Algebraic Topology 6: Seifert-Van Kampen Theorem 

Yes He is a great teacher. I m learning a lot from his lectures ❤

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

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.

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.

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)

Yep 😅

@@MathatAndrews ❤️

No