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By: gonmedare on Nov 16th, 2011  |  syntax: None  |  size: 11.56 KB  |  views: 281  |  expires: Never
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1. %%
2. %% This is the file 'contest.tex'
3. %%
4. \documentclass[twocolumn,12pt]{article}
5. \usepackage[english]{babel}
6. \usepackage[utf8]{inputenc}
7. \usepackage[T1]{fontenc}
8. \usepackage[fontsize=12pt,baseline=14pt]{grid}
9. \usepackage[top=5.5cm,
10.   bottom=2.5cm,
11.   left=2.5cm,
12.   right=2.5cm,
13. ]{geometry}
14. \usepackage{xparse}
15. \usepackage{fourier}
16. \usepackage{amsmath}
17. \usepackage{amssymb}
18. \usepackage{MnSymbol}
19. \usepackage{gridleno}
20. \usepackage{hyperref}
21. \hypersetup{
24.   citecolor=grlnblue,
25.   urlcolor=grlngray}
26.
27. \usetikzlibrary{calc,decorations.markings}
28.
29. \NewDocumentCommand\fgroup{O{1}O{0}mm}{%
30.   \pi_#1(#3,#4_#2)}
31. \NewDocumentCommand\erren{O{n}O{R}}{%
32.   \mathbb{#2}^#1}
33. \NewDocumentCommand\oper{O{f}O{g}}{%
34.   #1\ast#2}
35. \NewDocumentCommand\clase{O{f}}{%
36.   [#1]}
37. \NewDocumentCommand\opercl{O{f}O{g}}{%
38.   \clase[#1]\ast\clase[#2]}
39. \NewDocumentCommand\invclase{O{f}}{%
40.   \clase[\bar{#1}]}
41.
42. \course{General Topology}
43. \courseid{5348-01}
44. \professor{James R. Munkres}
45. \term{Trimester 3 2011-2012}
46. \topic{The Fundamental Group}
47. \date{November 14, 2011 (Mon)}
48.
49. \hbadness=99999 % Make TeX remain silent
50.
51. \begin{document}
52.
54.
55. \begin{summary}
56. The set of path-homotopy classes of paths in a space $$X$$ does not form a group under the operation $$\ast$$, only a groupoid. But if we pick out a point $$x_0$$ of $$X$$ to serve as a base point'' and restrict ourselves to those paths that begin and end at $$x_0$$, the set of these path-homotopy classes does form a group under $$\ast$$. It will be called the \emph{fundamental group} of $$X$$. We study the basic properties of the fundamental group and prove that it is a topological invariant.
57. \end{summary}
58.
59. \section{Definitions and basic properties}
60.
61. We shall prove several properties of the fundamental group. In particular, we shall show that the group is, up to isomorphism, independent of the choice of base point (provided that $$X$$ is path connected). We shall also show that the group is a topological invariant of the space $$X$$, the fact that is of crucial importance in using it to study homeomorphism problems.
62.
63. \begin{definition}
64. Let $$X$$ be a space; let $$x_0$$ be a point of $$X$$. A path in $$X$$ that begins and ends at $$x_0$$ is called a \emph{loop} based at $$x_0$$. The set of path homotopy classes of loops based at $$x_0$$, with the operation $$\ast$$, is called the \emph{fundamental group} of $$X$$ relative to the \emph{base point} $$x_0$$. It is denoted by $$\fgroup{X}{x}$$.
65. \end{definition}
66.
67. The operation $$\ast$$, when restricted to this set, satisfies the axioms for a group. Given two loops $$f$$ and $$g$$ based at $$x_0$$, the composition $$\oper$$ is always defined and it is a loop based at $$x_0$$. Associativity, the existence of an identity element $$\clase[e_{x_0}]$$, and the existence of an inverse $$\invclase$$ for $$\clase$$ are immediate.
68.
