Smooth deformations and the Gauss-Manin connection

TL;DR

This paper develops the Gauss-Manin connection for smooth deformations of associative topological algebras and uses it to establish rigidity results for cyclic cohomology in Banach algebras.

Contribution

It introduces a new application of the Gauss-Manin connection to analyze the cyclic homology and cohomology of deformed algebras, extending previous theoretical frameworks.

Findings

Defined Gauss-Manin connection on cyclic homology and cohomology

Proved rigidity of periodic cyclic cohomology for certain Banach algebras

Established foundational properties of the connection in the context of algebra deformations

Abstract

Given a smooth one parameter deformation of associative topological algebras, we define Getzler's Gauss-Manin connection on both the periodic cyclic homology and cohomology of the corresponding smooth field of algebras and investigate some basic properties. We use the Gauss-Manin connection to prove a rigidity result for periodic cyclic cohomology of Banach algebras with finite weak bidimension.