数理解析研究所講究録 1270 巻 2002 年 38-50

38Elliptic Operators and Finite Groups
東京水産大学

坪井 堅二 (Kenji Tsuboi)

TOKYO UNIVERSITY OF FISHERIES,
4-5-7 KOUNAN, TOKYO 108-8477 , JAPAN

0. DIRAC OPERATOR
and the Lie group Spin(n) are defined by
Definition 0.1. The Clifford algebra$C_{n}$$C_{n}= \sum_{k=0}^{n}\otimes^{k}\mathbb{R}^{n}/\{v\otimes v+|v|^{2}.1\}$$C_{n}\supset Spin(n)${$=$$v_{1}\cdots v_{m}$$(v_{1}\cdots v_{m}=[v_{1}\otimes\cdot...\otimes v_{m}]\in C_{n})$,; $|v:|=1$ (Vi) and $m$ : even} ,: Spin(n) $arrow SO(n)$ (universal covering if $n\geq 3$) is defined by. The Lie group Spirl(n): $Spin^{c}(n)arrow S^{1}$ are defined: $Spin^{c}(n)arrow SO(n)$ ,and the homomorphisms$(n)=(Spin(n)\crossS^{1})/\mathbb{Z}_{2}$: $(h, a)\sim(-h, -a))$ , $\pi^{c}([h, a])=\pi(h)$ ,where
by Spirl

and the double coverring$\pi$$\pi(v_{1}\cdots \mathrm{v}\mathrm{m})(\mathrm{w})=v_{1}\cdots v_{m}\cdot w\cdot v_{m}\cdots v_{1}\in \mathbb{R}^{n}\subset C_{n}(\forall w\in \mathbb{R}^{n})$$\pi^{c}$$\rho$$\mathrm{Z}$$\rho([h, a])=a^{2}$.

Now assume that $n=2m$ and that $M$ is the $2m$-dimensional closed smooth oriented
manifold with aRiemannian metric.

generated-dimensional -subspace ofdenote theDefinition 02. Let
by -elements $\{(1\pm e_{2m})\cdots(1\pm e_{4})(1\pm e_{2})(1+c_{2m-1})\cdots(1+c_{3})(1+c_{1})\}$ where,. Since :,:standard basis of. Moreover, it is known that $C_{2m}\otimes \mathbb{C}=Enk(\Delta)$ , and hence Spin(2m)$Enk(\Delta)$ . $Spin^{c}(2m)$ also acts on.
via Clifford multiplication$\tau=i^{m}e_{1}e_{2}\cdots
e_{2m}(\tau^{2}=1)$$the\pm1$.
-eigenspaces of whereare defined to be$Spin^{c}(2m)$.
and-representations,
are irreducible$2^{m}$$\Delta$$C_{2m}\otimes \mathbb{C}$$\mathbb{C}$$2^{m}$$\mathbb{R}^{2m}$$\{e_{i}\}$$\mathrm{C}2\mathrm{k}-\mathrm{i}=i^{k}e_{1}e_{2}\cdots$$e_{2k-1}$$e$$\Delta\subset\Delta$$\cdot$$(/or \forall i)$$C_{2m}\otimes$$\mathbb{C}\cdot\Delta\subset\Delta$$\subset$$[(h, a)]\cdot\delta=ah\cdot\delta for\delta\in\Delta$$\Delta$$\Delta\supset\Delta_{\pm}$$\Delta_{\pm}$$\tau$$v\cdot$$\Delta_{+}\subset\Delta_{-}for$$\forall v\in \mathbb{R}^{2m}$Definition 0.3. Assume that $w_{2}(M)\in Image\{H^{2}(M;\mathbb{Z})arrow H^{2}(M;\mathbb{Z}_{2})\}$ . Then there
eists a Spin $c(2m)$ -structure $Parrow M$ which is a principal SpinO(2m)-bundle such that
7’ $\cross_{Spin^{c}(2m)}\mathbb{R}^{2m}=TM$ . Then the associated complex line bundle$P\mathrm{x}_{Sp\cdot n^{\mathrm{c}}(2m),\rho}.\mathbb{C}$.