69. \begin{example}
70. Let $$\erren$$ denote the standard euclidean $$n$$\nobreakdash-space. Then the group $$\fgroup{\erren}{x}$$ is the trivial group consisting of the identity alone. For if $$f$$ is a loop in $$\erren$$ based at $$x_0$$, the straight line homotopy
71. \begin{gridenv}
72. \begin{equation*}
73.   F(s,t) = tx_0 + (1 - t)f(s)
74. \end{equation*}
75. \end{gridenv}
76. is a path homotopy between $$f$$ and the constant loop $$e_{x_0}$$.
77. \end{example}
78.
79. \begin{example}
80. More generally, if X is any $$\emph{convex}$$ subset of $$\erren$$, then $$\fgroup{X}{x}$$ is the trivial (one single element) group. The straight-line homotopy will work once again, for convexity of $$X$$ means that for any two points $$x$$ and $$y$$ of $$X$$, the straight-line segment
81. \begin{gridenv}
82. \begin{equation*}
83.   \{\, tx_0 + (1 - t)y \mid 0\leq t\leq 1\,\}
84. \end{equation*}
85. \end{gridenv}
86. between them lies in $$X$$. In particular, the \emph{unit ball} $$\erren[n][B]$$ in $$\erren$$,
87. \begin{gridenv}
88. \begin{equation*}
89.   \erren[n][B] = \{\, x \mid  x_1^2 + \cdots + x_n^2\leq 1 \,\},
90. \end{equation*}
91. \end{gridenv}
92. has trivial fundamental group.
93. \end{example}
94. An immediate question one asks is the ex\-tent to which the fundamental group depends on the base point. The answer is given in Corollary~\ref{cor:indbase}, which follows.
95.
96. \begin{definition}
97. Let $$\alpha$$ be a path in $$X$$ from $$x_0$$ to $$x_1$$. We define a map
98. \begin{gridenv}
99. \begin{equation*}
100.   \hat{\alpha}:\fgroup{X}{x} \rightarrow \fgroup[1][1]{X}{x}
101. \end{equation*}
102. \end{gridenv}
103. by the equation
104. \begin{gridenv}
105. \begin{equation*}
106.   \hat{\alpha}(\clase) = \opercl[\bar{\alpha}][f]\ast\clase[\alpha].
107. \end{equation*}
108. \end{gridenv}
109. \end{definition}
110. The map $$\hat{\alpha}$$ is pictured in Figure~\ref{fig:indbase}. It is well-defined because the operation $$\ast$$ is well-defined. If $$f$$ is a loop based at $$x_0$$, then $$\oper[\bar{\alpha}][(\oper[f][\alpha])]$$ is a loop based at $$x_1$$. Hence $$\hat{\alpha}$$ maps $$\fgroup{X}{x}$$ into $$\fgroup[1][1]{X}{x}$$, as desired.
111.
112. \begin{figure}
113. \centering
114. \begin{tikzpicture}[
115.   point/.style={circle,inner sep=2pt,fill},
116.   decoration={markings,
117.   mark=at position 0.5 with {\arrow{stealth}}}
118.   ]
119. \draw[fill=summarybg,draw=none] (3,3) rectangle (-3,-3);
120.
122. \node[point,label=left:$x_0$] at (-1,-0.2) (a) {};
123. \node[point,label=right:$x_1$] at (1.2,1.3) (b) {};
124.
125. \draw[postaction={decorate},shorten <= 1pt,shorten >= 1pt] (b.south) .. controls (-0.5,2) and (0.5,1.8) .. node[below=7pt] {$\bar{\alpha}$} (a.south);
126.
127. \draw[postaction={decorate}] ($(a.north)+(-1pt,0)$) .. controls ($(0.5,1.8)+(0,10pt)$) and ($(-0.5,2)+(0,2pt)$) .. node[left=4pt] {$\alpha$} (b.north west);
128.
129. \draw[postaction={decorate}] (a.south east) .. controls (-0.5,-3)  and (2,1) .. node[label=below:$f$] {} (a.south west);
130. \end{tikzpicture}
131. \caption{}
132. \label{fig:indbase}
133. \end{figure}
134.
