Retract

Abstract: In [23], this author began a study of so-called lifting and approximation problems for Galois extensions. One primary point was the connection between these problems and Noether’s problem. In [24], a similar sort of study was begun for central simple algebras, with a connection to the center of generic matrices. In [25], the notion of retract rational field extension was defined, and a connection with lifting questions was claimed, which was used to complete the results in [23] and [24] about Noether's problem and generic matrices. In this paper we, first of all, set up a language which can be used to discuss lifting problems for very general “linear structures”. Retract rational extensions are defined, and proofs of their basic properties are supplied, including their connection with lifting. We also determine when the function fields of algebraic tori are retract rational, and use this to further study Noether’s problem and cyclic 2-power Galois extensions. Finally, we use the connection with lifting to show that ifp is a prime, then the center of thep degree generic division algebra is retract rational over the ground field.

Abstract: The notion of hyperconvexity is due to Aronszajn and Panitchpakdi [1] (1956) who proved that a hyperconvex space is a nonexpansive absolute retract, i.e. it is a non-expansive retract of any metric space in which it is isometrically embedded. The corresponding linear theory is well developed and associated with the names of Gleason, Goodner, Kelley and Nachbin (see for instance [19, 29, 42, 46]). The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine [54] and Soardi [57] who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces.

Abstract: We extend Lacher's result [6,7] that a closed t/V°°-maρ between locally compact, finite dimensional ANRs is a fine homotopy equivalence to the case of arbitrary separable ANRs. It is hoped that this theorem will be useful in studying manifolds modelled on the Hubert Cube. (See [1], section PF3. Added in proof. See also [9]). A set A CX has property UV if for each open set U of X containing A, there is an open V, with A C V C U such that V is null-homotopic in U. A mapping /: X -» Y of X onto Y is a l/Vmap if for each y G F , f~\y) is a C/V subset of X. The mapping / is said to be closed if the image of every closed set is closed and proper if the inverse image of every compact set is compact. An absolute neighborhood retract for metric spaces is denoted an ANR. If a is a cover of Y and gι and g2 are maps of a space A into Y,g{ is α-near g2 if for each aEA there is a U Ea containing g\(a) and g2(a). The map gi is α-homotopic to gl9gι~g2, if there is a homotopy λ: A xI-*Y taking gι to g2 with the property that for each aEA there exists UEa containing λ({α}x/). A map /: X-» Y is a fine homotopy equivalence if for each open cover, α, of Y there exists a map g: Y-+X such

Abstract: The theorem mentioned in the title is proved.