \(X\) can be respectively approximated by maps \(f'\), \(g'\) such that \(f'([0,1]^n)\cap g'([0,1]^n)=\emptyset\). A parametric version of this occurs when \(M\) is a space, \(f\), \(g\) are maps of \(M\times[0,1]^n\) to \(X\), and one requires that for all \(z\in M\), \(f'(\{z\}\times[0,1]^n)\cap g'(\{z\}\times[0,1]^n)=\emptyset\). This property is called \(M\)-\(\mathrm{DD}^n\). When \(M=[0,1]^m\), one writes \(m\)-\(\mathrm{DD}^n\).

The property \(m\)-\(\overline{\mathrm{DD}}^n\) goes this way. It is required that for any maps \(f\), \(g:[0,1]^m\times[0,1]^n\to X\) and open cover \(\mathcal{U}\) of \(X\), there are maps \(f'\), \(g' :[0,1]^m\times[0,1]^n\to X\) with \(f'\) being \(\mathcal{U}\)-homotopic to \(f\), \(g'\) being \(\mathcal{U}\)-homotopic to \(g\), and \(f'(\{z\}\times[0,1]^n)\cap g'(\{z\}\times [0,1]^n)=\emptyset\) for all \(z\in [0,1]^m\). Sometimes the \(\mathcal{U}\)-homotopy concept is replaced by the notion of an \(\epsilon\)-homotopy (page 9). Proposition 3.2 describes a setting in which \(m\)-\(\overline{\mathrm{DD}}^n\) is equivalent to \(m\)-\(\mathrm{DD}^n\).

On page 7, the authors note that the set of one-to-one maps (i.e., embeddings) of a metrizable \(n\)-dimensional compactum \(K\) into a completely metrizable \(\mathrm{LC}^{n-1}\)-space \(X\) possessing the \(\mathrm{DD}^n\)-property is a dense \(\mathrm{G}_\delta\)-set in the function space \(C(K,X)\), the latter being provided with the source limitation topology. They obtain a parametric version of this result, the main output of Section I.1:

{ Theorem 1.1.} A completely metrizable \(\mathrm{LC}^{m+n}\)-space \(X\) has the \(m\)-\(\mathrm{DD}^n\)-property if and only if for every perfect map \(p:K\to M\) between finite-dimensional metrizable spaces with dim\(M\leq m\) and dim\((p)\leq n\), the function space \(C(K,X)\) contains a dense \(\mathrm{G}_\delta\)-set of maps \(f:K\to X\) that are injective on each fiber \(p^{-1}(z)\), \(z\in M\).

Section I.2 deals with the \(\Delta\)-dimension of a map between Tychonoff spaces. Proposition 2.1 compares this dimension with the classical dimension of a map and also lists some properties of this dimension. Theorem 2.2 is a version of Theorem 1.1 with \(\Delta\)-dimension in place of dimension of the given map \(p\).

In Section I.3 we are provided with a “general fiber embedding theorem”, Theorem 3.3, which involves the \(m\)-\(\overline{\mathrm{DD}}^n\)-property. This result is said to imply Theorem 1.1 as well as some of the other ones in I.1. On page 10, the notion of a Lefschetz \(\mathrm{ANE}[n]\)-space is defined. Proposition 3.4 lists 10 items related to this property. With help from this and some other propositions, Theorem 3.6 is obtained, and it is another generalization of Theorem 1.1.

Section I.4 gives us some results concerning approximating perfect maps by perfect PL-maps. A new, 3-parameter property of the \(m\)-\(\overline{\mathrm{DD}}^n\) type is introduced in I.5 as a useful generalization.

{ Definition 5.1.} A space \(X\) is defined to have the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-property if for any open cover \(\mathcal{U}\) of \(X\) and maps \(f:[0,1]^m\times[0,1]^n \to X\), \(g:[0,1]^m\times[0,1]^k \to X\), there exist maps \(f':[0,1]^m\times[0,1]^n \to X\), \(g':[0,1]^m\times[0,1]^k \to X\), \(f'\) being \(\mathcal{U}\) homotopic to \(f\) and \(g'\) being \(\mathcal{U}\) homotopic to \(g\) and having the property that \(f'(\{z\}\times[0,1]^n)\cap g'(\{z\}\times[0,1]^k) =\emptyset\) for all \(z\in[0,1]^m\).