135. \begin{theorem}
136. The map $$\hat{\alpha}$$ is a group homomorphism.
137. \end{theorem}
138.
139. \begin{Proof}
140. To prove that $$\hat{\alpha}$$ is a homomorphism, we compute
141. \begin{gridenv}
142. \begin{multline*}
143.   \hat{\alpha}(\clase) \ast \hat{\alpha}(\clase[g]) \\
144.   = ( \clase[\bar{\alpha}]\ast \clase \ast \clase[\alpha]) \ast (\clase[\bar{\alpha}]\ast \clase[g] \ast \clase[\alpha])\\
145. = ( \clase[\bar{\alpha}]\ast \clase \ast \clase[g] \ast \clase[\alpha]) = \hat{\alpha}(\clase \ast \clase[g]).
146. \end{multline*}
147. \end{gridenv}
148. This proof uses the groupoid properties of $$\ast$$. To show that $$\hat{\alpha}$$ is an isomorphism, we show that if $$\beta$$ denotes the path $$\bar{\alpha}$$, which is the reverse of $$\alpha$$, then $$\hat{\beta}$$ is the inverse for $$\hat{\alpha}$$. We compute, for each element $$\clase[h]$$ of $$\fgroup[1][1]{X}{x}$$,
149. \begin{gridenv}
150. \begin{equation*}
151.   \hat{\alpha}(\hat{\beta}(\clase[h])) = \clase[\bar{\alpha}] \ast (\clase[\alpha] \ast \clase[h] \ast \clase[\bar{\alpha}]) \ast \clase[\alpha] = \clase[h]
152. \end{equation*}
153. \end{gridenv}
154. A similar computation shows that $$\hat{\beta}(\hat{\alpha}(\clase))=\clase$$, for each class $$\clase$$ in $$\fgroup{X}{x}$$.
155. \end{Proof}
156.
157. \begin{corollary}\label{cor:indbase}
158. If X is path connected and $$x_0$$ and $$x_1$$ are two points of $$X$$, then $$\fgroup{X}{x}$$ is isomorphic to $$\fgroup[1][0]{X}{x}$$.
159. \end{corollary}
160.
161. Suppose that $$X$$ is a topological space. Let $$C$$ be the path component of $$X$$ containing $$x_0$$. It is easy to see that $$\fgroup[1][0]{C}{x}=\fgroup{X}{x}$$, since all loops and homotopies in $$X$$ that are based at $$x_0$$ must lie in the subspace $$C$$. Thus $$\fgroup{X}{x}$$ depends only on the path component of $$X$$ containing $$x_0$$, and gives us no information whatever about the rest of $$X$$. For this reason, it is usual to deal only with path-connected spaces when studying the fundamental group.
162.
163. If $$X$$ is path connected, then all the groups $$\fgroup[1][]{X}{x}$$ are isomorphic, so it is tempting to try to identify'' all these groups with one another, and to speak simply of the fundamental group of the space $$X$$, without reference to base point. The difficulty with this approach is that there is no \emph{natural} way of identifying $$\fgroup{X}{x}$$ with $$\fgroup[1][1]{X}{x}$$; different paths $$\alpha$$ and $$\beta$$ from $$x_0$$ to $$x_1$$ may give raise to different isomorphisms between these groups. For this reason, onitting the base point can lead to error.
164.
165. \section{Simply Connected Spaces}
166.
167. \begin{definition}
168. A space $$X$$ is said to be simply connected if it is a path-connected space and if $$\fgroup{X}{x}$$ is the trivial group for some $$x_0$$ in $$X$$, and hence for every $$x$$ in $$X$$.
169. \end{definition}
170.
171. \begin{lemma}
172. In a simply connected space $$X$$, any two paths having the same initial and final points are path homotopic.
173. \end{lemma}
174.
