NAGASAKI, Ikumitsu

USHITAKI, Fumihiro

Kyoto University Research Information Repository

Title ON BORSUK-ULAM GROUPS (Developments in Geometryof Transformation Groups)Author(s)NAGASAKI, Ikumitsu; USHITAKI, FumihiroCitation数理解析研究所講究録 (2012), 1816: 36-43Issue Date2012-10URL http://hdl.handle.net/2433/194590RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityON BORSUK-ULAM GROUPSIkumitsu NAGASAKI ( )Department of MathematicsFaculty of MedicineKyoto Prefectural University of MedicineFumihiro USHITAKI ( )Department of MathematicsFaculty of ScienceKyoto Sangyo UniversityABSTRACT. A Borsuk-Ulam group is a group for which the isovariant Borsuk-Ulam theorem holds. $A$ fundamental question is: which groups are Borsuk-Ulamgroups? In this article, we shall recall some properties and previous results on aBorsuk-Ulam group. After that, we provide a new family of Borsuk-Ulam groups.We also pose some open questions.1. NOTATION AND TERMINOLOGYLet $G$ be a compact Lie group and $V$ an (orthogonal or unitary) representationspace of $G$ . We denote by $SV$ the unit sphere of $V$ , called a $G$-representationsphere. $AG$-equivariant map (or $G$-map for short) $f$ : $Xarrow Y$ is a continuous mapbetween $G$-spaces satisfying$f(gx)=gf(x), \forall x\in X, g\in G.$It is easy to see that if $f$ is $G$-equivariant, then(1) $f(X^{H})\subset Y^{H}$ , so we have the restriction map$f^{H}:X^{H}arrow Y^{H}$(2) $G_{x}\leq G_{f(x)}(\forall x\in X)$ .Definition. $A$ continuous map $f$ : $Xarrow Y$ is called a $G$-isovariant map if $f$ is a$G$-equivariant map satisfying $G_{x}=G_{f(x)}(\forall x\in X)$ .It is easy to see that $f$ : $Xarrow Y$ is $G$-isovariant if and only if $f$ is a $G$-equivariantmap such that $f_{|G(x)}$ : $G(x)arrow Y$ is injective for any $x\in X$ , where $G(x)$ is theorbit of $x$ . Similarly we define an isovariant homotopy as follows.2000 Mathematics Subject Classification. $57S17,55M20,$The first author was partially supported by JSPS KAKENHI Grant Number 23540101.1816 2012 36-43 36Definition. Let $f,$ $g$ be $G$-isovariant maps. We call $f$ and $g$ isovariantly G-homotopic if there exists a $G$-isovariant map $H$ : $X\cross Iarrow Y$ , called a $G$-isovarianthomotopy, such that $H(-, 0)=f$ and $H(-, 1)=g.$Let $[X, Y]_{G}^{isov}$ denote the set of $G$-isovariant homotopy classes of $G$-isovariantmaps.By the definition of isovariance, we easily see the following.(1) Let $X$ and $Y$ be free $G$-spaces. Then $G$-equivariance is equivalent to G-isovariance.(2) If $f:Xarrow Y$ is an injective $G$-map, then $f$ is $G$-isovariant.(3) If there exists a $G$-isovariant map $f$ : $Xarrow Y$ , then Iso (X) $\subset$ Iso $(Y)$ ,where Iso (X) is the set of isotropy subgroups of $X.$Example 1.1. Let $X=G/H$ and $Y=G/K.$(1) There exists a $G$-map $f$ : $G/Harrow G/K$ if and only if $(H)\leq(K)$ , i.e.,$H\leq aKa^{-1}$ for some $a\in G.$(2) There exists a $G$-isovariant map $f$ : $G/Harrow G/K$ if and only if $(H)=(K)$ .In this case, a $G$-isovariant map $f$ is defined by $f(gH)=gaK,$ $H=aKa^{-1}.