$\eta$

is defined by$\eta=$

It is known that any $2m$-dimensional Spin or almost complex manifold has a-structure
structure and that any closed oriented -dimensional manifold has a$n\leq4$. On the other hand, it is known that the 5-dimensional homogeneous space
ifdoes not admit any Spinc-structure.
$\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(2m)-$$n$$\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(n)$$\mathrm{S}\mathrm{U}(3)/\mathrm{S}\mathrm{O}(3)$Definition 0.4. Since $(h(v))\cdot(h\cdot\Delta)=h(v\cdot\Delta)$ for any $h\in Spirl(2m)$ , we can define thewhere
Clifford multiplication cm :is given by the Riemannian metr$ric$ . Assume that there exists a SpinO(2m)-structure$Parrow M$ with a connection. For any complex vector bundle $E$ with a connection, the-valued $(Spin^{c}-)Dimc$ operator $D$ is defined by$TM\otimes S_{+}\simeq T^{*}M\otimes S_{+}arrow S_{-}$$S\pm=P\cross S\mu n^{\mathrm{c}}(2m)\Delta\pm and$$\simeq$$E$$D_{E}$:$\Gamma(S_{+}\otimes E)arrow\Gamma(\nabla T^{*}M\otimes S_{+}\otimes E)aarrow\Gamma n(S_{-}\otimes E)$

39
where

$\nabla$$TM,$ $D_{E}$is the tensor product connection. In terms of a local orthonomal basis
is expressed as.

$\{e_{k}\}$of$D_{E}( \gamma)=\sum_{k}e_{k}\cdot\nabla_{e_{k}}\gamma$

Definition 0.5. Let $E$ and $F$ be complex vector bundles over $M$ and $\Gamma(E)(\Gamma(F))$ theset of all smooth sections of $E(F)$ . A linear operator $D$ : $\Gamma(E)arrow\Gamma(F)$ is called
differential operator of order $m$ iff$a$$D( \sum_{j}u^{j}\epsilon_{j})(x)=\sum_{i,j}\sum_{|\alpha|\leq m}a_{\alpha}^{ij}(x)(D_{1}^{\alpha_{1}}\cdots D_{n^{n}}^{\alpha}u^{j})(x)f_{\dot{l}}(x)$whereare local basis of $E,$ $F$ on $U\subset M,$ $x=$
is a local coordinate
system on ,, a
, $|\alpha|=\alpha_{1}+\cdots+\alpha_{n}$ andisa smooth function onsuch thatfor some , , with $|\alpha|=m$ . For any
differential operator $D$ , any $q\in M$ and any $\xi=\sum_{k}\xi_{k}(dx^{k})_{q}\in T_{q}^{*}M$ , a linear map:
is defined by$\{\epsilon_{j}\},$$\{f_{i}\}$$U$$(x^{1}, \cdots, x^{n})$$D_{k}=-i(\partial/\partial x^{k})$$=(\alpha_{1}, \cdots, \alpha_{n})$$U$$\sigma(D)\xi$$a_{\alpha}^{ij}(x)\neq 0$$a_{\alpha}^{ij}(x)$

$i$

$j$

$\alpha$$E_{q}arrow F_{q}$$\sigma(D)_{\xi}(\sum_{j}u^{j}(q)\epsilon_{j}(q))=\sum_{i,j}(\sum_{|\alpha|=m}a_{\alpha}^{ij}(q)\xi_{1}^{\alpha_{1}}\cdots\xi_{n^{n}}^{\alpha})u^{j}(q)f_{i}(q)$.

