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Richard Melrose

Publications and source records attributed to Richard Melrose.

At least 19 recordsLinked to original sources
arXiv2020

Equivariant K-theory and Resolution II: Non-Abelian actions

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type corresponding to the open stratum and also in an iterated sense, with connecting equivariant fibrations over the boundary hypersurfaces covering the resolutions of the other strata. This structure descends to a resolution of the quotient as a stratified space. For an Abelian group action the equivariant K-theory can then be described in terms of bundles over the bases `dressed' by the representations of the isotropy types with morphisms covering the connecting maps. A similar model is given here covering the non-Abelian case. Now the reduced objects are torsion-twisted bundles over finite covers of the bases, corresponding to the projective action of the normalizers on the representations of the isotropy groups, again with morphisms over all the boundaries. This leads to a closely related iterated deRham model for equivariant cohomology and, now with values in forms twisted by flat bundles of representation rings over the bases, for delocalized equivariant cohomology. We show, as envisioned by Baum, Brylinksi and MacPherson, that the usual equivariant Chern character, mapping to equivariant cohomology, factors through a natural Chern character from equivariant K-theory to delocalized equivariant cohomology with the latter giving an Atiyah-Hirzebruch isomorphism.

math.KT↗
arXiv2019

Compactification of semi-simple Lie groups

We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary face fibering over two copies of the corresponding flag variety with fiber modeled on the (compactification of the) reductive part. On the hd-compactification Harish-Chandra's Schwartz space is identified with a space of conormal functions of rapid-logarithmic decay relative to square-integrable functions.

math.DG↗
arXiv2018

Compactification of SL(2)

We discuss `hd-compactifications' of $\SL(2,\bbK)$ for $\bbK=\bbC$ or $\bbR.$ These are compact manifolds with boundary on which both the Schwartz and the Harish-Chandra Schwartz spaces are shown to be relatively standard spaces of conormal functions relative to the boundary. Closure under convolution and other module properties are shown to follow from the structure of appropriate generalized product spaces and the functorial properties of conormal functions and smooth maps between manifolds with corners. It is anticipated that a similar approach applies to general real reductive Lie groups, with the additional complications for $\SL(n,\bbK)$ being essentially combinatorial.

math.GR↗
arXiv2018

Equivariant K-theory and Resolution I: Abelian actions

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type, in an iterated sense, with connecting fibrations and this structure descends to a resolution of the quotient. For an abelian group action the equivariant K-theory can then be described in terms of bundles over the base with morphisms covering the connecting maps. A similar model is given, in terms of appropriately twisted deRham forms over the base as an iterated space, for delocalized equivariant cohomology in the sense of Baum, Brylinski and MacPherson. This approach allows a direct proof of their equivariant version of the Atiyah-Hirzebruch isomorphism.

math.AT↗
arXiv2017

Boundary behaviour of Weil-Petersson and fiber metrics for Riemann moduli spaces

The Weil-Petersson and Takhtajan-Zograf metrics on the Riemann moduli spaces of complex structures for an $n$-fold punctured oriented surface of genus $g,$ in the stable range $g+2n>2,$ are shown here to have complete asymptotic expansions in terms of Fenchel-Nielsen coordinates at the exceptional divisors of the Knudsen-Deligne-Mumford compactification. This is accomplished by finding a full expansion for the hyperbolic metrics on the fibers of the universal curve as they approach the complete metrics on the nodal curves above the exceptional divisors and then using a push-forward theorem for conormal densities. This refines a two-term expansion due to Obitsu-Wolpert for the conformal factor relative to the model plumbing metric which in turn refined the bound obtained by Masur. A similar expansion for the Ricci metric is also obtained.

math.DG↗
arXiv2015

Loop-fusion cohomology and transgression

`Loop-fusion cohomology' is defined on the continuous loop space of a manifold in terms of \vCech cochains satisfying two multiplicative conditions with respect to the fusion and figure-of-eight products on loops. The main result is that these cohomology groups, with coefficients in an abelian group, are isomorphic to those of the manifold and the transgression homomorphism factors through the isomorphism.

math.AT↗
arXiv2015

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.

math.DG↗
arXiv2013

Equivalence of string and fusion loop-spin structures

The importance of the fusion relation of loops was recognized in the context of spin structures on the loop space by Stolz and Teichner and further developed by Waldorf. On a spin manifold M the equivalence classes of `fusive' spin structures on the loop space LM, incorporating the fusion property, strong regularity and reparameterization-invariance, are shown to be in 1-1 correspondence with equivalence classes of string structures on M. The identification is through the affine space of `string' cohomology classes considered by Redden.

math.DG↗
arXiv2013

Equivariant cohomology and resolution

The `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution tower' which projects to a resolution, with iterated boundary fibration, of the quotient. Equivariant K-theory and the Cartan model for equivariant cohomology are tracked under the resolution procedure as is the delocalized equivariant cohomology of Baum, Brylinski and MacPherson. This leads to resolved models for each of these cohomology theories, in terms of relative objects over the resolution tower and hence to reduced models as flat-twisted relative objects over the resolution of the quotient. As a result an explicit equivariant Chern character is constructed, essentially as in the non-equivariant case, over the resolution of the quotient.

