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2 results

  • 2 - The Language of Homology

  • Félix Cabello Sánchez, Universidad de Extremadura, Spain, Jesús M. F. Castillo, Universidad de Extremadura, Spain
  • Book:
    Homological Methods in Banach Space Theory
    Published online:
    19 January 2023
    Print publication:
    26 January 2023, pp 46-127

  • 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

  • Nonsingular bilinear maps revisited

  • Part of
  • Carlos Domínguez, Kee Yuen Lam
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 1 / February 2021
    Published online by Cambridge University Press:
    17 March 2020, pp. 377-390
    Print publication:
    February 2021

  • A bilinear map $\varPhi :\mathbb {R}^r\times \mathbb {R}^s\to \mathbb {R}^n$ is nonsingular if $\varPhi (\overrightarrow {a},\overrightarrow {b})=\overrightarrow {0}$ implies $\overrightarrow {a}=\overrightarrow {0}$ or $\overrightarrow {b}=\overrightarrow {0}$. These maps are of interest to topologists, and are instrumental for the study of vector bundles over real projective spaces. The main purpose of this paper is to produce examples of such maps in the range $24\leqslant r\leqslant 32,\ 24\leqslant s\leqslant 32,$ using the arithmetic of octonions (otherwise known as Cayley numbers) as an effective tool. While previous constructions in lower dimensional cases use ad hoc techniques, our construction follows a systematic procedure and subsumes those techniques into a uniform perspective.