Mathematics

3:13 AM

@BalarkaSen kiwisbybeat.netlify.com/verse3.html

@BalarkaSen What this argument shows me is that it's equivalent to $(S^n \times D^n)/(S^n \times S^{n-1})$ but I'm not sure I see that this is what you think it is

But maybe this is just a generalization of the torus case as you say

Balarka Sen

3:41 AM

@MikeMiller this is A+

@MikeMiller yeah this is easier to argue; it's the Thom construction on the trivial bundle on S^n

I do not see how to spell out the torus argument in this case but I'm too sleepy and on the airport so not in the best shape

Mike Miller

4:11 AM

@BalarkaSen Oh this is a nice observation. The Thom space you get when you add a rank one trivial bundle just is the suspension of the previous Thom space

So this is $\Sigma^n(S^n \vee S^0)$

You can see the two factors in the torus case

Secret

4:50 AM

ams.org/journals/notices/201907/rnoti-p1034.pdf

Leaky Nun

5:02 AM

@BalarkaSen $(S^n \times S^n)/(\ast \times S^n) = (\Bbb R^n \times S^n + \ast \times S^n)/(\ast \times S^n) = \Bbb R^n \times S^n + \ast = \Bbb R^n + \ast + \Bbb R^{2n} = (\Bbb R^n + \ast) \vee (\ast + \Bbb R^{2n}) = S^n \vee S^{2n}$

lol no

ok suppose $H_n = L_n H_0$, which is the case for nice enough spaces, then we are interested in $H_n(-;G) = L_n(H_0 - \otimes G)$

Grothendieck says that $L_p (- \otimes G) L_q H_0 - \implies H_{p+q}(-;G)$

i.e. $\operatorname{Tor}_p(H_q-,G) \implies H_{p+q}(-;G)$

as you noted, over PID we have $\operatorname{Tor}_{\ge 2} = 0$

7 hours ago, by Balarka Sen

$\cdots \to E^2_{n, 0} \to E^2_{n-2, 1} \to H_{n-1} \to E^2_{n-1, 0} \to E^2_{n-3, 1} \to \cdots$

how does this work

no this is wrong, the LES above is for concentrated in q=0,1

here we have p=0,1

so $E^2 = E^\infty$

so $0 \to E^2_{0,n} \to H_n \to E^2_{1,n-1} \to 0$

i.e. $0 \to H_n(-) \otimes G \to H_n(-;G) \to \operatorname{Tor}(H_{n-1}(-),G) \to 0$

let's check the UCT...

(aha, this is why you said UCT!)

2 days ago, by Leaky Nun

$0 \to H_i(X; \mathbf{Z})\otimes A \, \overset{\mu}\to \, H_i(X;A) \to \operatorname{Tor}(H_{i-1}(X; \mathbf{Z}),A)\to 0$

praise Grothendieck!

wait, if in general $H_n \ne L_n H_0$ then what is $L_n H_0$?

oh wait this makes no sense in general, because $\mathsf{Top}$ isn't an abelian category!

so above I should replace $L_n H_0$ with $L_n$ of the global section of the constant sheaf

the correct statement is that for nice enough spaces, $H_n$ is a left derived functor

since it coincides with the sheaf cohomology of the constant sheaf $\Bbb Z$

Balarka Sen

8:28 AM

@MikeMiller oh good point

Mike Miller

8:53 AM

@BalarkaSen I think the distinction between geometry and homotopy theory is that this is the unreduced suspension.

MatheinBoulomenos

10:40 AM

@Balarka is there a nice abstract reason why $\mathrm{vol}(\Bbb{H}/\mathrm{SL}_2(\Bbb Z)) < \infty$ or do you need to use a fundamental domain?

loch

11:23 AM

- sheaf cohomology gives you singular cohomology of your space (not homology)
- I don't think anyone uses the notation L_nH_0, usually L_nF or R_nF, for whatever your functor F is
- I don't think singular homology is realised as left derived functors for some functor on sheaves (at least ive never heard of such thing) .. you can get something called borel moore homology using sheaves though

hm based on this link mathoverflow.net/questions/66401/… maybe you can

MatheinBoulomenos

11:38 AM

@Leaky exercise: X is a compact Hausdorff space, then C(X) is Jacobson iff X is totally disconnected

Perturbative

12:33 PM

Suppose I have a fibration $p : E \to B$ with fiber $F$, let $G$ be an abelian group, what is the action of $\pi_1(B)$ on $H_{\bullet}(F; G)$?

Balarka Sen

1:07 PM

@MatheinBoulomenos Yeah just use the fundamental domain. It's a $(2, 3, \infty)$-triangle, so has area $\pi - (\pi/2 + \pi/3 + \pi/\infty) = \pi/6$.

By Gauss-Bonnet if you wish

@Perturbative $\pi_1 B$ acts on $F$ by monodromy. Namely, given a loop on $B$ based at a point $x \in B$, you can lift it to a path starting at any point $f \in F$ and ending at $\tilde{f} \in F$.

If the homotopy class of this loop is $\alpha$, we define the action by $\alpha \cdot f = \tilde{f}$. This is a continuous action of $\pi_1 B$ on $F$.

Look at the induced action on $H_*(F)$

@loch The sheafification of the singular cochains presheaf gives an acyclic resolution of the constant sheaf, so you can prove that the singular cohomology agrees with the Cech cohomology. That shows it's left derived functors for the global section functor on the constant sheaf, right?

loch

1:31 PM

@BalarkaSen I think that sounds about right - but it's right derived functors not left derived functors since the global sections functor is left exact :p

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