Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ: ∂X − → Y, and let x ∈ C ∞ (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 V x = O(x 2) and which are tangent to the fibres of φ; it is a Lie algebra and C ∞ (X) module. This Lie algebra is quantized to the ‘small calculus ’ of pseudodifferential operators Ψ ∗ Φ (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.

Abstract. We prove a local index formula for cusp-pseudodifferential operators on a manifold with boundary. This is known to be equivalent to an index formula for manifolds with cylindrical ends, and hence we obtain a new proof of the classical Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary, as well as an extension of Melrose’s b-index theorem. Our approach is based on an unpublished paper by Melrose and Nistor “Homology of pseudo-differential operators I. Manifolds with boundary ” [39]. We therefore take the opportunity to review some of the results from that paper from the perspective of subsequent research on the Hochschild and cyclic homologies of algebras of pseudodifferential operators and of their applications to index theory.

Abstract. We consider complexes (X, d) of nuclear Fréchet spaces and continuous boundary maps dn with closed ranges and prove that, up to topological isomorphism, (Hn(X, d)) ∗ ∼ = H n (X ∗ , d ∗), where (Hn(X, d)) ∗ is the strong dual space of the homology group of (X, d) and H n (X ∗ , d ∗ ) is the cohomology group of the strong dual complex (X ∗ , d ∗). We use this result to establish the existence of topological isomorphisms in the Künneth formula for the cohomology of complete nuclear DF-complexes and in the Künneth formula for continuous Hochschild cohomology of nuclear ˆ⊗-algebras which are Fréchet spaces or DF-spaces for which all boundary maps of the standard homology complexes have closed ranges. We describe explicitly continuous Hochschild and cyclic cohomology groups of certain tensor products of ˆ⊗-algebras which are Fréchet spaces or nuclear DF-spaces.