The (co)shape and (co)homological properties of continuous maps

Abstract

The purpose of this paper is to investigatecontinuous maps from the standpoint of geometric topology andalgebraic topology. Using a direct system approach and an inversesystem approach of continuous maps, we study the (co)shape and(co)homological properties of continuous maps. Applications of theobtained results include the constructions of long exact sequencesof continuous maps for the (co)homology pro-groups and(co)homology inj-groups, spectral Čech (co)homology groupsand spectral singular (co)homology groups.