Cyclic theory for commutative differential graded algebras and s-cohomology

TL;DR

This paper introduces and compares three homotopy functors on manifolds and commutative differential graded algebras, exploring their relationships and potential topological applications, especially focusing on a new functor inspired by cyclic homology.

Contribution

It defines a new functor sH* and analyzes its properties, establishing connections with existing homology theories and topological constructs.

Findings

The functors HH*, CH*, and SH* agree on 1-connected manifolds with de-Rham algebra.

SH* cannot be directly identified with classical cyclic homology.

The paper suggests potential topological roles for sH*.

Abstract

In this paper one considers three homotopy functors on the category of manifolds, $hH^\ast, cH^\ast, sH^\ast,$ and parallel them with other three homotopy functors on the category of connected commutative differential graded algebras, $HH^\ast, CH^\ast, SH^\ast.$ If $P$ is a smooth 1-connected manifold and the algebra is the de-Rham algebra of $P$ the two pairs of functors agree but in general do not. The functors $ HH^\ast $ and $CH^\ast$ can be also derived as Hochschild resp. cyclic homology of commutative differential graded algebra, but this is not the way they are introduced here. The third $SH^\ast ,$ although inspired from negative cyclic homology, can not be identified with any sort of cyclic homology of any algebra. The functor $sH^\ast$ might play some role in topology. Important tools in the construction of the functors $HH^\ast, CH^\ast $and $SH^\ast ,$ in addition to the linear algebra suggested by cyclic theory, are Sullivan minimal model theorem and the "free loop" construction described in this paper.