Higher homotopy normalities in topological groups

The purpose of this paper is to introduce (Formula presented.) -maps ((Formula presented.)), which describe higher homotopy normalities, and to study their basic properties and examples. An (Formula presented.) -map is defined with higher homotopical conditions. It is shown that a homomorphism is an (Formula presented.) -map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an (Formula presented.) -map is shown to be an (Formula presented.) -space if its LS category is not greater than (Formula presented.). As an application, we investigate when the inclusions (Formula presented.) and (Formula presented.) are (Formula presented.) -locally (Formula presented.) -maps.