AKAIKE, YUJI

CHINEN, NAOTSUGU

TOMOYASU, KAZUO

Kyoto University Research Information Repository

TitleCOLORINGS OF FIXED-POINT FREEHOMEOMORPHISMS ON FINITE CONNECTED GRAPHS(Research trends on general and geometric topology and theirproblems)Author(s)AKAIKE, YUJI; CHINEN, NAOTSUGU; TOMOYASU,KAZUOCitation数理解析研究所講究録 (2009), 1634: 13-19Issue Date2009-04URL http://hdl.handle.net/2433/140453RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityCOLORINGS OF FIXED-POINT FREE HOMEOMORPHISMSON FINITE CONNECTED GRAPHS(YUJI AKAIKE)( )(KURE NATIONAL COLLEGE OF TECHNOLOGY)(NAOTSUGU CHINEN)( )(OKINAWA NATIONAL COLLEGE OF TECHNOLOGY)(KAZUO TOMOYASU)( )(MIYAKONOJO NATIONAL COLLEGE OF TECHNOLOGY)ABSTRACT. In this paper, we give backgrounds why we study the coloringsof the fixed-point free maps, and give an announcement for our recent resultswhich calculates the exact value of the color number of a periodic homeomor-phism without fixed-points on a finite connected graph.1. INTRODUCTIONLet $f$ : $Xarrow X$ be a fixed-point free map. A subset (resp. closed subset) $A$ of$X$ is called a color (resp. closed color) of (X, f) if $f(A)\cap A=\emptyset$ . A coloring (resp.closed coloring) of (X, f) is a finite cover $\mathcal{U}$ of $X$ consisting of colors (resp. closedcolors). This notion was introduced in $[$2$]$ and $[$ 10$]$ , but the idea of the coloringwas appeared in the $1950s$ . Since finite open covers can be shrunk to closedcovers, and finite closed covers can be swelled to open covers, the closedness ofthe coloring is irrelevant. Finite open covers do equally well. Here, we can easilyverify the following facts.Proposition 1.1. Let $X$ be a regular space and $f$ : $Xarrow X$ a fixed-point freemap.(1) For every $x\in X$ there exists a closed neighborhood $N_{x}$ of $x$ such that $N_{x}$is a closed color of $(X, f)$ .(2) If $X$ is compact, then we can take a closed coloring of $(X, f)$ .By Proposition 1.1, every fixed-point free map admits a possibly infinite coverconsisting closed colors. This explains that we are interested in finite covers only.The minimal cardinality of a closed coloring is called the color number of $(X, f)$ ,denoted by col(X, f) (see [1] or [3]), i.e.,col$( X, f)=\min${ $|U|$ : $U$ is a closed coloring of (X, $f)$ }.1634 2009 13-19 13Let $C=\{C_{1},$ $\ldots,$ $C_{p}\}$ is a coloring $($ resp. closed coloring) of $(X, f)$ . If weemphasize the number of colors of $C$ , we say that $C$ is a p-coloring (resp. p-closedcoloring) of $(X, f)$ . Now, we recall old results concerning the coloring. Here,if $X$ is a Tychonoff space for which any autohomeomorphism $f$ with no fixedpoints has an extension $\beta f$ without fixed points, we say that $X$ is fixed-point freeautohomeomorphisms extends or $FAE$ (see [15]). At first, we give the followingfigure which may explain backgrounds why we study the coloring of the fixed-point free self maps, where ” col $(X, f)<\aleph_{0}arrow\varphi$ ’ means that for some suitable(X, f) we have col $(X, f)<\aleph_{0}$ implies $\varphi$ .col $(X,$ $f)<\aleph_{0}$ $FAE$$Non\downarrow$-homogeneity of $\mathbb{N}^{*}=\beta \mathbb{N}\backslash \mathbb{N}$$\ovalbox{\tt\small REJECT}$Fixed-point free propertyof a map from $\beta \mathbb{N}$ into $\mathbb{N}^{*}$1.1. Non-homogeneity of $\mathbb{N}^{*}$ versus coloring. During a seminar at the Uni-versity of Wisconsin in 1955, the following question was raised: