Great, thanks. AT is just too nice, so I have an easy job ;-) This series was meant to cover the topics in Hacher's AT book. Hatcher's book is awesome, but not complete. Of course not, no book is - no offense to Hatcher ;-) In particular, three main topics of AT are omitted - vector bundles, K-theory and spectral sequences. Hatcher writes somewhere in the text (I forgot where) that it is better to give these "a fresh start", and I tend to agree. I haven't decide yet (in mid October 2021) what the next series is meant to be. K-theory (and with it, vector bundles) would be fun, thanks for the idea, but I have no clear picture in my mind right now how to organize a series "What is...K-theory?".
I don't understand how this shouldn't be a graded commutative ring. The way I know ithe definition, if R decomposes as Σ R_i, for it to be called „ℕ-graded“ we reduire R_i R_j \subseteq R_{i+j}. But that does not exclude zero because 0 is present in every of the subrings R_i. Or are there multiple definitions which don't quite line up?
That confused me several times myself. The point is that what you define is a graded ring - and yes, it is definitely a graded ring. But we are talking about graded commutative which additionally means X*Y=(-1)^degXdegY YX. So when X is of odd degree XX=-XX and the polynomial ring collapses. I agree that this is confusing: the terminology is not chosen well. Mind the difference between "graded, commutative" (graded + commutative) and "graded commutative".
@@lukasjuhrich503 That is right. Do not quote me on it, but I think "graded commutative" is used by topologist, while "supercommutative" is more common in the physics/representation theory literature. The notions appear from two different context, and it took a while until community A realized that community B did the same thing. A classical example of a clash of notation ;-)
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I am sincerely enjoying this series. I hope it continues. Would be great if vector bundles and characteristic classes were covered at some point!
Great, thanks. AT is just too nice, so I have an easy job ;-)
This series was meant to cover the topics in Hacher's AT book. Hatcher's book is awesome, but not complete. Of course not, no book is - no offense to Hatcher ;-)
In particular, three main topics of AT are omitted - vector bundles, K-theory and spectral sequences. Hatcher writes somewhere in the text (I forgot where) that it is better to give these "a fresh start", and I tend to agree.
I haven't decide yet (in mid October 2021) what the next series is meant to be. K-theory (and with it, vector bundles) would be fun, thanks for the idea, but I have no clear picture in my mind right now how to organize a series "What is...K-theory?".
I don't understand how this shouldn't be a graded commutative ring. The way I know ithe definition, if R decomposes as Σ R_i, for it to be called „ℕ-graded“ we reduire R_i R_j \subseteq R_{i+j}. But that does not exclude zero because 0 is present in every of the subrings R_i.
Or are there multiple definitions which don't quite line up?
That confused me several times myself.
The point is that what you define is a graded ring - and yes, it is definitely a graded ring.
But we are talking about graded commutative which additionally means X*Y=(-1)^degXdegY YX.
So when X is of odd degree XX=-XX and the polynomial ring collapses.
I agree that this is confusing: the terminology is not chosen well. Mind the difference between "graded, commutative" (graded + commutative) and "graded commutative".
@@VisualMath ohhh, yes, thank you! I forgot about that property. Isn't it sometimes called supercommutativity?
@@lukasjuhrich503
That is right.
Do not quote me on it, but I think "graded commutative" is used by topologist, while "supercommutative" is more common in the physics/representation theory literature.
The notions appear from two different context, and it took a while until community A realized that community B did the same thing. A classical example of a clash of notation ;-)