$2. ISOVARIANT MAPS BETWEEN REPRESENTATIONSThe following result says that isovariant maps between representations are es-sentially same as those between representation spheres.Proposition 2.1. Let $V,$ $W$ be (orthogonal) $G$ -representations. The following areequivalent.(1) There exists a $G$ -isovariant map $f$ : $Varrow W.$(2) There exists a $G$-isovariant map $f$ : $V^{G^{\perp}}arrow W^{G^{\perp}}$(3) There exists a $G$ -isovariant map $f$ : $S(V^{G^{\perp}})arrow S(W^{G^{\perp}})$ .Here $V^{G^{\perp}}$ is the orthogonal complement of $V^{G}$ in V. In particular, if $V^{G}=$$W^{G}=0$ , then there exists a $G$-isovariant map $f$ : $Varrow W$ if and only if $f$ : $SVarrow$$SW.$Proof. (1) $\Rightarrow$ (2) $\Rightarrow$ (3) Composing the inclusion $i$ and the projection $p$ with$f$ : $Varrow W$ , we have an isovariant map$\overline{f}:V^{G^{\perp}}arrow iVarrow fWarrow pW^{G^{\perp}}$Composing the inclusion $j$ and the normalization map with $\overline{f}$, we have an iso-variant map$\overline{\overline{f}}:S(V^{G^{\perp}})arrow jV^{G^{\perp}}\backslash \{0\}arrow\overline{f}W^{G^{\perp}norm}\backslash \{0\}arrow. S(W^{G^{\perp}})$.(1) $\Leftarrow(2)\Leftarrow(3)$37Let $g:S(V^{G^{\perp}})arrow S(W^{G^{\perp}})$ be an isovariant map. By the radial extension, wehave an isovariant map$\tilde{g}:V^{G^{\perp}}arrow W^{G^{\perp}}$By adding the zero map to $g$ , we have an isovariant map$h$ $:=\tilde{g}\oplus 0$ : $V=V^{G^{\perp}}\oplus V^{G}arrow W^{G^{\perp}}\oplus W^{G}=W$By further arguments, we also obtainProposition 2.2. When $V^{G}=W^{G}=0$ , there is $a$ one-to-one correspondence$[V, W]_{G}^{isov}\cong[SV, SW]_{G}^{isov}.$We here provide some examples. Let $G=C_{n}=\langle c\rangle$ be a cyclic group of order $n,$where $c$ is a generator of $C$ . Consider the irreducible representations of $C$ . Let$\ovalbox{\tt\small REJECT}(=\mathbb{C})(0\leq k\leq n-1)$denote the irreducible representation with the linear action:$c \cdot z=\xi_{n}^{k}z(z\in U_{k}) , \xi_{n}=\exp(\frac{2\pi\sqrt{-1}}{n})$ .Assume $n=pq$ , where $p,$ $q$ are distinct primes and $G=C_{pq}.$Example 2.3. If $(k,pq)=(l,pq)=1$ , then there exist a $G$-isovariant map $f$ :$SU_{k}arrow SU_{l}.$In fact, fix $s$ such that $ks\equiv$ lmod $pq$ . We define a map $f$ by$f(z)=z^{sl}, z\in SU_{k}.$Then one can check that(1) $f$ is $G$-equivariant,(2) $G$ acts freely on $SU_{k}$ and $SU_{l}.$Hence $f$ is $G$-isovariant.Further arguments show that the degree of maps classifies isovariant homotopyclasses, and we have$[U_{k}, U_{l}]_{C_{pq}}^{isov}\cong[SU_{k}, SU_{l}]_{C_{pq}}^{isov}\cong \mathbb{Z},$and the representatives are given by$f_{m}(z)=z^{sl+mpq}, z\in SU_{k}, m\in \mathbb{Z}.$See [3], [4] for the detail.38Example 2.4. There do not exist isovariant maps $f$ : $U_{p}arrow U_{q}$ and $g:U_{1}arrow U_{q}.