Surgery without tears
This photo is of the board of Rob Kirby, who recently visited the University of Edinburgh to give a series of talks on the topology of manifolds. It shows a version of the Thom-Pontrjagin construction for connecting two components of the pre-image (under the map g) of a point (0,0), but doing so in a neighbourhood of the blue arc connecting the two components. The two components are in red, and as the homotopy progresses the red components are pushed towards each other until they meet at a point, like a saddle, and then become a cylinder joining the two components. This is basically surgery on a 0-sphere.
With thanks to Andrew Ranicki for taking the photo and providing the title for this post.