Let $X$ be some sort of sufficiently nice space, e.g. a (connected) cell complex. Then $X$ has a universal cover $\tilde{X}$. This is simply connected by definition and it is easy to show that $\pi_n(\tilde{X})\cong\pi_n(X)$ for all $n>1$. We have, in effect, killed off a homotopy group of $X$ with a construction that seems fundamentally different than merely attaching cells to $X$ like crazy.
I wonder if there is a way to continue this process. Obviously, taking covering spaces won't get us anywhere, so we allow ourselves some more leeway. Specifically, I'm asking whether there exist spaces $X_n$ and maps $f_{n+1}\colon X_{n+1}\to X_n$ such that:
- $X_0=X$ and $X_1=\tilde{X}$, the universal cover
- $f_1\colon\tilde{X}\to X$ is the covering projection
- $f_n$ is a (either Serre or Hurewicz) fibration for all $n$
- $\pi_i(X_n)=1$ for $i\leq n$ and $\pi_i(X_n)\cong \pi_i(X)$ for $i>n$
I've look around a bit for answers to this, but I don't really know what to search for. I did find out about Postnikov towers, which seem related; I'm asking for some sort of "reverse Postnikov tower", if that makes any sense.