Index in K-theory for families of fibred cusp operators

A families index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred cusp, pseudodifferential operators on the fibres (with boundary) of a fibration; a version of Poincare duality is also shown in this setting, identifying the stable Fredholm families with elements of a bivariant K-group.Comment: 64 pages, corrected typo

Bigerbes

The bigerbes introduced here give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form 'bundle 2-gerbes' in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski-McLaughlin bigerbe associated to a principal G-bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated principal bundle over the loop space to the structure group consisting of a central extension of the loop group; in particular, trivializations of this bigerbe for a spin manifold are in bijection with string structures on the original manifold. Other natural examples represent 'decomposable' 4-classes arising as cup products, a universal bigerbe on K(Z,4) involving its based double loop space, and the representation of any 4-class on a space by a bigerbe involving its free double loop space. The generalization to 'multigerbes' of arbitrary degree is also described.Comment: 56 pages. Version 2 includes the free loop version of the Brylinski-McLaughlin bigerbe and its relation to string structures, as well as a discussion of multigerbes of arbitrary orde

Spectral and scattering theory for symbolic potentials of order zero

The spectral and scattering theory is investigated for a generalization, to scattering metrics on two-dimensional compact manifolds with boundary, of the class of smooth potentials on the Euclidean plane which are homogeneous of degree zero near infinity. The most complete results require the additional assumption that the restriction of the potential to the circle(s) at infinity be Morse. Generalized eigenfunctions associated to the essential spectrum at non-critical energies are shown to originate both at minima and maxima, although the latter are not germane to the $L^2$ spectral theory. Asymptotic completeness is shown, both in the traditional $L^2$ sense and in the sense of tempered distributions. This leads to a definition of the scattering matrix, the structure of which will be described in a future publication.Comment: 69 page

Generalized Products and Semiclassical Quantization

The notion of a generalized product, refining that of a (symmetric and smooth) simplicial space is introduced and shown to imply the existence of an algebra of pseudodifferential operators. This encompasses many constructions of such algebras on manifolds with corners. The main examples discussed in detail here are related to the semiclassical (and adiabatic) calculus as used in the approach to a twisted form of the Atiyah-Singer index theorem in work with Is Singer and Mathai Varghese

Adiabatic Limit, Heat Kernel and Analytic Torsion

We study the uniform behavior of the heat kernel under the adiabatic limit using microlocal analysis and apply it to derive a formula for the analytic torsion. Keywords: adiabatic limit; heat kernel; singularity; microlocal analysis; analytic torsio

Resolution of the canonical fiber metrics for a lefschetz fibration

We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e., polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the logarithm of the length of the shrinking geodesic