Understanding the underlying ideas first, and exploring the history, the philosophy, and the intent behind the technology is absolutely crucial to mastering Algebraic Topology. Asking the right questions constitutes the indispensable first step. What is it that we hope to achieve? Curiously enough, one way of doing that would be to start with ‘a practical application’, albeit a problem requiring an answer in the abstract domain. Hence, recognizing the necessity for the systematic development of the necessary tools and technologies, prior to devising and comprehending the mechanics of the solution. That’s how it would all come together in a way that would make sense and be memorable.
50:00 The Fundamental Group as the set of classes of maps distinct under homotopy, with concatenation as group multiplication. 55:00 Base Point of the Fundamental Group. Invariance of the Fundamental Group under changes of base point if the space is path-connected. 57:00 Proof. Constructing a homomorphism or group isomorphism to show the Invariance Result^. 1:08:00 A space is simply connected iff it is path-connected. 1:09:00 Simply connected spaces any two loops with the same base point are homotopic. 1:11:00 Fundamental Group of the Circle. Set Theoretically Isomorphic to the integers.
Apologies if this is a trivial question. In building up CW complexes dimension by dimension, are we assuming that certain separation axioms continue to hold (e.g., Hausdorff, normal, regular) and that nothing pathological happens (e.g. Peano curves)? Thanks. Great lecture.
According to Lee's ITM, we always assume that cell complexes always lie in Hausdorff spaces. Additionally, the conditions of C & W impose further restrictions on infinite complexes. Hope that helps!
Using equivalence classes to form the connection between X and Y at the ends of the cylinder is a relativized or nonreference frame dependent way to connect two topological spaces. A subset is open iff it’s inverse image is open.
I believe that the (C) property is missing from his definition for it to be describing a CW-complex, meaning that the closure of every cell in the complex is covered by at most finitely many cells from the complex.
So, from the discussion at the end, can we say that fundamental group shows that how many times can we repeat our structure to wrap a simply connected space with that structure and the isomorphic group to the fundamental group determines how to repeat our structures to achieve a wrapping?
Here you defined the quotient topology, but I think one small point that seems to have been skipped over is the product topology (unless this class is already familiar with category theory).
This is obviously too late for a comment like "hey, cameraman, don't follow the lector, follow the BOARD instead", but just in case no conclusion was made, here it is.
Lmao why da heck you cant learn english even on the level on understanding such language-easy lectures like this one(I'm a non-native speaker but I find this lectures be pretty easy to comprehend)
@@altair2899 i read different math papers in french (galois theory) and italian (ring theory), despite i cant the language, but the symbols tell me a lot already. So if u got a basic understanding of topology it should tell u already alot whats going on, without understand the language already.
@@AmeyahOfficialTV You kinda misunderstood me, my message was directed to Rachna, I just meant that the language that Pierre Albin uses is pretty easy and if you know english I guess on B1 or even A2 level and you know some topology, it won't be a problem to understand this lecture
@@altair2899 ohhh i see! yeah thats true, but even if u know Topology in Hindi, then u can probably learn from this. Its maybe not easy, but its doable i think.
Hey, the "Last Time" segment is extremely helpful as a reference! Thank you!
Understanding the underlying ideas first, and exploring the history, the philosophy, and the intent behind the technology is absolutely crucial to mastering Algebraic Topology. Asking the right questions constitutes the indispensable first step. What is it that we hope to achieve? Curiously enough, one way of doing that would be to start with ‘a practical application’, albeit a problem requiring an answer in the abstract domain. Hence, recognizing the necessity for the systematic development of the necessary tools and technologies, prior to devising and comprehending the mechanics of the solution. That’s how it would all come together in a way that would make sense and be memorable.
Hi, I completely agree with what you write. I was thinking about taking this course here on yt. Did you see it? Do you recommend it?
50:00 The Fundamental Group as the set of classes of maps distinct under homotopy, with concatenation as group multiplication.
55:00 Base Point of the Fundamental Group. Invariance of the Fundamental Group under changes of base point if the space is path-connected.
57:00 Proof. Constructing a homomorphism or group isomorphism to show the Invariance Result^.
1:08:00 A space is simply connected iff it is path-connected.
1:09:00 Simply connected spaces any two loops with the same base point are homotopic.
1:11:00 Fundamental Group of the Circle. Set Theoretically Isomorphic to the integers.
Apologies if this is a trivial question. In building up CW complexes dimension by dimension, are we assuming that certain separation axioms continue to hold (e.g., Hausdorff, normal, regular) and that nothing pathological happens (e.g. Peano curves)? Thanks. Great lecture.
According to Lee's ITM, we always assume that cell complexes always lie in Hausdorff spaces. Additionally, the conditions of C & W impose further restrictions on infinite complexes. Hope that helps!
Using equivalence classes to form the connection between X and Y at the ends of the cylinder is a relativized or nonreference frame dependent way to connect two topological spaces.
A subset is open iff it’s inverse image is open.
Question: is what you call "cell complex" the same as what Wikipedia calls "CW complex"? Thanks for the great talk
I believe that the (C) property is missing from his definition for it to be describing a CW-complex, meaning that the closure of every cell in the complex is covered by at most finitely many cells from the complex.
its exactly the same
Yup hatcher it's the same. Hatcher calls it cell complex as well as CW complex
So, from the discussion at the end, can we say that fundamental group shows that how many times can we repeat our structure to wrap a simply connected space with that structure and the isomorphic group to the fundamental group determines how to repeat our structures to achieve a wrapping?
Question: What if the attaching map is not surjective? And some part of the skeleton is left out? How would that fit into the intuition?
Here you defined the quotient topology, but I think one small point that seems to have been skipped over is the product topology (unless this class is already familiar with category theory).
Product and quotient topology is typically touched on in the point-set topology course taken before this class
Very helpful, I already taken once in my university but it more energetic.
c est quoi embedding en francais ? merci d avance
un plongement
What part of the NB depended on X being convex?
That the linear combination of f and g would be guaranteed to be in X as well due to convexity.
@@felicote ye, makes a lot more sense now :)
@@The2378AlpacaMan lol sorry it took 2 years to answer
Lol can some one tell me what book he is referring here. P.S. great video
Hatcher, Algebraic Topology
Thanks sir, from India
38:16
50:21
This is obviously too late for a comment like "hey, cameraman, don't follow the lector, follow the BOARD instead", but just in case no conclusion was made, here it is.
cool
Every lecture in English pl z give option to convert language in hindi
learn english?!
Lmao why da heck you cant learn english even on the level on understanding such language-easy lectures like this one(I'm a non-native speaker but I find this lectures be pretty easy to comprehend)
@@altair2899 i read different math papers in french (galois theory) and italian (ring theory), despite i cant the language, but the symbols tell me a lot already. So if u got a basic understanding of topology it should tell u already alot whats going on, without understand the language already.
@@AmeyahOfficialTV You kinda misunderstood me, my message was directed to Rachna, I just meant that the language that Pierre Albin uses is pretty easy and if you know english I guess on B1 or even A2 level and you know some topology, it won't be a problem to understand this lecture
@@altair2899 ohhh i see! yeah thats true, but even if u know Topology in Hindi, then u can probably learn from this. Its maybe not easy, but its doable i think.