diff options
Diffstat (limited to 'ts')
| -rw-r--r-- | ts/n.lyx | 75 | ||||
| -rw-r--r-- | ts/n4.lyx | 293 | ||||
| -rw-r--r-- | ts/n5.lyx | 2762 |
3 files changed, 3044 insertions, 86 deletions
@@ -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 @@ -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 +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\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 +\is_math_indent 0 +\math_numbering_side default +\quotes_style french +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\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 |