Characterization of precompact shape and homology properties of remainders

In this paper a precompact shape theory is investigated. Necessary and sufficient conditions are found for which the precompact shapes of remainders are coinsided. An intrinsically characterization of Čech (co)homology groups of remainders is given. Border cohomological dimension, dimA∞X, and coefficient of border cyclicity, ηA∞X, are defined and the inequality dimA∞X⩽dimA(cX⧹X) and the equality ηA∞X=ηA(cX⧹X) are proved for a space X normally adjoined to its remainder.