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I am trying to prove this relatively simple statement from 'Nilpotence and periodicity in Stable Homotopy Theory'

Suppose $X$ as in the periodicity theorem has type $n$. Then the cofibre of the map 1.54 has type $n+1$

The definitions needed:

  • $X$ is a $p$-local finite CW-complex of type $n$
  • A finite $p$-local CW complex is a CW complex $X_{(p)}$ associated with a finite CW complex $X$ (this is the topological analogue of localising at a prime)
  • A $p$-local finite complex $X$ has type $n$ if $n$ is the smallest integer such that $\overline{K(n)_*}(X)$ is non-trivial (where $\overline{K(n)_*}(X)$ is the $n$-th Morava K-theory)
  • The map of 1.54 is a map $f:\Sigma^dX \to X$ such that $K(n)_*(f)$ is an isomorphism and $K(m)_*(f)$ is trivial for $m > n$
  • Let $W$ be the cofibre of the map $f$

We know that from the cofibre sequence $X \to \Sigma^d X \to W \to \Sigma X \to \Sigma^{d+1}X \to ...$ that we get a long exact sequence in the Morava K-homology $$ \cdots \to \overline{K(m)}_t(\Sigma^d X) \stackrel{f^*}{\to} \overline{K(m)}_t(X) \to \overline{K(m)}_t(W) \to \overline{K(m)}_{t-1}(\Sigma^d X)\stackrel{f^*}{\to} \cdots$$

For $m>n$ we get $f^*=0$ and we know that $\overline{K(m)_*}(X) \ne 0$ (by another result) and so we get short exact sequences

$$ 0 \to \overline{K(m)}_t(X) \to \overline{K(m)}_t(W) \to \overline{K(m)}_{t-1}(\Sigma^d X) \simeq \overline{K(m)_{t}}(\Sigma^{d+1}X) \to 0$$

The result follows if the sequence splits (and indeed this is what Ravanel claims),that: $$\overline{K(m)}_*(W) = \overline{K(m)}_*(X) \oplus \overline{K(m)}_{*}(\Sigma^{d+1}X)$$

I can't see a justification for this splitting - is there any obvious reason this sequence splits?

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up vote 3 down vote accepted

By Prop 1.5.2 on p. 7, every graded module over $K(n)_*$ is free, and $K(n)_*(X)$ is a graded module over $K(n)_*$. That gives your splitting.

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Thank you, much simpler then I was imagining! –  Juan S Oct 20 '11 at 1:56
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