Fundamental groups

3 files changed, 3044 insertions, 86 deletions

diff --git a/ts/n.lyx b/ts/n.lyx
index bf8b555..0fe0bd7 100644
--- a/ts/n.lyx
+++ b/ts/n.lyx
@@ -136,9 +136,14 @@ Diapositivas de clase, Pascual Lucas (2019–20), Departamento de Matemáticas,
 \end_layout
 
 \begin_layout Itemize
-Modelling CPV, Ian Richard Cole (2015), 
+Ian Richard Cole (2015).
+ 
+\emph on
+Modelling CPV
+\emph default
+, 
 \begin_inset Flex URL
-status collapsed
+status open
 
 \begin_layout Plain Layout
 
@@ -151,8 +156,13 @@ https://repository.lboro.ac.uk/articles/Modelling_CPV/9523520
 \end_layout
 
 \begin_layout Itemize
-Essential Topology, Martin D.
+Martin D.
  Crossley (2005), Springer.
+ 
+\emph on
+Essential Topology
+\emph default
+.
 \end_layout
 
 \begin_layout Itemize
@@ -170,6 +180,37 @@ https://en.wikipedia.org/
 .
 \end_layout
 
+\begin_layout Itemize
+Klint Qinami.
+ 
+\emph on
+Algebraic Topology
+\emph default
+, 
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+https://www.cs.princeton.edu/~kqinami/pdfs/algebraic_topology_notes.pdf
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+James R.
+ Munkres (2000).
+ 
+\emph on
+Topología
+\emph default
+ (segunda edición).
+ 
+\end_layout
+
 \begin_layout Chapter
 Espacios topológicos
 \end_layout
@@ -226,5 +267,33 @@ filename "n4.lyx"
 
 \end_layout
 
+\begin_layout Chapter
+El grupo fundamental
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n5.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+El número de Euler
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n6.lyx"
+
+\end_inset
+
+
+\end_layout
+
 \end_body
 \end_document
diff --git a/ts/n4.lyx b/ts/n4.lyx
index b292e28..2a486a1 100644
--- a/ts/n4.lyx
+++ b/ts/n4.lyx
@@ -551,7 +551,7 @@ Dos espacios
 \begin_inset Formula $X$
 \end_inset
 
- es 
+ e 
 \begin_inset Formula $Y$
 \end_inset
 
@@ -700,7 +700,7 @@ Si
 \begin_inset Formula $Y$
 \end_inset
 
- son homeomorfismos, 
+ son homeomorfos, 
 \begin_inset Formula $X\simeq Y$
 \end_inset
 
@@ -1326,20 +1326,31 @@ Circunferencia
 \end_layout
 
 \begin_layout Standard
-Llamamos 
+Una 
 \series bold
-aplicación exponencial
+aplicación recubridora
 \series default
- a 
-\begin_inset Formula $e:\mathbb{R}\to\mathbb{S}^{1}$
+ es una función 
+\begin_inset Formula $r:X\to Y$
 \end_inset
 
- dada por 
-\begin_inset Formula $e(\theta):=(\cos(2\pi\theta),\sin(2\pi\theta))$
+ sobreyectiva tal que para todo 
+\begin_inset Formula $x\in X$
 \end_inset
 
-.
- Sean un camino 
+ existe 
+\begin_inset Formula $U\in{\cal E}(x)$
+\end_inset
+
+ con 
+\begin_inset Formula $r:U\to r(U)$
+\end_inset
+
+ homeomorfismo.
+\end_layout
+
+\begin_layout Standard
+Sean un camino 
 \begin_inset Formula $\alpha:[0,1]\to\mathbb{S}^{1}$
 \end_inset
 
@@ -1380,91 +1391,207 @@ levantamiento
 \end_inset
 
 .
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $r:X\to Y$
+\end_inset
+
+ una aplicación recubridora, 
+\begin_inset Formula $\alpha:[0,1]\to Y$
+\end_inset
+
+ un camino, 
+\begin_inset Formula $x_{0}\in X$
+\end_inset
+
+ e 
+\begin_inset Formula $y_{0}:=r(x_{0})$
+\end_inset
+
+, existe un único camino 
+\begin_inset Formula $\tilde{\alpha}:[0,1]\to X$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\alpha=r\circ\tilde{\alpha}$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{\alpha}(0)=r_{0}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
 \begin_inset Note Comment
 status open
 
 \begin_layout Plain Layout
-Claramente $e$ es continua y sobreyectiva.
- Sea un abierto $U
-\backslash
-subsetneq
-\backslash
-mathbb{S}^1$, existe $V
-\backslash
-subseteq
-\backslash
-mathbb R$ tal que $e|_V:V
-\backslash
-to U$ es un homeomorfismo, y como esto es periódico, $e^{-1}(U)=
-\backslash
-bigcup_{n
-\backslash
-in
-\backslash
-mathbb Z}V_n$ con $e|_{V_n}:V_n
-\backslash
-to U$ homeomorfismo.
- 
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $V$
+\end_inset
+
+ un abierto de 
+\begin_inset Formula $Y$
+\end_inset
+
+, para cada 
+\begin_inset Formula $x\in r^{-1}(V)$
+\end_inset
+
+ existe 
+\begin_inset Formula $U_{x}\in{\cal E}(x)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $r|_{U_{x}}$
+\end_inset
+
+ es un homeomorfismo, luego 
+\begin_inset Formula $V_{x}:=V\cap f(U_{X})$
+\end_inset
+
+ es abierto, con lo que 
+\begin_inset Formula $f^{-1}(V_{x})$
+\end_inset
+
+ es abierto con 
+\begin_inset Formula $x\in f^{-1}(V_{x})\subseteq f^{-1}(V)$
+\end_inset
+
+ y 
+\begin_inset Formula $r^{-1}(V)=\bigcup_{x\in r^{-1}(V)}V_{x}$
+\end_inset
+
+, que es abierto.
 \end_layout
 
