Mini-course: an introduction to \(\ell_2\)-homology. (English) Zbl 1113.57012

Zvengrowski, Peter

Čadek, Martin (ed.), The proceedings of the 25th winter school “Geometry and physics”, Srní, Czech Republic, January 15–22, 2006. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 79, 39-66 (2006).

From the introduction: The inspiration for this mini-course comes from similar lectures given by Beno Eckmann during one of his visits to western Canada, and the author’s subsequent attempts to understand this fascinating subject. In 2000 a substantial paper of B. Eckmann appeared in [Isr. J. Math. 117, 183–219 (2000; Zbl 0948.55006)], based on the notes (by Guido Mislin) from a mini-course he gave in 1997 at the Mathematical Research Institute, ETH Zürich. The present notes are completely based on these notes of Eckmann, with very little, if any, claim to originality. An introductory chapter (Chapter II) on basic Hilbert space theory has been added, since the subsequent material is completely based on this.
Table of contents: Chapter I, Introduction; Chapter II, Hilbert space, a Brief review; Chapter III, Hilbert \(G\)-modules and von Neumann dimension; Chapter IV, Real homology of finite complexes and harmonic chains; Chapter V, Infinite complexes and \(\ell_2\)-homology; Chapter VI, Properties of \(\ell_2\)-homology; Chapter VII, Applications.
For the entire collection see [Zbl 1103.53001].

MSC:

57R19	Algebraic topology on manifolds and differential topology
55N35	Other homology theories in algebraic topology
55-02	Research exposition (monographs, survey articles) pertaining to algebraic topology
57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes

Citations:

Zbl 0948.55006