## Background

Let $\mathcal{A} = \lbrace A_0, \ldots, A_M \rbrace$ be an arbitrary sequence of finitely generated Abelian groups. It is well-known that a finite CW complex $X_\mathcal{A}$ may be constructed so that its $m$-th homology group $H_m(X_\mathcal{A})$ equals $A_m$ for all $0 \leq m \leq M$. In particular, one can build wedge sums of Moore spaces.

On the other hand, any CW complex is an Euclidean neighborhood retract (see Hatcher corollary A.10) and hence embeds into some Euclidean space. By a *geometric complex* of dimension $n$ I mean a CW complex whose minimal Euclidean embedding dimension is $n$.

## Question

What $\mathcal{A}$'s arise as homology sequences for $n$-dimensional geometric complexes as a function of $n$?

## Notes and Considerations

Some obstructions to having a CW complex embed in Euclidean space are outlined in the answers to this older question and may be helpful although I have not been able to use them efficiently. What one can say immediately about the sequences $\mathcal{A}$ arising from an $n$-dimensional geometric complex is that all but the first $n-1$ groups must be trivial by dimension considerations, and that the $(n-1)$-st group must be torsion-free. Alexander duality imposes constraints as well, but it is unlikely that this list of conditions characterizes the $\mathcal{A}$'s that are possible homology groups of geometric complexes. Is there a complete answer available somewhere?