 \begin_layout Plain Layout
-Como $
-\backslash
-alpha$ es continua, para $
-\backslash
-theta
-\backslash
-in[0,1]$ existe un intervalo $I_
-\backslash
-theta$ con $
-\backslash
-alpha(I_
-\backslash
-theta)
-\backslash
-subseteq U_
-\backslash
-theta$ para un cierto $U_
-\backslash
-theta
-\backslash
-ni
-\backslash
-alpha(
-\backslash
-theta)$ que queramos (por ejemplo, $B(
-\backslash
-alpha(
-\backslash
-theta),
-\backslash
-varepsilon)$.
- Como $
-\backslash
-alpha([0,1])$ es compacto, existe un subrecubrimiento finito $
-\backslash
-{I_{
-\backslash
-theta_1},
-\backslash
-dots,I_{
-\backslash
-theta_n}
-\backslash
-}$ (podemos suponer $
-\backslash
-theta_1<
-\backslash
-dots<
-\backslash
-theta_n$).
- En cada $I_k$, $e$ es biyectiva definida salvo suma de un entero, luego
- vamos <<enganchando>> y sale.
+Para 
+\begin_inset Formula $t\in[0,1]$
+\end_inset
+
+ existe 
+\begin_inset Formula $U_{t}\in{\cal E}(t)$
+\end_inset
+
+ con 
+\begin_inset Formula $r|_{U_{t}}$
+\end_inset
+
+ homeomorfismo, y como 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ es continua, existe 
+\begin_inset Formula $I_{t}\subseteq[0,1]$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\alpha(I_{t})\subseteq r(U_{t})$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $\alpha([0,1])$
+\end_inset
+
+ es compacto, existe un subrecubrimiento finito 
+\begin_inset Formula $\{I_{t_{1}},\dots,I_{t_{n}}\}$
+\end_inset
+
+ del recubrimiento 
+\begin_inset Formula $\{I_{t}\}_{t\in[0,1]}$
+\end_inset
+
+ de 
+\begin_inset Formula $[0,1]$
+\end_inset
+
+, y podemos suponer 
+\begin_inset Formula $t_{1}<\dots<t_{n}$
+\end_inset
+
+.
+ En cada 
+\begin_inset Formula $I_{t_{k}}$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha(I_{t_{k}})\subseteq r(U_{k})$
+\end_inset
+
+, pero 
+\begin_inset Formula $\alpha(I_{t_{k}})$
+\end_inset
+
+ estará en una componente conexa de 
+\begin_inset Formula $r(U_{k})$
+\end_inset
+
+ y, por el homeomorfismo, si 
+\begin_inset Formula $s\in I_{t_{k-1}}\cap I_{t_{k}}$
+\end_inset
+
+, existe una componente conexa de 
+\begin_inset Formula $r^{-1}(r(U_{k}))$
+\end_inset
+
+ que contiene a la componente conexa de 
+\begin_inset Formula $r^{-1}(\alpha(I_{t_{k}}))$
+\end_inset
+
+ donde se encuentra el elemento de 
+\begin_inset Formula $r^{-1}(\alpha(s))$
+\end_inset
+
+ elegido al establecer un levantamiento de 
+\begin_inset Formula $\alpha|_{I_{t_{k-1}}}$
+\end_inset
+
+ (si 
+\begin_inset Formula $k=1$
+\end_inset
+
+, tomamos una componente arbitraria), luego en esta componente conexa definimos
+ un levantamiento de 
+\begin_inset Formula $\alpha|_{I_{t_{k}}}$
+\end_inset
+
+ que concatenamos a la anterior, y concatenando sucesivamente obtenemos
+ un levantamiento de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
+\begin_layout Standard
+La 
+\series bold
+aplicación exponencial
+\series default
+ es la aplicación recubridora 
+\begin_inset Formula $e:\mathbb{R}\to\mathbb{S}^{1}$
 \end_inset
 
+ dada por 
+\begin_inset Formula $e(\theta):=(\cos(2\pi\theta),\sin(2\pi\theta))$
+\end_inset
 
