Whenever I’ve met low-dimensional topologists I’ve been dazzled by the effortless way these experts can mentally manipulate the subtlest of geometric configurations. Twisting and pulling manifolds about, and sewing in projective spaces all over the place is simply the air that they breathe.
Anyone who aspires to this world, or would simply like to gaze at it in wonder, can’t do better than to browse the archives of Sketches of topology where various topological constructions have been beautifully rendered by the former conservative cabinet minister Kenneth Baker.
This post, for example, perfectly illustrates the notion of blowing up a point on a manifold.
Richard Elwes - mathematician and writer 