Related papers: Index theorey and Non-Commutative Geometry. II. Di…

This is a survey paper, starting from the general notion of coordinate bundle taken from Steenrod. Its aim is to provide a motivation for the introduction of cyclic homology (and the closely related noncommutative de Rham cohomology) by…

We study an index of a transversal Dirac operator on an odd-dimensional manifold $X$ with locally free $\mathbb{S}^1$-action. One difficulty of using heat kernel method lies in the understanding of the asymptotic expansion as $t\to 0^+$. By…

We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that…

This paper expresses the Chern character for topological K-theory based on the formulation of the family of Fredholm operators, by using the points at which the Fredholm operator becomes singular (Fermi points). In particular, we explain…

We explain how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations. Since the twisting of the commutators turns the…

We investigate the index of the Neuberger's Dirac operator in abelian gauge theories on finite lattices by numerically analyzing the spectrum of the hermitian Wilson-Dirac operator for a continuous family of gauge fields connecting…

We introduce a differential extension of algebraic K-theory of an algebra using Karoubi's Chern character. In doing so, we develop a necessary theory of secondary transgression forms as well as a differential refinement of the smooth…

We introduce a slight modification of the usual equivariant $KK$-theory. We use this to give a $KK$-theoretical proof of an equivariant index theorem for Dirac-Schrodinger operators on a non-compact manifold of nowhere positive curvature.…

We define the Chern character of the index class of a $G$-invariant family of $G$-transversally elliptic operators, see [6]. Next we study the Berline-Vergne formula for families in the elliptic and transversally elliptic case.

In this paper, we define the eta cochain form and prove its regularity when the kernel of a family of Dirac operators is a vector bundle. We decompose the eta form as a pairing of the eta cochain form with the Chern character of an…

Odd $K$-theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential $K$-theory using the caloron correspondence and…

For an orbifold X and $\alpha \in H^3(X, Z)$, we introduce the twisted cohomology $H^*_c(X, \alpha)$ and prove that the Connes-Chern character establishes an isomorphism between the twisted K-groups $K_\alpha^* (X) \otimes C$ and twisted…

We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators $D$ starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of $M_2(\Bbb C)$, and also applies…

We describe the second integral cohomology group of a surface bundle as the group of Chern classes of fiberwise holomorphic complex line bundles and use this to obtain information on this group.

We define a map of simplicial presheaves, the Chern character, that assigns to every sequence of composable non connection preserving isomorphisms of vector bundles with holomorphic connections an appropriate sequence of holomorphic forms.…

In these survey lectures, we investigate the geometric and analytic properties of transverse Dirac operators. In particular, we define a transverse Dirac operator associated to a distribution that is essentially self-adjoint (Prokhorenkov-R…

Let $E$ be a holomorphic vector bundle over a compact K\"{a}hler manifold $(X,\omega)$ with negative sectional curvature $sec\leq -K<0$, $\Delta_{E}$ be the Chern connection on $E$. In this article we show that if…

We extend the groundbreaking results of Gromov and Lawson on positive scalar curvature and the Dirac operator on complete Riemannian manifolds to Dirac operators defined along the leaves of foliations of non-compact complete Riemannian…

One describes, using a detailed analysis of Atiyah--Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This…

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…