+.
 \end_layout
 
 \begin_layout Standard
diff --git a/ts/n5.lyx b/ts/n5.lyx
new file mode 100644
index 0000000..00531c4
--- /dev/null
+++ b/ts/n5.lyx
@@ -0,0 +1,2762 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
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+\spacing single
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+\use_package amsmath 1
+\use_package amssymb 1
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+\use_package mathdots 1
+\use_package mathtools 1
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+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
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+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
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+\quotes_style french
+\dynamic_quotes 0
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+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Dos caminos 
+\begin_inset Formula $\alpha,\beta:[0,1]\to X$
+\end_inset
+
+ son 
+\series bold
+homotópicos por caminos
+\series default
+, 
+\begin_inset Formula $\alpha\simeq_{p}\beta$
+\end_inset
+
+, si tienen el mismo punto inicial 
+\begin_inset Formula $x$
+\end_inset
+
+, el mismo punto final 
+\begin_inset Formula $y$
+\end_inset
+
+ y existe 
+\begin_inset Formula $F:[0,1]\times[0,1]\to X$
+\end_inset
+
+ continua tal que para 
+\begin_inset Formula $s,t\in[0,1]$
+\end_inset
+
+, 
+\begin_inset Formula $F(s,0)=\alpha(s)$
+\end_inset
+
+, 
+\begin_inset Formula $F(s,1)=\beta(s)$
+\end_inset
+
+, 
+\begin_inset Formula $F(0,t)=x$
+\end_inset
+
+ y 
+\begin_inset Formula $F(1,t)=y$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $F$
+\end_inset
+
+ es una 
+\series bold
+homotopía de caminos
+\series default
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula ${\cal C}(X,x,y)$
+\end_inset
+
+ (
+\begin_inset Formula $x,y\in X$
+\end_inset
+
+) al conjunto de los caminos en 
+\begin_inset Formula $X$
+\end_inset
+
+ que unen 
+\begin_inset Formula $x$
+\end_inset
+
+ a 
+\begin_inset Formula $y$
+\end_inset
+
+.
+ Un camino 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ es un 
+\series bold
+lazo
+\series default
+ si 
+\begin_inset Formula $\alpha(0)=\alpha(1)$
+\end_inset
+
+, y llamamos 
+\begin_inset Formula ${\cal L}(X,x):={\cal C}(X,x,x)$
+\end_inset
+
+.
+ Dos lazos 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ y 
+\begin_inset Formula $\beta$
+\end_inset
+
+ son 
+\series bold
+homotópicos
+\series default
+ si son homotópicos por caminos.
+\end_layout
+
+\begin_layout Standard
+La relación 
+\begin_inset Formula $\simeq_{p}$
+\end_inset
+
+ es de equivalencia, y llamamos 
+\begin_inset Formula $\pi_{1}(X,x,y):={\cal C}(X,x,y)/\simeq_{p}$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi_{1}(X,x):={\cal L}(X,x)/\simeq_{p}$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Si 
+\begin_inset Formula $\alpha:[0,1]\to X$
+\end_inset
+
+, 
+\begin_inset Formula $F:[0,1]\times[0,1]\to X$
+\end_inset
+
+ dada por 
+\begin_inset Formula $F(s,t):=\alpha(s)$
+\end_inset
+
+ es una homotopía de caminos de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\alpha$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\alpha\simeq_{p}\beta$
+\end_inset
+
+, sea 
+\begin_inset Formula $F$
+\end_inset
+
+ una homotopía de caminos de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\beta$
+\end_inset
+
+, 
+\begin_inset Formula $G(s,t):=F(s,1-t)$
+\end_inset
+
+ es una homotopía de caminos de 
+\begin_inset Formula $\beta$
+\end_inset
+
+ a 
+\begin_inset Formula $\alpha$
+\end_inset
+
+, luego 
+\begin_inset Formula $\beta\simeq_{p}\alpha$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\alpha\simeq_{p}\beta\simeq_{p}\gamma$
+\end_inset
+
+, sean 
+\begin_inset Formula $F$
+\end_inset
+
+ una homotopía de caminos de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\beta$
+\end_inset
+
+ y 
+\begin_inset Formula $G$
+\end_inset
+
+ una de 
+\begin_inset Formula $\beta$
+\end_inset
+
+ a 
+\begin_inset Formula $\gamma$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+H(s,t):=\begin{cases}
+F(s,2t), & t\in[0,\tfrac{1}{2}];\\
+G(s,2t-1), & t\in[\tfrac{1}{2},1]
+\end{cases}
+\]
+
+\end_inset
+
+es una homotopía de caminos de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\gamma$
+\end_inset
+
+, luego 
+\begin_inset Formula $\alpha\simeq_{p}\gamma$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $\alpha\in{\cal C}(X,x,y)$
+\end_inset
+
+ y 
+\begin_inset Formula $\beta\in{\cal C}(y,z)$
+\end_inset
+
+, llamamos 
+\series bold
+yuxtaposición
+\series default
+ o 
+\series bold
+producto
+\series default
+ de caminos a 
+\begin_inset Formula $\alpha\land\beta\in{\cal C}(X,x,z)$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+(\alpha\land\beta)(s):=\begin{cases}
+\alpha(2s), & s\in[0,\tfrac{1}{2}];\\
+\beta(2s-1), & s\in[\tfrac{1}{2},1].
+\end{cases}
+\]
+
+\end_inset
+
+La operación 
+\begin_inset Formula $*:\pi_{1}(X,x,y)\times\pi_{1}(X,y,z)\to\pi_{1}(X,x,z)$
+\end_inset
+
+ dada por 
+\begin_inset Formula $[\alpha]*[\beta]:=[\alpha\land\beta]$
+\end_inset
+
+ está bien definida.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, sean 
+\begin_inset Formula $\alpha\simeq_{p}\alpha'$
+\end_inset
+
+, 
+\begin_inset Formula $\beta\simeq_{p}\beta'$
+\end_inset
+
+ y 
+\begin_inset Formula $F,G:[0,1]\times[0,1]\to X$
+\end_inset
+
+ homotopías de caminos respectivas de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\alpha'$
+\end_inset
+
+ y de 
+\begin_inset Formula $\beta$
+\end_inset
+
+ a 
+\begin_inset Formula $\beta'$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+H(s,t):=\begin{cases}
+F(2s,t), & s\in[0,\tfrac{1}{2}];\\
+G(2s-1,t), & s\in[\tfrac{1}{2},1]
+\end{cases}
+\]
+
+\end_inset
+
+es una homotopía de caminos de 
+\begin_inset Formula $\alpha\land\beta$
+\end_inset
+
+ a 
+\begin_inset Formula $\alpha'\land\beta'$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+Asociatividad.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Sean 
+\begin_inset Formula $\alpha\in{\cal C}(X,x,y)$
+\end_inset
+
+, 
+\begin_inset Formula $\beta\in{\cal C}(X,y,z)$
+\end_inset
+
+ y 
+\begin_inset Formula $\gamma\in{\cal C}(X,z,w)$
+\end_inset
+
+.
+ Entonces
+\begin_inset Formula 
+\begin{align*}
+\alpha\land(\beta\land\gamma) & =\begin{cases}
+\alpha(2s), & s\in[0,\tfrac{1}{2}];\\
+\beta(4s-2), & s\in[\tfrac{1}{2},\tfrac{3}{4}];\\
+\gamma(4s-3), & s\in[\tfrac{3}{4},1];
+\end{cases} &  & \text{y} & (\alpha\land\beta)\land\gamma & =\begin{cases}
