Evaluation Fibrations and Path-Components of the Mapping Space $ M\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right) $ for 8 ≤ k ≤ 13

On path-components of the mapping spaces $$M(\mathbb {S}^m,\mathbb {F}P^n)$$ M ( S m , F P n )

“…The aim of [21] was to describe an approach which fits in well with Hansen's and leads to some further progress. In [6] the authors made use of Gottlieb groups of spheres to deal with path-components of the spaces M(S n+k , S n ) for 8 ≤ k ≤ 13.…”

“…If γ m * G m+i (S m ) = P m+i (RP m ) then f ∈ P m+i (RP m ). In particular, it holds if m ≥ 1 is odd and i ≥ 0, or m = 2 and i ≥ 0, or m ≥ 4 is even and i ≤ m − 2, or (i, m) = (6, 6),(7,4),(7,6).…”