Normal subgroup is generalized to crossed module. A crossed module consists of a homomorphism H -> G and a compatible ``conjugation'' action G -> Aut(H). Generalizing the fact that the quotient by a normal subgroup becomes a group, Farjoun and Segev have shown that the homotopy quotient of a topological crossed module is canonically a topological group. We introduce higher homotopy theoretic generalizations called N_k(l)-maps (k,l ≥ 1), which are more restrictive classes of classical homotopy normlaities defined by McCarty and James. N_k(l)-map has a good characterization using fiberwise projective spaces. The homotopy quotient of an N_k(l)-map is shown to be an H-space if it has small LS category. As an application, we study when the inclusion SU(m) -> SU(n) is p-locally an N_k(l)-map.

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