Algebraic Topology 2: Introduction to Fundamental Group

Playlist: • Algebraic Topology
We give a quick review of group theory then discuss homotopy of paths building up to the definition of the fundamental group.
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/...

[@-minushyphen1two379]

00:00 Review of groups, homomorphisms, and isomorphisms
18:45 Return to topology: path homotopy
22:55 Why must two paths with the same endpoints in R2 be homotopic?
30:20 Homotopy is an equivalence relation
42:15 Different equivalence classes of paths in the annulus
45:20 Loops
58:00 definition of the fundamental group

[@gustavogonzalez7707]

Wonderful lecture.

[@rolandscherer1618]

The topic was didactically perfectly motivated. Thank you very much!

[@joshuad.furumele365]

Another excellent lecture! Thanks

[@hanselpedia]

Thanks, lots of stuff explained in a intuitive way

[@imthebestmathematician7477]

Thank you

[@ompatel9017]

Gem

[@richardchapman1592]

Can you make a loop that approaches infinity or indeed any surface that approaches the infinities of it's orthogonality plus one?

[@xanderlewis]

45:00 “When I use a word, it means just what I choose it to mean - neither more nor less.” - Humpty Dumpty. You can tell Lewis Carroll was a mathematician.

[@fslakoh]

Great suit. Big effort on the outfit. Well done

[@paulwary]

At 24:30, the explicit linear interpolation formula is given for one possible homotopy, to show that there is always a homotopy of paths in R2, correct? The language suggest that this is THE homotopy (ie the one and only)

[@enpeacemusic192]

I think so, yeah, homotopy of paths is ány continuous deformation of paths afaik

[@forheuristiclifeksh7836]

17:11

[@forheuristiclifeksh7836]

6:10

[@bengrange]

at 39:00, when you said f and g are homotopy equivalent, did you mean to say homotopic?

[@bengrange]

and at 53:16, you meant "equivalence classes" not relations. Thank you for the great lectures!!

[@forheuristiclifeksh7836]

18:29 surjection=onto= heat everything to image. Onetoone. Man to one. Bikection

[@richardchapman1592]

In attempting to use topology in sociological circumstances, are therrighte different winding numbers for thought streams of what are commonly termed the

[@John-js2uj]

What on earth are you trying to say?

[@richardchapman1592]

@@John-js2uj have an egoistic humility that my partial understanding can use these precise mathematical concepts in the imprecise social sciences. Worries me tho that mathematics applied to human circumstance can lead to a kind of cyber fascism if AI is taken too far too fast.

[@John-js2uj]

@@richardchapman1592 You’ve got to be a bot

[@richardchapman1592]

@@John-js2uj so trained in logic and emotionally damaged couldn't refute that unless you saw me in flesh and blood.

[@richardchapman1592]

@@John-js2uj would ask of you an email address so I could send you a photo that you could possibly accept as not a fraud, but then there are Trojan horses on mails to worry about.

[@richardchapman1592]

Last comment on my editor needed a vector from the centre of a word to the end.

[@randomcandy1000]

isnt S^1 x [0,1] the cylinder?

[@DogeMcShiba]

Yes, the annulus is homeomorphic to the surface of a cylinder.

[@wipetywipe]

Great lecture. Camera work needs improvement.

[@hyornina]

39:59 😂😂

[@joshuad.furumele365]

I see you, and i raise you 29:03

[@turtle926]

I raise further with 44:44 😎