Algebraic Topology 6: Seifert-Van Kampen Theorem
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- čas přidán 18. 10. 2023
- Playlist: • Algebraic Topology
The Seifert-Van Kampen Theorem gives a way to calculate the fundamental group of a space by calculating the amalgamated product of two open spaces that cover the space and that have path connected intersection. Confused? Don't worry! We start simple with the problem of finding the fundamental group of a wedge of circles and build up to finding the fundamental group of a torus as well as 2-holed torus (and more!).
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/...
What a great tracher. A youtuber scaredmonger about math classes difficulty, and the topic they saw at the math class he attended was VanKampen's theorem. To prove him wrong I was determined to learn the same topic with no previous background knowledge, and thanks to this lecture I understood the theorem. The sign of a an amazing teacher
Yes He is a great teacher. I m learning a lot from his lectures ❤
Amazing, regards from Spain, you are helping me a lot, thanks.
Best playlist of video lectures! Really loving to follow these classes from this professor.
lovely, clear presentation!
Very clear and neat quality
Damn, 3000 views for this quality, math is crual. Ty !
thank you so much, super intuitive explanations, wich i often miss in topology
Amazing currently learning
lovely presentation
Very helpful! :)
What is a Union of Union to Learn more about the Ambient Spaces?
To see that ab =/= ba for the figure-8 space note that any homotopy of paths between a and b would have to pass through the basepoint, which means the loop would be contractible. Since the fundamental group of the circle is not trivial this is impossible.
Amazing
Is this the doctor who revealed the magician's lies using the knot theory?
Yep 😅
@@MathatAndrews ❤️
Great video!
What do you think about the textbook introduction to topology from UCLA?
if you are looking for topology textbooks, you can try Intro to Metric and Topological Spaces by Sutherland, which motivates topology well, or Munkres’s Topology, which is more comprehensive
Definition of "free product" (*)? I think the wedge (v) product of spaces was covered earlier, but I'm drawing a blank on that, too. Looks like a union of spaces?
The wedge sum takes one point of each space, typically the basepoint, and glues them together. Formally to construct it we take x in X and y in Y then take the disjoint union of X and Y and quotient it with the relation x ~ y. Once you've internalized the formal definition you should forget it and think gluing the base points together.
The free product of two groups G and H is a lot like the free product on a set where you construct words from the generators and reduce them. However for the free product the relations on G and H will continue to hold in the product. So say G = {g | g^3=1} and H = {h | h^4=1} then G * H = {g,h | g^3=h^4=1}. Some typical element of G*H would be gh, gh^3gh^2gh, g^2h, g^2h^2gh^3, etc. So even with finite groups the free product is typically infinite just like the free group is infinite with a finite set of generators.
1:05:00 Seifert was a German mathematician*
36:37 can anyone elaborate on this deformation retract im really not seeing it, i'm imagining the circle expanding onto the equator circle but then how does this leave a line through the middle?
similarly for 1:06:51
Take the solid diameter going exactly through the center of the carved out S¹ and perpendicular.
You get the first retraction to solid diameter by projecting all the points of S¹ downwards to the center, which is also a point on the diameter.
Now use a small angles to project (retract) more and more down to the solid diameter, until the angle is so big that it starts to make small concentric circles on S². Thus you can fill out step by step all of S² until you come back on the other side (negative angle) down to the solid diameter again and then the same procedere as in the beginning, just from S² towards the center of S¹, retractions along the solid diameter.
This way you have projected (contracted) all(!!!) points of D³ - S¹ either to the solid diameter or S², straight, without crossings of any lines. So this is what is left after contraction, as claimed.
For the second question, I would use rather the arguments from the last 10 minutes of the lecture, which are much simpler (almost trivial) to follow and much more generell.
So use the polygons with vertices identified etc. and you see immediately what's going on, without need of extreme Imagination power.
Removing an S^1 can be deformation retracted to removing a torus. Now pump air into it so it fills up the sphere and you'll be left with a solid diameter running through the middle of the torus.
Starting with R^3 - S^1 choose the basepoint to be the center of the circle we removed. Now create a loop that goes around the removed circle. This creates two disjoint circles that are linked together like a chain. You could make this with some twine if you want to build more intuition. Now take a second loop that goes around the removed circle and through the same base point. First deform these loops into circles of the same radius. Then rotate one around the removed circle until it aligns with the other loop. This shows that any two such loops are homotopy equivalent. Now notice we can stretch out the basepoint into a diameter without altering the construction. We can further ballon out the diameter to fill as much of the circle as we want.
@13:58 Is it true we can assume if two spaces are homotopic equivalent they have the same Fundamental group ?
yes, the fundamental group is a homotopy invariant
Yes, this was one of the main topics towards the end of the last lecture (lecture number 5)
nice
Could we use algebraic topology to multiply matrices of incompatible dimension?
No
Got homeomorphisms in legal concepts?