Nagami, Seiji

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Title On The Borsuk-Ulam Theorem and Bordism (Newtransformation groups and its related topics)Author(s)Nagami, SeijiCitation数理解析研究所講究録 (2017), 2016: 175-177Issue Date2017-01URL http://hdl.handle.net/2433/231690RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityOn The Borsuk‐Ulam Theorem and BordismSeiji Nagami1 IntroductionIn [1], Crabb, Goncalves, Libardi, and Perche studied the Borsuk‐Ulam prop‐erty by using bordism relation. Let  $\tau$\overline{:M}\rightarrow\overline{M} be a free involution be‐tween m‐dimensional manifold M , and Y a topological space. Then the pair((\overline{M},  $\tau$), Y) satisfies the Borsuk‐Ulam property (the BUP) iff for any continu‐ous map f : \overline{M}\rightarrow Y there exists x\in M such that f(x)=f( $\tau$(x))([[1]]) . Set M=\overline{M}/\langle T\rangle . Then the quotient map  q : \overline{M}\rightarrow M is a double covering map. Let $\lambda$ denote the real line bundle associated with the covering map. Then by usinfthe obstruction theory and cobordism theory, Crabb, Goncalves, Libardi, andPerche obtained the following ([[1]]) , where R_{ $\eta$}. denotes the unoriented cobordismgroup of dimension i, R_{i}(X)\mathrm{t}\mathrm{h}\mathrm{e} unoriented Uordism group of a topological spaceX , and R(\mathrm{Z}_{2}) the unoriented bordism group of free \mathrm{Z}_{2} action on closed smoothmanifolds. Note that R(\mathrm{Z}_{2})\cong R(B\mathrm{Z}_{2}) holds, and that R(\mathrm{Z}_{2}) is generated bythe classes [A_{i}]\in R(\mathrm{Z}_{2}) represented by the untipodal maps A_{i} : \mathrm{S}^{n}\rightarrow \mathrm{S}^{n} . Forgeneral references about bordism theory, see [2].Theorem 1.1. ((\overline{M},  $\tau$), \mathrm{R}^{m}) satisfies the BUP if and only if w_{1}( $\lambda$)^{m}\neq 0Theorem 1.2. Suppose that m>1 . Let  $\alpha$\in R_{7n}(\mathrm{Z}_{2}) . Then  $\alpha$ is wrriten as $\alpha$=a_{0}p_{m}+a_{1}\underline{p_{m-1}}+\cdots+a_{m-1}p_{1}+a_{m}p_{0} for some a_{i}\in R_{i}.(i) If  $\alpha$=\underline{[(}M,  $\tau$ then  a_{0}=\langle w_{1}( $\lambda$)^{m}, [M]\underline{\rangle.}(ii)  $\alpha$=[(M,  $\tau$)] holds for some connected M.(iii) If a_{0}=1 and  $\alpha$=[(\underline{\overline{M},} $\tau$)] , then \underline{((}\overline{M},  $\tau$), \mathrm{R}^{m} ) satisfies the BUP.(iv) If a_{0}=0 and  $\alpha$=[(M,  $\tau$)] with M connected, then ((\overline{M},  $\tau$), \mathrm{R}^{m}) doesnot satisfy the BUP.Theorem 1.3. Supose that 1<n<m . Then,(i) There is a pair (\overline{M},  $\tau$) such that  $\alpha$=[(M,  $\tau$)] with \overline{M} connected, and that((M,  $\tau$), \mathrm{R}^{n}) satisfies the BUP.(ii) If ( a_{0} , al, . . . a_{m-n} ) \neq(0,0, \ldots 0) and  $\alpha$=[(\overline{M},  $\tau$ then (\overline{M},  $\tau$) satisfiesthe BUP.