The Gauss–Manin connection for the cyclic homology of smooth deformations, and noncommutative tori

Abstract

Given a smooth deformation of topological algebras, we define Getzler's Gauss-Manin connection on the periodic cyclic homology of the corresponding smooth field of algebras. Basic properties are investigated including the interaction with the Chern-Connes pairing with K-theory. We use the Gauss-Manin connection to prove a rigidity result for periodic cyclic cohomology of Banach algebras with finite weak bidimension. Then we illustrate the Gauss-Manin connection for the deformation of noncommutative tori. We use the Gauss-Manin connection to identify the periodic cyclic homology of a noncommutative torus with that of the commutative torus via a parallel translation isomorphism. We explicitly calculate the parallel translation maps and use them to describe the behavior of the Chern-Connes pairing under this deformation. * This research was partially supported under NSF grant DMS-1101382. Preliminaries 2.1 Locally convex algebras and modulesSee [34] for background in the theory of locally convex topological vector spaces. More details concerning topological tensor products can be found in [14,15,34].We shall work in the category LCTVS of complete, Hausdorff locally convex topological vector spaces over C and continuous linear maps. All bilinear/multilinear maps we work with will be assumed to be jointly continuous. For example, by a locally convex algebra, we mean a space A ∈ LCTVS with a jointly continuous associative multiplication. Similarly, a locally convex module over a locally convex algebra has a jointly continuous module action.Dealing with joint continuity leads naturally to projective tensor products. If R is a unital commutative locally convex algebra and M and N are locally convex R-modules, then M ⊗ R N denotes the completed projective tensor product over R, which is universal for jointly continuous R-bilinear maps from M × N into locally convex R-modules, see [15, Chapter II]. Given X, Y ∈ LCTVS, we shall simply write X ⊗Y for X ⊗ C Y .Given X, Y ∈ LCTVS, we equip the space Hom(X, Y ) of continuous linear maps from X to Y with the topology of uniform convergence on bounded subsets of X. It is a Hausdorff locally convex topological vector space, but it may not be complete. However, it is for many nice cases, for example if X is a Fréchet space or an LF -space. The strong dual of X is X * = Hom(X, C). We remark that the strong dual of a Banach space is a Banach space, but the strong dual of a Fréchet space is never a Fréchet space, unless the original space is actually a Banach space.Given two locally convex R-modules M and N , we topologize Hom R (M, N ) as a subspace of Hom C (M, N ). When N = R, we obtain the topological R-linear dual M := Hom R (M, R).

Invariants of the $\mathbb{Z}_2$ orbifolds of the Podleś two spheres

“…This can be related to that fact that there is no "dimension drop" in the periodic cyclic cohomology. Though Hochschild cohomology does depend on the parameter Â, but the periodic cyclic cohomolgy is  invariant [19]. In other words, we can safely say that the noncommutative torus is an ideal noncommutative manifold and the dimension drop for the Podleś quantum sphere indicates that we need to look for other invariants which may characterise the properties of spaces like Podleś quantum 2-spheres, e.g.…”

Smooth Connes--Thom isomorphism, cyclic homology, and equivariant quantisation

“…This result can also be viewed as an equivariant generalization of the homotopy invariance of periodic cyclic homology to locally convex algebras, c.f. [Con85,Get93,Goo85,Yas17].…”