数理解析研究所講究録 1290 巻 2002 年 112-120

112
On the first homology of the group of equivariant Lipschit

homeomorphisms of the plane with circle action

信州大学・理学部 阿部 孝順 (K\={o}jun Abe)

Department of Mathematical Sciences,Shinshu University,
京都産業大学・理学部 福井 和彦 (Kazuhiko Fukui)

Department of Mathematics, Kyoto Sangyo University,
山形大学・工学部 三浦 毅 (Takeshi Miura)

Department of Basic Technology, Applied Mathematics and Physics,
Yamagata University,

Introduction and statement of the result\S 1.

Let $L_{G}(M)$ denote the group of equivariant Lipschitz homeomorphisms of a-manifold $M$ which are isotopic to the identity through equivariant Lipschitz
homeomorphisms with compact supports. In the previous papers [AF3],[AF4],$G$

we treated the subgroupof $L_{G}(M)$ whose elements are isotopic to
the identity with respect to the compact open Lipschitz topology, and proved’is perfect when $M$ is aprincipal -manifold or $M$ is asmooth
that-manifold for afinite group .
In this paper we consider the case of the complex plain with canonical$U(1)$ -action. We shall prove that the groupis not perfect by calculating the the first homology groupwhich is defined as the quotientofby its commutator subgroup.
Let
be the set of real valued functions on $(0, 1]$ such that there$\mathcal{H}_{LIP,G}(M)$$G$$\mathcal{H}_{LIP,G}(M)$

$G$

$G$

$\mathrm{C}$$L_{U(1)}(\mathrm{C})$$H_{1}(L_{U(1)}(\mathrm{C}))$$L_{U(1)}(\mathrm{C})$$f$$C(\mathrm{R})$exists apositive number $M$ satisfying$|f(x)-f(y)| \leq\frac{M}{x}(y-x)$

Thenis avector space over R. Let
denote the subspace of thosewith bounded on $(0, 1]$ . Then we shall prove the following$C(\mathrm{R})$$f\in C(\mathrm{R})$for $0<x\leq y\leq 1$ .
$C_{0}(\mathrm{R})$

$f$

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Theorem 1$H_{1}(L_{U(1)}(\mathrm{C}))\cong C(\mathrm{R})/C_{0}(\mathrm{R})$.

aHere the isomorphism is induced from the map assigning eachtendpoint
as
theofofrotation
which stand for the degree
function
is fairly large group sinceto zero (see \S 2). We note that the groupit contains linearly independent family of elements parameterized by $(0, 1]$ .
denoteThe situation is quite different in smooth category. Let
the group of equivariant diffeomorphism group of which are equivariantly
diffeomorphic to the identity through compact supports. By [AF2], Theorem
3.2, we have that there exists an isomorphism
induced from the map assigning each $h\in D_{U(1)}(\mathrm{C})$ the differential of atis contained in
0. Then it follows from Theorem 1that the groupthe commutator subgroup of $L_{U(1)}(D)$ , which implies that the first homology
detect absolutely different geometric property.
group of$h\in L_{U(1)}(\mathrm{C})$$h$$\hat{a}_{h}\in C(\mathrm{R})$$C(\mathrm{R})/C_{0}(\mathrm{R})$$D_{U(1)}(\mathrm{C})$$\mathrm{C}$$H_{1}(D_{U(1)}(\mathrm{C}))\cong \mathrm{R}\cross \mathrm{U}(1)$$h$$D_{U(1)}(\mathrm{C})$$D_{U(1)}(\mathrm{C})$

Orbit preserving equivariant Lipschitz homeomorphisms\S 2.

denote the unit disc in
and $L_{U(1)}(D)$ denote the group of $U(1)-$Letequivariant Lipschitz homeomorphisms of which are isotopic to the identity.
through $U(1)$ -equivariant homeomorphisms with identity on the boundary
Since $U(1)$ acts freely except for the origin, by combining Theorem 5.1 withis isomorphic to $H_{1}(L_{U(1)}(D))$ .
Corollary 5.5 in [AF3], the group
Let $L([0,1])$ denote the group of Lipschitz homeomorphisms of the unit
interval $[0, 1]$ which are isotopic to the identity through Lipschitz homeomorphisms. Then we have agroup homomorphism $P.arrow Lu(1)(D)arrow L([0,1])$ givenby$P(h)(x)=|h(x)|$ for $h\in L_{U(1)}(D)$ , $x\in[0,1]$ .
$C$

$D$

$D$

$\partial D$$H_{1}(L_{U(1)}(\mathrm{C}))$

There exists aright invers$\mathrm{e}$$\Psi(f)(xz)=f(x)z$$\Psi$: $\mathrm{U}(1)1])arrow L_{U(1)}(D)$ offor$f\in L([0,1])$$P$

defined by, $x\in[0,1]$ , $z\in U(1)$ .

