Algebraic Topology 5: Homeomorphic Spaces have Isomorphic Fundamental Groups

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.

[nahomdejene]

I love this channel so much❤😊. Keep doing all pure mathematics courses👍

[-minushyphentwo]

at 36:40, what about space-filling curves? Wouldn’t they be continuous surjections from the closed unit interval to the sphere?

[richardchapman]

Is learning a meaningful function topologically and in what sense can it be be considered continuous?

[MarioPucci_mamio]

As for the last proof, wouldn't one actually need to prove that there are no such phi and psi while you only proved the two shown are not what you'd be looking for?