Algebraic Topology 5: Homeomorphic Spaces have Isomorphic Fundamental Groups

Playlist: • Algebraic Topology
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: www.andrews.edu/cas/math/
In this course we are following Hatcher, Algebraic Topology: pi.math.cornell.edu/~hatcher/...

[@nahomdejene903]

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

[@OedipusLee]

Thank you for this amazing course.

[@JennieXie-rg2xd]

amazing!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@-minushyphen1two379]

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

[@enzoregna3521]

No

[@enpeacemusic192]

The inverse wouldn't be continuous

[@richardchapman1592]

Guessing that in R3 space can be filled continuously with membranes from R2. That would involve a discontinuity at infinity when mapping to S1.

[@forheuristiclifeksh7836]

0:07

[@forheuristiclifeksh7836]

14:07

[@richardchapman1592]

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

[@abebuckingham8198]

A function must have a domain and codomain to be defined. I don't see any obvious domain or codomain for learning.

[@richardchapman1592]

@@abebuckingham8198 could look at that the words used in various topics of learning to define several domains. The scopes of those words would have to be defined by other words and not sure if hierarchy could be handled with winding numbers in Z.

[@abebuckingham8198]

@@richardchapman1592 Math is a social discipline and it's up to you to explain your ideas to others in a way that's convincing. If you think there's a domain then articulate it. Vague notions of what might be possible aren't enough.

[@richardchapman1592]

Apologies for being inadequate. I can only attend to thought matters sporadically due to dissemblement by a range of other matters associated with what is loosely termed, reality. Am trying to plough through the Andrews lectures though and maybe will be, more cogent at a later date. I do however appreciate your feedback at my confusions of concept. Thanks.

[@richardchapman1592]

@@abebuckingham8198 My original query concerned the extent of which the use of words to describe concepts may be considered as elements of topological groups of those concepts. Whether the notions of homology and rings apply rather depends upon if there are inverse functions that take similar concepts from one to the other and back. One responder to the lectures has already briefly alluded to this.

[@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?

[@hbghlyj]

consider a point and R.

[@xanderlewis]

You're right, but the two shown are sufficiently general (because S^2 has a lot of symmetry) so as to be equivalent to having done the proof for 'any two maps'. It's like starting a proof by saying 'without loss of generality, let's assume we have...'. The only choice we have in such maps is which point of S^2 to pick out with psi.