To answer this question we'll first talk informally about manifolds, with special care given to surfaces (2-dimensional manifolds). Then we'll discuss a special class of maps called approximate fibrations. These maps form a useful class of maps because of their nice properties. In order to use these properties it is important to be able to detect them quickly. We'll address this question of detecting approximate fibrations by giving sufficient conditions under which maps with point preimages spheres can be classified as approximate fibrations and reveal other surfaces having this property as well.