Masters student here. I wanted to thank you so much! Your lectures are so clear and motivating. I stopped going to AT class and just learnt through this. You make me believe in teaching and love math again!! High regards hope you keep your great job!
Great lecture! I think it may have been cool to see two different representations of S^1 for example, and show that they still produced the same results. I tried it on my own for practice and to verify, which I think was a good exercise
Kedves Professzor úr! Én messziről, Európából, Magyarországról követtem az Ön briliáns előadásait. A Hatcher könyvet is olvasom párhuzamosan, természetesen. Nagyon várom a következő előadásokat. Az elmúlt több mint két hét jó volt az ismétlésre, de már ideje a továbbhaladásnak. Remélem hamarosan itt lesz az újabb lecke.😊😊😊
H2(S2) left me slightly confused with regards to orientation. I understand that because the outward surface consisting of T and B is connected, the orientation should stay the same, so we use the right hand rule to make sure of that. However, what if I were to construct a sphere that has one more D2 glued to the rims, splitting the interior into two "cavities"? (A shape that seems homotopy equivalent to a S2 v S2) With regards to connectedness and holes, the cavities look exactly symmetrical, but if the surfaces are oriented, they aren't symmetrical. How am I supposed to represent that with simplices when calculating homology groups?
Oh, I think I got it. When you are defining a cycle, your orientation should stay the same, so similar when doing a 1-cycles, if you encounter a reverse-orientation simplex, just have it part of the cycle as a negative part. So with the original construction of the sphere, it doesn't matter which way the surface is oriented, but when doing the cycle, be sure to take that into account. As for the D2 glued to its rims three times, you have just two cycles, and for one of them, you count the middle D2 in one orientation, and in other, the other orientation.
I followed along with the method of computing homology groups, and then counted the cycles in the surface, which matched. Now I know how to find the groups, and part of what they mean, but no idea how it works or why the quotient of the kernel by the image must be taken in particular. I feel like how I did when I learnt how to differentiate a function
The best explanation I've seen of this is the section "The Idea of Homology" in Hatcher's book (p. 98 of the free online book). The simple examples show why we want an abelian group, what the kernel really means, and why it is important to mod out the boundaries. Also see Pierre Albin's lecture explaining this section, ua-cam.com/video/KudAvmWUx8s/v-deo.htmlsi=yH-MpjeS3e-OHs5P&t=3642
I don't understand how all these delta basis form groups. What are the points in delta 0 acting on, and what is the operation? What's the inverse and identity? What does it mean to multiply them by integers, and is that different from exponentiation in the normal group sense? Why are they all Z and not R? I'm so lost on this one.
That is a wonderful question! We prove this in the later lectures in the series (or, at least, sketch a proof). In particular, this follows from the fact that simplicial and singular homologies are equivalent -- and hence, two different simplicial representations give rise to the same homology.
Sir, could you please upload a lecture on "Singular homology groups" before this coming Thursday ? it is my humble request to you, please sir . Most needed.
An One hour and a half video is better than my Top2/Alg Top prof ever could be.
Masters student here. I wanted to thank you so much! Your lectures are so clear and motivating. I stopped going to AT class and just learnt through this. You make me believe in teaching and love math again!! High regards hope you keep your great job!
I can't endorse skipping class, but thank you for sharing the words of encouragement! All the best in your masters!
Great lecture! I think it may have been cool to see two different representations of S^1 for example, and show that they still produced the same results. I tried it on my own for practice and to verify, which I think was a good exercise
Excellent lecture
thankyou so much, your lectures are very clear, easy to grasp. it really helped me.
Kedves Professzor úr! Én messziről, Európából, Magyarországról követtem az Ön briliáns előadásait. A Hatcher könyvet is olvasom párhuzamosan, természetesen. Nagyon várom a következő előadásokat. Az elmúlt több mint két hét jó volt az ismétlésre, de már ideje a továbbhaladásnak. Remélem hamarosan itt lesz az újabb lecke.😊😊😊
1:06:37 Homology groups of the sphere and the torus
i really love this playlist
When will the next lecture be??
That is the most epic sweater viz ever
H2(S2) left me slightly confused with regards to orientation. I understand that because the outward surface consisting of T and B is connected, the orientation should stay the same, so we use the right hand rule to make sure of that. However, what if I were to construct a sphere that has one more D2 glued to the rims, splitting the interior into two "cavities"? (A shape that seems homotopy equivalent to a S2 v S2) With regards to connectedness and holes, the cavities look exactly symmetrical, but if the surfaces are oriented, they aren't symmetrical. How am I supposed to represent that with simplices when calculating homology groups?
Oh, I think I got it. When you are defining a cycle, your orientation should stay the same, so similar when doing a 1-cycles, if you encounter a reverse-orientation simplex, just have it part of the cycle as a negative part. So with the original construction of the sphere, it doesn't matter which way the surface is oriented, but when doing the cycle, be sure to take that into account.
As for the D2 glued to its rims three times, you have just two cycles, and for one of them, you count the middle D2 in one orientation, and in other, the other orientation.
In minute 35, is the S1 represented as simplicial complex? Two edges intersect at two points, not single simplex
Excellent lecturer and material. I find the Hatcher book detailed but uninspiring
I followed along with the method of computing homology groups, and then counted the cycles in the surface, which matched. Now I know how to find the groups, and part of what they mean, but no idea how it works or why the quotient of the kernel by the image must be taken in particular. I feel like how I did when I learnt how to differentiate a function
Forthcoming lectures will get into this all much deeper. For this one, I was just introducing how to do the calculations.
The best explanation I've seen of this is the section "The Idea of Homology" in Hatcher's book (p. 98 of the free online book). The simple examples show why we want an abelian group, what the kernel really means, and why it is important to mod out the boundaries. Also see Pierre Albin's lecture explaining this section, ua-cam.com/video/KudAvmWUx8s/v-deo.htmlsi=yH-MpjeS3e-OHs5P&t=3642
@@Huffer7 Thanks, I’ll check those out!
I don't understand how all these delta basis form groups. What are the points in delta 0 acting on, and what is the operation? What's the inverse and identity? What does it mean to multiply them by integers, and is that different from exponentiation in the normal group sense?
Why are they all Z and not R? I'm so lost on this one.
Why are these groups well dedined ? I mean there are many ways to glue together lines to ontain a circle.
That is a wonderful question! We prove this in the later lectures in the series (or, at least, sketch a proof). In particular, this follows from the fact that simplicial and singular homologies are equivalent -- and hence, two different simplicial representations give rise to the same homology.
Ok thanks for the answer. By the war I really like this lecture style. It makes the subject much more approachable and understandable.
Can you please elaborate a little how about modding out these abelian groups
Can you be more specific?
@@-minushyphen1two3791:04:38
Like if you also want more lectures. Dear Prof. please upload further lectures after 11.
When will the next video be uploaded
Sir, could you please upload a lecture on "Singular homology groups" before this coming Thursday ? it is my humble request to you, please sir . Most needed.
We're off for Thanksgiving holiday this week. 🦃 Uploads will resume next week when we will begin to get into Singular homology.
Please try to answer that question as soon as possible because my exam is coming up