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.



Map from homotopy sphere with lifting property induces surjections on homotopy groups. Is it weak equivalence?
Yes, we can conclude that either $q$ is an equivalence or $X$ is contractible. Since any cycle lives in a compact subset of $X$, $q$ will also induce surjections on homology. It follows that $X$ is a Moore space $M(\mathbb{Z}/n,k)$ for some $n$, and $q$ is homotopy equivalent to the unique map $S^k\to M(\mathbb{Z}/n,k)$ that induces the quotient map $\mathbb{Z}\to\mathbb{Z}/n$ on homology. In particular, $X$ is homotopy equivalent to a finite CW-complex, and lifting such a homotopy equivalence to $E$ we find that $X$ is a retract of $E$ up to homotopy. In particular, this means $\mathbb{Z}\to\mathbb{Z}/n$ must split, so either $n=0$ and $X\simeq S^k$ or $n=1$ and $X$ is contractible.


