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$BC+@n@6N4(B $BLu(B
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{\it Houston Journal of Mathematics} {\bf10} (1984), 35-41.

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{\Large
{\bf $BJ?9T0\F08I$K4X$9$k%V%i%&%o!<$NJdBj$N?7$?$J>ZL@(B \\
A new proof of Brouwer's lemma on translation arcs}
}

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\begin{center}
Morton Brown \\ 
University of Michigan, Ann Arbor, Michigan 48109 
\end{center}

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{\bf $B=x(B.} $BJ?LL$NITF0E@$r;}$?$J$$J}8~J]B8F1Aj<LA|$N8&5f$O(BBrouwer$B$NO@J8(B[3]$B$+$i;O$^$C$?!%$+$l$O$=$NO@J8$G!VJ?LLJ?9T0\F0DjM}!W$r>ZL@$7$?!%2??M$+$NCx<T(B[1]$B!"(B[4]$B!"(B[6]$B!"(B[7]$B!"(B[8]$B$,(BBrouwer$B$NO@J8$K$"$k[#Kf$5$d%.%c%C%W$K$D$$$F%3%a%s%H$7$?$j!"4JC12=$r<(:6$7$?!%$3$N$&$A2??M$+$OF1MM$NHcH=$rLH$l$J$$(B([1,p.335]$B!"(B[5,p.226]$B;2>H(B)$B!%$7$+$7$J$,$i!"=0L\$N0lCW$9$k$H$3$m!"(BBrouwer$B$NM}O@$N4pK\E*=PH/E@$OJ?9T0\F08I$K4X$9$k(BBrouwer$B$NJdBj$K$"$k(B([3]$B$N(BSatz 1)$B!%$?$@$7(B[7]$B$NCx<T$O(BBrouwer$B$N>ZL@$O4V0c$C$F$$$k$H=R$Y$F$$$k!%(B[1]$B!"(B[4]$B!"(B[7]$B$NCx<T$O(B
Brouwer$B$NJdBj$NJL>ZL@$rM?$($F$$$k!%:#2s$NO@J8$NL\E*$O!"(BBrouwer$B$NJdBj$N$5$i$KJL$N>ZL@$rM?$($k$3$H$K$"$k!%$=$N>ZL@$O(Bindex$B$N35G0$*$h$S<+M3F1CM@-(B(free equivalence)$B$N35G0$K0M$k!%8e<T$O$3$NO@J8$GF3F~$9$k!%>ZL@$N?7$7$5$O(BAlexander$B%$%=%H%T!<$r;H$&$H$3$m$K$"$k!%JL$NO@J8$G!"J?LL$NF1Aj<LA|$NNO3X$r$b$C$H?<$/D4$Y$k$?$a$K<+M3F1CM@-$rMxMQ$9$k!%(B

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{\bf $BDj5A(B.} $J$$B$,J?LL(B$R^2$$B$NC1=cJD6J@~$G(B$f$$B$,(B$J$$B$+$i(B$R^2$$B$NCf$X$NITF0E@$r;}$?$J$$<LA|$J$i!"(B$index (f,J)$$B$O(B($B@5$K8~$-$E$1$i$l$?(B)$BC1=cJD6J@~(B$J$$B$+$iC10L1_$NCf$X$N<LA|(B$(f(x) - x ) / \parallel f(x) - x \parallel$$B$N4,?t$HDj5A$9$k!%;X?t(B(index)$B$N4pK\E*@-<A$K$D$$$F$O(B[2]$B$r;2>H$5$l$?$$!%(B

