I feel like I'd rather try to continue with Bredon if anything, and before that even, maybe Sakai, and before that, well, I plan to learn a lot from general topology
I want to continue to read about topics adjacent to what I've already learned in Gillman and Jerison, and I have a sort of idea of which direction to go towards
Generally speaking I want to study Tychonoff spaces and adjacent issues such as compactifications, dimension etc. but also $C(X)$ and specially embedded sets
@Thorgott Yes for basic homology cohomology and homotopy theory. But I didn't manage them well, just learnt them from class. I am now intending reading the book talking with someone doing more exercises to make me really understand these basic theory for future study.
I have taken 2 and half semester courses about algebraic topology which covered homology, cohomology and half semester's homotopy theory. I just found out I didn't know these theory very well now. So I decided to go through them again before too late.
@Thorgott Yes you are right. Rotman's book is my first book. Now I feel Dieck's book is quite concise though.
@BalarkaSen if memory serves correctly, a degree one back pulling back a given bundle to the tangent bundle (up to stable equivalence) or something like that?
@Thorgott This must be an equivalent description, yeah. The one I know is the following: $f : M \to N$ is a normal map if $f^*\nu(N) \oplus TM$ is stably trivial, where $\nu(N)$ is the stable normal bundle of $N$ (embed $N$ in a sufficiently high dimensional Euclidean space, take normal bundle)
@Jakobian No, the worst case is that your filter will get all clogged up, and you won't get your coffee. Again, depending on what kind of filter you have.
@Jakobian It is more about the size of the holes. In a French press, the holes are large enough that the cinnamon powder will likely go straight through. But in an espresso machine, the holes are very small, and could end up clogged. And if you are using a paper filter, good luck. You are doomed.
I was trying to verify that for a cobordism $W : M \Longrightarrow N$, if there exists a retract $r : W \to M$ then it is a normal map. It makes intuitive sense but I cannot quite prove it, embarrassingly.
If I embed $W$ in a sufficiently high dimensional Euclidean slab (relative to the boundaries), the retract should send the normal bundle to the normal bundle.
@Thorgott it essentially contains all the spectral sequences you might ever need to do computations in the stable homotopy category
which is a few, and they're all so much fun to work with :^)
it's the kind of field where people develop spectral sequences converging to the $E_2$-term of another spectral sequence just to facilitate computing that
