What can be said about the map $K(n)_\ast(X) \to K(n)_\ast(\tau_{\leq n}X)$ when $X$ is a finite complex?

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?

• You do have the Serre Spectral sequence, but I'm not sure how helpful it is – Denis Nardin Apr 17 at 18:22
• I wouldn't think that there is any meaningful relationship. For $E$ stable, there is an equivalence $L_{K(n)} \tau_{\ge k} E \simeq L_{K(n)} E$, so that all of the information that $L_{K(n)}$ (or $K(n)_*$) sees is attached to the "germ at infinity" of $E$ (a memorable slogan credited to Dwyer). Since the parts of $\Sigma^\infty_+ X$ and $\Sigma^\infty_+ \tau_{\le k} X$ near infinity can be arbitrarily far apart without very strong hypotheses on $X$, their $K(n)$-homologies are not likely to bear any particular relation. – Eric Peterson Apr 17 at 20:31
• You can make 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/… . – Eric Peterson Apr 17 at 20:31
• I don't know a good answer, but I suspect that this could be a fruitful area to investigate. One possible approach: the above comments tell us about $K(n)_*(\Omega^\infty X)$ for $X$ in the thick subcategory generated by $H=BP\langle 0\rangle$, and some of the proofs can be given in terms of the Bousfield-Kuhn functor $\Phi_n$ which satisfies $\Phi(\Omega^\infty X)=L_{K(n)}X$. So you can proceed to investigate $K(n)_*(\Omega^\infty X)$ when $X$ is in the thick subcategory generated by $BP\langle m\rangle$ with $m>0$, using the theory of Wilson spaces. – Neil Strickland Apr 17 at 21:44
• @NeilStrickland Neil and others: I wrote a long paper on the Morava K-theory of infinite loopspaces [Adv. Math. 201 (2006), 318-378]. It is quite definitive, and, yes, does use the $\Phi_n$ functors. Folks are encouraged to look at this to get a sense of weird stuff that can happen. – Nicholas Kuhn Apr 17 at 22:16

Let me make more concrete a comment I already wrote. In [Adv. Math. 201 (2006), 318-378], my example 2.22, illustrating a theorem just before it when $$n=1$$, says that, for any spectrum $$Y$$, there is a short exact sequence of Hopf algebras over $$K(1)_*$$ as follows: $$K(1)_*(\mathbb PY) \rightarrow K(1)_*(\Omega^{\infty}Y) \rightarrow K(1)_*(\tau_{\leq 2}\Omega^{\infty}Y).$$ Here $$\mathbb PY$$ is the free $$E_\infty$$--algebra generated by $$Y$$: the wedge of all the extended powers of $$Y$$.
So if $$X = \Omega^{\infty}Y$$, then $$K(1)_*(X) \rightarrow K(1)_*(\tau_{\leq 2}X)$$ is onto. One could wonder if one has an epimorphism for other $$X$$. Okay, lets try something else: Bousfield gave a cute short argument that if $$X$$ is $$E_*$$--acyclic, so is $$K(\pi_j(X), j)$$ for any $$j$$. So if $$X$$ is $$K(1)_*$$--acyclic, so is $$\tau_{\leq 2}X$$, and thus our map is still an epi.
• @TimCampion See [Bousfield, A. K. On homology equivalences and homological localizations of spaces. Amer. J. Math. 104 (1982), no. 5, 1025–1042.] The cute idea: $SP^{\infty}$ preserves homology isomorphisms. – Nicholas Kuhn Apr 19 at 2:46