math.DG↗
arXiv2013

Delocalized equivariant cohomology and resolution

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established by the authors in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution resulting in an `equivariant resolution structure'. Here, equivariant K-theory and the Cartan model for equivariant cohomology are tracked under the resolution procedure as is the delocalized equivariant cohomology of Baum, Brylinski and MacPherson. This leads to resolved models for each of these cohomology theories, in terms of relative objects over the resolution structure and hence to reduced models as flat-twisted relative objects over the resolution of the quotient. An explicit equivariant Chern character is then constructed, essentially as in the non-equivariant case, over the resolution of the quotient.

math.AT↗
arXiv2011

Eta forms and the odd pseudodifferential families index

Let $A(t)$ be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration $ϕ$ with base $Y.$ The standard example is $A+it$ where $A$ is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and $t\in\bbR$ is the `suspending' parameter. Let $π_{\cA}:\cA(ϕ)\longrightarrow Y$ be the infinite-dimensional bundle with fibre at $y\in Y$ consisting of the Schwartz-smoothing perturbations, $q,$ making $A_y(t)+q(t)$ invertible for all $t\in\bbR.$ The total eta form, $η_{\cA},$ as described here, is an even form on $\cA(ϕ)$ which has basic differential which is an explicit representative of the odd Chern character of the index of the family: % dη_{\cA}=π_{\cA}^*γ_A, \Ch(\ind(A))=[γ_{A}]\in H^{\odd}(Y). \tag{*}\label{efatoi.5} % The 1-form part of this identity may be interpreted in terms of the $τ$ invariant (exponentiated eta invariant) as the determinant of the family. The 2-form part of the eta form may be interpreted as a B-field on the K-theory gerbe for the family $A$ with \eqref{efatoi.5} giving the `curving' as the 3-form part of the Chern character of the index. We also give `universal' versions of these constructions over a classifying space for odd K-theory. For Dirac-type operators, we relate $η_{\cA}$ with the Bismut-Cheeger eta form.

math.KT↗
arXiv2011

Diffraction of singularities for the wave equation on manifolds with corners

We consider the fundamental solution to the wave equation on a manifold with corners of arbitrary codimension. If the initial pole of the solution is appropriately situated, we show that the singularities which are diffracted by the corners (i.e., loosely speaking, are not propagated along limits of transversely reflected rays) are smoother than the main singularities of the solution. More generally, we show that subject to a hypothesis of nonfocusing, diffracted wavefronts of any solution to the wave equation are smoother than the incident singularities. These results extend our previous work on edge manifolds to a situation where the fibers of the boundary fibration, obtained here by blowup of the corner in question, are themselves manifolds with corners.

math.AP↗
arXiv2010

Relative Chern character, boundaries and index formulae

For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have `geometric K-theory', namely the `transmission algebra' introduced by Boutet de Monvel, the `zero algebra' introduced by Mazzeo and the `scattering algebra' from [MR95k:58168] we give explicit formulae for the Chern character of the index bundle in terms of the symbols (including normal operators at the boundary) of a Fredholm family of fibre operators. This involves appropriate descriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bundle over a manifold with boundary in which the Chern character, mapping from the corresponding realization of K-theory, naturally takes values.

math.DG↗
arXiv2010

Fredholm realizations of elliptic symbols on manifolds with boundary II: fibered boundary

We consider two calculi of pseudodifferential operators on manifolds with fibered boundary: Mazzeo's edge calculus, which has as local model the operators associated to products of closed manifolds with asymptotically hyperbolic spaces, and the phi calculus of Mazzeo and the second author, which is similarly modeled on products of closed manifolds with asymptotically Euclidean spaces. We construct an adiabatic calculus of operators interpolating between them, and use this to compute the `smooth' K-theory groups of the edge calculus, determine the existence of Fredholm quantizations of elliptic symbols, and establish a families index theorem in K-theory.

math.DG↗
arXiv2010

Resolution of smooth group actions

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution structure' consisting of equivariant iterated fibrations of the boundary faces. This structure projects to give a similar resolution structure for the quotient. In particular these results apply to give a canonical resolution of the radial compactification, to a ball, of any finite dimensional representation of a compact Lie group; such resolutions of the normal action of the isotropy groups appear in the boundary fibers in the general case.

math.DG↗
arXiv2010

Fredholm realizations of elliptic symbols on manifolds with boundary

We show that the existence of a Fredholm element of the zero calculus of pseudodifferential operators on a compact manifold with boundary with a given elliptic symbol is determined, up to stability, by the vanishing of the Atiyah-Bott obstruction. It follows that, up to small deformations and stability, the same symbols have Fredholm realizations in the zero calculus, in the scattering calculus and in the transmission calculus of Boutet de Monvel.

math.DG↗