For 10 points - identify these objects, familiar as sets of points in n-dimensional Euclidean space a fixed distance from a given point.
The generalized Schoenflies Theorem concerns embedding one of these in another one dimension higher.  For n=4 and higher, it is known that any compact n-manifold is homotopy equivalent to one of them if and only if it is homeomorphic to one, and that statement for n=3 is the original Poincar?conjecture.  In high numbers of dimensions, most of their area is concentrated in a small band in coordinate space, and the numerical value of the volume of a unit one is largest when enclosing a five-space volume.  For 10 points - identify these objects, familiar as sets of points in n-dimensional Euclidean space a fixed distance from a given point.
The generalized Schoenflies Theorem concerns embedding one of these in another one dimension higher.  
n-sphere or hypersphere or sphere in n-space or equivalents