Searcharxiv⌕ Search

arXiv subjects

Richard B. Melrose

Publications and source records attributed to Richard B. Melrose.

11 recordsLinked to original sources
arXiv2025

Index and small bundle gerbes

By a small bundle gerbe we mean a bundle gerbe in the sense of Murray defined on a smooth, finite-dimensional, fibre bundle over a manifold. We construct such gerbes over compact oriented aspherical 3-manifolds, as well as in higher dimensions, generalizing the construction of decomposable bundle gerbes in earlier work with Singer. For these small bundle gerbes there is a direct index map given in terms of either fibrewise pseudodifferential operators, or more conveniently fibrewise semiclassical smoothing operators, twisted by the simplicial line bundle. We prove the Atiyah-Singer type theorem that this realizes the push-forward into twisted K-theory. We also give an application via the index of projective families of Spin_c Dirac operators, to show the existence of obstructions to metrics with large positive scalar curvature.

math.DG↗
arXiv2024

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.

math.DG↗
arXiv2022

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.

math.AT↗
arXiv2013

Generalized blow-up of corners and fiber products

Real blow-up, including inhomogeneous versions, of boundary faces of a manifold (with corners) is an important tool for resolving singularities, degeneracies and competing notions of homogeneity. These constructions are shown to be particular cases of `generalized boundary blow-up' in which a new manifold and blow-down map are constructed from, and conversely determine, combinatorial data at the boundary faces in the form of a refinement of the `basic monoidal complex' of the manifold. This data specifies which notion of homogeneity is realized at each of the boundary hypersurfaces in the blown-up space. As an application of this theory, the existence of fiber products is examined for the natural smooth maps in this context, the b-maps. Transversality of the b-differentials is shown to ensure that the set-theoretic fiber product of two maps is a `binomial variety'. Properties of these (extrinsically defined) spaces, which generalize manifolds but have mild singularities at the boundary, are investigated and a condition on the basic monoidal complex is found under which the variety has a smooth structure. Applied to b-maps this additional condition with transversality leads to a universal fiber product in the context of manifolds with corners. Under the transversality condition alone the fiber product is resolvable to a smooth manifold by generalized blow-up and then has a weaker form of the universal mapping property requiring blow-up of the domain.

math.GT↗
arXiv2011

A remark on distributions and the de Rham theorem

We show that the de Rham theorem, interpreted as the isomorphism between distributional de Rham cohomology and simplicial homology in the dual dimension for a simplicial decomposition of a compact oriented manifold, is a straightforward consequence of elementary properties of currents. The explicit construction of this isomorphism extends to other cases, such as relative and absolute cohomology spaces of manifolds with corners.

math.DG↗
arXiv2006

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.

math.DG↗
arXiv2001

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.

math.AP↗
arXiv2000

Singularities and the wave equation on conic spaces

Let $X$ be a manifold with boundary, endowed with a metric with conic singularities at the boundary components of $X$. Let $u$ be a solution to the wave equation on $\mathbb{R} \times X$. When a singularity of $u$ strikes a cone point of $X$, it undergoes a mixture of diffractive spreading and geometric propagation.

math.AP↗
arXiv1998

Pseudodifferential operators on manifolds with fibred boundaries

Let $X$ be a compact manifold with boundary. Suppose that the boundary is fibred, $ϕ:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary defining function. This data fixes the space of `fibred cusp' vector fields, consisting of those vector fields $V$ on $X$ satisfying $Vx=O(x^2)$ and which are tangent to the fibres of $ϕ;$ it is a Lie algebra and $\CI(X)$ module. This Lie algebra is quantized to the `small calculus' of pseudodifferential operators $\PsiF*(X).$ Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an `exact fibred cusp' metric is analyzed as is the wavefront set associated to the calculus.

math.DG↗
arXiv1996

Homology of pseudodifferential operators I. Manifolds with boundary

The Hochschild and cyclic homology groups are computed for the algebra of `cusp' pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and evaluated in terms of extensions of the trace functionals on the two natural ideals, corresponding to the two filtrations by interior order and vanishing degree at the boundary, together with the exterior derivations of the algebra. This leads to an index formula which is a pseudodifferential extension of that of Atiyah, Patodi and Singer for Dirac operators; together with a symbolic term it involves the `eta' invariant on the suspended algebra over the boundary previously introduced by the first author.

funct-an↗