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.
Tabella dei contenuti
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 forms
Bibliography
Circa l’autore
Liviu Nicolaescu is Professor at the University of Notre Dame, Indiana, USA.
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