Cyclic Singular Homology

Abstract

In this study, we develop cyclic singular homology, which is a modification of singular homology in much the same way cyclic homology is a modification of Hochschild homology, and which is a generalized homology theory in terms of the Eilenberg-Steenrod axioms. To achieve this, we take advantage of a cyclic action on the relative singular chain complex induced by the cyclic category of Connes. This results in a precyclic object for which we construct a cyclic double complex. The cyclic singular homology groups of a pair of topological spaces are the total homology groups of this double complex. Having said it is a generalized homology theory, the cyclic singular homology of a point has a copy of the integers in every even dimension, and is trivial otherwise. Reduced cyclic singular homology is defined and the cyclic singular homology groups of some classical, but simple, spaces are calculated. The study is partly based on a paper from 2006 by Jinhyun Park, where he reviews cyclic homology in terms of precyclic objects. As an example application, he shows that the singular chain complex of a topological space is precyclic, and defines cyclic singular homology. He also calculates the homology groups of a point.