Homotopic Extension Problem

This lecture focuses on solving the homotopic extension problem for composite maps by decomposing it into two problems: finding a homotopy and a vibration. The instructor explains how an infinite composition in a CW complex gives a cofibration operation, and demonstrates the process of constructing a relative CW complex. Through detailed examples and proofs, the lecture covers the concept of attaching cells to a subspace and how inclusions in CW complexes are cuff vibrations. The lecture concludes with a discussion on turning any map into a vibration using the cylinder of the map, and the application of the hamotopic extension property to ensure uniqueness in CW approximations.