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We estimate the number of homotopy types of path-components of the mapping spaces M(S m , FP n) from the m-sphere S m to the projective space FP n for F being the real numbers R, the complex numbers C, or the skew algebra H of quaternions. Then, the homotopy types of path-components of the mapping spaces M(E m , FP n) for the suspension E m of a homology m-sphere m are studied as well.

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Porter's approach is used to derive some properties of higher order Whitehead products, similar to those ones for triple products obtained by Hardie.Computations concerning the higher order Whitehead product for spheres and projective spaces are presented as well.

Given a map {f\colon X\to Y} , we extend Gottlieb’s result to the generalized Gottlieb group {G^{f}(Y,f(x_{0}))} and show that the canonical isomorphism {\pi_{1}(Y,f(x_{0}))\overset{\approx}{\to}\mathcal{D}(Y)} restricts to an isomorphism {G^{f}(Y,f(x_{0}))\overset{\approx}{\to}\mathcal{D}^{\tilde{f}_{0}}(Y)} , where {\mathcal{D}^{\tilde{f}_{0}}(Y)} is some subset of the group {\mathcal{D}(Y)} of deck transformations of Y for a fixed lifting {\tilde{f}_{0}} of f with respect to universal coverings of X and Y, respectively.