The Reidemeister Torsion Of 3-manifolds (de Gruyter Studies In Mathematics)

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. Cover ......Page 1 de Gruyter Studies in Mathematics Series......Page 3 Title Page ......Page 4 Publication Data......Page 5 Introduction......Page 8 Notations and conventions......Page 12 Contents......Page 14 1.1 The torsion of acyclic complexes of vector spaces......Page 16 1.2 The determinant line of a chain complex......Page 22 1.3 Basic properties of the torsion......Page 31 1.4 Some generalizations......Page 35 1.5 Abelian group algebras......Page 36 1.6 Abelian harmonic analysis......Page 45 2.1 The Reidemeister torsion of a CW-complex......Page 59 2.2 Fitting ideals......Page 74 2.3 The Alexander function and the Reidemeister torsion......Page 78 2.4 The Reidemeister torsion of 3-manifolds......Page 82 2.5 Computing the torsion of 3-manifolds using surgery presentations......Page 85 2.6 Plumbings......Page 98 2.7 Applications......Page 117 3.1 Combinatorial Euler structures......Page 121 3.2 Smooth Euler structures......Page 123 3.3 U(2) and Spinc(3)......Page 130 3.4 Euler structures on 3-manifolds......Page 135 3.5 The Reidemeister–Turaev torsion of Euler structures......Page 141 3.6 Arithmetic properties of the Reidemeister–Turaev torsion of 3-manifolds......Page 142 3.7 Axiomatic description of the Reidemeister–Turaev torsionof 3-manifolds......Page 146 3.8 The torsion of rational homology 3-spheres. Part 1......Page 150 3.9 Quadratic functions, spinc structures and charges......Page 163 3.10 The torsion of rational homology 3-spheres. Part 2......Page 179 4.1 A gauge theoretic interpretation: Seiberg–Witten invariants......Page 189 4.2 A Morse theoretic interpretation......Page 202 4.3 A spectral interpretation: the Ray–Singer analytic torsion......Page 208 A.1 Formal Hodge theory......Page 212 A.2 Determinants and zeta functions......Page 217 A.3 Extensions of Abelian groups......Page 220 B.1 How to compute the Alexander polynomial of a knot......Page 225 B.2 Dehn surgery and linking forms......Page 231 Bibliography......Page 254 Symbols......Page 260 Index......Page 262 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. "This book is addressed to mathematicians with geometric interests who want to become comfortable users of this versatile invariant. It should be accessible to graduate students who are familiar with the fundamentals of algebraic topology taught in a first year graduate course."--BOOK JACKET This work discusses the theoretical foundations of torsion, one of the oldest topological variants. It presents the work of Reidmeister, Taubes, Turaev and the author, focusing particularly on diverse examples and techniques rather than abstract generalizations The notion of torsion is a multifaceted generalization of the concept of determinant of an isomorphism of vector spaces.