数理解析研究所講究録 1248 巻 2002 年 50-58 50 HOMOTOPY TYPES OF DIFFEOMORPHISM GROUPS OF NONCOMPACT 2-MANIFOLDS 矢ヶ崎 達彦 (TATSUHIKO YAGASAKI) 京都工芸繊維大学 工芸学部 1. INTRODUCTION This is areport on the study of topological properties of the diffeomorphism groups of noncompact smooth 2-manifolds endowed with the compact-0pen $C^{\infty}$ topology [18]. When $M$ is compact smooth 2-manifold, the diffeomorphism group $D(M)$ with the compact- $C^{\infty}-\mathrm{t}\mathrm{o}\mathrm{p}\mathrm{o}1\mathrm{e}$ $\mathrm{s}\mathrm{m}\infty \mathrm{t}\mathrm{h}$ open )$\mathrm{y}$ is a R&het manifold [6, Saetion $\mathrm{I}.4$], $\mathrm{m}\mathrm{d}$ the homotopy tyPe of the $V(M)q$ identity component has been classified by S. Smale [15], C. J. Earle and J. $\mathrm{E}\mathrm{e}\mathrm{U}[4]$ , et. al. In $C^{0}-\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{y}$ $M$ the , for any compact 2- anifold , the homeomorphism group $\mathcal{H}(M)$ with the compact-0pen topology is atopological R&het manifold [6, 11, 1.4], and the homotopy type of the identity component $\mathcal{H}(M)0$ has been classified by M. E. Hamstrom [7]. Recently we have shown that $H(M)0$ is atopological R&het-manifold even if $M$ is anon- compact connected 2-manif0ld, and have classified its homotopy type [17]. The argument in [17] is based on the following ingredients: (i) the ANR-property and the contractibilty of $\mathcal{H}(M)_{0}$ for $M$ compact , (ii) the bundle theorem connecting the homeomorphism group $\mathcal{H}(M)0$ and the embedding spaces of submanifolds into $M$ [$16$ , Corolary 1.1], and (hi) aresult on the relative isotopies of 2-manif0ld [17, Theorem 3.1]. The same strategy based on the $C^{\infty}$ -versions of these results implies acorresponding conclusion for the diffeomorphism groups of noncompact smooth 2-manif0lds Suppose $M$ is asmooth 2-manif0ld and $X$ is aclosed subset of $M$ . We denote by $Dx(M)$ the group of $C^{\infty}-\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\infty \mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{m}\mathrm{s}h$ of $M$ onto itself with $h|\mathrm{x}=\dot{l}d_{X}$ , endowed with the compact- $C^{\infty}- \mathrm{t}\mathrm{o}\mathrm{p}\mathrm{o}1\mathrm{o}\mathrm{y}$ $9,$ $\mathrm{C}\mathrm{h}.2$ $\mathrm{m}\mathrm{d}$ open [ , Section 1], by $D_{X}(M)0$ the identity $\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{e}\alpha \mathrm{d}$ component of $D_{X}(M)$ . The following is our main result: Theorem 1.1. Suppose $M$ is a noncompact connected smooth 2-manifold without boundary. (1) $D(M)0$ is a topological $\ell_{2}$ -manifold. (2) (i) $D(M)_{0}\simeq \mathrm{S}^{1}$ if $M=a$ plane, an $opm$ M\"obius band or an open annulus. $(|.i)D(M)0\simeq*in$ all other cases. $Mad\iota emah.cs$ 2000 Subject Classification. $57\mathrm{S}05;58\mathrm{D}05$; $57\mathrm{N}05;57\mathrm{N}20$. Key words and phrases. Diffeomorphism groups, hfinite dimensional manifolds, 2manifolds, 51 Any separable infinite-dimensional R\'echet space is homeomorphic to the Hilbert space $\ell_{2}\equiv$ $\{(x_{n})\in \mathbb{R}^{\infty} : \sum_{n}x_{n}^{2}<\infty\}$ . Atopological $\ell_{2}$ -manifold is aseparable metrizable space which is $\ell_{2}$ locally homeomorphic to $\ell_{2}$ . Topological types of -manifolds are classified by their homotopy types. Theorem 1.1 implies the following conclusion: Corollary 1.1. (i) $D(M)_{0}\cong \mathrm{S}^{1}\mathrm{x}\ell_{2}$ if M $=a$ plane, an open M\"obius band or an open annulus. (ii) $D(M)_{0}\underline{\simeq}\ell_{2}$ in all other cases. For the subgroup of diffeomorphisms with compact supports, we have the foUowing results: $h\in D(M)$ $C^{\infty}$ -isotopy Let $D(M)_{0}^{c}$ denote the subgroup of $D(M)0$ consisting of which admits a $h_{0}=id_{M}\mathrm{m}\mathrm{d}$ with acompact support, $h_{t}$ : $Marrow M$ such that $h_{1}=h$. We say that asubspace $A$ of a space $X$ has the homotopy negligible $(\mathrm{h}.\mathrm{n}.)