Note that the kernel KerP of coincides with the set of those $h\in L_{U(1)}(D)$which are orbit preserving. Next we shall investigate the relation between the.
groups KerP and$h\in
KerP$, let
be the map satisfying:For$P$$C(\mathrm{R})$$a_{h}$$(0, 1]arrow \mathrm{C}/(1)$$h(xz)=xza_{h}(x)$for$x\in(0,1]$, $z\in U(1)$ .

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Now we investigate the properties of those maps . For amap : $(0, 1]arrow$, we define maps : $[0, 1]arrow D$ and: $Darrow D$ as follows.
$a_{h}$$U(1)\subset \mathrm{C}$$\alpha$$F_{\alpha}$$\overline{\alpha}$$\overline{\alpha}(x)=\{$$x\alpha(x)$$(0<x\leq 1)$0$(x=0)$$(0\leq x\leq 1, z\in U(1))$$F_{\alpha}(xz)=z\overline{\alpha}(x)$.

Lemma 2The following conditions (1), (2) and (3) are equivalent.
(1) $T/iere$ exists a positive number $K$ such that$| \alpha(x)-\alpha(y)|\leq\frac{K}{x}(y-x)$

(2)
(3)

$\overline{\alpha}$$F_{\alpha}$

for$0<x\leq y\leq \mathrm{L}$.

is a Lipschitz map.
is a Lipschitz map.

Proof.

First assume the condition (1). Then, for $0<x\leq y\leq 1$ , we have$|\overline{\alpha}(x)-\overline{\alpha}(y)|\leq x|\alpha(x)-\alpha(y)|+|\alpha(y)||x-y|\leq(K+1)|x-y|$

Since
for $0<x\leq 1$ , the condition (2) is satisfied.
Secondly assume the condition (2). Then, for $0<x\leq y\leq 1$ ,.

$|\overline{\alpha}(x)|\leq x$$|F_{\alpha}(xz_{1})-F_{\alpha}(yz_{2})|$$z_{1}$,$z_{2}\in \mathrm{U}(1)$$\leq$$|z_{1}(\overline{\alpha}(x)-\overline{\alpha}(y)|+|(z_{1}-z_{2})\overline{\alpha}(y)|$$\leq$$M(|x-y|+|z_{1}(y-x) +(z_{1}x-z_{2}y)|)$$\leq$$3M|xz_{1}-yz_{2}|$.

,where $M$ is aLipschitz constant of . Since $|F_{\alpha}(xz)|\leq M|xz|$ , the condition(3) is satisfied.
Finally assume the condition (3). Then, for $0<x\leq y\leq 1$ , we have$\overline{\alpha}$$\leq$$\frac{1}{x}(|x\alpha(x)-y\alpha(y)|+|(y-x)\alpha(y)|)$$=$$\frac{1}{x}(|F_{\alpha}(x)-F_{\alpha}(y)|+|y-x|)\leq\frac{L+1}{x}(y-x)$$|\alpha(x)-\alpha(y)|$where is aLipschitz constant ofLemma 2follows.
$L$$F_{\alpha}$,. Thus the condition (1) is satisfied anddenote the exponential map given by $E(x)=e^{\sqrt{-1}x}$ .
Let $h\in KerP$ . Since is identity on,. Let $ah$ :bethe lifting of for with \^a (1) $=0$ . Then.
Let

$E$:

$\mathrm{R}arrow U(1)$$h$$a_{h}$$E$$3\mathrm{D}$$\mathrm{a}\mathrm{h}(1)=1$$E\mathrm{o}\hat{a}_{h}=a_{h}$$(0, 1]arrow \mathrm{R}$115
Lemma 3is contained in. Conversely ifthecondition (1) in Lemma 2.
satisfies$C(\mathrm{R})$$a\wedge h$

Proof$\alpha\wedge\in C(\mathrm{R})_{f}$then E$\mathrm{o}\hat{\alpha}$

By Lemma 2, there exists apositive number $K$ such that$|a_{h}(x)-a_{h}(y)| \leq\frac{K}{x}(y-x)$for$0<x\leq y\leq 1$.

Note that, for each , $y\in(0,1]$ with $x<y$ , the restriction
is Lipschitz.
Then we can choose an increasing series of points $x=x_{0}<x_{1}<\cdots<x_{n-1}<$$x_{n}=y$ such that$a_{h}|[x,y]$$x$$|a_{h}(x_{i-1})-a_{h}(x_{i})|\leq\sqrt{3}$$(i=1, \ldots, n)$.

