In 3D, it would be somthing like : if you have a closed, simply-connected surface, then it is more or less a sphere. For lay(wo)man's vocabulary : simply-connected means that if you put a a rope and closed it on your surface, you can reduce it to a point, e.g. on a donut you quickly see there are possiblities such that you cannot tight the rope without breaking the donut, if you really want your rope to become a point like object (ideally). However, it is not known in 4 dimension if this is true for what is sometimes called a 3-sphere : the generalization of a sphere surface (you have 2 angles to parametrize everypoint on a sphere) to three dimension (three such angles, which is, i heard, quite hard to visualize or intuitiv. understand). However, I think it was proven that for higher dimensions this was true, so that for not mixing : for 2,-,4,5....dimensional varieties (number of free parameter on your object), this was true : i.e. the shape is deformable to a n-sphere if it has some properties like closedness and simple-connectedness...but Poincaré made at first wrong assumptions and corrected himself his mistake, but could solve after it...Technically it's quite complicated, with Homology and Homotopy groups, and other math. stuff..(which i personnally don't know even only the surface of those concepts)
its an attempt to describe a sphere by simple properties. for instance the usual 2-sphere, x^2 + y^2 + z^2 = 1, can be described by saying it is a smooth, compact, connected surface, has no boundary, and every loop in it can be shrunk to a point on the surface.
so having settled this case, we go up one dimension to the "3 sphere", defined by the analogous equation x^2 + y^2 + z^2 + w^2 = 1, and we ask if it is the only three dimensional, compact connected, smooth 3 dimensional gadget, in which again all loops on it can be shrunk toa point.
no one knows for sure, but a solution (yes) has been propsed recently, and a conference on the topic will be held presently in france.
A mathematician named Smale back in the 1960s settled the question for dimensions greater than 4, with the answer yes - an n-dimensional manifold which is compact, connected, and simply-connected is topologically equivalent to an n-dimensional sphere, when n > 4.