Let $E$ be a homology theory whose coefficients $E_*(pt)$ are concentrated in even dimensions. This could be complex bordism $MU$, but also complex $K$-theory, $BP\langle n \rangle$, $E(n)$, ...

Let now $X$ be a $CW$-complex whose cells are concentrated in even dimensions. Then the Atiyah-Hirzebruch spectral sequence

$H_*(X; E(pt)) \Rightarrow E(X)$

collapses and we get that $E(X)$ is free as a $E_*(pt)$-module. Examples include $X = \mathbb{CP}^\infty$ or $X = BU$.

If $E_*(pt)$ is torsionfree for a homology theory $E$, then there are also other constructions of spaces with free $E$-homology. For example, one can choose a torsion-element $x\in \pi_{n+k}(S^n)$ and take the cone $C(x)$. Since every simply-connected finite $CW$-complex has torsion in its homotopy groups, this construction can be iterated arbitrarily far. The problem is that this procedure is not explicit at all since homotopy groups are hard to calculate.

So my question is the following:

Is there an explicit infinite $CW$-complex $X$ such that the following three conditions are fulfilled?

- Its skeleta have free $E_*$-homology (for one of the examples of $E$ above).
- Its skeleta are indecomposable in the stable category.
- $X$ is not concentrated in even degrees.