Algebraic Topology 3: Fundamental Group is a Group!

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We recall the definition of the fundamental group develop in the previous lecture then prove that it is indeed a group. Finally, we show that the fundamental group of the circle is isomorphic to Z, the integers.

Presented by Anthony Bosman, PhD.

Math at Andrews University

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The isomorphism at 31:00 is independent of the path if the fundamental group is abelian. Not in general.

kapoioBCS

Yoooo banger video!! good explanations and examples. Have been using as a supplemental for my topology class to try to look over key ideas before lectures.

franciscogomez-paz

Intuitively the paths of one dimension on a 2dim surface look sensible for π[S¹] when stretched into 3dim. Is a 2dim path a vector and it possible to generalise the meaning of path? Into something of ndimensions on an m>n brane in an s>mdimensional space?

richardchapman

Also, does this way of mathematising spaces exhaust the possibilities. ie. Is there anything sensible in working topology beyond rationals?

richardchapman

A little slow on the uptake of this due partially to social factors (a mixed blessing) but so far I see the 'paths' as precursors of vectors and have never seen a treatment of >1dim vectors. Also we have the use of Z and R closed by using mods. Cannot see how irrationals with their infinite series definitions can be included in the metrics of mod fields.

richardchapman

You make wonders of projection possible, like how you make your parameters irrational and what is the meaning of shape when they all approach infinity. Also how many types of "n" are there?

richardchapman

Has anybody tried projecting winding numbers on to a unit circle orthogonal to π[S¹]? You would need a function that took zero to π/2 such that it converges to 1 using an infinite series which may be quantised by values of wn according to the normalised series chosen?

richardchapman

May i ask why do we need three pages of math to justify lifting paths and homotopies? I thought it follows from compositions of continuous maps are continuous, right? (in that composing paths and homotopies with lifts or projections preserves continuity.)

lll

22:00 how do we know that [f] * ([g] * [h] ) even exists though? We should probably prove it.

kapoioBCS

Suggestion to the camera operator for future videos... Stop zooming in. Its extremely useful to see more of the blackboard at once.

wipetywipe

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