/sci/ - Science & Math

>>12026346
>>12026372
So instead of doing algebra and related things, you think w- they should only be treated as tools for pleasure and housekeeping?

>>11708061
Stuff. All sorts of homotopy equivalences you get when you apply the loop space functor. For example [math] \Omega( A\vee B) \simeq \Omega(A) \times \Omega(B) \times \Omega \Sigma( \Omega(A) \wedge\Omega(B))[/math] or [math]\Omega( A\rtimes B)\simeq \Omega(A) \times \Omega\Sigma( \Omega(A)\wedge B)[/math]. Combining those, one gets a nice chain of isomorphisms: [math] \pi_n (A \vee B) \cong \pi_{n+1} (\Omega(A \vee B)) \cong \pi_{n+1} (\Omega(A)\times \Omega(B)\times \Omega\Sigma( \Omega(A)\wedge \Omega(B))) \cong \pi_{n+1}(\Omega(A)) \oplus \pi_{n+1} (\Omega(A \rtimes \Omega(B))) \cong \pi_n (A)\oplus \pi_n (A\rtimes \Omega(B))[/math]. Please don't explode...

>>11696491
I hope you have less than 24 fingers, anon. Otherwise you will not make it...

>>11679295
s/[math]\chi [/math]/izopfrenia.

>>11679328
Consider a model for the hyperbolic plane, then take its Möbius group. It can be shown that the Möbius group is a topological group, and then you can consider its discrete subgroups. Then you get all sorts of nice things like how every commutative Fuchsian group is cyclic, and Dirichlet fundamental domains etc. This one's pretty short, so go ahead and give it a read: http://www.f.waseda.jp/ykomori/umemoto-master.pdf