If a space $X$ ishomogeneous, does it necessarily follow that the growth $X^{*}=\beta X\backslash X$ is alsohomogeneous? Under CH, W. Rudin proved that $\mathbb{N}^{*}$ is not homogeneous (see[13] and [14] $)$ , but the question above still had remained at that time. Afterward,in the $1960s$ , Frol\’ik showed the following theorem $($see $[$ 16, Theorem 6.25$])$ :Theorem 1.2 $(Froli’k[8])$ . If $f$ is $a$ one-to-one mapping of an extremally dis-connected compact space $X$ into itself, then there exists a 3-clopen coloring of$(X\backslash$ Fix$(f),$ $f|_{X\Psi i_{J}(f)})and$ Fix$(f)=\{x$ : $f(x)=x\}$ is clopen. In particular,col $(X\backslash$ Fix $(f),$ $f|_{X*ix(f)})\leq 3$ .Applying Theorem 1.2, Frol\’ik showed the following (see [16, Theorem 6.33]):Corollary 1.3 (Frol\’ik [8]). No infinite closed subspace of $N^{*}$ is homogeneous.141.2. Fixed-point free property versus coloring. It is well-known that $\mathbb{N}^{*}$contains a copy of $\beta \mathbb{N}$ . In the $1960s$ , Kat\v{e}tov was interested in the followingquestion: Let $f$ be a homeomorphism of $\beta \mathbb{N}$ into $\mathbb{N}^{*}$ . Does such an $f$ havefixed-points? In [11], Kat\v{e}tov showed the following:Theorem 1.4 $($de Bruijn and Erd\"os $[$4$]$ and Kat\v{e}tov $[$ 11 $])$ . Let $X$ be a set and$f$ : $Xarrow X$ a fixed-point free map (not necessarily continuous). Then there existsa 3-coloring of $(X, f),$ $i.e$ ., col $($X, $f)\leq 3$ .Applying Theorem 1.4, Kat\v{e}tov showed the following:Corollary 1.5 (Kat\v{e}tov [11]). Let $f$ be a homeomorphism of $\beta \mathbb{N}$ into $\mathbb{N}^{*}$ . Then$f$ has no fixed-point.1.3. FAE versus coloring. By de $Bruijn- Erd\ddot{o}s- Kat\check{e}tov$ ’s theorem, it is natu-rally to ask whether we can have colors as closed sets whenever $X$ is a topologicalspace? In the $1980s$ , Blaszczyk and Kim gave the following partial answer:Theorem 1.6 $($Blaszczyk and Kim, $[$5$])$ . Let $X$ be a 0-dimensional paracompactspace and $f$ : $Xarrow X$ a fixed-point free homeomorphism. Then there exists a3-clopen coloring of $($X, $f)$ , i. e., col $(X, f)\leq 3$ .Afterward, van Douwen (see [6, Theorem 1.1]) showed the following:Theorem 1.7 (van Douwen [6]). Let $X$ be an n-dimensional paracompact spaceand $f$ : $Xarrow X$ a fixed-point free homeomorphism. Then there exists a $(2n+3)-$closed coloring of $($X, $f)_{2}$ i. e., col $($X, $f)\leq 2n+3$ .By Theorem 1.7, van Douwen pointed out that every finite dimensional para-compact space is FAE:Corollary 1.8 (van Douwen [6]). Let $X$ be a n-dimensional pamcompact spaceand $f$ : $Xarrow X$ a fixed-point free homeomorphism. Then $\beta f$ is fixed-point free.In particular, every finite dimensional paracompact space is $FAE$.Furthermore, the following fact is known:Theorem 1.9 (Douwen [6], Hartskamp-Mill [9]). Let $X$ be a normal space andlet $f$ : $Xarrow X$ be a fixed-point free map. Then the following conditions areequivalent:(1) $\beta f$ is fixed-point free.