$| \hat{a}_{h}(x_{i-1})-\hat{a}_{h}(x_{i})|\leq\frac{2\pi}{3}$$(i=1, \ldots, n)$.

It follows that

Then we have$|a_{h}(x_{i-1})-a_{h}(x_{i})|$$=$$|e^{\sqrt{-1}\hat{a}(x_{i-1})}-e^{\sqrt{-1}\hat{a}(x_{i})}|$$=$$2| \sin\frac{\hat{a}_{h}(x_{i-1})-\hat{a}_{h}(x_{i})}{2}|$$=$$| \cos\frac{\theta(\hat{a}_{h}(x_{i-1})-\hat{a}_{h}(x_{i}))}{2}||\hat{a}_{h}(x:_{-1})-\hat{a}_{h}(x_{i})|$,for some $0<\theta<1$ . Thus$| \hat{a}_{h}(x_{i-1})-\hat{a}_{h}(x_{i})|\leq 2|a_{h}(x_{i-1})-a_{h}(x_{i})|\leq\frac{2K}{x_{i-1}}|x_{i-1}-x.\cdot|$.

Therefore we have$| \hat{a}_{h}(x)-\hat{a}_{h}(y)|\leq.\cdot\sum_{=1}^{n}\frac{2K}{x_{i-1}}|x_{i-1}-x_{i}|\leq\frac{2K}{x}(y-x)$

and then we have thatSince$\hat{a}_{h}\in C(\mathrm{R})$,.

$|E(x)-E(y)|=|e^{\sqrt{-1}x}-e^{\sqrt{-1}y}|\leq(y-x)$for$0<x\leq y\leq 1$ ,it is clear that, for each,satisfies the condition (1) in Lemma
2. This completes the proof of Lemma 3.
$\hat{\alpha}\in C(\mathrm{R})$$E\mathrm{o}\hat{\alpha}$

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\S 3.

Basic homomorphisms

By Lemma 3we can define ahomomorphism$T:KerParrow C(\mathrm{R})/C_{0}(\mathrm{R})$,$T(h)=\hat{a}_{h}$$\mathrm{m}\mathrm{o}\mathrm{d} C_{0}(\mathrm{R})$.

Now we have amap

0: $L_{U(1)}(D)arrow L([0,1])\cross C/C_{0}$defined by$\mathrm{T}(\mathrm{h})=(P(h), T(\Psi(P(h))^{-1}\mathrm{o}h))$.

Proposition 40is an onto group homomorphism.
First we prove that 0is agroup homomorphism. For each $h\in$$L_{U(1)}(D)$ , we set. Let $h_{i}\in L_{U(1)}(D)(i=1,2)$ . Sinceis agroup homomorphism, in order to prove 0agroup homomorphism it issufficient to prove thatProof.

$\tilde{h}=\Psi(P(h))^{-1}\mathrm{o}h$$\hat{a}_{\overline{h_{1}\mathrm{o}h_{2}}}=\hat{a}_{\tilde{h}_{1}}+\hat{a}_{\tilde{h}_{2}}$

$P$

mod

$C_{0}(\mathrm{R})$.

If $0<x\leq 1$ , $z\in U(1)$ , then$h_{i}(xz)=P(h_{i})(x)za_{\tilde{h}:}(x)^{-1}$$(i=1,2)$ ,and$(h_{1}\mathrm{o}h_{2})(xz)=P(h_{1}\mathrm{o}h_{2})(x)za_{\overline{h_{1}\mathrm{o}h_{2}}}(x)^{-1}$.

On the other hand we have.

$(h_{1}\mathrm{o}h_{2})(xz)=P(h_{1}\mathrm{o}h_{2})(x)za_{\tilde{h}_{2}}(x)^{-1}a_{\tilde{h}_{1}}(P(h_{2})(x))^{-1}.$

Then$a_{\overline{h_{1}\mathrm{o}h_{2}}}=(a_{\tilde{h}_{1}}\mathrm{o}P(h_{2}))\cdot a_{\tilde{h}_{2}}$Thus$\hat{a}_{\overline{h_{1}\mathrm{o}h_{2}}}=\hat{a}_{\tilde{h}_{1}}\mathrm{o}P(h_{2})+\hat{a}_{\tilde{h}_{2}}$.

.