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$f,g$$B$r(B$R^2$$B$NF1Aj<LA|$H$9$k!%(B$f$$B$,(B$g$$B$K(B\underline{$B6/F1CM(B}(strongly equivalent)$B$H$O!"(B2-$B1_HW(B$D^2 \subset R^2$$B$,$"$C$F(B$f(D^2) \cap D^2 = \emptyset$$B$G$"$j!"$^$?(B$g^{-1}f$$B$,(B$D^2$$B$KBf$r;}$D$H$-!"$9$J$o$A!"(B$x \not\in D^2$$B$KBP$7$F(B$f(x) = g(x)$$B$J$k$H$-$G$"$k!%(B$f$$B$,(B$g$$B$K(B\underline{$B<+M3F1CM(B}(freely equivalent)$B$H$O!"M-8BNs(B$f = f_1, \ldots, f_k = g$$B$,$"$C$F!"3F(B$f_i$$B$,(B$f_{i+1}$$B$K6/F1CM$N$H$-$G$"$k!%(B$f$$B$,(B$g$$B$K<+M3F1CM$J$i!"$3$l$i$OF10l$NITF0E@=89g$r;}$D$3$H$KCm0U$7$h$&!%<B:]!"(B$g^{-1}f$$B$OITF0E@=89g$N6aK5>e$G91Ey<LA|$G$"$k!%(B

$h$$B$r(B$R^2$$B$NF1Aj<LA|$H$9$k!%8I(B$\alpha = p_0p_1$$B$O(B$h(p_0) = p_1$$B$+$D(B$\alpha \cap h(\alpha) = p_1$$B$N$H$-(B$h$$B$N(B\underline{$BJ?9T0\F08I(B}$B$H8F$P$l$k!%(B($BJ88%$K$O$b$C$H<e$$Dj5A$b8+<u$1$i$l$k!%$?$H$($P!"(B$h(\alpha) \cap \alpha$$B$O(B$\alpha$$B$^$?$O(B$h(\alpha)$$B$N!VFbE@!W$r4^$^$J$$(B[$B$9$J$o$A!"(B$h(p_1)$$B$O(B$p_0$$B$K0lCW$7$F$b$h$$(B]$B!"$"$k$$$O(B$h(\alpha - p_1) \cap (\alpha - p_1) = \emptyset$$B!%8e<T$G$O(B$h(p_1) = x \in \alpha - (p_0 \cup p_1)$$B$,5v$5$l$k!%(B $h$$B$,ITF0E@$r;}$?$:J}8~J]B8$N$H$-!"$3$N$h$&$JFCJL$N>l9g$,$I$A$i$b5/$3$i$J$$$3$H$r$"$H$G<($9(B.)

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{\bf $BDjM}(B.}

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{\bf $BDjM}(B 1.}(Brouwer$B$NJdBj(B) $f$$B$r(B$R^2$$B$NITF0E@$J$7$NJ}8~J]B8F1Aj<LA|$H$7!"(B$\alpha = p_0p_1$$B$r(B$f$$B$NJ?9T0\F08I$H$9$k!%$3$N$H$-3F@0?t(B$n \geq 2$$B$KBP$7$F(B$f^n(\alpha) \cap \alpha = \emptyset$$B$G$"$k!%(B

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$BDjM}(B1$B$N>ZL@$r;O$a$kA0$K(B$R^2$$B$N0LAj$K4X$7$F$$$/$D$+JdBj$r=R$Y$k!%(B

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{\bf $BJdBj(B 1.} $f$$B$rJ?LL$NJ}8~J]B8F1Aj<LA|$G(B$f \mid \alpha = {\rm id}$$B$J$k$b$N$H$9$k!%$3$3$G(B$\alpha$$B$O8I$G$"$k!%$3$N$H$-(B$f$$B$O(B$\alpha$$B$KAjBPE*$K(B(relative to $\alpha$)$B91Ey<LA|$K%$%=%H!<%W$G$"$k!%(B

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$B$3$NDjM}$N>ZL@$O<!$N$H$*$j$G$"$k!%8I(B$\alpha$$B$r!V@Z$j3+$$$F!W1_HW$K$7!"1_HW$N30B&$K(BAlexander$B$N%$%=%H%T!<$rE,MQ$9$k!%(B

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{\bf $BJdBj(B 2.} $F$$B$r(B$R^2$$B$N1_HW$H$7(B$b_0 x$$B$r(B$R^2$$B$N8I(B$b_0 b_1$$B$N8GM-(B(proper)$BItJ,8I$G(B$F$$B$NFbIt$K4^$^$l$k$b$N$H$9$k!%$3$N$H$-(B$R^2$$B$NF1Aj<LA|(B$\phi$$B$G(B$F$$B$rBf(B(support)$B$K$9$k$b$N$,$"$C$F!"(B$\phi(b_0 b_1) = x b_1$$B$G$"$k!%(B(Fig.1$B;2>H(B.)

