Active contour models are an efficient, accurate, and robust tool for the segmentation of 2D and 3D image data. In particular, geometric deformable models (GDM) that represent an active contour as the level set of an implicit function have proven to be very effective. GDMs, however, do not provide any topology control, i.e. contours may merge or split arbitrarily and hence change the genus of the reconstructed surface. This behavior is inadequate in settings like the segmentation of organic tissue or other objects whose genus is known beforehand. In this paper we describe a novel method to overcome this limitation while still preserving the favorable properties of the GDM setup. We achieve this by adding (sparse) topological information to the volume representation at locations where it is necessary to locally resolve topological ambiguities. Since the sparse topology information is attached to the edges of the voxel grid, we can reconstruct the interfaces where the deformable surface touches itself at sub-voxel accuracy. We also demonstrate the efficiency and robustness of our method on synthetic as well as on real (MRT) scan data.