Generalizing work of W. Müller we investigate the spectral theory for the Dirac operator D on a noncompact manifold X with generalized fibred cusps $$ C(M)=M\times [A,\infty[_r, g= d r^2+ ϕ^*g_Y+ e^{-2cr}g_Z, $$ at infinity. Here $ϕ:M^{h+v}\to Y^h$ is a compact fibre bundle with fibre Z and a distinguished horizontal space HM. The metric $g_Z$ is a metric in the fibres and $g_Y$ is a metric on the base of the fibration. We also assume that the kernel of the vertical Dirac operator at infinity forms a vector bundle over $Y$.
Using the ``$ϕ$-calculus'' developed by R. Mazzeo and R. Melrose we explicitly construct the meromorphic continuation of the resolvent $G(λ)$ of D for small spectral parameter as a special ``conormal distribution''. From this we deduce a description of the generalized eigensections and of the spectral measure of D.
Complementing this, we perform an explicit construction of the heat kernel $[\exp(-tD^2)]$ for finite and small times t, corresponding to large spectral parameter $λ$. Using a generalization of Getzler's technique, due to R. Melrose, we can describe the singular terms in the heat kernel expansion and prove an index formula for D, calculating the extended $L^2$-index of D in terms of the usual local expression, the family eta invariant for the family of vertical Dirac operators at infinity and the eta invariant for the horizontal ``Dirac'' operator at infinity.
On a smoothly stratified space, we identify intersection cohomology of any given perversity with an associated weighted $L^2$ cohomology for iterated fibred cusp metrics on the smooth stratum. In particular given a Witt space, we identify the $L^2$ cohomology of iterated fibred cusp metrics with the middle perversity intersection cohomology of the corresponding stratified space.
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.
In the framework of fibred cusp operators on a manifold $X$ associated to a boundary fibration $Φ: \pa X\to Y$, the homotopy groups of the space of invertible smoothing perturbations of the identity are computed in terms of the K-theory of $T^{*}Y$. It is shown that there is a periodicity, namely the odd and the even homotopy groups are isomorphic among themselves. To obtain this result, one of the important steps is the description of the index of a Fredholm smoothing perturbation of the identity in terms of an associated K-class in the K-theory of $T^{*}Y$.
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. Th…
For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of $L^2$ harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of…
We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends.
One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the Φ-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Melrose, that a stratified pseudomanifold can be `resolved' into a manifold with fibred corners. This allows us to define pseudodifferential operators as conormal distributions on a suitably blown-up double space. Various symbol maps are introduced, leading to the notion of full ell…
In this paper we continue the analysis of spectral problems in the setting of complete manifolds with fibred boundary metrics, also referred to as $ϕ$-metrics, as initiated in our previous work. We consider the Hodge Laplacian for a $ϕ$-metric and construct the corresponding heat kernel as a polyhomogeneous conormal distribution on an appropriate manifold with corners. Our discussion is a generalization of an earlier work by Albin and Sher, and provides a fundamental first step towards analysis of Ray-Singer torsion, eta-invariants and index theorems in the setting.
This paper is concerned with pseudodifferential calculus on manifolds with fibred corners. Following work of Connes, Monthubert, Skandalis and Androulidakis, we associate to every manifold with fibred corners a longitudinally smooth groupoid which algebraic and differential structure is explicitely described. This groupoid has a natural geometric meaning as a holonomy groupoid of singular foliation, it is a singular leaf space in the sense of Androulidakis and Skandalis. We then show that the associated compactly supported pseudodifferential calculus coincides with Mazzeo and Melrose's phi-cal…
We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior and a term that comes from the boundary. This answers the index problem formulated by Mazzeo and Melrose. We give a more precise answer in the case where the base of the boundary fiber bundle is the circle. In particular, for Dir…
Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the family index theorem of Melrose and Piazza and by treating the index of families of such operators. We also extend the index theorem of Moroianu and Leichtnam-Mazzeo-Piazza to families of perturbed Dirac-type operators associated to fibered cusp metrics (sometimes known as fibered b…
The cobordism invariance of the index on closed manifolds is reproved using the calculus of cusp pseudodifferential operators on a manifold with boundary. More generally, on a compact manifold with corners, the existence of a symmetric cusp differential operator of order 1 and of Dirac type near the boundary implies that the sum of the indices of the induced operators on the hyperfaces is null.
We prove that the indices of fibered-cusp and $d$-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an index formula for the Dirac operator of horn-cone and of fibered horn metrics.
An analytic index is defined for a family of cusp pseudodifferential operators, $P_b,$ on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation $Q_b$ by a family of cusp operators of order $-\infty$ such that each $P_b+Q_b$ is invertible. Thus any elliptic family of symbols has a realization as an invertible family of cusp pseudodifferential operators, which is a form of the cobordism invariance of the index. A crucial role is played by the weak contractibil…
We study the space of L^2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincaré metrics on Q-rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in Kähler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The L^2 signatur…
Let $X$ be a smooth compact manifold with corners which has two embedded boundary hypersurfaces $\partial_0 X , \partial_1 X$, and a fiber bundle $ϕ:\partial_0 X \to Y$ is given. By using the method of blowing up, we define a pseudodifferential culculus $Ψ^* _{Φ,b} (X)$ generalizing the $Φ$-calculus of Mazzeo and Melrose and the (small) $b$-calculus of Melrose. We discuss the Fredholm condition of such operators and prove the relative index theorem. And as its application, the index theorem of "non-closed" $\mathbb{Z}/k$ - manifolds is proved.
We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.
Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration `resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta inva…
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; togeth…
