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Is there an example of a simply connected (included following Michael's answer) CW complex with homology/cohomology concentrated in even degrees and levelwise a free abelian group, which does not admit a cell structure with cells in only even dimensions?

I believe the rational version of this question with rational CW-complexes has every example admitting such a cell structure.

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    $\begingroup$ Hatcher's Algebraic Topology has a section devoted to this problem $\endgroup$
    – user17786
    Jan 9, 2020 at 12:39
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    $\begingroup$ Wow that result is much stronger than I would have expected. For anyone interested, a simply connected CW complex with finitely generates homology groups can always have a CW structure chosen that has as few cells as the homology allows. $\endgroup$ Jan 9, 2020 at 20:56

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Let $M$ be an integral homology sphere of dimension $2n$. Then

$$H_i(M; \mathbb{Z}) \cong \begin{cases} \mathbb{Z} & i = 0, 2n\\ 0 & \text{otherwise}\end{cases}$$

and likewise for cohomology, so $M$ satisfies your requirements. However, unless $M$ is homeomorphic to a sphere, $\pi_1(M) \neq 0$ so any CW complex homeomorphic to $M$ must contain one-cells.

By a similar argument, presentation complexes of perfect groups are also counterexamples.


If you assume the CW complex is simply connected, then what you hope for is true. As user17786 indicates, the answer can be found in Hatcher's Algebraic Topology.

Proposition $4C.1$. Given a simply-connected CW complex $X$ and a decomposition of each of its homology groups $H_n(X)$ as a direct sum of cyclic groups with specified generators, then there is a CW complex $Z$ and a cellular homotopy equivalence $f : Z \to X$ such that each cell of $Z$ is either:

(a) a 'generator' $n$-cell $e^n_{\alpha}$, which is a cycle in cellular homology mapped by $f$ to a cellular cycle representing the specified generator $\alpha$ of one of the cyclic summands of $H_n(X)$; or

(b) a 'relator' $(n + 1)$-cell $e^{n+1}_{\alpha}$, with cellular boundary equal to a multiple of the generator $n$-cell $e^n_{\alpha}$, in the case that $\alpha$ has finite order.

In particular, there is one $n$-cell for each $\mathbb{Z}$ summand of $H_n$, and a pair of cells of dimension $n$ and $n + 1$ for each $\mathbb{Z}_k$ summand of $H_n$. So a simply connected CW complex with homology concentrated in even degrees admits a cell structure with cells in only even dimensions, one for each $\mathbb{Z}$ summand of homology.

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    $\begingroup$ Of course this is a good example. I guess if I want to draw the analogy with the rational case, I should ask that the space is simply connected. Any thoughts on this version? $\endgroup$ Jan 9, 2020 at 10:47

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