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2 - The Language of Homology

Published online by Cambridge University Press:  19 January 2023

Félix Cabello Sánchez  and
Jesús M. F. Castillo
Félix Cabello Sánchez
Affiliation:
Universidad de Extremadura, Spain
Jesús M. F. Castillo
Affiliation:
Universidad de Extremadura, Spain
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Summary

The chapter introduces the basic elements of the homological language and translates the statements about complemented and uncomplemented subspaces presented in Chapter 1 into this language. The reader will find everything they need to know at this stage about exact sequences, categorical and homological constructions for absolute beginners and injective and projective Banach and quasi-Banach spaces. The chapter describes the basic homological constructions appearing in nature: complex interpolation, the Nakamura-Kakutani, Foia\c{s}-Singer, Pe\l czy\’nski-Lusky and Bourgain’s $\ell_1$ sequences, the Ciesielski-Pol, Bell-Marciszewski and Bourgain-Pisier constructions, the Johnson-Lindenstrauss spaces and so on. A good number of advanced topics are included: diagonal and parallel principles for exact sequences, the Device, 3-space results, extension and lifting of operators, $M$-ideals and vector-valued Sobczyk’s theorems

Keywords

Exact sequences of Banach and quasi-Banach spacespullbackpushoutinjective spacesprojective spacesextension and lifting of operatorsdiagonal principlesparallel principlesM-ideals.
Type
Chapter
Information
Homological Methods in Banach Space Theory , pp. 46 - 127
Publisher: Cambridge University Press
Print publication year: 2023

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