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Let $M$ and $M’$ be Lipschitz constants of $P(h_{2})$ and, respectively.
Let $x\in(0,1]$ . For the case $x\leq P(h_{2})(x)$ , by Lemma 3there exists apositive
number $K$ such that
$P(h_{2})^{-1}$$| \hat{a}_{\tilde{h}_{1}}(P(h_{2})(x))-\hat{a}_{\tilde{h}_{1}}(x)|\leq\frac{K}{x}|P(h_{2})(x)-x|\leq K(M+1)$.

By definition $x\leq M’P(h_{2})(x)$ . Then, for the case $P(h_{2})(x)<x$ , we have$| \hat{a}_{\tilde{h}_{1}}(P(h_{2})(x))-\hat{a}_{\overline{h}_{1}}(x)|\leq\frac{K}{P(h_{2})(x)}|P(h_{2})(x)-x|\leq K(1+M’)$.

Then we have$\hat{a}_{\tilde{h}_{1}}\mathrm{o}P(h_{2})-\hat{a}_{\tilde{h}_{1}}\in C_{0}(\mathrm{R})$

.

Thus
mod

$\hat{a}_{\overline{h_{1}\mathrm{o}h_{2}}}=\hat{a}_{\tilde{h}_{1}}+\hat{a}_{\tilde{h}_{2}}$.

$C_{0}(\mathrm{R})$

Therefore 0is agroup homomorphism.
Let. Combining Lemma 2with Lemma 3, we havethat. Set
$f\in L([0,1]),\hat{\alpha}\in C(\mathrm{R})$$F_{E\mathrm{o}\hat{\alpha}}\in KerP$$h(xz)=f(x)F_{E\mathrm{o}\hat{\alpha}}(xz)$for$0\leq x\leq 1$, $z\in U(1)$ .

Then we see that $h\in L_{U(1)}(D)$ andonto. This completes the proof of Proposition 4.

$\ominus(h)=(f,\hat{\alpha}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{C}0(\mathrm{R})\cdot$\S 4
Proposition 5

Proof.

Ker$\Theta$$\ominus \mathrm{i}\mathrm{s}$Proof of main theorem
is contained in the commutator subgroup $ofL_{U(1)}(D)$ .

If $h\in Ker\Theta$ , then $h\in KerP$ andnumber , there exists an integer$\epsilon$$n$Note that $a_{h}=E\{nhh$ )number , we can assume that$=E(\hat{a}_{h})^{n}$$\hat{a}_{h}\in C_{0}(\mathrm{R})$such that$| \frac{\hat{a}_{h}(x)}{n}-\frac{\hat{a}_{h}(y)}{n}|\leq\frac{\epsilon}{x}(y-x)$$\epsilon$. Thus. Thus, for any positive$| \frac{\hat{a}_{h}(x)}{n}|\leq\epsilon$

for

for $0<i\leq 1$ and$0<x\leq y\leq 1$.

. Then, for asufficiently small positive$|\hat{a}_{h}(x)|\leq\epsilon$$| \hat{a}_{h}(x)-\hat{a}_{h}(y)|\leq\frac{\epsilon}{x}(y-x)$

for $0<x\leq 1$ andfor$0<x\leq y\leq 1$.

118
Letthat$v$

be areal valued smooth monotone increasing function on $(0, 1]$ such$v(x)=\{$

Then it is easy to see
defined by

10g
0$v\in C(\mathrm{R})$$(0<x\leq 1/2)$ ,$x$$(3/4\leq x\leq 1)$. Let$f$$xe^{\hat{a}_{h}(x)}$$f(x)=\{$0.

.

be areal valued function on $[0, 1]$$(0<x\leq 1)$ ,$(x=0)$ .

Note that $f(1)=1$ . We shall prove that $f\in L([0,1])$ for sufficiently small .
If $0<x\leq y\leq 1$ , then we have$\epsilon$$|(f(y)-y)-(f(x)-x)|$$=$$|(y-x)(e^{\hat{a}_{h}(y)}-1)$ $+$ $x(e^{\hat{a}_{h}(y)}-e^{\hat{a}_{h}(x)})|$$\leq$$(y-x)|e^{|\hat{a}_{h}(y)|}-1|+x|\hat{a}_{h}(y)-\hat{a}_{h}(x)|e^{\hat{a}_{h}(x)+\theta(\hat{a}_{h}(y)-\hat{a}_{h}(x))}$$\leq$$((e^{\epsilon}-1)+\epsilon e^{3\epsilon})(y-x)$,

for some $0<\theta<1$ . Here we take the positive number$(e^{\epsilon}-1)+\epsilon e^{3\epsilon}<1$$\epsilon$

satisfying.

