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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?

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    $\begingroup$ You do have the Serre Spectral sequence, but I'm not sure how helpful it is $\endgroup$
    – Denis Nardin
    Commented Apr 17, 2019 at 18:22
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    $\begingroup$ 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. $\endgroup$
    – Eric Peterson
    Commented Apr 17, 2019 at 20:31
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    $\begingroup$ 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/… . $\endgroup$
    – Eric Peterson
    Commented Apr 17, 2019 at 20:31
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    $\begingroup$ 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. $\endgroup$
    – Neil Strickland
    Commented Apr 17, 2019 at 21:44
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    $\begingroup$ @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. $\endgroup$
    – Nicholas Kuhn
    Commented Apr 17, 2019 at 22:16

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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.

This is enough fun for one answer, so I'll leave things here.

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  • $\begingroup$ Great, thanks! It's good to at least know that this map is decidedly nonzero for a good class of nontrucated spaces. Btw, where can I find Bousfield's cute argument? $\endgroup$
    – Tim Campion
    Commented Apr 18, 2019 at 21:50
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    $\begingroup$ @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. $\endgroup$
    – Nicholas Kuhn
    Commented Apr 19, 2019 at 2:46

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