SIMULTANEOUSLY VANISHING HIGHER DERIVED LIMITS

Jeffrey Bergfalk

Centro de Ciencias Matemáticas. UNAM, Morelia.

Date: Wednesday, September 18.
Time: 12:00
Place: Seminar room of the Institut de Matemàtica de la Universitat de Barcelona, School of Mathematics and Computer Science (UB). (IMUB)
Gran Via 585.

Abstract

In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system A with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that limnA (the n th derived limit of A) vanishes for every n > 0. Since that time, the question of whether it is consistent with the ZFC axioms that limnA = 0 for every n > 0 has remained open. In recent joint work with Chris Lambie-Hanson we show that, assuming the existence of a measurable cardinal μ, it is indeed consistent with the ZFC axioms that limn A = 0 for all n > 0. More precisely, we show that in a length-μ iteration of Hechler forcings, a condition equivalent to limn A = 0 holds for each n > 0. This condition is of set-theoretic interest in its own right: in the case of n = 1, it is the well-studied triviality of coherent families of functions indexed by ωω. This work raises a number of further questions in turn. In this talk we will outline the "limn A" problem’s passage from homological contexts through a number of independence results to the questions of the present, including recent indications that the system A is only the "least" of a wide family of inverse systems whose higher derived limits articulate deep questions in both set theory and category theory at once.

Supported by:

Organized by Joan Bagaria, Enrique Casanovas and Rafel Farré

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