(iii) If \underline{(a}_{0} , al, . . . a_{\underline{m-}n} ) =(0,0, \ldots 0) , then there exisis a pair (\overline{M},  $\tau$) suchthat  $\alpha$=[(M,  $\tau$)] with M connected, and that (\overline{M},  $\tau$) does not satisfy the BUP.It seems interesting to restrict  $\tau$ to spin stucture preserving case. Followingcorollary is obtained by Theorem 1.1.数理解析研究所講究録第2016巻 2017年 175-177175Corollarly 1.1. Suppese that \overline{M} is a spin manifold and that  $\tau$ is spin structurepreserving. Then ((X,  $\tau$), \mathrm{R}^{2}) satisfies the BUP if and only if the types of spinstructures that are preserved by  $\tau$ are unique.Our aim of this paper is to generalize the BUP property to compact Liegroups and to consider similar result for the case  G=\mathrm{U}(1) . We work insmooth category.2 Definition of the BUP for compact Lie gruopLet \tilde{M} be a closed smooth m‐dimensional manifolds such that a compact Liegroup G acts \mathrm{f}x\mathrm{e}\mathrm{e}1_{X_{\vee}}. Suppose that G acts on \mathrm{R}^{n} via a representation  $\rho$ :  G\rightarrow GL(n, \mathrm{R}) Let f : M\rightarrow \mathrm{R}^{n} a continuous map.If G is finite (resp. not finite), set f_{G}(x)=\displaystyle \frac{1}{\#(G)}\sum_{g\in G}g^{-1}f(g(x)) (resp.f_{G}(x)=\displaystyle \int_{G}g^{-1}f(g(x))dx) . Note that f_{G}(gx)=gf_{G}(x) holds for all g\in G.Definition 2.1. Let  $\rho$ :  G\rightarrow GL(n, \mathrm{R}) be a representation. Then ((\overline{M}, G),  $\rho$)satisfies the BUP if and only if for any continuous function f:M\rightarrow \mathrm{R}^{n} thereexists x\in M such that f_{G}(x)=0.3 Borsuk Ulam property for (\mathrm{G} $\rho$)Let $\rho$_{2} : \mathrm{U}(1)\rightarrow GL(2, n) denote the representation defined by $\rho$_{2}(e^{ $\theta$ i})=\left(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s} $\theta$ & -\mathrm{s}\mathrm{i}\mathrm{n} $\theta$\\\mathrm{s}\mathrm{i}\mathrm{n} $\theta$ & \mathrm{c}\mathrm{o}\mathrm{s} $\theta$\end{array}\right) . Then define the representation  $\rho$ : \mathrm{U}(n)\rightarrow GL(n, \mathrm{R}) asfollows; $\rho$=\left\{\begin{array}{ll}p_{2}\oplus\cdots\oplus$\rho$_{2} & (n: even)\\$\rho$_{2}\oplus\cdots\oplus$\rho$_{2}\oplus 1 & (n: odd)\end{array}\right.From now on, we consider the case for G=\mathrm{U}(1) together with the aboverepresentation. In this case, the quotient map M\rightarrow M=\overline{M}/\mathrm{U}(1) is theprojection map of a principal \mathrm{U}(1)‐bundle  $\lambda$ equipped with the classifying map f:M\rightarrow B\mathrm{U}(1) .In case for odd n , we have the nowhere‐zero section s : M\rightarrow E(n $\lambda$)= M\times \mathrm{R}^{n} $\rho$ given by  s(x)=[\tilde{x}, (0,0, \ldots, 0,1 Therefore we obtain;Proposition 3.1. If n is odd, then ((\overline{M},  $\rho$), \mathrm{R}^{n}) does not satisfy the BUP.We also have;Proposition 3.2. ((\overline{M}^{2n+1},  $\rho$);\mathrm{R}^{2n}) satisfies the BUP if and only if  c_{1}( $\lambda$)^{n}\neq 0\in H^{2n}(M;\mathrm{Z}) holds.