User:Juan Marquez

$D<\Sigma \subset M^3$

$O_1\cong S^1\times S^1$, $O_2=O_1\# O_1$, ..., $O_g=O_{g-1}\# O_1$, ...



$N_1=M\ddot{o}\cup D^2$, $N_2=N_1\# N_1$, ..., $N_k=N_{k-1}\# N_1$, ...

$M\ddot{o}\times I,\qquad$ $M\ddot{o}\stackrel{\sim}\times I$

$M\ddot{o}\subset M\ddot{o}\times S^1\to S^1$

$O_1\times S^1$

$S^1\subset M\ddot{o}\times S^1\to M\ddot{o}$

$I\subset O_1\times I\to O_1$

$I\subset O_1\stackrel{\sim}\times I\to O_1$

$O_1\stackrel{\sim}\times I$ is $M\ddot{o}\times S^1$

$N_2\times I\ ,\qquad N_2\stackrel{\sim}\times I^{NO}\, \qquad N_2\stackrel{\sim}\times I^O=M\ddot{o}\stackrel{\sim}\times S^1$


 * $I\subset N_2\stackrel{\sim}\times I^O\to N_2$

N_3-bundles
...

$f\in {\cal{MCG}}(N_3)=GL_2(\Z)$


 * $=pA(N_3)\cup per(N_3)\cup red(N_3)$


 * $=per(N_3)\cup red(N_3)$


 * cuz'

$pA(N_3)=\varnothing$, (1983).

$\# per(N_3)=6$

$N_3\subset N_3\times_f S^1\to S^1$

$E_f=N_3\times_f S^1$

$w_1(E_f)\neq0$ and $\beta(w_1(E_f))\neq0$