Algebraic Topology 5: How Homeomorphic Spaces Share Isomorphic Fundamental Groups 🧩

Playlist: https://www.youtube.com/playlist?list=PLOROtRhtegr7DmeMyFxfKxsljAVsAn_X4

We show that a continuous map between topological spaces induces a homomorphism between the fundamental groups. Then we prove that if the map is a homeomorphism, the induced homomorphism is in fact an isomorphism. This fact lets us prove some neat facts such as the fundamental group of a sphere S^n (for n at least 2) is trivial. We also show that it is enough for the spaces to be homotopy equivalent for the induced homomorphism to be an isomorphism (though the converse fails).

Presented by Anthony Bosman, PhD.
Learn more about math at Andrews University: https://www.andrews.edu/cas/math/
In this course we are following Hatcher, Algebraic Topology: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf