Can homology groups be used to find the location of holes in addition to their number? Very new to this subject, apologies if this question is obvious or not. 
As the title says, we can compute the homology of a chain complex of abelian groups from Smith normal form of an integer matrix. Can we compute their location or does this defeat the purpose? 
I realize, the first response, would be with regard to what metric. 
 A: I assume what you mean is that given some space $X$, can we compute the location of the "holes" in $X$ that elements of the homology $H_k(X)$ are supposed to correspond to?
You can try to do this as follows: given a homology class $\alpha \in H_k(X)$, for any subspace $S \subseteq X$ there's an induced map $H_k(S) \to H_k(X)$, and you can ask whether $\alpha$ comes from a class in $H_k(S)$ via this map. If so, you've localized the "support" of $\alpha$ in some sense to be contained in $S$, and then you can further subdivide $S$ to try to pin down exactly where the "hole" $\alpha$ describes is. In general $\alpha$ will be a linear combination of cycles around multiple "holes," though, and it's not at all clear how to select the subspaces $S$. 
The clearest example where this "holes" idea really pays off is considering $X$ given by $\mathbb{R}^n$ with either a finite number of points or a finite number of disjoint balls removed (the two choices are homotopy equivalent); intuitively these are the "holes" and the homology calculation exactly bears this out. $H_{n-1}(X)$ is free abelian with a basis given by cycles around each of these holes, and one way of making this precise is that there are maps $S^{n-1} \to X$ from the $(n-1)$-sphere to arbitrarily small neighborhoods of each of these holes and the generators of $H_{n-1}(X)$ come from the generators of $H_{n-1}(S^{n-1})$ pushed along these maps. 
More generally, see Alexander duality.

I might argue that the notion of "holes" is not quite intrinsic and really depends on a choice of embedding into an ambient space which trivializes some of the homology; this corresponds to filling in some of the "holes" and one can ask exactly what holes are being "filled in." For example, where are the holes in the torus $T^2$? Abstractly there are a $GL_2(\mathbb{Z})$'s worth of bases of $H_1(T^2)$ and no reason to prefer one over the others. The "obvious" visual answer depends on a particular choice of embedding of $T^2$ into $\mathbb{R}^3$ and one can choose others. 
