Homology Functors with Cubical Bars

Abstract

This work arose from efforts to generalise the usual cubical boundary by using different 'weights' for opposite faces, but still to obtain a chain complex, and this method was found to generalise. We describe a variant of the classical singular homology theory, in which the usual boundary (n-1)-cubes of each n-cube are replaced by combinations of internal (n-1)-cubes parallel to the boundary. This defines a generalised homology theory, but the usual singular homology can be recovered by taking the quotient by the degenerate singular cubes.