This is a state-of-the-art introduction to the work of Franz Reidemeister, Meng Taubes, Turaev, and the author on the concept of torsion and its generalizations. Torsion is the oldest topological (but not with respect to homotopy) invariant that in its almost eight decades of existence has been at the center of many important and surprising discoveries. During the past decade, in the work of Vladimir Turaev, new points of view have emerged, which turned out to be the "right ones" as far as gauge theory is concerned. The book features mostly the new aspects of this venerable concept. The theoretical foundations of this subject are presented in a style accessible to those, who wish to learn and understand the main ideas of the theory. Particular emphasis is upon the many and rather diverse concrete examples and techniques which capture the subleties of the theory better than any abstract general result. Many of these examples and techniques never appeared in print before, and their choice is often justified by ongoing current research on the topology of surface singularities. The text is addressed to mathematicians with geometric interests who want to become comfortable users of this versatile invariant.

Autorentext

Liviu Nicolaescu is Professor at the University of Notre Dame, Indiana, USA.

Klappentext

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.

Zusammenfassung

"It is very well written, with many examples and also many exercises."
EMS Newsletter

"This very readable book provides an introduction to the theory of torsion."
St. Haller in: Monatshefte für Mathematik 3/2004

"This useful and well-written book gives an introduction to current developments in the theory and applications of Reidemeister torsion. [...] The particular strength of Nicolaescu's book is perhaps in the examples and its nature as a `user's guide' to torsion. These aspects will make it equally valuable to both established researchers and graduate students trying to gain experience in the field." Mathematical Reviews

Inhalt
Algebraic Preliminaries · The torsion of acyclic complexes of vector spaces · The determinant line of a chain complex · Basic properties of the torsion · Some generalizations · Abelian group algebras · Abelian harmonic analysis · The Reidemeister Torsion · The Reidemeister torsion of a CW-complex · Fitting ideals · The Alexander function and the Reidemeister torsion · The Reidemeister torsion of 3-manifolds · Computing the torsion of 3-manifolds using surgery presentations · Plumbings · Applications · Turaev's Refined Torsion · Combinatorial Euler structures · Smooth Euler structures · U(2) and Spinc(3) · Euler structures on 3-manifolds · The Reidemeister-Turaev torsion of Euler structures · Arithmetic properties of the Reidemeister-Turaev torsion of 3-manifolds · Axiomatic description of the Reidemeister-Turaev torsion of 3-manifolds · The torsion of rational homology 3-spheres. Part 1 · Quadratic functions, Spinc structures and charges · The torsion of rational homology 3-spheres. Part 2 · Alternative Interpretations of the Reidemeister Torsion · A gauge theoretic interpretation: Seiberg-Witten invariants · A Morse theoretic interpretation · A spectral interpretation: the Ray-Singer analytic torsion · Algebra · Formal Hodge theory · Determinants and zeta functions · Extensions of Abelian groups · Topology · How to compute the Alexander polynomial of a knot · Dehn surgery and linking formsBibliography