Some years ago V. Goryunov and the speaker found a method for computing the homology of the image of a map from information on the homology of the sets of points identified by the map (its "multiple point spaces"). The method was derived by "general nonsense" - sheaf cohomology, injective resolutions, a spectral sequence . . . , although there are many simple examples of its application. This talk approaches the topic from a completely elementary point of view; all that is required to follow it is the most basic notion of the homology of a topological space. The constructions are motivated by pictures. Although the subject matter is algebraic topology, there is an intriguing analogy with free resolutions in algebra. The talk is based on joint work with an undergraduate student at Warwick, Ewan Lawson.