Cohomology concentrated in even degrees implies cells concentrated in even degrees?

Question

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.

Answer 1

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.

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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.