+\alpha(4s), & s\in[0,\tfrac{1}{4}];\\
+\beta(4s-1), & s\in[\tfrac{1}{4},\tfrac{1}{2}];\\
+\gamma(2s-1), & s\in[\tfrac{1}{2},1].
+\end{cases}
+\end{align*}
+
+\end_inset
+
+Así, 
+\begin_inset Formula $F:[0,1]\times[0,1]\to X$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+F(s,t):=\begin{cases}
+\alpha({\textstyle \frac{4s}{2-t}}), & s\in[0,\tfrac{2-t}{4}];\\
+\beta(4s-2+t), & s\in[\tfrac{2-t}{4},\tfrac{3-t}{4}];\\
+\gamma(\tfrac{4s-3+t}{1+t}), & s\in[\tfrac{3-t}{4},1]
+\end{cases}
+\]
+
+\end_inset
+
+es una homotopía de caminos de 
+\begin_inset Formula $\alpha\land(\beta\land\gamma)$
+\end_inset
+
+ a 
+\begin_inset Formula $(\alpha\land\beta)\land\gamma$
+\end_inset
+
+, luego 
+\begin_inset Formula $[\alpha]*([\beta]*[\gamma])=[\alpha\land(\beta\land\gamma)]=[(\alpha\land\beta)\land\gamma]=([\alpha]*[\beta])*[\gamma]$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Llamamos 
+\series bold
+camino constante
+\series default
+ en 
+\begin_inset Formula $x$
+\end_inset
+
+ a 
+\begin_inset Formula $c_{x}\in{\cal L}(X,x)$
+\end_inset
+
+ dado por 
+\begin_inset Formula $c_{x}(s):=x$
+\end_inset
+
+.
+ Entonces, si 
+\begin_inset Formula $\alpha\in{\cal C}(X,x,y)$
+\end_inset
+
+, 
+\begin_inset Formula $[c_{x}]*[\alpha]=[\alpha]*[c_{y}]=[\alpha]$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Tenemos
+\begin_inset Formula 
+\begin{align*}
+(c_{x}\land\alpha)(s) & =\begin{cases}
+x, & s\in[0,\tfrac{1}{2}];\\
+\alpha(2s-1), & s\in[\tfrac{1}{2},1];
+\end{cases} &  & \text{y} & (\alpha\land c_{y})(s) & =\begin{cases}
+\alpha(2s), & s\in[0,\tfrac{1}{2}];\\
+y, & s\in[\tfrac{1}{2},1].
+\end{cases}
+\end{align*}
+
+\end_inset
+
+Entonces 
+\begin_inset Formula 
+\[
+F(s,t):=\begin{cases}
+x, & s\leq\tfrac{1-t}{2};\\
+\alpha(\tfrac{2s-1+t}{1+t}), & s\geq\tfrac{1-t}{2}
+\end{cases}
+\]
+
+\end_inset
+
+es una homotopía de 
+\begin_inset Formula $c_{x}\land\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ y 
+\begin_inset Formula 
+\[
+G(s,t):=\begin{cases}
+\alpha(\tfrac{2s}{1+t}), & s\leq\tfrac{1+t}{2};\\
+y, & s\geq\tfrac{1+t}{2}
+\end{cases}
+\]
+
+\end_inset
+
+ lo es de 
+\begin_inset Formula $\alpha\land c_{y}$
+\end_inset
+
+ a 
+\begin_inset Formula $\alpha$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Llamamos 
+\series bold
+camino inverso
+\series default
+ de 
+\begin_inset Formula $\alpha\in{\cal C}(X,x,y)$
+\end_inset
+
+ a 
+\begin_inset Formula $\overline{\alpha}\in{\cal C}(X,y,x)$
+\end_inset
+
+ dado por 
+\begin_inset Formula $\overline{\alpha}(s):=\alpha(1-s)$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $[\alpha]*[\overline{\alpha}]=[c_{x}]$
+\end_inset
+
+ y 
+\begin_inset Formula $[\overline{\alpha}]*[\alpha]=[c_{y}]$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Tenemos
+\begin_inset Formula 
+\begin{align*}
+(\alpha\land\overline{\alpha})(s) & =\begin{cases}
+\alpha(2s), & s\in[0,\tfrac{1}{2}]\\
+\alpha(2-2s), & s\in[\tfrac{1}{2},1]
+\end{cases}=\alpha(1-|1-2s|)\in{\cal L}(X,x),
+\end{align*}
+
+\end_inset
+
+luego 
+\begin_inset Formula $F(s,t):=\alpha(t(1-|1-2s|))$
+\end_inset
+
+ es una homotopía de caminos de 
+\begin_inset Formula $c_{x}$
+\end_inset
+
+ a 
+\begin_inset Formula $\alpha\land\overline{\alpha}$
+\end_inset
+
+ y 
+\begin_inset Formula $[c_{x}]=[\alpha\land\overline{\alpha}]=[\alpha]*[\overline{\alpha}]$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $[\overline{\alpha}]*[\alpha]=[c_{y}]$
+\end_inset
+
+, basta ver que 
+\begin_inset Formula $\overline{\overline{\alpha}}=\alpha$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+De aquí que 
+\begin_inset Formula $(\pi_{1}(X,x),*)$
+\end_inset
+
+ es un grupo, llamado 
+\series bold
+grupo fundamental
+\series default
+ o 
+\series bold
+primer grupo de homotopía
+\series default
+ de 
+\begin_inset Formula $X$
+\end_inset
+
+ relativo al 
+\series bold
+punto base
+\series default
+ 
+\begin_inset Formula $x$
+\end_inset
+
+, con neutro 
+\begin_inset Formula $[c_{x}]$
+\end_inset
+
+ y 
+\begin_inset Formula $[\alpha]^{-1}=[\overline{\alpha}]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $\alpha\in{\cal C}(X,x,y)$
+\end_inset
+
+, 
+\begin_inset Formula $\hat{\alpha}:\pi_{1}(X,x)\to\pi_{1}(X,y)$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\hat{\alpha}([\gamma]):=[\overline{\alpha}]*[\gamma]*[\alpha]$
+\end_inset
+
+ es un isomorfismo de grupos.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, es un homomorfismo porque 
+\begin_inset Formula $[\beta],[\gamma]\in\pi_{1}(X,x)$
+\end_inset
+
+, 
+\begin_inset Formula $\hat{\alpha}([\beta])\hat{\alpha}([\gamma])=[\overline{\alpha}]*[\beta]*[\alpha]*[\overline{\alpha}]*[\gamma]*[\alpha]=[\overline{\alpha}]*[\beta]*[\gamma]*[\alpha]=\hat{\alpha}([\beta]*[\gamma])$
+\end_inset
+
+, y es biyectivo porque su inversa es 
+\begin_inset Formula $\hat{\overline{\alpha}}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+ Así, si 
+\begin_inset Formula $X$
+\end_inset
+
+ es conexo por caminos, el grupo fundamental no depende del punto base,
+ es decir, 
+\begin_inset Formula $\pi_{1}(X,x)\cong\pi_{1}(X,y)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $x,y\in X$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $x,y\in X$
+\end_inset
+
+, 
+\begin_inset Formula $\pi_{1}(X,x)$
+\end_inset
+
+ es abeliano si y sólo si 
+\begin_inset Formula $\hat{\alpha}=\hat{\beta}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $\alpha,\beta\in{\cal C}(X,x,y)$
+\end_inset
+
+.
+\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
+
+
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $\alpha,\beta\in{\cal C}(X,x,y)$
+\end_inset
+
+ y 
+\begin_inset Formula $\gamma\in{\cal L}(X,x)$
+\end_inset
+
+, 
+\begin_inset Formula $\hat{\alpha}(\gamma)*\hat{\beta}(\gamma)=[\overline{\alpha}]*[\gamma]*[\alpha]*[\overline{\beta}]*[\gamma]*[\beta]\overset{[\alpha]*[\overline{\beta}]\in{\cal L}(X,x)}{=}[\overline{\alpha}]*[\gamma]*[\gamma]*[\alpha]*[\overline{\beta}]*[\beta]=\hat{\alpha}(\gamma)*\hat{\alpha}(\gamma)$
+\end_inset
+
+.
+ Cancelando, 
+\begin_inset Formula $\hat{\beta}(\gamma)=\hat{\alpha}(\gamma)$
+\end_inset
+
+, y como 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ es arbitrario, 
+\begin_inset Formula $\hat{\alpha}=\hat{\beta}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $\gamma,\delta\in{\cal L}(X,x)$
+\end_inset
+
+, 
+\begin_inset Formula $[\overline{\gamma}]*[\delta]*[\gamma]=\hat{\gamma}(\delta)=\hat{\delta}(\delta)=[\overline{\delta}]*[\delta]*[\delta]=[\delta]$
+\end_inset
+
+, y multiplicando por 