175. \begin{proof}
176. Let $$f$$ and $$g$$ be two paths from $$x_0$$ to $$x_1$$. Then $$\oper[f][\bar{g}]$$ is defined and is a loop on $$X$$ based at $$x_0$$. Since $$X$$ is simply connected, $$\oper[f][\bar{g}]\sim_p e_{x_0}$$. Applying the groupoid properties, we see that
177. \begin{gridenv}
178. \begin{equation*}
179. \clase[(\oper[f][\bar{g}]) \ast g ] = \clase[\oper[e_{x_0}][g]] = \clase[g].
180. \end{equation*}
181. \end{gridenv}
182. But
183. \begin{gridenv}
184. \begin{equation*}
185. \clase[(\oper[f][\bar{g}]) \ast g ] = \clase[f \ast (\oper[\bar{g}][g]) ]=\clase[\oper[f][e_{x_0}]] = \clase[f].
186. \end{equation*}
187. \end{gridenv}
188. Thus $$f$$ and $$g$$ are path homotopic.
189. \end{proof}
190.
191. \section{The fundamental group is a topological invariant}
192.
193. It should be intuitively clear that the fundamental group is a topological invariant of the space $$X$$. A convenient way to prove this fact formally is to introduce the notion of the homomorphism induced by a continuous map''.
194.
195. Suppose that $$h:X\rightarrow Y$$ is a continuous map that carries the point $$x_0$$ of $$X$$ to the point $$y_0$$ of $$Y$$. We often denote this fact by writting
196. \begin{gridenv}
197. \begin{equation*}
198.   h: (X,x_0)\rightarrow (Y,y_0).
199. \end{equation*}
200. \end{gridenv}
201. If $$f$$ is a loop in $$X$$ based at $$x_0$$, then the composite $$h\circ f:I\rightarrow Y$$ is a loop in $$Y$$ based at $$y_0$$. The correspondence $$f\mapsto h\circ f$$ thus gives rise to a map carrying $$\fgroup{X}{x}$$ to $$\fgroup{Y}{y}$$.
202. \begin{exercises}
203. \begin{enumerate}
204.   \item A subset $$A$$ of $$\erren$$ is said to be \emph{star convex} if for some point $$a_0$$ of $$A$$, all the line segments joining $$a_0$$ to other points of $$A$$ lie in $$A$$.
205.   \begin{enumerate}
206.     \item Find a star convex set that is not convex.
207.     \item Show that if $$A$$ is star convex, $$A$$ is simply connected.
208.     \item Show that if $$A$$ is star convex, any two paths in $$A$$ having the same initial and final points are path homotopic.
209.   \end{enumerate}
210.   \item Let $$x_0$$ and $$x_1$$ be two given points of the path-connected space $$X$$. Show that the group $$\fgroup[1][1]{X}{x}$$ is abelian if and only if for every par $$\alpha$$ and $$\beta$$ of paths from $$x_0$$ to $$x_1$$, we have $$\hat{\alpha}=\hat{\beta}$$.
211.   \item Let $$A\subseteq X$$ and let $$r:X\rightarrow A$$ be a retraction. Given $$a_0\in A$$, show that
212. \begin{gridenv}
213. \begin{equation*}
214.   r_\ast:\fgroup[1][0]{X}{a}\rightarrow \fgroup[1][0]{A}{a}
215. \end{equation*}
216. \end{gridenv}
217. is surjective.
218.   \item Let $$A$$ be a subset of $$\erren$$; let $$h: (A,a_0)\rightarrow (Y,y_0)$$. Show that if $$h$$ is extendable to a continuous map of $$\erren$$ into $$Y$$, then $$h_\ast$$ is the zero homomorphism.
219. \end{enumerate}
220. \end{exercises}
221.
222. \begin{thebibliography}{9}
223.   \bibitem{munkres} Munkres, James R. \emph{Topology, A First Course}. Prentice Hall, Inc., 1975.
224.   \bibitem{willard} Willard, Stephen. \emph{General Topology}. Massa\-chusetts: Addison- Wesley, 1970.
225. \end{thebibliography}
226.
227. \end{document}
228.