diff options
Diffstat (limited to 'ts/n6.lyx')
| -rw-r--r-- | ts/n6.lyx | 155 |
1 files changed, 137 insertions, 18 deletions
@@ -683,6 +683,50 @@ Así, \end_inset es triangulable a un tetraedro. + +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +En efecto, sean +\begin_inset Formula $a:=(0,0,1)$ +\end_inset + +, +\begin_inset Formula $b:=(0,1,-1)$ +\end_inset + +, +\begin_inset Formula $c:=(-1,-1,-1)$ +\end_inset + + y +\begin_inset Formula $d:=(1,-1,-1)$ +\end_inset + +, entonces el complejo simplicial dado por +\begin_inset Formula +\begin{multline*} +\{\{a,b,c\},\{a,b,d\},\{a,c,d\},\{b,c,d\},\{a,b\},\{a,c\},\\ +\{a,d\},\{b,c\},\{b,d\},\{c,d\},\{a\},\{b\},\{c\},\{d\}\} +\end{multline*} + +\end_inset + +junto con el homeomorfismo +\begin_inset Formula $h(x,y,z):=\frac{(x,y,z)}{|(x,y,z)|}$ +\end_inset + + forman una triangulación de +\begin_inset Formula $\mathbb{S}^{2}$ +\end_inset + +. +\end_layout + +\end_inset + + \end_layout \begin_layout Standard @@ -1610,7 +1654,7 @@ Sea $K_1$ una triangulación de $S_1$ y $K_2$ una de $S_2$, y suponemos que \end_layout \begin_layout Standard -Así: +Entonces: \end_layout \begin_layout Enumerate @@ -1618,21 +1662,11 @@ Así: \end_inset . - \begin_inset Note Comment status open \begin_layout Plain Layout -\begin_inset Note Note -status open - -\begin_layout Plain Layout -Demostración. -\end_layout - -\end_inset - - +Por triangulación con un tetraedro. \end_layout \end_inset @@ -1645,21 +1679,106 @@ Demostración. \end_inset . - \begin_inset Note Comment status open \begin_layout Plain Layout -\begin_inset Note Note -status open +Sean +\begin_inset Formula $a_{0}:=(0,1,0)$ +\end_inset -\begin_layout Plain Layout -Demostración. -\end_layout +, +\begin_inset Formula $a_{1}:=(0,3,1)$ +\end_inset + +, +\begin_inset Formula $a_{2}:=(0,3,-1)$ +\end_inset + +, +\begin_inset Formula $b_{0}:=(-1,-1,0)$ +\end_inset + +, +\begin_inset Formula $b_{1}:=(-3,-3,1)$ +\end_inset + +, +\begin_inset Formula $b_{2}:=(-3,-3,-1)$ +\end_inset + +, +\begin_inset Formula $c_{0}:=(1,-1,0)$ +\end_inset + +, +\begin_inset Formula $c_{1}:=(3,-3,1)$ +\end_inset + + y +\begin_inset Formula $c_{2}:=(3,-3,-1)$ +\end_inset + +. + Usamos el complejo simplicial cuyas caras son +\begin_inset Formula +\begin{multline*} +\{\{a_{0},b_{0},a_{1}\},\{a_{1},b_{0},b_{1}\},\{a_{1},b_{1},a_{2}\},\{a_{2},b_{1},b_{2}\},\{a_{2},b_{2},a_{0}\},\{a_{0},b_{2},b_{0}\},\\ +\{c_{0},b_{0},c_{1}\},\{c_{1},b_{0},b_{1}\},\{c_{1},b_{1},c_{2}\},\{c_{2},b_{1},b_{2}\},\{c_{2},b_{2},c_{0}\},\{c_{0},b_{2},b_{0}\},\\ +\{a_{0},c_{0},a_{1}\},\{a_{1},c_{0},c_{1}\},\{a_{1},c_{1},a_{2}\},\{a_{2},c_{1},c_{2}\},\{a_{2},c_{2},a_{0}\},\{a_{0},c_{2},c_{0}\}\}, +\end{multline*} + +\end_inset + +y cuyas aristas y vértices son los subsímplices de estas caras. + Entonces, si +\begin_inset Formula $r:=\frac{29}{20}$ +\end_inset + +, la circunferencia +\begin_inset Formula $r\mathbb{S}^{1}\times\{0\}$ +\end_inset + está contenida en el interior del complejo, pues este contiene a +\begin_inset Formula $([(0,3),(-3,-3),(3,-3)]\setminus[(0,1),(-1,-1),(1,-1)])$ \end_inset +, pero el punto más alejado del origen del triángulo interior (uno de ellos) + es +\begin_inset Formula $(-1,1)$ +\end_inset + + con norma +\begin_inset Formula $\sqrt{2}<r$ +\end_inset + + y el más cercano al origen del triángulo exterior (uno de ellos) es +\begin_inset Formula $\frac{(0,3)+(-3,-3)}{2}=(-\frac{3}{2},0)$ +\end_inset + con norma +\begin_inset Formula $\frac{3}{2}>r$ +\end_inset + +. + Entonces, si +\begin_inset Formula $p(x,y):=r(\frac{x}{\sqrt{x^{2}+y^{2}}},\frac{y}{\sqrt{x^{2}+y^{2}}},0)$ +\end_inset + +, la función +\begin_inset Formula $h(x,y,z):=r(x,y)+\frac{(x,y,z)-r(x,y)}{|(x,y,z)-r(x,y)|}$ +\end_inset + + es un homeomorfismo del complejo al toro con circunferencia interior +\begin_inset Formula $r\mathbb{S}^{1}\times\{0\}$ +\end_inset + + y exterior de radio 1. + El complejo tiene 18 caras, 27 aristas y 9 vértices, por lo que +\begin_inset Formula $\chi(\mathbb{T}^{2})=9-27+18=0$ +\end_inset + +. \end_layout \end_inset |