+\begin_inset Formula $[\gamma]$
+\end_inset
+
+ a la izquierda, 
+\begin_inset Formula $[\delta]*[\gamma]=[\gamma]*[\delta]$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Funciones homotópicas
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f:X\to Y$
+\end_inset
+
+ cumple 
+\begin_inset Formula $f(x_{0})=y_{0}$
+\end_inset
+
+, escribimos 
+\begin_inset Formula $f:(X,x_{0})\to(Y,y_{0})$
+\end_inset
+
+.
+ Entonces llamamos 
+\series bold
+homomorfismo inducido
+\series default
+ por 
+\begin_inset Formula $f$
+\end_inset
+
+ relativo a 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+ a 
+\begin_inset Formula $(f_{x_{0}})_{*}:=f_{*}:\pi_{1}(X,x_{0})\to\pi_{1}(Y,y_{0})$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f_{*}([\alpha])=[f\circ\alpha]$
+\end_inset
+
+.
+ En efecto, si 
+\begin_inset Formula $\alpha\simeq_{p}\beta$
+\end_inset
+
+ y 
+\begin_inset Formula $F:[0,1]\times[0,1]\to X$
+\end_inset
+
+ es una homotopía de caminos de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\beta$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f\circ F$
+\end_inset
+
+ es una homotopía de caminos de 
+\begin_inset Formula $f\circ\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $f\circ\beta$
+\end_inset
+
+, y 
+\begin_inset Formula $f_{*}([\alpha]*[\beta])=f_{*}([\alpha\land\beta])=[f\circ(\alpha\land\beta)]=[(f\circ\alpha)\land(f\circ\beta)]=f_{*}([\alpha])*f_{*}([\beta])$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(1_{X})_{*}=1_{\pi_{1}(X,x)}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $f:(X,x_{0})\to(Y,y_{0})$
+\end_inset
+
+ y 
+\begin_inset Formula $g:(Y,y_{0})\to(Z,z_{0})$
+\end_inset
+
+, 
+\begin_inset Formula $(g\circ f)_{*}=g_{*}\circ f_{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $(g\circ f)_{*}([\alpha])=[g\circ f\circ\alpha]=[g\circ(f\circ\alpha)]=g_{*}([f\circ\alpha])=(g_{*}\circ f_{*})(\alpha)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $f:(X,x_{0})\to(Y,y_{0})$
+\end_inset
+
+ es un homeomorfismo, 
+\begin_inset Formula $f_{*}$
+\end_inset
+
+ es un isomorfismo
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues 
+\begin_inset Formula $(f^{-1})_{*}\circ f_{*}=(1_{X})_{*}=1_{\pi_{1}(X,x)}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f_{*}$
+\end_inset
+
+ es biyectiva
+\end_layout
+
+\end_inset
+
+, luego el grupo fundamental es un invariante topológico salvo isomorfismos.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $f,g:(X,x)\to(Y,y)$
+\end_inset
+
+ continuas, 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\series bold
+homotópica
+\series default
+ a 
+\begin_inset Formula $g$
+\end_inset
+
+ con 
+\begin_inset Formula $x\to y$
+\end_inset
+
+, 
+\begin_inset Formula $f\simeq_{x\to y}g$
+\end_inset
+
+, si existe una homotopía 
+\begin_inset Formula $F:X\times[0,1]\to Y$
+\end_inset
+
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ a 
+\begin_inset Formula $g$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall t\in[0,1],F(x,t)=y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $f\simeq_{x\to y}g$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f_{*}=g_{*}:\pi_{1}(X,x)\to\pi_{1}(Y,y)$
+\end_inset
+
+.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, sean 
+\begin_inset Formula $F:X\times[0,1]\to Y$
+\end_inset
+
+ una homotopía de 
+\begin_inset Formula $f$
+\end_inset
+
+ a 
+\begin_inset Formula $g$
+\end_inset
+
+ con 
+\begin_inset Formula $F(x,t)=y$
+\end_inset
+
+ para todo 
+\begin_inset Formula $t$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha\in{\cal L}(X,x)$
+\end_inset
+
+, 
+\begin_inset Formula $G(s,t):=F(\alpha(s),t)$
+\end_inset
+
+ es una homotopía de 
+\begin_inset Formula $f\circ\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $g\circ\alpha$
+\end_inset
+
+, luego 
+\begin_inset Formula $f_{*}([\alpha])=[f\circ\alpha]=[g\circ\alpha]=g_{*}([\alpha])$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una función 
+\begin_inset Formula $f:(X,x)\to(Y,y)$
+\end_inset
+
+ es una 
+\series bold
+equivalencia homotópica
+\series default
+ si existe 
+\begin_inset Formula $g:(Y,y)\to(X,x)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $g\circ f\simeq_{x\to x}1_{X}$
+\end_inset
+
+ y 
+\begin_inset Formula $f\circ g\simeq_{y\to y}1_{Y}$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $(X,x)$
+\end_inset
+
+ e 
+\begin_inset Formula $(Y,y)$
+\end_inset
+
+ son (
+\series bold
+equivalentes
+\series default
+) 
+\series bold
+homotópicos
+\series default
+, 
+\begin_inset Formula $(X,x)\simeq(Y,y)$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $(X,x)\simeq(Y,y)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\pi_{1}(X,x)\cong\pi_{1}(Y,y)$
+\end_inset
+
+.
+ 
+\series bold
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $f:(X,x)\to(Y,y)$
+\end_inset
+
+ y 
+\begin_inset Formula $g:(Y,y)\to(X,x)$
+\end_inset
+
+ las equivalencias homotópicas, como 
+\begin_inset Formula $g\circ f\simeq_{x\to x}1_{X}$
+\end_inset
+
+, 
+\begin_inset Formula $g_{*}\circ f_{*}=(g\circ f)_{*}=(1_{X})_{*}=1_{\pi_{1}(X,x)}$
+\end_inset
+
+, y como 
+\begin_inset Formula $f\circ g\simeq_{y\to y}1_{Y}$
+\end_inset
+
+, 
+\begin_inset Formula $f_{*}\circ g_{*}=1_{\pi_{1}(Y,y)}$
+\end_inset
+
+, luego 
+\begin_inset Formula $f_{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $g_{*}$
+\end_inset
+
+ son una inversa de la otra.
+ Como además 
+\begin_inset Formula $f_{*}$
+\end_inset
+
+ es un homomorfismo, es un isomorfismo de 
+\begin_inset Formula $\pi_{1}(X,x)$
+\end_inset
+
+ a 
+\begin_inset Formula $\pi_{1}(Y,y)$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Espacios simplemente conexos
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $X$
+\end_inset
+
+ es 
+\series bold
+simplemente conexo
+\series default
+ si es conexo por caminos y 
+\begin_inset Formula $\pi_{1}(X,x)$
+\end_inset
+
+ es el grupo trivial para algún 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, y por tanto para todo 
+\begin_inset Formula $x\in X$
+\end_inset
+
+.
+ Escribimos 
+\begin_inset Formula $\pi_{1}(X,x)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Todo espacio contráctil es simplemente conexo.
+ 
+\series bold
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es contráctil, toda función 
+\begin_inset Formula $\mathbb{S}^{1}\to X$
+\end_inset
+
+ es homotópica a una constante.
+ Sea 
+\begin_inset Formula $\gamma\in{\cal L}(X,x_{0})$
+\end_inset
+
+ para algún 
+\begin_inset Formula $x_{0}\in X$
+\end_inset
+
+, existe 
