The Entire Cyclic Cohomology of Noncommutative 3-Spheres

TL;DR

This paper calculates the entire cyclic cohomology of noncommutative 3-spheres, showing it is isomorphic to the de Rham homology of the ordinary 3-sphere, using Mayer-Vietoris sequences in a noncommutative setting.

Contribution

It establishes the entire cyclic cohomology of noncommutative 3-spheres and connects it to classical de Rham homology, extending previous noncommutative geometric results.

Findings

Cyclic cohomology of noncommutative 3-spheres matches classical de Rham homology.

Verification of Mayer-Vietoris sequence in the context of Fréchet *-algebras.

Application of noncommutative Heegaard decomposition to compute cohomology.

Abstract

In this paper, we compute the entire cyclic cohomology of noncommutative 3-spheres. First of all, we verify the Mayer-Vietoris exact sequence of entire cyclic cohomology in the framework of Fr\'echet $^*$-algebras. Applying it to their noncommutative Heegaard decomposition, we deduce that their entire cyclic cohomology is isomorphic to the d'Rham homology of the ordinary 3-sphere with the complex coefficients.