Ravenel and Wilson showed that $K(\mathbb Z / p^j,q)$ is $K(n)$-acyclic for any $q \geq n+1$, and that $K(\mathbb Z, q)$ is $K(n)$-acyclic for $q \geq n+2$. It follows that $K(A,q)$ is $K(n)$-acyclic for $q \geq n+2$ when $A$ is finitely-generated.

From here, a Serre spectral sequence argument reveals that the map $\tau_{\leq m} X \to \tau_{\leq n+1} X$ (where $\tau$ is Postnikov truncation) is a $K(n)$-local equivalence for all $m \geq n+1$ when $X$ has finitely-generated homotopy groups. (For $\pi$-finite spaces, $\tau_{\leq m} X \to \tau_{\leq n} X$ is in fact a $K(n)$-local equivalence, as observed by Carmeli,Schlank, and Yanovski).

It's tempting to conclude that $X \to \tau_{\leq n+1} X$ is a $K(n)$-local equivalence for any $X$ with finitely-generated homotopy groups, but this can't possibly be true. If it were true, then in particular $X \to \tau_{\leq n+1} X$ would be an equivalence for all simply-connected finite spaces $X$. Then we could conclude $K(n)_\ast(S^2) = K(n)_{\ast+n+1}(\Sigma^{n+1} S^2) = K(n)_{\ast+n+1}(pt)$, which is false.

Indeed, according to Bauer, the convergence of the spectral sequence for the Postnikov tower of $X$ only holds when $X$ is $n$-truncated. This leads to my

**Question:** If $X$ is a space with infinitely many nontrivial homotopy groups, is there any meaningful relationship between $K(n)_\ast(X)$ and $K(n)_\ast(\tau_{\leq n} X)$ or $K(n)_\ast(\tau_{\leq n+1} X)$? (Beyond the mere existence of a map -- for all I know, this map is zero!) How about if $X$ is finite? Or perhaps, what if $X$ is $(n-1)$-connected?

verystrong hypotheses on $X$, their $K(n)$-homologies are not likely to bear any particular relation. $\endgroup$ – Eric Peterson Apr 17 '19 at 20:31canmake pretty strong statements about what $K(n)_* \tau_{\le k} X$ looks like, though, which you might find separately interesting: math.jhu.edu/~wsw/papers2/math/… . $\endgroup$ – Eric Peterson Apr 17 '19 at 20:312more comments