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\hspace*{6cm} Figure 1

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{\bf $BJdBj(B 3.} $f$$B$O(B$R^2$$B$NF1Aj<LA|$G(B$A \subset R^2$$B$GITF0E@$r;}$?$J$$$H$9$k!%$3$N$H$-!"(B$f$$B$O(B$f(A)$$B$KITF0E@$r;}$?$:!"$7$?$,$C$F(B$A \cup f(A)$$B$KITF0E@$r;}$?$J$$!%(B

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\noindent
{\bf $BJdBj(B 4.} $J$$B$r(B$R^2$$B$NC1=cJD6J@~$H$7!"(B$n+1$$B8D$NAjB3$/D:E@(B$x_0, x_1, \ldots, x_n$$B$*$h$S8I(B$\alpha_0 = x_0 x_1, \ldots,$ $\alpha_n = x_n x_0,$($n \geq 1$)$B$K$h$C$FJ,3d$5$l$F$$$k$H$9$k!%$^$?(B$f$$B$O(B$\alpha_0 \cup \cdots \cup \alpha_{n-1}$$B$+$i(B$\alpha_1 \cup \cdots \cup \alpha_n$$B$N>e$X$NF1Aj<LA|$G!"(B$f(x_i) = x_{i+1}, i=0,1, \ldots, n-1$$B$H$9$k!%$3$N$H$-!"(B$R^2$$B$N(B($BJ}8~J]B8(B)$BF1Aj<LA|(B$\rho$$B$,$"$C$F(B$\rho | \alpha_0 \cup \cdots \cup \alpha_{n-1} = f | \alpha_0 \cup \cdots \cup \alpha_{n-1}$$B$*$h$S(B$\rho(\alpha_n) = \alpha_0$$B$G$"$k!%$5$i$K!"(B$\rho$$B$O(B$J$$B>e$KITF0E@$r;}$?$:!"(Bindex$(\rho, J) =1$$B$G$"$k!%(B(index$(\rho, J)$$B$O(B$J$$B$rH?;~7W$^$o$j$K$^$o$C$F7W;;$9$k!%$3$l$O(B$
\alpha_0, \alpha_1, \ldots, \alpha_n$$B$N=g=x$HF1$8J}8~$H$O8B$i$J$$(B.)

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$BJdBj(B2$B!"(B3$B!"(B4$B$N>ZL@$OFI<T$N1i=,$H$7$F$*$/!%(B

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\hspace*{6cm} Figure 2

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\noindent
\underline{$BDjM}(B 1$B$N>ZL@(B}. ($B:G>.$N(B)$B@0?t(B$n \geq 2$$B$,$"$C$F(B$f^n(\alpha) \cap \alpha \neq \emptyset$$B$G$"$k$H2>Dj$9$k!%(B$\alpha_k = p_kp_{k+1} = f^k(\alpha), k=0,1,2, \ldots,n$$B$H$7!"(B$x$$B$r(B$\alpha_n$$B$H(B$\alpha = \alpha_0$$B$N:G=i$N8rE@$H$9$k(B(Fig.2$B;2>H(B)$B!%(B

