We overview recent results on the mathematical foundations of Cutkosky rules. We emphasize
that the two operations of shrinking an internal edge or putting internal lines on the mass-shell are
natural operation on the cubical chain complex studied in the context of geometric group theory.
This together with Cutkosky’s theorem regarded as a theorem which informs us about variations
connected to the monodromy of Feynman amplitudes allows for a systematic approach to normal
and anomalous thresholds, dispersion relations and the optical theorem. In this report we follow
[1] closely.