Families Torsion and Morse Functions
The relation between classical Ray-Singer torsion and Reidemeister torsion was studied by many mathematicians (including J. Cheeger, W. Müller, J. Lot, M. Rothenberg, J.-M. Bismut, W. Zhang and U. Bunke). In this book, the authors construct an equivariant version of the theory of analytic torsion, including a suitable normalization of analytic torsions forms. They show that equivariant torsion forms in higher degrees are essentially invariant under a deformation of the considered flat connection. They evaluate equivariant torsion forms (up to coboundaries) for fibrations satisfying suitable assumptions and they prove a formula for equivariant torsion forms in the case of unit sphere bundles. Tools and methods used in the book include superconnections and Chern-Simons theory, the (extended) de Rham map, the Witten deformation of the de Rham operator and instanton calculus, local families index techniques, the Berezin integration and localization of estimates using finite propagation speed. The book contains a full presentation of results announced earlier by the authors in Compt.Rend.Acad.Sci.