Terminado superficies

1 files changed, 79 insertions, 85 deletions

diff --git a/ts/n4.lyx b/ts/n4.lyx
index 52c7efa..c315d68 100644
--- a/ts/n4.lyx
+++ b/ts/n4.lyx
@@ -2500,103 +2500,97 @@ Si
 \begin_inset Note Comment
 status open
 
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
 \begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
 
-\backslash
-begin{itemize}
 \end_layout
 
-\begin_layout Plain Layout
+\end_inset
 
-\backslash
-item[$
-\backslash
-implies]$] Sea $F:
-\backslash
-mathbb{S}^1
-\backslash
-times[0,1]
-\backslash
-to
-\backslash
-mathbb{S}^1$ la homotopía de una cierta función constante $c(z):=z_0$ a
- $f$ y definimos $
-\backslash
-hat f:
-\backslash
-mathbb{D}^2
-\backslash
-to
-\backslash
-mathbb{S}^1$ como $
-\backslash
-hat f(0):=z_0$ y $
-\backslash
-hat f(z):=F(
-\backslash
-frac z{
-\backslash
-Vert z
-\backslash
-Vert},
-\backslash
-Vert z
-\backslash
-Vert)$ para $z
-\backslash
-neq0$.
- Para ver que $
-\backslash
-hat f$ es continua también en $z_0$, $
-\backslash
-lim_{z
-\backslash
-to0}
-\backslash
-hat f(z)=
-\backslash
-lim_{z
-\backslash
-to0}F({z
-\backslash
-over
-\backslash
-Vert z
-\backslash
-Vert},
-\backslash
-Vert z
-\backslash
-Vert)=z_0$.
- Para $z
-\backslash
-in
-\backslash
-mathbb{S}^1$, $
-\backslash
-hat f(z)=F(z,1)=f(z)$.
+Si 
+\begin_inset Formula $\deg f=0$
+\end_inset
+
+, existe una homotopía 
+\begin_inset Formula $F:\mathbb{S}^{1}\times[0,1]\to\mathbb{S}^{1}$
+\end_inset
+
+ de una función constante en 
+\begin_inset Formula $z_{0}\in\mathbb{S}^{1}$
+\end_inset
+
+ a 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Entonces tomamos 
+\begin_inset Formula $\hat{f}:\mathbb{D}^{2}\to\mathbb{S}^{1}$
+\end_inset
+
+ como 
+\begin_inset Formula $\hat{f}(re(\theta)):=F(e(\theta),r)$
+\end_inset
+
+, que está bien definida porque, si 
+\begin_inset Formula $re(\theta)=r'e(\theta')$
+\end_inset
+
+ con 
+\begin_inset Formula $(r,\theta)\neq(r',\theta')$
+\end_inset
+
+, debe ser 
+\begin_inset Formula $r=r'=0$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $F(e(\theta),r)=z_{0}=F(e(\theta'),r')$
+\end_inset
+
+.
 \end_layout
 
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
 \begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
 
-\backslash
-item[$
-\backslash
-impliedby]$] Definimos $F(z,t):=
-\backslash
-tilde f(tz)$.
- Entonces $F(z,1)=
-\backslash
-hat f(z)=f(z)$ y $F(z,0)=
-\backslash
-hat f(0)$ que es constante, y como $F$ es continua, define una homotopía
- entre $f$ y una constante.
 \end_layout
 
-\begin_layout Plain Layout
+\end_inset
 
-\backslash
-end{itemize}
+Basta ver que 
+\begin_inset Formula $F:\mathbb{S}^{1}\times[0,1]\to\mathbb{S}^{1}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $F(z,t):=\hat{f}(zt)$
+\end_inset
+
+ es una homotopía de la función constante en 
+\begin_inset Formula $\hat{f}(0)$
+\end_inset
+
+ a 
+\begin_inset Formula $f$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene el mismo grado que una función constante, 0.
 \end_layout
 
 \end_inset