((\overline{M}^{4n+1},  $\rho$);\mathrm{R}^{4_{7} $\iota$}) satisfies the BUP if and only if p_{1}( $\lambda$)^{7b}\neq 0\in H^{4n}(M;\mathrm{Z})holds.1763.1 Unoriented bordism category\mathrm{W} first consider in the unoriented bordism category R_{*}.Theorem 3.1. Let m=2n+1.Suppose that  $\alpha$=a_{0}p_{n}+a_{2}p_{n-1}+\cdots+a_{2n}p_{0}\in R_{2_{7} $\iota$}(B\mathrm{U}(1)) , where a_{i}\in R_{i}(i) If  $\alpha$=[(M^{2n}, f then a_{0}\equiv\langle(c_{1}( $\lambda$)^{n})_{2}, [M]_{2}\rangle holds modulo 2.(ii) There exists (M^{2n}, f) such that  $\alpha$=[(M, f)]- and that M is connected.(iii) If a_{0}\neq 0 and  $\alpha$=[(M^{2n}, f then ((X,  $\rho$);\mathrm{R}^{2n}) satisfies the BUP.It seems natural to consider in the oriented bordism category rather than inthe unoriented bordism category,3.2 Oriented bordism categoryNext we consider in the oriented bordism category $\Omega$_{*} . Our consequences arefollowing;Theorem 3.2. Suppose that  $\alpha$\in$\Omega$_{2n+1}(\mathrm{U}(1))\cong$\Omega$_{2n}(B\mathrm{U}(1))(i) If [(M^{2n}, f)]= $\alpha$ holds, then  a_{0}=\langle c_{1}( $\lambda$)^{n}, [M]\rangle\in \mathrm{Z}\cong$\Omega$_{0}(ii) There exists (M^{2n}, f) such that  $\alpha$=[M, f\underline{]}and that M is connected.(iii) If a_{0}\neq 0 and  $\alpha$=[(M^{2n}, f)] hold, then ((M,  $\rho$);\mathrm{R}^{2n}) satisfies the BUP.(iv) If a_{0}=0 , then there exists (M^{2n}, f) such that [(M, f)]= $\alpha$ , that \overline{M} isconnected, and that ((\overline{M}^{2n+1},  $\rho$);\mathrm{R}^{2n}) does not satisfies the BUP.Theorem 3.3. Suppose that m\geq 2n+2 and that  $\alpha$=a_{0}p_{m}+a_{1}p_{m-1}+\cdots+a_{m}p_{0}\in$\Omega$_{i}(B\mathrm{Z}) .(i) There exists \exists(\overline{M},  $\tau$) such that [(\tilde{X},  $\tau$)]=a , and that \tilde{X} is connected,and that ((\overline{X},  $\tau$);\mathrm{R}^{n}) satisfies the BUP.(ii) If( a_{0} , al, . . . a_{m-n} ) \neq(0,0, \ldots, 0)) and ([(\overline{M},  $\tau$)]= $\alpha$ hold, then ((\overline{M},  $\tau$);\mathrm{R}^{n}) )does not satisfies the BUP.(iii) f ( a_{0} , al, . . . a_{m-n} )-=(0,0, \ldots, 0) ) holds, then there exists (\overline{X},  $\tau$) suchthat [(M,  $\tau$)]= $\alpha$ , that  M is connected and that ((\overline{M},  $\tau$);\mathrm{R}^{n}) satisfies the BUP.References[1] \mathrm{Z}_{2} ‐Bordism and the Borsuk‐Ulam Theorem, M. C. Crabb. D.L. Goncalves,A. K. M. Libardi, L. Q. Percher, \mathrm{a}\mathrm{r}\mathrm{X}\mathrm{i}\mathrm{v}:1504.03929\mathrm{v}2[\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{A}\mathrm{T}].[2] Differentiable periodic maps, P. E. Conner and E. E. Floyd, Springer, 1964,Berlin.Academic Support CenterSetsunan UniversityNeyagawasi ikeda‐nakamati 17‐8JAPAN\mathrm{E}‐mail address: nagami@atf.setsunan.ac.jp177

On The Borsuk-Ulam Theorem and Bordism (New transformation groups and its related topics)

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On The Borsuk-Ulam Theorem and Bordism (New transformation groups and its related topics)

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