How to visualize Homology groups? I've been studying Algebraic Topology recently (Hatcher). I have always been very good at Topology, since I am chiefly a visual type of person. I've found that most things in Topology can be thought of visually, and then translated into rigorous mathematics.
However, I am having a very difficult time visualizing Homology groups. When studying simplicial homology, I can see how the groups correspond in a way to the number of "n-holes" in a space. However, when I make the transition to singular homology, this intuition doesn't make as much sense to me anymore. 
I feel I will never be able to truly understand what is going on with these groups until I can visualize whats happening with them. How do you visualize homology groups?
 A: Although it is an old question I think is worthwhile to try to give some insight of the non-intuitive formal definition of singular homology.
Singular homology is not easy to visually interpret it as simplicial or cellular homology, i.e. as triangulations of an n dimensional space (as in the link provided by Martin).
In general singular homology is a continuos (not injective) map from the ensemble of all possible n-dimensional simplexes to points X of the target topological space, and the standard answer by the book is that it is not possible to visualize it. Singular homology is a tool for abstraction and axiomatics, simplicial homology is a tool for computation and visualization.
First problem is that this map is continuos but not differentiable, therefore there may be points where there are multiple derivatives and the extreme case of a fractal where there are none.
Yet another example of such a continuos map can be found here https://www.youtube.com/watch?v=1AYqqbhG6jQ (around 12 min from the start)
the original simplex is squeezed into the space X in singular manner squeezing the triangle by squeezing two points in the middle of two edges, where this last example show a case where the map is not injective, and the intuition of the name "singular".
Second problem is that this map is applicable to any topological space including discrete ones, which may have a discrete topology from a point set standpoint (Yet another area that is problematic to visually interpret).
Last problem is that simplexes are in general defined on the basis of their embedding in the same space X.
Finally the topological spaces themselves are easy to visualize only as far as they are in a 3D embedding, and even then only in the simplest cases they are orientable and smooth, already complex line bundles are not trivial to visualize.
If you now restrict then yourself to smooth maps and locally smooth manifolds I imagine singular homology as a set of imploded delta complexes, or I imagine them as infinitesimal complexes embedded in the same space X. But the embedding is somehow microscopic and the actual full simplex is actually mapped to a single point of X. Being the space topological you still have a notion of limit and neighborhood, you can then imagine the mapping of an n-simplex to a point in X as a set of n points converging together to the target point of X. Being the space smooth you can still imagine that the local differentials are meaningful, and that they can be computed on this local "microscopic" simplexes. Here you see how also notions of higher order differentials, and by abstraction, differential forms and de rham cohomology, may still makes sense locally.
From a second corner you can also imagine the singular homology in the scope of algebraic geometry also here the maps are continuous. In this case the maps are decomposition of the algebraic variety (i.e. extra internal variables of an initial system of algebraic equations), or if you want pieces of continuos transformations from one subset of the space to another subset, or if you want in two dimension from one arc of an algebraic curve to another arc. Yet again the maps are not injective and they may map multiple points to one point and may have points where they are not differentiable (cusps, points with double derivatives or with multiple arcs converging in one point such as a lemniscate).
Finally from an historic standpoint I associate the singular homology to the name of Whitney, and I associate Whitney to a paper on Matroids where he makes the dualities across linear dependence, matrixes, graphs, circuits, ranks and "grassmann alike" vector subspaces, see "On the Abstract Properties of Linear Dependence". It's curious how in that paper there are many intuitions of low dimensional combinatorics and low dimensional algebraic topology and how they relate to totally different areas of mathematics (linear algebra, graph theory, dual graphs, functorial mappings between graphs and matrixes, or matrixes and ranks, circuits and homotopies and functorial mappings from matrixes to vectorial subspaces and in general other analogies with topological spaces/subspaces). I interpret this paper as his entry point into Algebraic Topology and a way to understand singular homology as central to the deep parallels across the different buildings of mathematics, which goes down to Chern & Characteristic classes and more recently to motivic cohomology and the HOTT book.