Then it follows. from [AF3], Lemma 4:1 that the function is aLipschitz
homeomorphism of $[0, 1]$ which is isotopic to the identity through Lipschitz
homeomorphisms.
, then we haveIf$f$

$0<x \leq\frac{1}{2e^{\epsilon}}$$v(f(x))-v(x)=\log(xe^{\hat{a}_{h}(x)})-\log x=\hat{a}_{h}(x)$Then, for$\langle$$0<x \leq\frac{1}{2e^{\epsilon}},$$z\in U(1)$.

we have$F_{E\mathrm{o}v}^{-1}\mathrm{o}\Psi(f)^{-1}\mathrm{o}F_{E\mathrm{o}v}^{-1}\mathrm{o}\Psi(f))(xz)$$=$$(F_{E\mathrm{o}v}^{-1}\mathrm{o}\Psi(f)^{-1}\mathrm{o}F_{E\mathrm{o}v}^{-1})(f(x)z)$$=$$(F_{E\mathrm{o}v}^{-1}\mathrm{o}\Psi(f)^{-1})(f(x)ze^{\sqrt{-1}v(f(x))})$$=$$F_{E\mathrm{o}v}^{-1}(xze^{\sqrt{-1}v(f(x))})$

$=$

$=$

$xze^{\sqrt{-1}v(f(x))}e^{-\sqrt{-1}v(x)}$$h(xz)$

Set$g=ho\Psi(f)^{-1}\mathrm{o}F_{E\mathrm{o}v}^{-1}\mathrm{o}\Psi(f)0$ $F_{E\mathrm{o}v}$.

119

for

$g(xz)=xz$$0 \leq x\leq\frac{[perp]}{2e^{\epsilon}}$, $z\in U(1)$ .

. From [AF3], Theorem 5.1, isThus the support of is contained in$L_{U(1)}(D)$. Hence is also contained
contained in the commutator subgroup of
in the commutator subgroup. This completes the proof of Proposition 5.
$D\backslash \{0\}$$g$$g$$\mathrm{h}$Let : $Ker\Thetaarrow L_{U(1)}(D)$ denote the inclusion. By
Proposition 4we have the following exact sequence.

Proof of Theorem 1.

$\iota$$arrow i_{*}$$Ker\Theta/[Ker\Theta, L_{U(1)}(D)]$ASinceby Proposition 5,$L([0,1])$ is perfect. Thus we have$\iota_{*}=0$$\Theta_{*}$$H_{1}(L_{U(1)}(D))$$H_{1}(L([0,1])\cross C(\mathrm{R})/C_{0}(\mathrm{R}))arrow 1$

is isomorphic. By [TS], [AF4], the group$H_{1}(L_{U(1)}(D))\cong C(\mathrm{R})/C_{0}(\mathrm{R})$Remark.
that.

Let $v_{c}(0<c\leq 1)$ be real valued smooth functions on $(0, 1]$ such$(-\log x)^{c}$$v_{\mathrm{c}}(x)=\{$Thenfamilies.

$v_{c}\in C(\mathrm{R})$0$(3/4\leq x\leq 1)$. Thus the group$\{v_{\mathrm{c}}\mathrm{m}\mathrm{o}\mathrm{d} C_{0} ; 0<c\leq 1\}$$(0<x\leq 1/2)$ ,$C(\mathrm{R})/C_{0}(\mathrm{R})$.

contains linearly independent.

References[AF1] K. Abe and K. Fukui, On commutators of equivariant diffeomorphisms,Proc. Japan Acad., 54 (1978), 52-54.

[AF2] K. Abe and K. Fukui, On the stmcture of the group of equivariant
diffeomorphisms of -manifolds with codimension one orbit, Topology, 40(2001), 1325-1337.
$G$[AF3] K. Abe and K. Fukui, On the structure of the group of Lipschitz homeomorphisms and its subgroups, J. Math. Soc. Japan, 53 (2001), 501-511

120[AF4] K. Abe and K. Fukui, On the structure of the group of Lipschitz homeomorphisms and its subgroups II, preprint.

[B1] A. Banyaga, On the structure of the group of equivariant diffeomorphisms,
Topology, 16(1977), 279-283.

[F] K. Fukui, Homologies of the group of $Diff^{\infty}(R^{n},$0) and its subgroups, J.
Math. Kyoto Univ., 20(1980), 475-487.

[TS] T. Tsuboi, On the perfectness of groups of diffeomorphisms of the intervaltangent to the identity at the endpoints, preprint.
[TH] W. Thurston, Foliations and group of diffeomorphisms,
Math. Soc, 80(1974), 304-307

Bull. Amer.