I need to prove that A is diagonalizable when I already know this:
A is 7x7
Rank(A) = 4
The characterized polynom is t^3(t^2-2)(t^2-5)
Now I know that 0 is eigenvalue with both algebraic multiplicity and geometric multiplicity of 3,
and I now that sqrt(2) is eigenvalue with algebraic multipli...
Is there an example for a one-one functions that maps from set of all square integrable continous functions of the from (0,1) to \mathbb{R} to set of all square integrable Lipschitz continous functions of the form (0,1) to \mathbb{R}. I am curious and want to know any web references/clues
@Danu, @Balarka, (@MikeM, @PVAL): I've checked details on the spectral sequence. $d_2$ does in fact map the generator of $H^1(S^1)$ to the Euler class in $H^2(T_g)$. So we get $H^1 = \Bbb Z^{2g}$ and $H^2 = \Bbb Z^{2g}\oplus \Bbb Z_{|e|}$. Dualizing gives what PVAL deduced about $H_1$. There is torsion.
I had this pedagogy question : why is what we call maths in early classes mostly calculation ? Is maturity the only reason we don't teach children maths that doesn't need computation ?
@TedShifrin Yeah, I talked with PVAL earlier today. $\pi_1(T_1 \Sigma)$ is a $\Bbb Z$-extension of $\pi_1(\Sigma)$, but abelianization does in fact introduce torsion.
@Astyx: There is a trend to try to get kids to think a bit more conceptually (not abstractly, necessarily). But many of the teachers can't handle it, and parents certainly can't.
You can do Serre on this, right? Monodromy should be trivial along the generator loops of $\pi_1(\Sigma)$ because the bundle is orientation, and restricts to torii along those.
I just thought it through in terms of the Čech construction. It is the usual zigzag chase, @Balarka (for me, getting the curvature 2-form from the connection 1-form).
It's the same spectral sequence, @Balarka, just a matter of how one understands it. For me it's a double complex with d going vertically and Čech coboundary going horizontally.
@TedShifrin Just to check, am I right to think that you're talking about cohomology with integer coefficients? It's been a while since I looked at cohomology stuff so I may just be mangling things.
I need to prove that A is diagonalizable when I already know this:
A is 7x7
Rank(A) = 4
The characterized polynom is t^3(t^2-2)(t^2-5)
Now I know that 0 is eigenvalue with both algebraic multiplicity and geometric multiplicity of 3,
and I now that sqrt(2) is eigenvalue with algebraic multipli...