+\begin_inset Formula $\Gamma:\mathbb{S}^{1}\to X$
+\end_inset
+
+ continua con 
+\begin_inset Formula $\gamma=\Gamma\circ e$
+\end_inset
+
+, luego existe una homotopía 
+\begin_inset Formula $F:\mathbb{S}^{1}\times[0,1]\to X$
+\end_inset
+
+ de una cierta constante 
+\begin_inset Formula $c_{z}$
+\end_inset
+
+ a 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ y por tanto una extensión 
+\begin_inset Formula $\Gamma:\mathbb{D}^{2}\to X$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\Gamma(re(\theta))=F(e(\theta),r)$
+\end_inset
+
+, que es continua.
+ Sea entonces 
+\begin_inset Formula $G:[0,1]\times[0,1]\to X$
+\end_inset
+
+ dada por 
+\begin_inset Formula $G(s,t):=\Gamma((1-t,0)+te(s))$
+\end_inset
+
+, 
+\begin_inset Formula $G$
+\end_inset
+
+ es continua, 
+\begin_inset Formula $G(0,t)=G(1,t)=\Gamma(1,0)=\gamma(0)=x_{0}$
+\end_inset
+
+, 
+\begin_inset Formula $G(s,0)=\Gamma(1,0)=x_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $G(s,1)=\Gamma(e(s))=\gamma(s)$
+\end_inset
+
+, luego 
+\begin_inset Formula $G$
+\end_inset
+
+ es una homotopía de caminos de 
+\begin_inset Formula $c_{x_{0}}$
+\end_inset
+
+ a 
+\begin_inset Formula $\gamma$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $[\gamma]=[c_{x_{0}}]$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi_{1}(X,x_{0})$
+\end_inset
+
+ es unipuntual.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+En particular, todo subespacio estrellado de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ es simplemente conexo, y por tanto también todo subespacio convexo no vacío
+ de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es simplemente conexo y 
+\begin_inset Formula $\alpha,\beta\in{\cal C}(X,x,y)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\alpha\simeq_{p}\beta$
+\end_inset
+
+.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, como 
+\begin_inset Formula $\pi_{1}(X,x)=\{[c_{x}]\}$
+\end_inset
+
+, 
+\begin_inset Formula $[\alpha]*[\overline{\beta}]=[c_{x}]$
+\end_inset
+
+, luego 
+\begin_inset Formula $[\alpha]=[\alpha]*[\overline{\beta}]*[\beta]=[c_{x}]*[\beta]=[\beta]$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Algunos grupos fundamentales
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $r:X\to Y$
+\end_inset
+
+ una aplicación recubridora, 
+\begin_inset Formula $y_{0}\in\mathbb{S}^{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $r(x_{0}):=y_{0}$
+\end_inset
+
+, si para 
+\begin_inset Formula $[\alpha]\in\pi_{1}(Y,y_{0})$
+\end_inset
+
+ llamamos 
+\begin_inset Formula $\tilde{\alpha}$
+\end_inset
+
+ al levantamiento de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ por 
+\begin_inset Formula $r$
+\end_inset
+
+ con 
+\begin_inset Formula $\tilde{\alpha}(0)=x_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi([\alpha]):=\tilde{\alpha}(1)$
+\end_inset
+
+, llamamos 
+\series bold
+correspondencia del levantamiento
+\series default
+ asociada a 
+\begin_inset Formula $r$
+\end_inset
+
+ a
+\begin_inset Formula 
+\[
+\phi:\pi_{1}(Y,y_{0})\to r^{-1}(y_{0}),
+\]
+
+\end_inset
+
+que está bien definida.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Si 
+\begin_inset Formula $|r^{-1}(y_{0})|\leq1$
+\end_inset
+
+, es obvio.
+ En otro caso, como para cada 
+\begin_inset Formula $x\in r^{-1}(y_{0})$
+\end_inset
+
+ existe un entorno 
+\begin_inset Formula $U_{x}$
+\end_inset
+
+ de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ donde 
+\begin_inset Formula $r$
+\end_inset
+
+ es biyectiva y por tanto 
+\begin_inset Formula $U_{x}\cap r^{-1}(y_{0})$
+\end_inset
+
+ es unipuntual, hay una separación por abiertos 
+\begin_inset Formula $\{U_{x}\cap r^{-1}(y_{0}),\bigcup_{p\in r^{-1}(y_{0})\setminus\{x\}}U_{p}\cap r^{-1}(y_{0})\}$
+\end_inset
+
+, luego 
+\begin_inset Formula $r^{-1}(y_{0})$
+\end_inset
+
+ es totalmente disconexo.
+ Entonces, si 
+\begin_inset Formula $\alpha\simeq_{p}\alpha'$
+\end_inset
+
+, sean 
+\begin_inset Formula $F:[0,1]\times[0,1]\to\mathbb{S}^{1}$
+\end_inset
+
+ la homotopía de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\alpha'$
+\end_inset
+
+ y 
+\begin_inset Formula $\tau:[0,1]\to r^{-1}(z_{0})$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\tau(t):=\phi(s\mapsto F(s,t))$
+\end_inset
+
+, como 
+\begin_inset Formula $[0,1]$
+\end_inset
+
+ es conexo y 
+\begin_inset Formula $r^{-1}(z_{0})$
+\end_inset
+
+ es totalmente disconexo, 
+\begin_inset Formula $\tau$
+\end_inset
+
+ es constante y 
+\begin_inset Formula $\phi(\alpha)=\tau(0)=\tau(1)=\phi(\alpha')$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+La correspondencia del levantamiento asociada a 
+\begin_inset Formula $e$
+\end_inset
+
+ es biyectiva.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Para 
+\begin_inset Formula $x\in e^{-1}(y_{0})$
+\end_inset
+
+, sea 
+\begin_inset Formula $k:=x-\theta_{0}\in\mathbb{Z}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\alpha(s):=e(ks+\theta_{0})$
+\end_inset
+
+ cumple 
+\begin_inset Formula $\phi(\alpha)=k+\theta_{0}=x$
+\end_inset
+
+, luego 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es suprayectiva.
+ Por otro lado, si 
+\begin_inset Formula $\phi(\alpha)=\phi(\beta)$
+\end_inset
+
+, sean 
+\begin_inset Formula $\hat{\alpha},\hat{\beta}:\mathbb{S}^{1}\to\mathbb{S}^{1}$
+\end_inset
+
+ con 
+\begin_inset Formula $\alpha=:\hat{\alpha}\circ e$
+\end_inset
+
+ y 
+\begin_inset Formula $\beta=:\hat{\beta}\circ e$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\deg\hat{\alpha}=\phi(\alpha)-\theta_{0}=\phi(\beta)-\theta_{0}=\deg\hat{\beta}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\hat{\alpha}\simeq\hat{\beta}$
+\end_inset
+
+ y existe una homotopía 
+\begin_inset Formula $F:\mathbb{S}^{1}\times[0,1]\to\mathbb{S}^{1}$
+\end_inset
+
+ de 
+\begin_inset Formula $\hat{\alpha}$
+\end_inset
+
+ a 
+\begin_inset Formula $\hat{\beta}$
+\end_inset
+
+, de donde 
+\begin_inset Formula $F'(s,t):=F(s,t)\frac{y_{0}}{F((1,0),t)}$
+\end_inset
+
+ es otra homotopía de 
+\begin_inset Formula $\hat{\alpha}$
+\end_inset
+
+ a 
+\begin_inset Formula $\hat{\beta}$
+\end_inset
+
+ con 
+\begin_inset Formula $F((1,0),t)=y_{0}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $t\in[0,1]$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $G(s,t):=F(e(s),t)$
+\end_inset
+
+ es una homotopía de caminos de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ a 
+\begin_inset Formula $\beta$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $[\alpha]=[\beta]$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es inyectiva.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, como 