Several parts of this section are devoted to establishing when a space has the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-property. Section I.6 contains (Theorem 6.1) a selection theorem which the authors note is useful for obtaining results concerning \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties. The property of a subset of a topological space being a \(Z_n\)-set or a homotopical \(Z_n\)-set is defined. In I.7 relations between homotopical \(Z_n\)-sets and \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties are explored. A certain relation (shorthand if you will) is developed there in the 6 parts of Theorem 7.1. Again, in I.8, more notation is developed in that a type of “arithmetic” of \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties is developed. For example, Theorem 8.1 involves multiplication formulas, and expresses in an arithmetic way how these properties persist in products. Section I.9 deals with \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties of products.

The notion of homological \(Z_n\)-sets associated with an abelian group \(G\) is defined in I.10. Some basic facts are listed; the authors state that all these results have been established in [T. Banakh, R. Cauty, and A. Karassev, Topol. Proc. 38, 29–82 (2011; Zbl 1227.57029)]. In this section, the Bockstein family \(\sigma(G)\) is described. These appear in particular in Theorem 1.2. The notion of the class of Tychonoff spaces of type \(\mathcal{Z}_n^G\) is introduced. In this section, the concept of transfinite separation dimension also comes into play. In I.11 there is a discussion of the interplay between the classes \(\mathcal{Z}_n^G\) and \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\). Section I.12 contains what the authors call a “quantified” version of the homological characterization of the \(0\)-\(\overline{\mathrm{DD}}^{\{\infty,\infty\}}\)-property due to Daverman and Walsh.

A property called local rectifiability, a type of homogeneity condition, is defined in I.13. The section then shows the interplay with this property and the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-property. In I.14, there is more development of the arithmetic ideas mentioned above; in particular there are \(k\)-root and division formulas for the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties. Section I.15 is entitled, “Characterizing \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties with \(\{m,n,k\}\subset\{0,\infty\}\).” Section I.16 studies the dimensional properties of spaces possessing the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-property. Finally, Section I.17 contains some examples and lists some open problems.

This work covers 120 pages; it includes a Table of Contents, a list of 77 references, and an index.

The work is written in two parts, the first being a survey of certain results that appear in the second part. The authors of course give in the first part various definitions; they discuss some applications and interplay of their theory with existing knowledge in this area. The second part is dedicated to providing proofs of what was given in the first part.

A few definitions are in order. We presume in the following that “approximation” means approximation up to some open cover (of the given space). A space \(X\) has the \(\mathrm{DD}^n\)-property if any two maps \(f\), \(g\) of \([0,1]^n\) to \(X\) can be respectively approximated by maps \(f'\), \(g'\) such that \(f'([0,1]^n)\cap g'([0,1]^n)=\emptyset\). A parametric version of this occurs when \(M\) is a space, \(f\), \(g\) are maps of \(M\times[0,1]^n\) to \(X\), and one requires that for all \(z\in M\), \(f'(\{z\}\times[0,1]^n)\cap g'(\{z\}\times[0,1]^n)=\emptyset\). This property is called \(M\)-\(\mathrm{DD}^n\). When \(M=[0,1]^m\), one writes \(m\)-\(\mathrm{DD}^n\).

The property \(m\)-\(\overline{\mathrm{DD}}^n\) goes this way. It is required that for any maps \(f\), \(g:[0,1]^m\times[0,1]^n\to X\) and open cover \(\mathcal{U}\) of \(X\), there are maps \(f'\), \(g' :[0,1]^m\times[0,1]^n\to X\) with \(f'\) being \(\mathcal{U}\)-homotopic to \(f\), \(g'\) being \(\mathcal{U}\)-homotopic to \(g\), and \(f'(\{z\}\times[0,1]^n)\cap g'(\{z\}\times [0,1]^n)=\emptyset\) for all \(z\in [0,1]^m\). Sometimes the \(\mathcal{U}\)-homotopy concept is replaced by the notion of an \(\epsilon\)-homotopy (page 9). Proposition 3.2 describes a setting in which \(m\)-\(\overline{\mathrm{DD}}^n\) is equivalent to \(m\)-\(\mathrm{DD}^n\).

On page 7, the authors note that the set of one-to-one maps (i.e., embeddings) of a metrizable \(n\)-dimensional compactum \(K\) into a completely metrizable \(\mathrm{LC}^{n-1}\)-space \(X\) possessing the \(\mathrm{DD}^n\)-property is a dense \(\mathrm{G}_\delta\)-set in the function space \(C(K,X)\), the latter being provided with the source limitation topology. They obtain a parametric version of this result, the main output of Section I.1:

{ Theorem 1.1.} A completely metrizable \(\mathrm{LC}^{m+n}\)-space \(X\) has the \(m\)-\(\mathrm{DD}^n\)-property if and only if for every perfect map \(p:K\to M\) between finite-dimensional metrizable spaces with dim\(M\leq m\) and dim\((p)\leq n\), the function space \(C(K,X)\) contains a dense \(\mathrm{G}_\delta\)-set of maps \(f:K\to X\) that are injective on each fiber \(p^{-1}(z)\), \(z\in M\).

