Sturmian sequences, Maslov index and Bott periodicity. (Suites de Sturm, indice de Maslov et périodicité de Bott.) (French) Zbl 1151.55001

Barge, Jean; Lannes, Jean

Progress in Mathematics 267. Basel: Birkhäuser (ISBN 978-3-7643-8709-9/hbk). vii, 199 p. (2008).

Using the mechanism and techniques of a generalization of Sturm sequences, the authors give a notion of Maslov index for algebraic loops of Lagrangians defined on commutative rings, and provide a proof of what they call the fundamental theorem of Hermitian \(K\)-theory along the lines of Karoubi-Villamayor. They also provide a proof of Bott periodicity. The contents of the book are as follows. Chapter 1 gives an interpretation of Sturm sequences in terms of signatures and Maslov indices. This chapter also includes a description of the contents of the rest of the book. Chapter 2 gives an intoduction to symplectic algebra. It gives the definitions and fundamental notions, describing Lagrangians, Sturm sequences, Sturm forms associated with Sturm sequences, and symplectic reductions. Chapter 3 contains further information on Lagrangians. Chapter 4 introduces the Maslov index and proves the fundamental theorem of Hermitian \(K\)-theory along the lines of Karoubi-Villamayor. It gives several versions, each successively more precise, and proves the topological version, which in effect is Bott periodicity. Chapter 5 shows how to use some of the ideas of the previous chapters to get variants of some results of Sharpe. It gives interpretations of certain ideas in terms of group homology. It discusses group homology and central extensions and characteristic classes. Chapter 6 consists of various generalizations. The heart is 6.2, which gives analogues of the Karoubi-Villamayor result. Also included is the algebraic version of Bott periodicity. There are 4 appendices. Appendix A gives the mechanism of using Sturm sequences and Sturm forms, and calculations of determinants. Appendix B provides a proof of a result in Chapter 2. Appendix C discusses bipartite graphs associated with transversal relations of Lagrangians. Appendix D gives a proof of an important result used in Chapter 3 regarding a theorem of Karoubi.

Reviewer: C. S. Hoo (Edmonton)

MSC:

55-02	Research exposition (monographs, survey articles) pertaining to algebraic topology
19-02	Research exposition (monographs, survey articles) pertaining to \(K\)-theory
55R45	Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity
19G38	Hermitian \(K\)-theory, relations with \(K\)-theory of rings
19G24	\(L\)-theory of group rings
19C09	Central extensions and Schur multipliers

Keywords:

Sturm sequences; Maslov index; Bott periodicity