Reactions of linear topology are the most common, and consist of all transformations whose transition states are acyclic, including addition, elimination, substitution, and (some types of) fragmentation reactions. By contrast, in pericyclic reactions, the atoms under chemical change form a single closed cycle, and include reactions like the Diels–Alder reaction and Cope rearrangement, among many others.
In contrast to these types of reactions, a coarctate reaction is characterized by a doubly cyclic transition state, in which at least one atom undergoes the simultaneous making and breaking of two bonds.  Thus, the topology of the transition state of a coarctate reaction is a constricted cycle that meets with itself (resembling a figure eight) while the topology of pericyclic and linear reactions are a circle (or Möbius strip) and line segment, respectively.  The concept was first proposed by Rainer Herges.