(2) There exists a closed coloring of $(X, f)$ , i. e., col $(X, f)<\aleph_{0}$ .On the other hand, Theorem 1.9 makes us have the following question: Is thereany “nice” space which is not FAE? For the question above, we need the followingfolklore (see [7]).Theorem 1.10 (Lusternik-Schnirelmann). If $\iota$ : $S^{n}arrow S^{n}$ is the antipodal map,then every closed cover $y$ of $S^{n}$ such that $F\cap\iota(F)=\emptyset$ for $F\in y$ has at least$n+2$ elements. In particular, col $(S^{n}, \iota)\geq n+2$ .15By Theorem 1.10, van Douwen found the following example:Example 1.11 (van Douwen [6]). Let $X$ be $\oplus_{n\in N}S^{n}$ , the topological sum of then-sphere $S^{n}$ , and let $f$ be the topological sum of the antipodal maps, i.e., $f|_{S^{n}}$ isthe antipodal map of $S^{n}$ for each $n\in N$ . Then $f$ is a fixed-point free autohome-omorphism on $X$ such that $\beta f$ is not fixed-point free because col $(X, f)=\aleph_{0}$ . Inparticular, $X$ is not FAE.2. MOTIVATIONIn the $1990s$ , an upper bound of the color number is improved as follows.Theorem 2.1 (van Hartskamp and Vermeer [10]). Let $X$ be a paracompact Haus-dorff space with $\dim X\leq n$ . If $f$ : $Xarrow X$ is a fixed-point free homeomorphism,then col $($X, $f)\leq n+3$ .In [12], van Mill gives a simple proof of the theorem above. By Theorem 2.1,we can easily verify the following:Corollary 2.2. Let $X$ be a 0-dimensional paracompact space and $f$ : $Xarrow X$ afixed-point free homeomorphism. If there exists an $x\in X$ such that $f^{3}(x)=x$ ,then col $($X, $f)=3$ .Furthermore, for a fixed-point free involution, the upper bound of the colornumber can be improved.Theorem 2.3 (Aarts, Fokkink, and Vermeer [2]). Let $X$ be a paracompact Haus-dorff space with $\dim X\leq n$ and $f$ : $Xarrow X$ a fixed-point free homeomorphism.If $f$ is an involution, i. e., $f^{2}(x)=x$ for all $x\in X$ , then col $($X, f $)$ $\leq n+2$ .By Theorem 2.3, we have the following:Corollary 2.4. Let $X$ be a 0-dimensional paracompact space and $f$ : $Xarrow X$ afixed-point free involution. Then col$(X, f)=2$ .Furthermore, Theorem 2.3 indicates that Lusternik-Schnirelmann’s theoremcan be improved as follows:Corollary 2.5 (Lusternik-Schnirelmann). If $\iota$ : $S^{n}arrow S^{n}$ is the antipodal map,then col $(S^{n},$ $\iota)=n+2$ .More generally, the extension of Theorem 2.1 to fixed-point free continuousmaps. However, this requires extra conditions on the space.Theorem 2.6 (van Hartskamp and Vermeer [10]). Let $X$ be a compact Hausdorffspace with $\dim X\leq n$ . If $f$ : $Xarrow X$ is a fixed-point free continuous map, thencol$(X, f)\leq n+3$ .16For example, the color number of the rotation through 120 degree on a circleis 4, and the color number of the rotation through 90 degree on a circle is 3.Moreover, let $S_{Y}^{n}$ be the n-dimensional Y-sphere and $\gamma^{n+1}$ : $S_{Y}^{n}arrow S_{Y}^{n}$ the period3 homeomorphism defined in $[$3, p.258$]$ . Then col $(S_{Y}^{n}, \gamma^{n+1})=n+3([3$ , Theorem4$])$ . Here, $S_{Y}^{1}$ is the bipartite cubic graph on six nodes $K(3,3)$ .By Theorem 2.1, it is naturally to ask the question whether col$(X, f)=n+3$or not. Then, we have concentrated the following question.Question 2.7. Let $X$ be a finite connected graph and $f$ : $Xarrow X$ a fixed-pointfree homeomorphism on X. Which is true, col $(X, f)=3$ or col$(X, f)=4$ ?In the rest of paper, we are going to give an announcement of our recent resultswhich give exact values of color numbers of periodic homeomorphisms.3. FIXED-POINT FREE HOMEOMORPHISMS WITH A PERIOD THREE POINTLet $X$ be a connected space and $f$ : $Xarrow X$ a fixed-point free homeomorphism.Clearly, col $(X, f)\geq 3$ . Moreover, if $f^{3}(x)=x$ for each $x\in X$ , then col $($X, $f)\geq 4$(cf. [2, Example $7(1)]$ ). In fact, suppose that there is a coloring $\{U_{1},$ $U_{2},$ $U_{3}\}$of $(X, f)$ . We may assume that $U_{1}\cap U_{2}\neq\emptyset$ , and let $a\in U_{1}\cap U_{2}$ . Then wehave $f(a)\in U_{3}$ , so $f^{2}(a)\in U_{1}\cup U_{2}$ . However, $f^{3}(a)=a\in U_{1}\cap U_{2}$ , we have acontradiction.The next proposition below asserts that if a fixed-point free homeomorphismon an arcwise-connected space has a point of period 3, then its color number isat least 4.Proposition 3.1. Let $X$ be an arcwise-connected space and $f$ : $Xarrow X$ a fixed-point free homeomorphism with $f^{n}=id_{X}$ for some $n\in \mathbb{N}$ . If $f$ has a period 3point in $X$ , then col $($X, f $)$ $\geq 4$ .By Theorem 2.1 and Proposition 3.1, we have the following.Corollary 3.2. Let $X$ be a l-dimensional arcwise-connected space and $f$ : $Xarrow$$X$ a fixed-point free homeomorphism with $f^{n}=id_{X}$ for some $n\in \mathbb{N}$ . If $f$ has aperiod 3 point in $X$ , then col$(X, f)=4$ .Example 3.3. Let $Z_{n}=\{x_{0},$ $x_{1},$ $\ldots,$ $x_{n-1}\}$ be an n-points discrete space, $Z_{m}*Z_{n}$a join of $Z_{m}$ and $Z_{n}$ . Define $f_{n}$ : $Z_{n}arrow Z_{n}$ by $f_{n}(x_{i})=x_{i+1}$ modulo $n$ for$i=0,$ $\ldots,$ $n-1$ and $f_{m}*f_{n}:Z_{m}*Z_{n}arrow Z_{m}*Z_{n}$ the natural map constructing$f_{m}$ and $f_{n}$ . By Corollary 3.2, col $(Z_{3}*Z_{n}, f_{3}*f_{n})=4$ for all $n\in \mathbb{N}$ with $n\geq 2$ .4. FIXED-POINT FREE HOMEOMORPHISMS WITHOUT PERIOD THREE POINTIn this section, we calculate the exact value of the color number for a fixed-pointfree homeomorphism without period 3 points on a finite connected graph.17For any homeomorphism $f$ : $Xarrow X$ and any periodic point $x\in X$ , we write$n_{x}= \min\{m$ : $f^{m}(x)=x\}$ . Set $P(f)=\{x$ : $x$ is a periodic point of $f\}$ andPer $(f)=\{n_{x} : x\in P(f)\}$ . Now, we give the following main lemma withoutproof.Lemma 4.1. Let $T$ be a triangulation of a finite connected graph $X$ and $f$ :$Xarrow X$ a fixed-point free homeomorphism with $P(f)\neq\emptyset$ . If there exists an$n\in \mathbb{N}\backslash \{1,3\}$ such that $n_{x}$ is a multiple of $n$ for each $x\in P(f)$ , then col$(X, f)=3$ .Let $\{a_{1},$ $\ldots,$ $a_{m}\}\subset \mathbb{N}$ , and let us denote by $gcd\{a_{1},$ $\ldots,$ $a_{m}\}$ the great commondivisor of $a_{1},$ $a_{2},$ $\ldots,$ $a_{m}$ .Theorem 4.2. Let $f$ : $Xarrow X$ be a fixed-point free homeomorphism on a finiteconnected graph $X$ with Per $(f)\neq\emptyset$ . If $gcd(Per(f))\neq 1,3,$ then col $(X,$ $f)=3$ .Proof. At first, we need the following fact:Fact. Let $\{a_{1},$ $\ldots,$ $a_{m}\}$ be a subset of natural numbers. Then the followingconditions are equivalent:(1) There exists an $n\in N\backslash \{1,3\}$ such that $a_{k}$ is a multiple of $n$ for each$k=1$ , . . . , $m$ .(2) $gcd\{a_{1},$ $\ldots,$ $a_{m}\}\neq 1,3$ .By Lemma 4.1 and the fact above, the proof is complete.Corollary 4.3. Let $X$ be a finite connected graph and $f$ : $Xarrow X$ a fixed-pointfree homeomorphism. If there exists an $m\in \mathbb{N}\backslash \{1,3\}$ such that $f^{p}(x)\neq x$ with$1\leq p<m$ and $f^{m}(x)=x$ for each $x\in X$ , then col $($X, $f)=3$ .Corollary 4.4. Let $X$ be a finite connected graph and $f$ : $Xarrow X$ a fixed-pointfree homeomorphism. Then col$(X, f)=3$ if either the following conditions isfulfilled:(1) Per $(f)$ consists of even numbers.