Higher homotopy normalities in topological groups

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The purpose of this paper is to introduce N_k(`)-maps (1 k; ` 1), which describe higher homotopy normalities, and to study their basic properties and examples. An N_k(`)-map is defined with higher homotopical conditions. It is shown that a homomorphism is an N_k(`)-map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an N_k(k)-map is shown to be an H-space if its LS category is not greater than k. As an application, we investigate when the inclusions SU(m) ! SU(n) and SO(2m + 1) ! SO(2n + 1) are p-locally N_k(`)-maps.

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