/sci/ - Science & Math

>>12148736
Have you tried it? A quick brainstorming gave me this idea: If we define [math]M := \Phi_a^{-1}( \pi^{-1}(N))[/math], then you could maybe define [math]H\colon M\times I\to M, H(x, t) = \Phi_a^{-1}(h(\Phi_a(x), t))[/math] for [math]x\not\in S^{p-1}, t<1[/math] and [math]H(x, t)=x[/math] otherwise. Could this be extended to [math]t=1[/math]? Perhaps. If so, then [math]H(x, 0) = \Phi_a^{-1}(h(\Phi_a(x), 0)) = (\Phi_a^{-1}\circ \Phi_a)(x) = x[/math] even for the interior points and this would then give you the desired homotopy. Here [math]h\colon N\times I\to N[/math] is the deformation he mentions in the text.

>>12111292
Just your basic ring theory, but if you wish to walk the AG path I will not stop you.

>>11765106
I think it's pretty much the same all the time. There are things you know and there are things you don't know. The things you know help you build the bridge between the stuff the same way as you would on lower levels, but that could depend on the individual. Finding the material is the actual challenge.

>>11703848
>how important
Yes. You will often be working with various quotient structures, and the elements of those are literally equivalence classes. Normal subgroups and ideals give you quotient groups and rings, resp., and yeah.