Let $E$ be homotopy equivalent to a $k$-sphere. Let $q\colon E\to X$ be a map such that given any continuous $f\colon C\to X$ from a compact space $C$, there exists (a non-unique) $\tilde{f}\colon C\to E$ with $q\tilde{f}=f$. Assume also that $X$ is a CW complex, but possibly infinite-dimensional.

Can we say anything at all about the homotopy groups of $X$ besides that they must be quotients of the homotopy groups of $\mathbb{S^k}$?

I apologize if this is too easy for the forum, this is not my area.

Let $E$ be homotopy equivalent to a $k$-sphere. Let $q\colon E\to X$ be a map such that given any continuous $f\colon C\to X$ from a compact space $C$, there exists (a non-unique) $\tilde{f}\colon C\to E$ with $q\tilde{f}=f$. Assume also that $X$ is a connected CW complex, but possibly infinite-dimensional.

Can we say anything at all about the homotopy groups of $X$ besides that they must be quotients of the homotopy groups of $\mathbb{S^k}$?

I apologize if this is too easy for the forum, this is not my area.

Let $q\colon \mathbb{S}^k\to X$ be a map such that given any continuous $f\colon K\to X$ from a compact space $K$, there exists (a non-unique) $\tilde{f}\colon K\to \mathbb{S}^k$ with $q\tilde{f}=f$. Assume also that $X$ is a CW complex, but possibly infinite-dimensional.

Can we say anything at all about the homotopy groups of $X$ besides that they must be quotients of the homotopy groups of $\mathbb{S^k}$?

I apologize if this is too easy for the forum, this is not my area.

Let $E$ be homotopy equivalent to a $k$-sphere. Let $q\colon E\to X$ be a map such that given any continuous $f\colon C\to X$ from a compact space $C$, there exists (a non-unique) $\tilde{f}\colon C\to E$ with $q\tilde{f}=f$. Assume also that $X$ is a CW complex, but possibly infinite-dimensional.

Can we say anything at all about the homotopy groups of $X$ besides that they must be quotients of the homotopy groups of $\mathbb{S^k}$?

I apologize if this is too easy for the forum, this is not my area.