On a classification of surjections and embeddings : surjective characterization of injective classes of compact spaces with applications to compact and locally compact groups

Thesis

This work has largely grown out of the papers of R. Haydon (Studia Math. 52 (1974), 23-31) and E.V. Shchepin (Russian Math. Surveys 31 : 5 (197&), 155-191) Among other results affirmative answers to problems 3, 18, 20 and 21 posed by A. Pełczyński in his monograph (Diss. Math. 58, Warszawa 1968) are given.

For pairs (m,n) of extended integers (-1≤m≤n≤∞) the surjective and injective notions of (m,n) -soft and (m,n)-hard maps respectively are introduced and investigated. It is shown that for a map X → Y of compact spaces (-1,0)-softness (resp. (0,0)-hardness) entails the existence of a regular averaging (resp. extension) operator and that the converse holds, if X is metrizable (resp. Y is a Dugundji space). (0,0)-hardness, (-1,0)-softness and (0,0)-softness are shown to be local properties and it is deduced that each point of a Dugundji space and of a quotient space of a locally compact group has a neighbourhood base of Dugundji spaces. Moreover, we prove that a space and its cone are Dugundji spaces simultaneously. (∞,∞)-soft group homomorphisms are isomorphisms. This follows from the result that a compact contractible group is trivial.

A compact space is classified as an absolute extensor for m- to n-dimensional spaces or AE(m,n), if the constant map is (m,n)~soft.

Equivalently, a compact space is an AE(m,n) if and only if every embedding into a compact space is (m,n)-hard. Haydon's paper entails coincidence of the notions of AE(0,0) and Dugundji space. Here an AE(0,0) (resp. AE(0,∞)) is surjectively characterized as (-1,0)-soft image of a generalized Cantor set (resp. Tychonoff cube). An in- jective characterization of the class of Poano spaces is given as metrizable AE(0,n), n (1≤n≤∞) being arbitrary.

Using combinatorial techniques we obtain some results concerning the countable chain condition on spaces of probability measures and spaces of ⩽ n-element subsets.

Authors: Hoffmann, B

• Peer review status: Peer reviewed
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford