Cellular Approximation Theorem

This lecture focuses on proving the Cellular Approximation Theorem, which states that every map is homotopic to a cellular map. The instructor introduces the concept of polyhedra and piecewise linear maps, explaining how to construct a homotopy between a given map and a piecewise linear map on a polyhedron. The lecture covers technical lemmas, construction of homographies, and the subdivision of cubes into polyhedra. By using induction on cells, the theorem is proven, showing how to modify a map inside a space to be piecewise linear on specific subsets. The lecture concludes with a detailed explanation of the construction and properties of the homotopy, ensuring that the map remains piecewise linear within defined boundaries.