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\hspace*{6cm} Figure 3

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\noindent
$B%1!<%9(B1. $B$$$A$P$s4JC1$J>l9g!"(B$x = p_0 = p_{n+1}$$B$r$^$:D4$Y$k(B(Fig.3$B;2>H(B)$B!%$3$N$H$-(B$f$$B$O(B$p_0$$B>e$G<~4|E*$G$"$j!"(B$\alpha_{n+1}$$B$O(B$p_0$$B$+$i(B$p_1$$B$X$N8I$G$"$k!%(B$J$$B$rC1=cJD6J@~(B$J = \alpha_0 \cup \alpha_1 \cup \cdots \cup \alpha_n$$B$H$7!"(B$\rho$$B$O(B$R^2$$B$NF1Aj<LA|$G$"$C$F(B$\rho|\alpha_i = f|\alpha_i, i=0,1, \ldots, n-1$$B$*$h$S(B$\rho(\alpha_n) = \alpha_1$$B$J$k$b$N$H$9$k(B($BJdBj(B4$B;2>H(B)$B!%(B$f|\alpha_0 \cup \cdots \cup \alpha_{n-1} = \rho|\alpha_0 \cup \cdots \cup \alpha_{n-1}$$B$G$"$k$+$i!"JdBj(B1$B$h$j!"(B$\gamma_t|\alpha_0 \cup \cdots \cup \alpha_{n-1} = f|\alpha_0 \cup \cdots \cup \alpha_{n-1}, 0 \leq t \leq 1$$B$J$k%$%=%H%T!<$K$h$C$F(B$f$$B$O(B$\rho$$B$K%$%=%H!<%W$G$"$k!%(B$\gamma_t$$B$O(B$\alpha_0 \cup \cdots \cup \alpha_{n-1}$$B>e$KITF0E@$r;}$?$J$$$+$i!"$3$l$O$^$?(B$\gamma_t(\alpha_0 \cup \cdots \cup \alpha_{n-1}) = \alpha_1 \cup \cdots \cup \alpha_n$$B>e$KITF0E@$r;}$?$J$$!%$f$($K(B($BJdBj(B3$B;2>H(B)$\gamma_t$$B$O(B$J$$B>e$KITF0E@$r;}$?$J$$!%$@$+$i(B$\gamma_t$$B$O!"(B$J$$B>e$KITF0E@$r;}$?$J$$F1Aj<LA|$rDL$7$F(B$f|J$$B$r(B$\rho|J$$B$K%$%=%H!<%W$9$k!%$@$+$i(B$index(f,J) = index(\rho,U) = 1$$B$G$"$k!%$f$($K(B$f$$B$O(B$J$$B$NFbIt$KITF0E@$r;}$A!"2>@b$KL7=b$9$k!%(B

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\hspace*{6cm} Figure 4

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\noindent
$B%1!<%9(B2. $B0lHL$N>l9g$rFCJL$J>l9g(B($B%1!<%9(B1)$B$K5"Ce$5$;$h$&!%(B$\alpha'_n,\alpha''_n$$B$r(B$\alpha_n$$B$NItJ,8I(B$p_nx$$B$*$h$S(B$xp_{n+1}$$B$H$7$h$&!%(B$\alpha'_0, \alpha''_0$$B$r(B$\alpha = \alpha_0$$B$NItJ,8I(B$p_0x$$B$*$h$S(B$xp_1$$B$H$7!"(B$J$$B$OC1=cJD6J@~(B$\alpha''_0 \cup \alpha_1 \cup \cdots \cup \alpha'_n$$B$H$9$k(B(Fig.4$B;2>H(B)$B!%(B$D$$B$O(B$f^{-1}(\alpha''_n)$$B$rFbIt$K4^$`1_HW$H$7!"(B$D \cap [f(D) \cup \alpha_0 \cup \cdots \cup \alpha_{n-2}] = \emptyset$$B$H$9$k!%(B$\phi$$B$O(B$R^2$$B$NF1Aj<LA|$G$"$C$F(B$f(D)$$B$KBf$r;}$A!"(B$\phi(\alpha_n \cap f(D)) = \alpha'_n \cap f(D)$$B$G$"$k$H$9$k(B($BJdBj(B2$B;2>H(B)$B!%(B$g = \phi \circ f$$B$H$9$k!%$9$k$H(B$g$$B$O(B$f, g(\alpha_0) = \alpha_1, \ldots, g(\alpha_{n-2}) = \alpha_{n-1}$$B$*$h$S(B$g(\alpha_{n-1}) = \alpha'_n$$B$K<+M3F1CM$G$"$k(B(Fig.5$B;2>H(B)$B!%$5$F(B$E$$B$O(B$\alpha'_0$$B$rFbIt$K4^$`1_HW$G!"(B$E \cap [g(E) \cup \alpha_1 \cup \cdots \cup \alpha_{n-1}] = \emptyset$$B$G$"$k$H$9$k!%(B$\psi$$B$O(B$R^2$$B$NF1Aj<LA|$G$"$C$F(B$E$$B$KBf$r;}$A!"(B$\psi(\alpha_0 \cap E) = \alpha''_0 \cap E$$B$G$"$k$H$9$k!%(B$h = g \circ \psi^{-1}$$B$H$*$/!%$3$N$H$-(B$h(\alpha''_0) = \alpha_1, h(\alpha_1) = \alpha_2, \cdots, h(\alpha_{n-1}) = \alpha'_n$$B$G$"$j!"(B$h$$B$O(B$g$$B$K<+M3F1CM$G$"$k!%$@$+$i(B$f$$B$O(B$h = \phi \circ f \circ \psi^{-1}$$B$K<+M3F1CM$G$"$j!"(B$h$$B$O(BCase 1$B$N2>@b$rK~$?$9!%(B

