Algebraic topology and differential topology distill geometric data into discrete topological quantities, which have become an indispensable part of quantum field theory language. This last appendix attempts a succinct overview of cohomologies, starting with de Rham and its compact counterpart, and eventually connects to Čech cohomology. These form the most immediate basis for index theorems and characteristic classes that were employed usefully in much of the latter part of the volume. We close with a bare minimum introduction to homotopy theory.