This is a video for the course Topologie en Meetkunde taught in Utrecht University. We explain how we can compute the fundamental group of a topological space by dividing it into pieces and combining the fundamental groups of each piece.
Along the way we introduce the free product of groups and show that the higher dimensional spheres have trivial fundamental group.
00:00 Informal statement of van Kampen
00:30 The particular case of the sphere
06:31 Summary of particular case
07:20 Formal statement of van Kampen and strategy of proof
11:58 Ilustrating the general case
14:43 Final summary
Website of the Utrecht Geometry Centre: http://utrechtgeometrycentre.nl/
Lecturer: Álvaro del Pino (https://alvarodelpino.com/)