/sci/ - Science & Math

Let [math]\operatorname{FiveBrane}(n) \rightarrow \operatorname{String}(n) \rightarrow \operatorname{Spin}(n) \rightarrow SO(n) \rightarrow O(n) [/math] be the Whitehead tower of [math]O(n)[/math], I wish to find the cohomology ring of [math]\operatorname{FiveBrane}(n)[/math]. This can be done by noting that [math]\pi_k(\operatorname{FiveBrane}(n)) = \begin{cases}0 &; k \leq 7 \\ \pi_k(O(n)) &; \text{otherwise}\end{cases}[/math] whence [math]\operatorname{FiveBrane}(n)[/math] is the 7-connected cover of [math]O(n)[/math], hence the fibration sequence [math]F \rightarrow \operatorname{FiveBrane}(n)\rightarrow O(n)[/math] has fibre [math]\pi_k (F) \cong \pi_k(O(n))[/math] for all [math]k \leq 7[/math], and is a CW complex with known cohomogloy. Using this fibration sequence, I construct the Leray-Serre spectral sequence [math]E^{p,q}_r[/math] whose second page reads [math]E^{p,q}_2 \cong H^p(O(n),H^p(F,\mathbb{Z}))[/math].
How do I show that the sequence [math]\{E^{p,q}_r\}_r[/math] converges/collapses and [math]\lim\limits_{r\rightarrow\infty}\bigoplus_{p}E^{n,p}_r \cong H^n(\operatorname{FiveBrane}(n),\mathbb{Z})[/math]?
>>11559463
There, a question on limits of sequences. You like?

>>11512745
I mean [math]technically[/math] up to Morita equivalence the irreps of Clifford groups are classified by their ranks up to Bott periodicity but anon obv wasn't aware of that.

>>11459070
Because it wants you to develop intuition. At the research level it's what allowed e.g. Bethe/Laughlin/Feynman to guess the ground states and correlations of non-exactly solvable many-body quantum theories without knowing what a Yangian/Virasoro/Hodge algebra is, let along their representation theory. This is also what gives my previous advisor the gall to shit on mathematicians constantly.
https://arxiv.org/abs/hep-th/9310158v1
Sure, proving that a certain ansatz is valid is nice, but knowing where to look in the first place takes a whole different training.

>>11451303
https://ncatlab.org/nlab/show/universal+coefficient+theorem

>>11417803
Wot?
Also add Urs Schrieber to meh tier.

>>11407933
>Abel and Jacobi
>AMS

>>11398111
>wrecks Fatio's boypuss on the daily
>incel

>>11248462
>in the sense of distribution theory
Well in the sense of currents actually.

>>11155701
How is it confusing? Exact sequences (fibration, homotopy, relative, Meyer-Vietoris, Gysin, etc.) appear everywhere in algebraic topology. I just used a very elementary example.

>>11108797
I was never any confession thread though.

>>11101310
?
My commentary definitely applies for general commentary on what a weak-* topology is, which is presumably what you replied to. Not sure why you're so defensive, I'm not here to condescend to nor patronize you.

>>10875378
To boost his confidence.

>>10709620
>made significant progress towards Kashiwara-Vergne
>hack
??

>>9471837
>inelegant
Wrong.