It is a well known and often touted fact that the singular homology groups 'count the k- dimensional holes' in a space (see: How does singular homology H_n capture the number of n-dimensional "holes" in a space? for an explanation), but has anyone succeeded in making this heuristic rigorous?

I am aware of the work of Rene Thom on a related subject (see: Cohomology and fundamental classes), but am dubious that this comes close to what I am asking.

Specifically what I *am* asking is whether there is or indeed ever could be a theorem of the form:

Suppose $M^n \subset \mathbb{R}^m$ is a hausdorff [insert category here] manifold of dimension n: let $Y_k$ be the set of generators of $H_k(M^n)\otimes \mathbb{Q}$ with $k\neq 0,n$ then for each $[x]\in Y_k$ $\exists x \in [x]$ and a point of $x$, $\hat{x}$, s.t. $\exists$ a convex neighbourhood $U$ of $\hat{x} $ with $U \cap M $ homotopic to $\mathbb{R}^{k+1} \setminus 0$.

(The torsion has been killed in an effort to make this a reasonable request)

If not, why not? (I am aware that there could be an arbitrary number of twists in the codimension of the fundamental class for example, but surely this could be sorted out!) Are there counter examples? Is there a situation where this is true?

Edit: After a counter example to the original conjecture, rather than retreating all the way back to manifolds Let's attempt to defend the intermediate bridge of $U \cap M $ homotopic to $S^{i_1} \times... \times S^{i_p}$ with $\Sigma_j i_j=k$

Edit2: $H_2(\mathbb{R}^3 \setminus T \wedge T)$ scuppers that. Compact orientable smooth manifolds it is then...