$ complement in $\varphi_{0}=id_{X}$ $\varphi_{t}(X)\subset A(0<t\leq 1)$ In $X$ if there exists ahomotopy $\varphi t:Xarrow X$ such that and . $\mathrm{A}\mathrm{N}\mathrm{R}$ $\mathrm{m}\mathrm{A}\mathrm{N}\mathrm{R}$ $\mathrm{m}\mathrm{d}$ $A$ this case, the inclusion $A\subset X$ is a homotopy equivalence, $X$ is $\mathrm{m}$ iff is . Theorem 1.2. Suppose M is a noncompact connected smooth 2-manifold wiihout boundary. Then $D(M)_{0}^{c}$ has the h.n. complement in $D(M)0$ Corollary 1.2. (1) $D(M)_{0}^{c}$ is an $ANR$ . (2) The inclusion $D(M)_{0}^{c}\subset D(M)_{0}$ is a homotopy equivalence. Section 2contains fundamental facts on diffeomorphism groups of 2-manif0lds and $\ell_{2}-$ manifolds. Section 3contains asketch of prooffi of Theorems 1.1 and 1.2. 2. FUNDAMENTAL PROPEBT1ES OF D1FFEOMORPH1SM GROUPS In this preliminary section we list fundamental facts on diffeomorphism groups of 2-manif0lds (general properties, bundle theorem, homotopy type, relative isotopies, etc) and basic facts on ANR’s and $\ell_{2^{\mathrm{r}}}\mathrm{m}\mathrm{a}\dot{\mathrm{m}}\mathrm{f}\mathrm{o}1\mathrm{d}\mathrm{s}$. Throughout the paper all spaces are separable and metrizable md maps are continuous. 2.1. General property of diffeomorphism groups. Suppose M is asmooth $n$-manifold possibly with boundary and X is aclosed subset of M. Lemma 2.1. ($c.f.$ $[9$ , Ch2., Section 4], etc) $D\mathrm{x}(M)$ is a topological group, which is separable, completely metrizable, infinite-dimensional and not locally compact 52 $N$ $M$ When is a smooth submmifold of , the symbol $\mathcal{E}x(N, M)$ denotes the space of $C^{\infty}-$ $f$ $Narrow M$ embeddings : $f|\chi=id_{X}$ $C^{\infty}- \mathrm{t}\mathrm{o}\mathrm{p}\mathrm{o}1\mathrm{o}\mathrm{y}$ with with the compact-0pen , and $\mathcal{E}_{X}(N, M)_{0}$ denotes the connected component of the inclusion : : $N\subset M$ in $\mathcal{E}_{X}$ ($N$, At). Lemma 2.2. (i) Suppose $M$ is a smooth manifold without boundary, $N$ is a comapct smooth $M$ $X$ submanifold of and is a closed subset of N. $Thm$ $\mathcal{E}x(N, M)$ is a Frichet manifold. (ii) Suppose $M$ $n$ $X$ is a compact mooh manifold and is a dosed subset of $M$ with $\partial M\subset X$ or $\partial M\cap X=\emptyset$ . Then $Dx(M)$ is a hdchet manifold. $\mathcal{E}_{X}(N, M)_{0}\mathrm{m}\mathrm{d}$ In Lemm 2.2 $D_{X}(M)_{0}$ are $\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}-\omega \mathrm{m}\mathrm{u}\mathrm{t}\mathrm{d}$ . Thus my h $\in Dx(M)0$ can be joined with idM by apath $h_{t}$ (t $\in[0,1])$ in $D_{X}(M)_{0}$ . 2.2. Bundle theorems. The bundle theorem asserts that the natural restriction maps from diffeomorphism groups to embedding spaces are principal bundles $[2, 12]$ . This has been used to study the homotopy types of diffeomorphism groups. This theorem also plays an essential role in our argument. Suppoee $M$ is a smooth m-manifold without boundary, $N$ is a compact smooth n-submanifold of $M$ and $X$ is aclosed subset of $N$. Case 1: n $<m$ [2,12] Let U be any open neighborhood of N in M. TheQrem 2.1. For any $f\in \mathcal{E}_{X}(N, U)$ $\mathcal{U}$ $f$ there exist a neighborhood of in Sx $(N, U)$ and $a$ map $\varphi:\mathcal{U}arrow D_{X\cup(M\backslash U)}(M)_{0}$ such Mat $\varphi(g)f=g(g\in \mathcal{U})$ and $\varphi(f)=:d_{M}$ . Corollary 2.1. The $\pi$ $D_{X\cup(M\backslash U)}(M)_{0}arrow \mathcal{E}_{X}(N, U)_{0}$ restriction rnap : , $\pi(h)=h|N$ , is $a$ principal bundle with fiber $D_{X\cup(M\backslash U)}(M)_{0}\cap D_{N}(M)$ . Case 2: $n=m$ In this case we have aweaker conclusion: Suppose $N’$ is acompact smooth n-submanifold $M$ $\mathrm{c}\mathrm{o}1\pi$ of $N$ $\partial N\mathrm{x}$ $\partial N$ obtained from by attaching aclosed $[0,1]$ to . Let $U$ be any open $N’$ neighborhood of in $M$ . We can apply Theorem 2.1 to $\partial N'$ to obtain the following result: $f\in \mathcal{E}_{X}(N', U)$ $\mathcal{U}'$ Theorem 2.2. For any $f$ $(\mathrm{N}, U)$ there exist a neighborhood of in Sx and $a$ map $\varphi:\mathcal{U}'arrow D_{X\cup(M\backslash U)}(M)0$ $\varphi(g)f|_{N}=g|N(g\in \mathcal{U}')$ such Mat and $\varphi(f)=:d_{M}$ . For the sake of simplicity, we set $D_{0}=D_{X\cup(M\backslash U)}(M)_{0}$, $\mathrm{f}\mathrm{i}$ $=\mathcal{E}_{X}(N, U)_{0}$ , $\mathcal{E}_{0}'=\mathcal{E}_{X}(N', U)_{0}$ . $p:\mathcal{E}_{0}'arrow\hslash$ Consider the restriction map , $p(f)=f|N$ and $\pi$ : $D_{0}arrow\hslash$ , $\pi(h)=h|N$ . We have the pulback diagram 53 $p^{*}(D_{0})arrow p_{*}D_{0}$ $arrow p\mathcal{E}_{0}\downarrow\pi$ $\pi_{2}\downarrow \mathcal{E}_{0}'$ , $\pi_{*}(f, h)=f$.