1-spheres and 2-spheres don't have centres? Can you explain?
Sure - a 1-sphere is a circle, a 1-dimensional line that loops back on itself. There is no unique point on the circle that can be considered "the" centre, but all points can equally be considered "a" centre. The 1-sphere is not to be confused with the 2-ball, which is a disc: a 2-dimensional area bounded by a 1-sphere (circle). The disc does possess a centre.
The 2-sphere is the 2-dimensional area that closes in on itself exemplified by the surface of the earth. Once again, there is no unique point on the 2-sphere that can be considered "the" centre, but all points can equally be considered "a" centre. The 2-sphere is not to be confused with the 3-ball, which is a solid 3-dimensional volume bounded by a 2-sphere. The earth is a reasonable 3-ball. The 3-ball does possess a centre.
The naive impression that n-spheres have centres occurs because n-spheres are most often visualised embedded in an n+1 dimensional space (circle drawn on piece of paper.) A point completely divorced from the n-sphere is viewed as the centre as it conincides with the centre of the corresponding n+1-ball.
Cool facts: the boundary of an n-ball is an n-1-sphere. An n-sphere has no boundary.