arXiv research

The paper solves a general case of the cohomological relative index problem for foliations.

We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold…

We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative inde…

We prove an Atiyah-Patodi-Singer index theorem for Dirac operators twisted by C*-vector bundles. We use it to derive a general product formula for eta-forms and to define and study new rho-invariants generalizing Lott's higher rho-form. The higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza can be recovered …

The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac ope…

We prove an analogue for even dimensional manifolds of the Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles. We show that the eta invariant appearing in this result coincides with the eta invariant by Dai and Zhang up to an integer. We also obtain the odd dimensional counterpart for manifolds wit…

Let $Γ$ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois $Γ$-coverings, thus providing an explicit formula for the higher index associated to a group cocycle $c\in Z^k (Γ;\mathbb{C})$ which is of polynomial growth wit…

Constructs small bundle gerbes and proves index theorems for manifolds.

Geometric models for representations up to homotopy using simplicial vector bundles.

We prove two geometric index theorems for a family of first-order elliptic operators over a manifold with boundary by computing eta form representatives for the Chern character classes of the index bundle. The eta forms occur as relative and regularized traces on infinite-dimensional vector bundles realized as the limi…

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external p…

Generalizes Novikov conjecture results to infinite-dimensional bundles.

We announce a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle $(X,\F)$ with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary foliation, that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the cl…

This article surveys the relations among local and nonlocal invariants in Atiyah-Singer index theory. We discuss the local invariants that arise from the heat equation approach to the index theorem for geometric operators, as well as the nonlocal invariants (the eta invariant, the determinant of the Laplacian/analytic …

Paper proves nonzero foliated Rosenberg index for noncompactly enlargeable foliations.

The paper explores higher fixed point theorems for foliations with applications to rigidity and integrality.

The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a rela…

This is the sequel of the first part math.DG/0611281. Here, the procedure of transgressing the families index theorem (the so-called $η$-form) is adapted to take in account the case of Dirac type operators with kernels of varying dimension. The constructed form is then used to define the direct image under proper subme…

Uniform K-homology theory applied to elliptic operators on manifolds with boundary.

Extends a theorem for first-order elliptic operators on manifolds.

We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah's…

Paper compares higher torsions and removes fiberwise Morse function assumption.

We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle $\GR$ of Lie groups. If the fibers of $\GR \to B$ are simply-connected solvable, we then compute the Chern character of the (equivariant family) index, the result being given by an Atiyah-Singe…

We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index…

In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-typ…

In this paper, we extend Roe's cyclic $1$-cocycle to relative settings. We also prove two relative index theorems for partitioned manifolds by using its cyclic cocycle, which are generalizations of index theorems on partitioned manifolds. One of these theorems is a variant of [M. Karami-A.H.S. Sadegh-M.E. Zadeh, arXiv:…

We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold $M$, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the $C^k$-topology, $k=2,...,\infty$, in the set of metrics of …

Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.

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 Melro…

Defines new Roe algebras for cylindrical spaces, solving metric curvature problems.

In this paper we prove a relative index theorem for pairs of generalized Dirac operators on orbifolds which are the same at infinity. This generalizes to orbifolds a celebrated theorem of Gromov and Lawson.

Higher index theorem for Dirac operators on finite-volume spaces.

We exhibit geometric situations, where higher indices of the spinor Dirac operator on a spin manifold $N$ are obstructions to positive scalar curvature on an ambient manifold $M$ that contains $N$ as a submanifold. In the main result of this note, we show that the Rosenberg index of $N$ is an obstruction to positive sc…

Defines discrete differential geometry concepts in homotopy type theory.

Absolute index theorem for warped product manifolds.

This is a sequel to the paper "The signature package on Witt spaces, I. Index classes" by the same authors. In the first part we investigated, via a parametrix construction, the regularity properties of the signature operator on a stratified Witt pseudomanifold, proving, in particular, that one can define a K-homology …

We consider a {\em Hamiltonian setup} $\sextuple$, where $(\mathcal M,ω)$ is a symplectic manifold, $\mathfrak L$ is a distribution of Lagrangian subspaces in $\mathcal M$, $\mathcal P$ a Lagrangian submanifold of $\mathcal M$, $H$ is a smooth time dependent Hamiltonian function on $\mathcal M$ and $Γ:[a,b]\to\mathcal…

We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin$^-$ manifolds. The analytic index is the reduced $η$ invariant of (twisted) Dirac operators and the topological index is defined through $KO$-theory. Our main result extends the mod 2 index theorem of Atiyan and Singer to non-o…

Proves a theorem for complex flat vector bundles using differential forms.

Two proofs of Melrose-Piazza theorem on spectral sections.

Abstract: Generalizes modular forms to family case and finds new anomaly cancellation formulas.

Proves an equivariant version of index theorem for geometric families.

For two complex vector bundles admitting a homomorphism, whose singularity locates in the disjoint union of some odd--dimensional spheres, we give a formula to compute the relative Chern characteristic number of these two complex vector bundles. In particular, for a spin manifold admitting some sphere bundle structure,…

This paper contains the constructions of a real manifold version of relative K-theory, and of an extension of Karoubi's multiplicative K-theory suggested by U. Bunke (which I call ``free multiplicative K-theory'' in the sequel). Chern-Simons-Nadel type classes on relative K-theory are constructed, while it is proved th…

The purpose of this paper is to give a proof of the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac op…

The paper proves drilled bundles over graphs are virtually special cubulable.

In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the…

Defines complex structure for families of Hilbert spaces with reasonable curvature.