+\series bold
+teorema
+\series default
+, el grupo fundamental de 
+\begin_inset Formula $\mathbb{S}^{1}$
+\end_inset
+
+ es isomorfo a 
+\begin_inset Formula $(\mathbb{Z},+)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sean 
+\begin_inset Formula $X=U\cup V$
+\end_inset
+
+ con 
+\begin_inset Formula $U$
+\end_inset
+
+, 
+\begin_inset Formula $V$
+\end_inset
+
+ y 
+\begin_inset Formula $U\cap V$
+\end_inset
+
+ abiertos en 
+\begin_inset Formula $X$
+\end_inset
+
+ conexos por caminos y 
+\begin_inset Formula $x\in U\cap V$
+\end_inset
+
+, todo 
+\begin_inset Formula $[\alpha]\in\pi_{1}(X,x)$
+\end_inset
+
+ se expresa como producto de elementos de 
+\begin_inset Formula $\pi_{1}(U,x)$
+\end_inset
+
+ o 
+\begin_inset Formula $\pi_{1}(V,x)$
+\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
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Van Kampen especial, versión 1:
+\series default
+ Sea 
+\begin_inset Formula $X=U\cup V$
+\end_inset
+
+ con 
+\begin_inset Formula $U$
+\end_inset
+
+, 
+\begin_inset Formula $V$
+\end_inset
+
+ y 
+\begin_inset Formula $U\cap V\neq\emptyset$
+\end_inset
+
+ abiertos en 
+\begin_inset Formula $X$
+\end_inset
+
+ conexos por caminos, si 
+\begin_inset Formula $U$
+\end_inset
+
+ y 
+\begin_inset Formula $V$
+\end_inset
+
+ son simplemente conexos, 
+\begin_inset Formula $X$
+\end_inset
+
+ también lo es.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, dado 
+\begin_inset Formula $x\in U\cap V$
+\end_inset
+
+, 
+\begin_inset Formula $[\alpha]\in\pi_{1}(X,x)$
+\end_inset
+
+ se expresa como 
+\begin_inset Formula $[\alpha]=[\beta_{1}]*\dots*[\beta_{n}]$
+\end_inset
+
+ con cada 
+\begin_inset Formula $[\beta_{i}]\in\pi_{1}(U,x)\cup\pi_{1}(V,x)=\{[c_{x}]\}\cup\{[c_{x}]\}$
+\end_inset
+
+, luego 
+\begin_inset Formula $[\alpha]=[c_{x}]$
+\end_inset
+
+ y 
+\begin_inset Formula $X$
+\end_inset
+
+ es simplemente conexo.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{S}^{n}$
+\end_inset
+
+ es simplemente conexa
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues si 
+\begin_inset Formula $N$
+\end_inset
+
+ es su polo norte y 
+\begin_inset Formula $S$
+\end_inset
+
+ es su polo sur, 
+\begin_inset Formula $U:=\mathbb{S}^{n}\setminus\{N\}$
+\end_inset
+
+ y 
+\begin_inset Formula $V:=\mathbb{S}^{n}\setminus\{S\}$
+\end_inset
+
+ son abiertos homeomorfos a 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ y por tanto simplemente conexos, y su intersección es conexa por caminos
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Además, 
+\begin_inset Formula $\mathbb{R}^{2}$
+\end_inset
+
+ no es homeomorfo a 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ para 
+\begin_inset Formula $n\neq2$
+\end_inset
+
+.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+De serlo, sería $A:=
+\backslash
+mathbb{R}^2
+\backslash
+setminus
+\backslash
+{0
+\backslash
+}$ homeomorfo a algún $B:=
+\backslash
+mathbb{R}^2
+\backslash
+setminus
+\backslash
+{p
+\backslash
+}$.
+ Entonces la circunferencia unidad en $A$ sería homotópicamente equivalente
+ a algún lazo en $B$ por el homeomorfismo, pero el lazo en $B$ sería homotópicam
+ente equivalente a un lazo constante y la circunferencia no lo es.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ dos grupos, una 
+\series bold
+palabra
+\series default
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ es una secuencia 
+\begin_inset Formula $s_{1}\cdots s_{n}$
+\end_inset
+
+ con cada 
+\begin_inset Formula $s_{i}\in G\amalg H$
+\end_inset
+
+.
+ 
+\series bold
+Reducir
+\series default
+ una palabra es aplicarle sucesivamente las siguientes acciones hasta no
+ poder aplicar ninguna:
+\end_layout
+
+\begin_layout Enumerate
+Eliminar un elemento identidad de 
+\begin_inset Formula $G$
+\end_inset
+
+ o 
+\begin_inset Formula $H$
+\end_inset
+
+ de la secuencia.
+\end_layout
+
+\begin_layout Enumerate
+Reemplazar una subsecuencia 
+\begin_inset Formula $s_{k}s_{k+1}$
+\end_inset
+
+ con 
+\begin_inset Formula $s_{k},s_{k+1}\in G$
+\end_inset
+
+ o 
+\begin_inset Formula $s_{k},s_{k+1}\in H$
+\end_inset
+
+ por su producto.
+\end_layout
+
+\begin_layout Standard
+El resultado de esto es una 
+\series bold
+palabra reducida
+\series default
+.
+ El 
+\series bold
+producto libre
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+, 
+\begin_inset Formula $G*H$
+\end_inset
+
+, es el conjunto de las palabras reducidas en 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ con la operación de concatenación seguida de reducción.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Van Kampen especial, versión 2:
+\series default
+ Sean 
+\begin_inset Formula $X=U\cup V$
+\end_inset
+
+ con 
+\begin_inset Formula $U$
+\end_inset
+
+ y 
+\begin_inset Formula $V$
+\end_inset
+
+ abiertos conexos por caminos y 
+\begin_inset Formula $U\cap V\neq\emptyset$
+\end_inset
+
+ simplemente conexo, si 
+\begin_inset Formula $x_{0}\in U\cap V$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\pi_{1}(X,x)\cong\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})$
+\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
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados dos espacios 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+, 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ e 
+\begin_inset Formula $y\in Y$
+\end_inset
+
+, llamamos 
+\series bold
+unión por un punto
+\series default
+ de 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+a 
+\begin_inset Formula $X\lor Y:=(X\amalg Y)/\{x,y\}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $Z=A\cup B$
+\end_inset
+
+ con 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ cerrados en 
+\begin_inset Formula $Z$
+\end_inset
+
+ y 
+\begin_inset Formula $A\cap B=\{x_{0}\}$
+\end_inset
+
+, decimos que 
+\begin_inset Formula $Z=A\lor B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+figura ocho
+\series default
+ es 
+\begin_inset Formula $E:=\mathbb{S}^{1}\lor\mathbb{S}^{1}$
+\end_inset
+
+, y 
+\begin_inset Formula $\pi_{1}(E)\cong\mathbb{Z}*\mathbb{Z}$
+\end_inset
+
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+por el teorema de Van Kampen especial, versión 2
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Cálculo de grupos fundamentales
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+monomorfismo
+\series default
+ es un homomorfismo inyectivo.
+ Si 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ es un retracto de 
