Why is the boundary of a boundary always null? What is a coboundary? Thayer Watkins answers these questions for a triangle, whereas Robert Ghrist applies sequences of boundary mappings (the “chain” in “graded chain complex”) to surfaces, singular homology, Čech homology, cycles, graphs, and Kirchhoff’s Law on electrical circuits. Terry Tao applies them to dynamical systems. Beware, some of the + signs in Ghrist’s Elementary Applied Topology are direct sums, which is pasting things together rather than ∫umming in the usual dimension-reducing sense.