I want to consider those $\eta_\bullet^t$ as natural transformations between functors. Those functors are the functors TOP->ABEL associated to $S_\bullet(X)$ and $S_\bullet(X\times I)$, right?
Hmm, I was thinking about the functors $S_q$ sending $X$ to $S_q(X)$ (the q-th group of chains) and $f:X\to Y$ to the induced map $S_q(X)\to S_q(Y)$
What you mean is instead the functor that sends $X$ to $S_\bullet(X)$ and $f:X\to Y$ to the associated chain map $S_\bullet(X)\to S_\bullet(Y)$, right?
Oh, lol, I didn't notice the square and was like "but doesn't it just follow from the cone construction? I guess those $\eta_\bullet^t$ must be useful for something!" feels dumb
Ok, that's very nice as far as 9.3.2 is concerned then! However I don't see why the $\eta_\bullet^t$ are natural transformations (probably because my experience with natural transformation is limited to our conversation from the other day)
So you have two functors from TOP to chain complexes of abelian groups, one sends $X$ to $S_\bullet(X)$ and the other one sends $X$ to $S_\bullet(X \times I)$
Right, and they send a continuous map $f:X\to Y$ maps between topological spaces to the induced chain maps $f_\bullet:S_\bullet(X)\to S_\bullet(Y)$ and $(f\times \mathrm{Id})_\bullet:S_\bullet(X\times I)\to S_\bullet(Y\times I)$
and tom Dieck claims that the $\eta_\bullet^t$ are natural transformations, the reason for naturality is basically just that the square $\require{AMScd} \begin{CD} X @>>{\eta^t_X}> X \times I\\ @V{f}VV @VV{f \times \mathrm{Id}}V\\ Y @>>{\eta^t_Y}> Y \times I \end{CD}$ commutes
I'm writing $\eta^t_X$ to keep track of which space where're considering the map $\eta$
So this diagram above says that $\eta^t$ (without going to the induced map on singular chains) is a natural transformation between the identity functor on TOP and the functor which sends $X \mapsto X \times I$ and $f:X \to Y$ to $f \times \mathrm{id}:X \times I \to Y \times I$
you have to check that it commutes, but that's basically obvious
and then you get that $\eta_\bullet^t$ is a natural transformation from the functor $X \mapsto S_\bullet(X)$ to the functor $X \mapsto S_\bullet(X \times I)$
that's a general principle: if $\eta: F \Rightarrow G$ is a natural transformation between parallel functors $F,G: \mathcal{C} \to \mathcal{D}$ and $H$ is any functor $H: \mathcal{D} \to \mathcal{E}$, then you can apply $H$ to each morphism in $\eta$ (since $\eta$ is a collection of morphisms satisfying some conditions) and you get that $H(\eta)$ is a natural transformation $H \circ F \Rightarrow H \circ G$
and the reason for this very abstract thing is just that functors preserve commutative diagrams, as you said
So in our example $\mathcal{C}=\mathcal{D}=\mathbf{TOP}$ and $\mathcal{E}=\mathbf{CH}(\Bbb Z)$, the category of chain complexes of abelian groups and $F=\mathrm{Id}_{\mathbf{Top}}$ and $G: X \mapsto X \times I, (f:X \to Y) \mapsto (f \times \mathrm{Id}: X \times I \to Y \times I)$ and $H: \mathbf{Top} \to \mathbf{CH}(\Bbb Z), X \mapsto S_\bullet(X)$
and $\eta=\eta^t$
that was quite an abstract explanation, but I think it's in line with tom Dieck's style :P
From skimming it looks like the point of 9.3.3 is that homotopic maps induce chain homotopic maps on the chain complexes and we'll use that to conclude that homotopic maps induce the same homomorphisms of homology groups, seems promising! I'll read the details now, thanks a lot for your help
@MatheinBoulomenos I'm afraid I don't know what those are but I'll reach them eventually
I'm kinda blackboxing the results he cites from Ch11 at the moment and postponing them for later
I looked up some of them from here, especially the snake lemma, because tom Dieck doesn't explain the connecting homomorphism when constructing the homology exact sequence in Ch9
Hi @Balarka, welcome back, this is the category theory chat room now
@AlessandroCodenotti yeah, showing that homotopic maps induce the same homomorphism has two parts: the algebraic part about chain complexes which is very easy. the more difficult part is that a continuous homotopy induces a chain homotopy but that's a consequence of 9.3.3
