n - cells

>>8172148
The beauty of CW complexes is in the attaching maps. So, you look at how the boundary of the disk (so, a 1-sphere) is attached to the wedge of circles. What map S^1 —> (S^1 V S^1) produces a torus when you fill in the boundary you just mapped onto the 1-skeleton?

Hatcher is a shit tier book.

>>8172179

so there is a single point on the boundary of disk which is common point for two circles? Feels like I need to turn disk inside out.

>>8172212
Hatcher gives explicitly the attaching map as the polygon on the right. Remember that every vertex of the polygon corresponds to the same point because of the lower attaching maps used to define the 1-skeleton. So, the disk fills in the polygon, and the quotient maps glue it into being a genus-n surface.

>>8172179
Can I think of it as a way of "wrapping" boundary of disk around wedge?

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>>8172195
I'd second this. It may be free, but I just don't like his exposition. I guess I'm a Rotman fanboy or something, but I'd really recommend Rotman's Algebraic Topology over Hatcher's.

>>8172179
S-senpai! Y-you probably don't remember me, but I'm a fan of yours. How are you proggressing with your hodology?

And on the topic of the thread, consider the usual way to construct a torus. You take the unit square (which is homeomorphic to the closed disk in [math]\mathbb{R}^2[/math]) and identify the edges the usual way. Now, take the corner points. Each of these is a 0-cell, and they get identified as a single point or a single 0-cell. Then, the rest four pieces of the edges, which are now 1-cells get identified into two 1-cells. Finally, the interior of the square is itself a 2-cell. This may help you somehow.

And to anyone complaining about me referring to the square without borders as its interior, I know I abuse language here. It's the only obvious way I can think of to call that thing.

>>8172523
Yes, exactly!

>>8172531
Hey Anon! I'm partway through a thorough exposition on hodology. Get this: hodological spaces naturally form a cohesive topos!! It's been very very cool. I am also studying chromatic homotopy theory and have been trying to learn Lurie's work on derived algebraic geometry because I have a cool new method for studying homotopy fibration sequences, which will greatly further my work on the homotopy groups of spheres. I have also been studying the philosophy of George Friedrick Hegel as well as a Liebnitz, and am trying to formalize ideas from Hermiticism using the language of flavour theory. How are you?

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>>8172542
Your method would not be a way to determine some of the yet mysterious homotopy groups, would it? It'd be pretty awesome if it was. I can't wait to read the paper named "On Hodological Properties" by OHP! Gosh, you're really awesome even if just half of the stuff your claim to do was true! But, as an evil europoor, I must correct your spelling: It's Georg Friedrich.

I'm doing pretty well, I guess. I was totally rekt by an exam on alg top I had prepared a lot to do, but somehow there was this mental lockdown in my mind when I saw the paper. Now I've recovered and regaining my strenght to tackle that shit again in August. Otherwise it's going ok, atleast I have no complaints about being alive or anything like that.

>>8172564
Hey man, you keep on studying the stuff you love, and the path to mathematical wisdom will be an easy one. If it makes you feel better, I haven't even been able to take a formal course on algebraic topology yet, so it's cool that you get to have access to a knowledgeable professor. Best wishes!

>>8172531
OP here, do you perhaps have a pdf of Rotman's boo? I would like to take a look at it cause I also have some problems with Hatcher since i got used on Munkres with general topology

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>>8172579
I enjoy homological algebra the most, I guess. I have a decent visual memory and many essential concepts are presented by diagrams making it easy for me to remember stuff like direct limits etc. I'd like to do my master's thesis on the subject, but there are no courses on the subject (except for one held a few years ago by some visitnig frenchman but I was too much of a noob to go there then), so I have no idea what is suitable for a topic. Do you think Freyd-Mitchell's embedding theorem would be good or would you see it as something too minimalistic?

>>8172580
http://gen.lib.rus.ec/book/index.php?md5=9289615E4E6A45A0DB368895DFAED341 I hope you live in the free world, otherwise I'll just put it in some cloud.

>>8172598
Make sure you learn category theory VERY thoroughly as soon as possible! I say it to most mathematicians, but for homological algebra it is the natural language. I guess you already know enough if you are asking about Freyd-Mitchell. On that, see if Freyd-Mitchell can be extended to unconventional settings. When is the embedding a localization? As far as I know, these are good areas of discourse.

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>>8172643
I've studied some category theory on my own already. I find it disturbingly intuitive somehow, though I haven't yet ventured into the real depths on the subject. I guess it has something to do with the degree of abstraction shaving of the unnecessary details and revealing the big picture, or something like that.

Thanks for your ideas! I'll check what a localization means in this context and so on. I've actually read Freyd's book on the theorem, it was rather interesting per se, and more interesting is the corollary that snake lemmas and such hold for every abelian category, as one can for the minimal abelian category with just the required stuff as its objects, and this is necessarily small and then the embedding gives the exactness in R-mod, and so also in the original category. This kind of stuff is pretty cool, imo.

>>8172697
In this case, since the embedding is additive, when does the embedding have a left adjoint?

Yes, higher category theory is profound. If you check out Lurie's work, all of abelian category theory comes from stuff going on between stable ∞-categories. The irritatingly extensive number of axioms in a lot of constructions turn out to all fall from simple axioms on these higher structures, as abelian categories are decategorifications of higher stuff. It's all on the nLab, if you dig it!

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>>8172724
>In this case, since the embedding is additive, when does the embedding have a left adjoint?
This is something I'll have to do some work on, but now is not the time. I woke up in the morning and can see the sunrise already, beautiful btw. nLab is something I've used a few times to get some addtitional info on stuff but never had the courage to really dig in to it yet. Maybe now that I finally understand more.

>>>/a/

>>8172785
pls post more, I need to refill my folder since I lost it

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>>8172867
I used to think losing folders was just a meme, but it has actually happened to me quite a few times this year (now that I updated to Windows10!), so I'll choose to believe you. Post something related to CW complexes or something like that, though. Otherwise my folder stays shut.