@BalarkaSen: I have a homomorphism $f: \pi_1(\Sigma_g) \to F_h$, the free group on $h$ letters. I have $f^{-1}([F_h, F_h]) \subset [\pi_1, \pi_1]$. Can you prove that this implies $f$ is injective? I think this is probably true, and I can't write down any counterexamples.
That's where this comes from, @Balarka. I don't think there is one.
@TedShifrin: It shouldn't be possible by geometric heuristics I have. It's almost a correct argument from topology, but it seems impossible to get it to actually work, so I gave up on that approach. I have no idea whether this should be true for all finitely generated groups of rank $g<h$.
I guess it probably "should", because $\pi_1(\Sigma_g)$ is "close" to the free group $F_g$ itself, and everything else is actually "closer" to the abelianization. (It's only got one relator.)
@TedShifrin: Consider the following statement. For any genus $g$, there is, up to equivalence, precisely one epimorphism $\pi_1(\Sigma_g) \to F_g \times F_g$ (equivalence is just a commutative diagram of epis where the vertical maps are isomorphisms)
This was only proved in 2003.
2006, if you count the myriad details that had to be filled into the original argument...
i still say one should lift stuff. you can get higher maps outta lifting lower maps. for example, a map $\Sigma_3 \to \vee_4 S^1$ can be obtained from lifting the covering map $\Sigma_3 \to \Sigma_1$ composed with some map $\Sigma_1 \to \vee_2 S^1$ to the cover $\vee_4 S^1$.
i don't know what can be accomplished by this, but one should get to know about the $\Sigma_g \to \vee_h S^1$ maps before seeing what they does to homology and \pi_1
@TedShifrin: When doing gauge theory, before passing to the moduli space, one first looks at a suitable $H^p$ completion of the solution space to your equations, and gets a (usually Hilbert) manifold. You need to know some analysis of Hilbert manifolds for this situation.
not sure what of, but projects involving 3D using WebGL (pretty much OpenGL if you've heard of it) as well some smaller things like games using physics engines
I've already taken through calc 3 and physics through electricity and magnetism
am curious if linear algebra would be helpful at all
@robjohn I managed to do those because I did research in that specific area (in general, I invest much time in research), so I started with an advantage. Otherwise it could have been more difficult ...
@DiscipleofBarney Hey it is <me> on another account(at uni in a common area)
Can you help me understand what they are doing after they show $A$ is diagonal here: http://www.math.toronto.edu/vtk/247Winter2012/247practicefinalsol.pdf
1. We let each $\lambda$ be unique to all others. 2. We now take $a_{11}\ne a_{22}$ and use a permutation matrix, that would need not be diagonal(which is the premise I believe)
And then we deduce that $a_{11}=a_{22}$ and so on
and then we say that all of the $\lambda$'s are the same
What they do is take a matrix $A$, that is diagonal, but has different entries along the diagonal, then they exibit matrix $B$ which it does not commute with, since $C= B A B^{-1}$ is not equal to $A$.
From that we get that anything in the center must be a multiple of the identity (which is easy to prove those are in the center)
@Marc-AntoineJacob You don't have to post twice, you would have better luck posting a well thought out question on the main site since a lot more eyes will be on the question
And it is easy to prove $ \lambda I$ is in the center, so they characterize all things in the centere essentially by elimination, first by showing anything that is in the center is diagonal, then anything that is not a scalar multiple of $I$ is not in the center because of that permuation argument.