@AlessandroCodenotti tbf tom Dieck probably learned the snake lemma in the Kindergarden
he defines a map $\Delta^{n-1} \times I \to \Delta^n$ the LHS is a prism
@BalarkaSen so you're saying nobody needs acyclic models except for that proof? Tom Dieck just formulates acyclic models as a precise theorem instead of doing the same arguments every time
Eh, I just find that reading the formal stuff from tom Dieck and then looking at Hatcher for intuition is better for me than looking at Hatcher directly, but of course different people will have different approaches
@Alessandro Notice that if you work with the chain complex $\square_\bullet(X)$ consisting of the free abelian group generated by maps $I^\bullet \to X$ (singular $\bullet$-cubes in $X$), then it's trivial that homotopic maps induce the same morphism in $H(\square_\bullet)$
Because all the prism formalism vanishes. A cube crossed with an interval is cube.
The cleanest proof would say that both of those homology theories are functors on the category of CW complexes satisfying all the relevant axioms of Eilenberg-Steenrod, and takes the same value at a point.
Then those are isomorphic theories by a small theorem.
@Alessandro There is a natural way to interpret a singular cube as a sum of singular simplices, by triangulating $I^n$
I wonder why aren't cubes used instead then, also in the $H_1\cong\pi_1^{\mathrm{ab}}$ proof there was a bit of annoying formalism to go from $\Delta^1\times I$ to $\Delta^2$
Well actually an homotopy of paths $I\times I\to X$ is a singular cube in $X$ and tom Dieck did quotient a face to get a $2$-complex instead so it's the same situation
@MatheinBoulomenos how much does/can Galois cohomology come up in doing elliptic curves? Especially when looking at the latter from a more Galois-ish point of view?
In the cubical singular homology story you need to quotient by 'degenerate' cubes. Without doing something like this check the homology groups you get if a point
I'd say Galois cohomology doesn't come up with elliptic curves doesn't come up until you go fairly deep
there are elementary books e.g. Silverman/Tate which don't mention it at all
Silverman's more advanced book "Arithmetic of Elliptic Curves" does the results he needs in a 7 page appendix
but when you do get deeper into the theory, it becomes more important, then you have stuff like the Weil–Châtelet group, the Tate-Shafarevich group and the Selmer group which are all defined in terms of group cohomology
and there's an indirect way how Galois cohomology can come up: you might want to look at the Galois representation associated to an elliptic curves and then Galois cohomology can become pretty important. I'm taking a class in Galois reps right now and Galois cohomology is pretty important
I may try to see if I can aim toward the Galois rep side of this. But yeah I feel like elliptic curves sorta give you some neat content right off the bat while group cohomology feels like it takes more plowing. I'll still try to include it since it seems neat but I think my principal thing for now is elliptic curves
that should be really accessible too, (gal group acting on your torsion points) - so you can see how things work there and see if you like the feel of that
i dont know group cohomology to say if i like it or not
it feels quite technical in the beginning, but when you get to Galois cohomology you realize that it's pretty cool. (that's how it felt for me) and it's a pretty important tool in algebraic number theory
It also comes up if you want to do computations in étale cohomology
@MikeMiller I found out that Milnor K-groups come up with higher local class field theory, so that's something topological I have a motivation from NT to learn
You two would love one of the beer skits this year
Sermon for the Church of Thurston
Asking him to protect Neves from becoming an analyst, Benson from getting hooked on p-adic Hodge theory, forgive Peter, the corruptor of youth, evil soul, etc
i found these lectures online on refined enumerative invariants which had something to do with milnor k theory etc which i might watch (if i can follow)