+\begin_inset Formula $X$
+\end_inset
+
+, la inclusión 
+\begin_inset Formula $i:A\to X$
+\end_inset
+
+ induce un monomorfismo 
+\begin_inset Formula $i_{*}:\pi_{1}(A,x_{0})\to\pi_{1}(X,x_{0})$
+\end_inset
+
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues claramente si 
+\begin_inset Formula $\phi([\alpha])=\phi([\beta])$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha=\beta$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+ es un retracto de deformación en 
+\begin_inset Formula $X$
+\end_inset
+
+ (por ejemplo, si 
+\begin_inset Formula $A=\mathbb{S}^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $X=\mathbb{R}^{n+1}\setminus\{0\}$
+\end_inset
+
+), la inclusión 
+\begin_inset Formula $i:A\to X$
+\end_inset
+
+ induce un isomorfismo 
+\begin_inset Formula $i_{*}$
+\end_inset
+
+ entre los grupos fundamentales.
+ 
+\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
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+La figura ocho 
+\begin_inset Formula $E$
+\end_inset
+
+ es un retracto de deformación de 
+\begin_inset Formula $\mathbb{R}^{2}\setminus\{p,q\}$
+\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
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+El 
+\series bold
+espacio theta
+\series default
+, 
+\begin_inset Formula $\theta:=\mathbb{S}^{1}\cup([-1,1]\times\{0\})$
+\end_inset
+
+, es un retracto de deformación de 
+\begin_inset Formula $\mathbb{R}^{2}\setminus\{p,q\}$
+\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
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $E$
+\end_inset
+
+ no es un retracto de deformación de 
+\begin_inset Formula $\theta$
+\end_inset
+
+ ni al revés.
+ 
+\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
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, 
+\begin_inset Formula $\pi_{1}(X\times Y,(x,y))\cong\pi_{1}(X,x)\times\pi_{1}(Y,y)$
+\end_inset
+
+.
+ En particular el grupo fundamental del toro, 
+\begin_inset Formula $\mathbb{T}\cong\mathbb{S}^{1}\times\mathbb{S}^{1}$
+\end_inset
+
+, es isomorfo a 
+\begin_inset Formula $\mathbb{Z}\times\mathbb{Z}$
+\end_inset
+
+.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Sea $
+\backslash
+phi([
+\backslash
+alpha]):=(p_*([
+\backslash
+alpha]),q_*([
+\backslash
+alpha]))$, queremos ver que $
+\backslash
+phi$ es un isomorfismo de $
+\backslash
+pi_1(X
+\backslash
+times Y,(x_0,y_0))$ a $
+\backslash
+pi_1(X,x_0)
+\backslash
+times
+\backslash
+pi_1(Y,y_0)$.
+\end_layout
+
+\begin_layout Plain Layout
+Dado $([
+\backslash
+beta],[
+\backslash
+gamma])
+\backslash
+in
+\backslash
+pi_1(X,x_0)
+\backslash
+times
+\backslash
+pi_1(Y,y_0)$, $
+\backslash
+phi([(
+\backslash
+beta,
+\backslash
+gamma)])=(
+\backslash
+beta,
+\backslash
+gamma)$, luego $
+\backslash
+phi$ es suprayectiva.
+ Para la inyectividad, si $
+\backslash
+phi([
+\backslash
+alpha])=([c_{x_0}],[c_{y_0}])$, como $p
+\backslash
+alpha
+\backslash
+cong_pc_{x_0}$ y $q
+\backslash
+alpha
+\backslash
+cong_pc_{y_0}$, $
+\backslash
+alpha
+\backslash
+cong_p(c_{x_0},c_{y_0})$, con lo que $[
+\backslash
+alpha]=[(c_{x_0},c_{y_0})]=[c_{(x_0,y_0)}]$.
+\end_layout
+
+\begin_layout Plain Layout
+Ver que $
+\backslash
+phi$ es un homomorfismo de grupos es fácil.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sea 
+\begin_inset Formula $p:E\to B$
+\end_inset
+
+ una aplicación recubridora con 
+\begin_inset Formula $p(e_{0})=b_{0}$
+\end_inset
+
+, si 
+\begin_inset Formula $E$
+\end_inset
+
+ es conexo por caminos, la correspondencia del levantamiento 
+\begin_inset Formula $\phi:\pi_{1}(B,b_{0})\to p^{-1}(b_{0})$
+\end_inset
+
+ es sobreyectiva, y si además 
+\begin_inset Formula $E$
+\end_inset
+
+ es simplemente conexo, 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es biyectiva.
+ 
+\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
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $\mathbb{RP}^{2}$
+\end_inset
+
+ es el plano proyectivo e 
+\begin_inset Formula $y\in\mathbb{RP}^{2}$
+\end_inset
+
+, 
+\begin_inset Formula $\pi_{1}(\mathbb{RP}^{2},y)\cong\mathbb{Z}_{2}$
+\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
+
+
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, el grupo fundamental de la figura ocho no es abeliano.
+ 
+\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
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dadas dos superficies 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ con subespacios respectivos 
+\begin_inset Formula $X_{0}$
+\end_inset
+
+ e 
+\begin_inset Formula $Y_{0}$
+\end_inset
+
+ y homeomorfos a un disco en 
+\begin_inset Formula $\mathbb{R}^{2}$
+\end_inset
+
+, dado un homeomorfismo 
+\begin_inset Formula $h:\partial X_{0}\cong\mathbb{S}^{1}\to\partial Y_{0}\cong\mathbb{S}^{1}$
+\end_inset
+
+, llamamos 
+\series bold
+suma conexa
+\series default
+ de 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+, 
+\begin_inset Formula $X\sharp Y$
+\end_inset
+
+, a 
+\begin_inset Formula $((X\setminus\text{\ensuremath{\mathring{X}_{0}}})\amalg(Y\setminus\mathring{Y}_{0}))/\sim$
+\end_inset
+
+, donde 
+\begin_inset Formula $x\sim y$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $x=y$
+\end_inset
+
+, o bien 
+\begin_inset Formula $x\in X_{0}$
+\end_inset
+
+ e 
+\begin_inset Formula $y\in Y_{0}$
+\end_inset
+
+ con 
+\begin_inset Formula $y=h(x)$
+\end_inset
+
+, o bien al revés.
+ Como 
+\series bold
+teorema
+\series default
+, el grupo fundamental del 
+\series bold
+doble toro
+\series default
+, 
+\begin_inset Formula $\mathbb{T}\sharp\mathbb{T}$
+\end_inset
+
+, no es abeliano.
+ 
+\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
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Llevar este párrafo al tema 6.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+
+\begin_inset Formula $m$
+\end_inset
+
+-variedad
+\series default
+ es un espacio Hausdorff 
+\begin_inset Formula $X$
+\end_inset
+
+ tal que todo 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ tiene un entorno homeomorfo a un abierto de 
+\begin_inset Formula $\mathbb{R}^{m}$
+\end_inset
+
+, aunque también se suele exigir que 
+\begin_inset Formula $X$
+\end_inset
+
+ sea 
+\begin_inset Formula $\text{1A}\mathbb{N}$
+\end_inset
+
+.
+ Una 
+\series bold
+curva
+\series default
+ es una 1-variedad, y una 
+\series bold
+superficie
+\series default
+ es una 2-variedad.
+ Así, la esfera, el toro, el plano proyectivo real y el doble toro son superfici
+es topológicamente distintas.
+\end_layout
+
+\end_body
+\end_document