Section I.2 deals with the \(\Delta\)-dimension of a map between Tychonoff spaces. Proposition 2.1 compares this dimension with the classical dimension of a map and also lists some properties of this dimension. Theorem 2.2 is a version of Theorem 1.1 with \(\Delta\)-dimension in place of dimension of the given map \(p\).

In Section I.3 we are provided with a “general fiber embedding theorem”, Theorem 3.3, which involves the \(m\)-\(\overline{\mathrm{DD}}^n\)-property. This result is said to imply Theorem 1.1 as well as some of the other ones in I.1. On page 10, the notion of a Lefschetz \(\mathrm{ANE}[n]\)-space is defined. Proposition 3.4 lists 10 items related to this property. With help from this and some other propositions, Theorem 3.6 is obtained, and it is another generalization of Theorem 1.1.

Section I.4 gives us some results concerning approximating perfect maps by perfect PL-maps. A new, 3-parameter property of the \(m\)-\(\overline{\mathrm{DD}}^n\) type is introduced in I.5 as a useful generalization.

{ Definition 5.1.} A space \(X\) is defined to have the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-property if for any open cover \(\mathcal{U}\) of \(X\) and maps \(f:[0,1]^m\times[0,1]^n \to X\), \(g:[0,1]^m\times[0,1]^k \to X\), there exist maps \(f':[0,1]^m\times[0,1]^n \to X\), \(g':[0,1]^m\times[0,1]^k \to X\), \(f'\) being \(\mathcal{U}\) homotopic to \(f\) and \(g'\) being \(\mathcal{U}\) homotopic to \(g\) and having the property that \(f'(\{z\}\times[0,1]^n)\cap g'(\{z\}\times[0,1]^k) =\emptyset\) for all \(z\in[0,1]^m\).

Several parts of this section are devoted to establishing when a space has the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-property. Section I.6 contains (Theorem 6.1) a selection theorem which the authors note is useful for obtaining results concerning \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties. The property of a subset of a topological space being a \(Z_n\)-set or a homotopical \(Z_n\)-set is defined. In I.7 relations between homotopical \(Z_n\)-sets and \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties are explored. A certain relation (shorthand if you will) is developed there in the 6 parts of Theorem 7.1. Again, in I.8, more notation is developed in that a type of “arithmetic” of \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties is developed. For example, Theorem 8.1 involves multiplication formulas, and expresses in an arithmetic way how these properties persist in products. Section I.9 deals with \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties of products.

The notion of homological \(Z_n\)-sets associated with an abelian group \(G\) is defined in I.10. Some basic facts are listed; the authors state that all these results have been established in [T. Banakh, R. Cauty, and A. Karassev, Topol. Proc. 38, 29–82 (2011; Zbl 1227.57029)]. In this section, the Bockstein family \(\sigma(G)\) is described. These appear in particular in Theorem 1.2. The notion of the class of Tychonoff spaces of type \(\mathcal{Z}_n^G\) is introduced. In this section, the concept of transfinite separation dimension also comes into play. In I.11 there is a discussion of the interplay between the classes \(\mathcal{Z}_n^G\) and \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\). Section I.12 contains what the authors call a “quantified” version of the homological characterization of the \(0\)-\(\overline{\mathrm{DD}}^{\{\infty,\infty\}}\)-property due to Daverman and Walsh.

A property called local rectifiability, a type of homogeneity condition, is defined in I.13. The section then shows the interplay with this property and the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-property. In I.14, there is more development of the arithmetic ideas mentioned above; in particular there are \(k\)-root and division formulas for the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties. Section I.15 is entitled, “Characterizing \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-properties with \(\{m,n,k\}\subset\{0,\infty\}\).” Section I.16 studies the dimensional properties of spaces possessing the \(m\)-\(\overline{\mathrm{DD}}^{\{n,k\}}\)-property. Finally, Section I.17 contains some examples and lists some open problems.

This work covers 120 pages; it includes a Table of Contents, a list of 77 references, and an index.

Reviewer: Leonard R. Rubin (Norman)

### MSC:

57N75 | General position and transversality |

57Q65 | General position and transversality |

55R70 | Fibrewise topology |

54C25 | Embedding |

54C55 | Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) |

54C60 | Set-valued maps in general topology |

54C65 | Selections in general topology |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

55M10 | Dimension theory in algebraic topology |

55M15 | Absolute neighborhood retracts |

55M20 | Fixed points and coincidences in algebraic topology |