Space of Non-Surjective maps between Spheres

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Let $\text{Map}_{ns}(S^n,S^n)$ denote the space of all continuous, non-surjective maps from $S^n$ to $S^n$ carrying the compact-open topology. There is the following fibration:$$ \text{Map}_{ns}((S^n,N),(S^n,N))\to \text{Map}_{ns}(S^n,S^n)\to S^n $$ where the last map is evaluation at the north pole $N$ and $\text{Map}_{ns}((S^n,N),(S^n,N))$ is the subspace of pointed non-surjective maps.

My question is wether evaluation at the north pole is a (weak) homotopy equivalence or in other words wether $\text{Map}_{ns}((S^n,N),(S^n,N))$ is (weakly) contractible. It might also be interesting wether it makes a difference if one restricts to smooth maps (I'm a bit hesistant to just cite smooth approximation in this case).

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Smooth approximation does indeed show that the homotopy type doesn't depend on smoothness. The point is that the set of surjective maps is closed by compactness, and so given any compact family $f: X \times S^n \to S^n$ of non-surjective maps, the infimum $$D = \inf_x d(f_x, \text{Map}_{\text{surj}})$$ is strictly positive. Here distance is measured in $C^0$.

Because you can apply smooth approximation to any compact family while changing the $C^0$ norm by an arbitrarily small amount, you can in fact smoothly approximate while moving the whole family by $C^0$ norm less than $D$, so that you can't reach a surjective map by such a small perturbation.

The actual question you ask strikes me as hard but I bet it is false for $n > 1$. It strikes me that it is probably equivalent to whether or not there is a continuous map $r: \text{Map}^*_{ns}(S^n, S^n) \to S^n$ (the domain being pointed non-surj maps) with $$r(f) \not \in \text{Im}(f).$$ I do not have an idea for constructing that map.