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.
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.