$In fact, if $f$ : $Xarrow Y$ is an isovariant map, then Iso $(X)\subset$ Iso $(Y)$ . HoweverIso $(U_{p})=\{C_{p}, G\}\not\subset$ Iso $(U_{q})=\{C_{q}, G\}$andIso $(U_{1})=\{1, G\}\not\subset Iso$ ( ) $=\{C_{q}, G\}.$Example 2.5. There exists an isovariant map $f:U_{1}arrow U_{p}\oplus U_{q}.$In fact there are isovariant maps$f_{\alpha,\beta}:SU_{1}arrow S(U_{p}\oplus U_{q})$defined by$f_{\alpha,\beta}(z)=(z^{(1+\alpha q)p}, z^{(1+\beta p)q}) , \alpha, \beta\in \mathbb{Z}, z\in SU_{1}.$These are isovariant maps since$G_{f_{\alpha,\beta}(z)}=G_{z^{(1+\alpha q)p}}\cap G_{z^{(1+\beta p)q}}=1(z\in SU_{1})$ .In this case, the multidegree classifies isovariant maps and one sees$[U_{1}, U_{p}\oplus U_{q}]_{C_{pq}}^{isov}\cong[SU_{1}, S(U_{p}\oplus U_{q})]_{C_{pq}}^{isov}\cong \mathbb{Z}\oplus \mathbb{Z}.$See [3], [4] for the detail.Example 2.6. There does not exist a $G$-isovariant map $f:U_{1}\oplus U_{1}arrow U_{p}\oplus U_{q}.$If there is an isovariant map, then the isovariant Borsuk-Ulam theorem statedin the next section shows$\dim U_{1}\oplus U_{1}-\dim(U_{1}\oplus U_{1})^{C_{p}}\leq\dim U_{p}\oplus U_{q}-\dim(U_{p}\oplus U_{q})^{C_{p}}$$|| ||$$4-0=4 4– 2=2.$This is a contradiction.Remark. There is a $G$-map $f$ : $S(U_{1}\oplus U_{1})arrow S(U_{p}\oplus U_{q})$ . In fact there are $G$-maps$f_{i}:SU_{1}arrow SU_{i}$ defined by $f_{i}(z)=z^{i}$ for $i=p$ and $q$ . Taking join of $f_{p}$ and $f_{q}$ , oneobtains a $G$-map $f=f_{p}*f_{q}:S(U_{1}\oplus U_{1})arrow S(U_{p}\oplus U_{q})$ .39Thus one can finally seeProposition 2.7. Let $G=C_{pq}$ , and $V,$ $W$ $G$-representations. There exists a$G$ -isovaMnt map $Varrow W$ if and only if$\{\begin{array}{l}\dim V-\dim V^{H}\leq\dim W-\dim W^{H}\dim V^{H}-\dim V^{G}\leq\dim W^{H}-\dim W^{G}\end{array}$for $H=C_{p},$ $C_{q}.$See [2] for the detail.Question (unsolved). How about $C_{n}$ for an arbitrary $n$?3. BORSUK-ULAM TYPE THEOREM FOR ISOVARIANT MAPSIn this section we discuss a Borsuk-Ulam type theorem for isovariant maps, whichprovides non-existence results on isovariant maps as mentioned in the previoussection.The Borsuk-Ulam theorem due to Borsuk [1] is generalized in various ways (see[6]. [7] $)$ . The following is one of them. Let $C_{p}$ be a cyclic group of prime order $p$and assume that $C_{p}$ acts freely on spheres $S^{m}$ and $S^{n}.$Theorem 3.1 ($mod p$ Borsuk-Ulam theorem).If there exists a $C_{p}$ -map ( $\Leftrightarrow C_{p}$ -isovariant map) $f$ : $S^{m}arrow S^{n}$ , then $m\leq n$ , (orequivalently, if $m>n$ , there does not exist a $C_{p}$ -map $f$ : $S^{m}arrow S^{n}$ ).Wasserman first studied the isovariant version of the Borsuk-Ulam theorem andintroduced the notion of the Borsuk-Ulam group.Definition (Wasserman). $A$ compact Lie group $G$ is called a Borsuk- Ulam group(BUG) if the following statement holds:For any pair of $G$-representations $V$ and $W$ , if there is a $G$-isovariant map $f$ :$Varrow W$ , then the Borsuk-Ulam inequality:$\dim V-\dim V^{G}\leq\dim W-\dim W^{G}$holds.Proposition 3.2 ([8]). $C_{p}$ and $S^{1}$ are BUGs.The following are fundamental properties of Borsuk-Ulam groups.Proposition 3.3 ([8]).