## 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][1]. 

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][2] 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?


  [1]: http://en.wikipedia.org/wiki/Moore_space_(algebraic_topology)
  [2]: https://mathoverflow.net/questions/19618/when-does-a-cw-complex-of-dimension-2-embedd-in-r4