In this paper, the energy landscape of the cyclo-octane molecule (foreign words to me!) is shown to be a Klein bottle and a 2-sphere fused along two circles. Using persistent homology techniques discussed in the Ghrist paper mentioned by Lucas, the homology has been experimentally recovered by Mikael Vejdemo-Johansson (I learned the above from this talk).
I'm under the impression that these questions are directly related to protein folding, which is a major concern of modern biology. The basic idea is that the space of possible ways the protein can be folded forms a manifold, with some function on the manifold governing which possible folds take place.