(1) If $1arrow Harrow Garrow Karrow 1$ is exact and $H,$ $K$ are BUGs, then $G$ is also aBUG.(2) $A$ quotient group of a BUG is also a BUG.40Question (unsolved). Is a subgroup of a BUG also a BUG?Using this result repeatedly, we haveCorollary 3.4. If$1=H_{0}\triangleleft H_{1}\triangleleft H_{2}\triangleleft\cdots\triangleleft H_{r}=G$and $H_{i}/H_{i-1}$ are BUGs $(1\leq i\leq r)$ , then $G$ is a BUG.We have the following.Theorem 3.5 (Isovariant Borsuk-Ulam theorem). Any solvable compact Lie group$G$ is a BUG.Proof. As is well-known, $G$ is solvable if and only if there exists a compositionseries$1=H_{0}\triangleleft H_{1}\triangleleft H_{2}\triangleleft\cdots\triangleleft H_{r}=G$such that $H_{i}/H_{i-1}=C_{p}$ or $S^{1}$ . By Proposition 3.4, $G$ is a BUG. $\square$So the next question is: how about non-solvable case? Wasserman also foundnon-solvable examples of BUGs using the prime condition.Definition (Prime condition ($PC$)). (1) We say that a finite simple group $G$satisfies the prime condition ( $PC$ ) if$\sum_{p|o(g)}\frac{1}{p}\leq 1$holds for any $g\in G$ , where $o(g)$ is the order of $g$ , and the sum is taken overall prime divisors of $o(g)$ .(2) We say that a finite group $G$ satisfies ( $PC$ ) if for a composition series$1=H_{0}\triangleleft H_{1}\triangleleft H_{2}\triangleleft\cdots\triangleleft H_{r}=G,$each simple $H_{i}/H_{i-1}$ satisfies ( $PC$ ) in the sense of (1).Theorem 3.6 ([8]). If a finite group $G$ satisfies $(PC)$ , then $G$ is a BUG.Remark. In the proof of [8], the fact that a cyclic group $C$ is a BUG is used.Example 3.7. Altemating groups $A_{5},$ $A_{6},$ $\ldots,$ $A_{11}$ satisfy ( $PC$), and hence BUGs.But $A_{n},$ $n\geq 12$ , does not satisfy ( $PC$). In fact $A_{n},$ $n\geq 12$ , has an element of order$30=2\cdot 3\cdot 5$ and $1/2+1/3+1/5=31/30>1.$Question (unsolved). Is $A_{n}$ a BUG for $n\geq 12$?Example 3.8. $PSL(2, p)$ satisfies ( $PC$ ) for $p$ : prime $\leq 53$ ; hence a BUG. But$PSL(2,59),$ $PSL(2,61)$ do not satisfy ( $PC$ ). Indeed there are infinitely many primes$p$ such that $PSL(2,p)$ does not satisfy ($PC$).414. A NEW FAMILY OF $BoRSUK-ULAM$ GROUPSIn this section $G$ is a finite group. Let $\mathbb{F}_{q}$ be a finite field of order $q=p^{r},$ $p$ :prime. Recall$PSL(2, q)=SL(2, q)/\{\pm I\}$$=\{A\in M_{2}(\mathbb{F}_{q})|\det A=1\}/\{\pm I\}.$Remark. $PSL(2,2^{r})=SL(2,2^{r})$ .Also recall:(1) If $q=p^{r}\geq 4$ , then $PSL(2, q)$ is simple. On the other hand $PSL(2,2)\cong S_{3}$and $PSL(2,3)\cong A_{4}$ , which are non-simple.(2) $|PSL(2, q)|=\{\begin{array}{ll}q(q-1)(q+1) p=2\frac{1}{2}q(q-1)(q+1) p: odd prime.\end{array}$We introduce the M\"obius condition in [5] and show the following.Theorem 4.1 ([5]). $PSL(2, q)$ is a BUG for any $q=p^{r}.$As a corollary,Corollary 4.2. $SL(2, q),$ $GL(2, q),$ $PGL(2, q)$ are BUGs.Proof. These are shown from the following exact sequences.