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\noindent
{\bf $B7O(B.} $h$$B$r(B$R^2$$B$NITF0E@$r;}$?$J$$J}8~J]B8F1Aj<LA|$H$7!"(B$\alpha = p_0 p_1$$B$r(B$h(p_0) = p_1$$B$+$D(B$(\alpha - \{ p \}) \cap h(\alpha - \{ p_1 \}) = \emptyset$$B$J$k8I$H$9$k!%$3$N$H$-!"(B$\alpha$$B$OJ?9T0\F08I$G$"$k!%$9$J$o$A!"(B$h(p_1) \not\in \alpha$$B!%(B

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\noindent
\underline{$B>ZL@(B}. $h(p_1) = p_0$$B$J$i%1!<%9(B1($n=1$)$B$N>u67$G$"$k!%;D$j$N5DO@$O%1!<%9(B1$B$N>l9g$KN`;w$G$"$k!%(B$h(p_1) = x \in \alpha - \{ p_0, p_1 \}$$B$J$i%1!<%9(B2$B$NBhFs$NItJ,$KN`;w$N>u67$G$"$k!%(B

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\hspace*{6cm} Figure 5

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\begin{center}
{\bf $B;29MJ88%(B}
\end{center}

\re
1. R. B. Barrar, Proof of the fixed point theorems of Poincar\'{e} and Birkhoff, {\it Cand. J. of Math.} {\bf 19}(1967), 333-343.

\re
2. M. Brown and W. Neumann, Proof of the Poincar\'{e}-Birkhoff fixed point theorem, {\it Mich Math. J.} {\bf 24} (1977), 21-31.

\re
3. L. E. J. Brouwer, Beweis des ebenen translationssatzes, {\it Math. Ann.} {\bf 72} (1912), 37-54.

\re
4. B. de K\'{e}rekj\'{a}rt\'{o}, The plane translation theorem of Brouwer and the last geometric theorem of Poincar\'{e}, {\it Acta Sci. Math. Szeged.} {\bf 4} (1928/1929), 86-102.

\re
5. B. de K\'{e}rekj\'{a}rt\'{o}, \"{U}ber die fixpunktfreien abbildungen der Ebenen, {\it Acta Sci. Math. Szeged.} {\bf 6} (1934).

\re
6. W. Scherrer, Translation \"{u}ber einfach zuzzamenhansenden Gebiete, {\it Viertelkschr Naturf. Ges Z\"{u}rich} {\bf 70} (1925), 77-84.

\re
7. E. Sperner, \"{U}ber die fixpunktfreien abbildungen der Ebene, {\it Hamberger Math. Einzelschr.} {\bf 14} (1933), 1-47.

\re
8. H. Terasaka, Ein Beweis des Brouwerschen ebenen Translationsatz, {\it Japan J. Math.} {\bf 7} (1930), 61-69.

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