(2) Per $(f)$ consists of the power of some prime number $p$ except 3.Example 4.5. (1) Let $S^{1}=\{(\cos\theta, \sin\theta)\in \mathbb{R}^{2}|0\leq\theta\leq 2\pi\}$ , and let $R_{m}$ : $S^{1}arrow$$S^{1}$ be defined by $R_{m}(\cos\theta, \sin\theta)=(\cos(\theta+2\pi/n), \sin(\theta+2\pi/n))$ for $n\in$ N. If$n\neq 1,3$ , by Corollary 4.3, col $(S^{1}, R_{n})=3$ . On the other hand, by Theorem 2.1and Proposition 3.1, col $(S^{1},$ $R_{3})=4$ .(2) Let $Z_{4}*Z_{4}$ be as in Example 3.3. By Corollary 4.4, col $(Z_{4}*Z_{4}, f)=3$for any fixed-point free homeomorphism $f$ : $Z_{4}*Z_{4}arrow Z_{4}*Z_{4}$ . This showsthat a condition that col$(X, f)\leq n+3$ for any fixed-point free homeomorphism$f$ : $Xarrow X$ of period $k$ for some $k\in \mathbb{N}$ does not imply $\dim X\leq n$ .REFERENCES[1] J. M. Aarts and R. J. Fokkink An addition theorem for the color number, Proc. Amer.Math. Soc. 129 (2001) no. 9, 2803-2807.[2] J. M. Aarts, R. J. Fokkink and H. Vermeer, Variations on a theorem of Lustemik andSchnirelmann, Topology 35 (1996) no. 4, 1051-1056.18$[$3] J. M. Aarts, R. J. Fokkink and H. Vermeer, Coloring maps of period three, Pacific J. Math.202 (2002) no. 2, 257-266.[4] N. G. de Bruijn and P. A. Erd\"os, colour problem for infinite graphs and a problem in thetheory of relations, Indagationes Math. 13, (1951), 369-373.$[$5$]$ A. Blaszczyk and D. Y. Kim, A topological version of a combinatorial theorem of Kat\v{e}tov,Comment. Math. Univ. Carolin. 29 (1988), no. 4, 657-663.[6] E. K. van Douwen, $\beta X$ and fixed-point free maps Topology Appl. 51 (1993), no. 2, 191-195.[7] J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass. (1966).[8] Z. Frol\’ik, Fixed points of maps of extremally disconnected spaces and complete Booleanalgebras, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 16 (1968), 269-275.[9] M. A. van Hartskamp and J. van Mill, Some examples related to colorings, Comment.Math. Univ. Carolin. 41 (2000) no. 4, 821-827.[10] M. A. van Hartskamp and J. Vermeer, On colorings of maps, Topology Appl. 73 (1996)no. 2, 181-190.$[$11$]$ M. Kat\v{e}tov, A theorem on mappings, Comment. Math. Univ. Carolinae 8 (1967) 431-433.$[$ 12] J. van Mill, Easier proofs of coloring theorems, Topology Appl. 97 (1999) no. 1-2, 155-163.[13] W. Rudin, Homogeneity problems in the theory of $\check{\text{\v{c}}}ech$ compactifications, Duke Math. J.23 (1956), 409-419.[14] W. Rudin, Note of correction, Duke Math. J. 23 (1956), 633.[15] S. Watson, Fixed points arising only in the growth of first countable spaces, Proc. Amer.Math. Soc. 122 (1994), no. 2, 613-617.[16] R. C. Walker, The Stone-\v{C}ech compactification, Springer-Verlag, New York-Berlin, 83(1974).KURE NATIONAL COLLEGE OF TECHNOLOGY, 2-2-11 $AGA$-MINAMI KURE-SHI HIROSHIMA737- 8506, JAPANE-mail address: akaike@kure-nct.ac.jpOKINAWA NATIONAL COLLEGE OF TECHNOLOGY, NAGO-SHI OKINAWA 905-2192, JAPANE-mail address: chinenQokinawa-ct. ac. jpMIYAKONOJO NATIONAL COLLEGE OF TECHNOLOGY, 473-1 YOSHIO-CHO MIYAKONOJO-$SHI$ MIYAZAKI 885-8567, JAPANE-mail address: tomoyasuQcc.miyakonoj o-nct. ac. jp19

COLORINGS OF FIXED-POINT FREE HOMEOMORPHISMS ON FINITE CONNECTED GRAPHS (Research trends on general and geometric topology and their problems)

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COLORINGS OF FIXED-POINT FREE HOMEOMORPHISMS ON FINITE CONNECTED GRAPHS (Research trends on general and geometric topology and their problems)

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