$1arrow\{\pm I\}arrow SL(2, q)arrow PSL(2, q)arrow 1$$1arrow SL(2, q)arrow GL(2, q)^{\det}arrow \mathbb{F}_{q}^{*}arrow 1$$(F_{q}^{*}\cong C_{q-1})$$PGL(2, q)=GL(2, q)$ /center$($ center $=\{aI|a\in \mathbb{F}_{q}^{*}\}\cong \mathbb{F}_{q}^{*})$ .As seen before, $PSL(2,59),$ $PSL(2,61)$ etc. do not satisfy ( $PC$ ). Our resultprovides the first example to be a BUG not satisfying ( $PC$ ).Finally we announce the following result which will be proved in the forthcomingpaper. Let $Sy1_{p}(G)$ denote a p–Sylow subgroup of $G.$Theorem 4.3 (N-$U$ ). If $G$ satisfies one of the following conditions, then $G$ is aBUG.(1) $Sy1_{2}(G)$ is a cyclic group $C_{2^{r}}$ of order $2^{r}.$(2) $Sy1_{2}(G)$ is a dihedml group $D_{2^{r}}$ of oder $2^{r}(r\geq 2)$ . As a convention,$D_{4}=C_{2}\cross C_{2}.$(3) $Sy1_{2}(G)$ is a genemlized quatemion group $Q_{2^{r}}$ of order $2^{r}(r\geq 3)$ .(4) $Sy1_{2}(G)$ is abelian and $Sy1_{p}(G)$ is cyclic for every odd prime $p.$42Example 4.4.(1) $PSL(2, q),$ $q$ : odd, is an example of (2).(2) $SL(2, q),$ $q$ : odd, is an example of (3).(3) $SL(2,2^{r})$ is an example of (4).(4) $A$ finite group with periodic cohomology is an example of (1), (3) or (4).For the proof, we use the fact that $PSL(2, q)$ is a BUG and several deep resultsof finite group theory.REFERENCES[1] K. Borsuk, Drei Satze uber die $n$ -dimensionale Sphare, Fund. Math, 20 (1933), 177-190.[2] I. Nagasaki, The converse of isovariant Borsuk- Ulam results for some abelian groups, Osaka.J. Math. 43 (2006), 689-710.[3] I. Nagasaki and F. Ushitaki, Isovariant maps from free $C_{n}$ -manifolds to representationspheres, Topology Appli., 155 (2008), 1066-1076.[4] I. Nagasaki and F. Ushitaki, A Hopf type classification theorem for isovariant maps fromfree $G$-manifolds to representation spheres, Acta Math. Sinica 27 (2011), 685-700.[5] I. Nagasaki and F. Ushitaki, New examples of the Borsuk-Ulam groups, to appear.[6] H. Steinlein, Borsuk’s antipodal theorem and its generalizations and applications: a survey,Topological methods in nonlinear analysis, 166-235, Montreal, 1985.[7] H. Steinlein, Spheres and symmetw: Borsuk’s antipodal theorem, Topol. Methods NonlinearAnal. 1 (1993), 15-33.[S] A. G. Wasserman, Isovariant maps and the Borsuk-Ulam theorem, Topology Appli. 38(1991), 155-161.DEPARTMENT OF MATHEMATICS, KYOTO PREFECTURAL UNIVERSITY OF MEDICINE, 13NISHITAKATSUKASA-CHO, TAISHOGUN KITA-KU, KYOTO 603-8334, JAPAN$E$-mail address: nagasakiQkoto. kpu-m. ac. jp (I. Nagasaki)DEPARTMENT OF MATHEMATICS, , FACULTY OF SCIENCE, KYOTO SANGYO UNIVERSITY,KAMIGAMO MOTOYAMA, KITA-KU, KYOTO 603-S555, JAPAN$E$-mail address: ushitaki@ksuvxO.kyoto-su.ac.jp (F. Ushitaki)43

ON BORSUK-ULAM GROUPS (Developments in Geometry of Transformation Groups)

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ON BORSUK-ULAM GROUPS (Developments in Geometry of Transformation Groups)

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