It is known that if any prime power branched cyclic cover of a knot in S is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in S whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of F(1.0)/F(1.5) for which Casson-Gordon invari...

We study the periodic cyclic homology groups of the cross-product of a finite type algebra A by a discrete group Γ. In case A is commutative and Γ is finite, our results are complete and given in terms of the singular cohomology of the sets of fixed points. These groups identify our cyclic homology groups with the “orbifold cohomology” of the underlying (algebraic) orbifold. The proof is based ...

We provide a new construction of the topological cyclic homology TC(C) of any spectrally-enriched ∞-category C, which affords a precise algebro-geometric interpretation of the cyclotomic trace map K(X) → TC(X) from algebraic K-theory to topological cyclic homology for any scheme X. This construction rests on a new identification of the cyclotomic structure on THH(C), which we find to be a conse...

A Fréchet algebra endowed with a multiplicatively convex topology has two types of invariants: homotopy invariants (topological K-theory and periodic cyclic homology) and secondary invariants (multiplicative Ktheory and the non-periodic versions of cyclic homology). The first aim of this paper is to establish a Riemann-Roch-Grothendieck theorem which describes direct images for homotopy and sec...

The central result of this paper is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby gener...