It is shown that the definition above is independent of the choice of the local coordinate
systems and local basis, and the homomorphism
is determined only by $D$ and.
is called the principal symbol of $D$ at 4.
Definition 0.6. A differential operator is called an elliptic operator iff
gives
an isomorphism
for any $q\in M$ and any $T_{q}^{*}M\ni\xi\neq 0$ .
Example 0.7. Let:
be the Dime operator. Let beany point in $M$ anda local coordinate system around such that $\{X_{k}=$is orthonormal. Then$\sigma(D)_{\xi}$$\sigma(D)\xi$$\xi\in T_{q}^{*}M$$D$$\sigma(D)_{\xi}$$E_{q}arrow F_{q}$$D_{E}$$\Gamma(S_{+}\otimes E)arrow\Gamma(S_{-}\otimes E)$$q$$(x^{1}, \cdots, x^{n})$$q$$( \frac{\partial}{\partial x^{k}})_{q}\}_{1\leq k\leq n}$$D_{E}( \gamma)(q)=\sum_{k}X_{k}\cdot$

and hence, for any$( \nabla x_{k}\gamma)(q)=\sum_{k}(\frac{\partial}{\partial x^{k}})_{q}\cdot$$( \frac{\partial\gamma}{\partial x^{k}})(q)+higher$

order terms,, eve have$T_{q}^{*}M \ni\xi=\sum_{k}\xi_{k}(dx^{k})_{q}\simeq\sum_{k}\xi_{k}(\partial/\partial x^{k})_{q}\in T_{q}M$$\sigma(D_{E})_{\xi}(\gamma(q))=\sum_{k}(\frac{\partial}{\partial x^{k}})_{q}\cdot\dot{\iota}\xi_{k}\gamma(q)=i\sum_{k}\xi_{k}(\frac{\partial}{\partial x^{k}})_{q}\cdot\gamma(q)=i\xi\cdot\gamma(q)$Thus$\sigma(D_{E})\xi$

is invertible for any$\xi\neq 0$

and$D_{E}$.

is an elliptic operator of order 1.

Elliptic operators are “almost” invertible operators and it is known that the kernel andthe cokernel of elliptic operators are finite dimensional.

1. MAIN THEOREM
Let $M$ be a $2m$-dimensional closed oriented Riemannian manifold with aSpinc-structure
and the associated complex line bundle over $M$ . Let
be afinite group. In thispaper, we define an action of as an orientation-preserving isometric faithful action of
on $M$ which lifts to an action on the Spinc-structure $P$ . Assume that there exists anaction of on $M$ . Then for any complex -vector bundle $E$ over $M$ we can define the-equivariant Spin-Dirac operator
$P$

$G$

$\eta$

$G$

$G$

$G$

$G$

$G$

$D_{E}$:$\Gamma(S_{+}\mathrm{C}\otimes E)$$arrow\Gamma(S_{-}\otimes E)$

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by using -invariant metric connections of the tangent bundle $TM$ and $E$ , whereis equal to the non-twisted
are the half spinor bundles. Note that the operator
Spin-Dirac operator$D$ :if is the trivial complex line bundle with the trivial -action. Then the determinantofevaluated at $g\in G$ is defined by$S\pm$$G$$D_{E}$$\Gamma(S_{+})arrow\Gamma(S_{-})$

$G$

$E$

$D_{E}$’(1)$\det(D_{E},g)=\det(g|\mathrm{k}\mathrm{e}\mathrm{r}D_{E})/\det(g|\mathrm{c}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}D_{E})\in S^{1}\subset \mathbb{C}$If $g^{p}=1(p\geq 2)$ , as was proved in Appendix of [9], we have
(2)

where

$\det(D_{E},g)=\exp\frac{2\pi i}{p}\sum_{k=1}^{p-1}\frac{1}{1-\xi_{\overline{p}}^{k}}$$\xi_{p}${Ind$(D_{E})$-Ind$(D_{E},g^{k})$}is the primitive -th root of unity,$p$$\mathrm{I}\mathrm{n}\mathrm{d}(D_{E},g^{k})=\mathrm{T}\mathrm{r}(g^{k}|\mathrm{k}\mathrm{e}\mathrm{r}D_{E})-\mathrm{R}(g^{k}|\mathrm{c}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}D_{E})\in \mathbb{C}$is the equivariant index of$D_{E}$

evaluated at $g^{k}\in G$ and$\mathrm{I}\mathrm{n}\mathrm{d}(D_{E})=\mathrm{I}\mathrm{n}\mathrm{d}(D_{E}, 1)=\dim \mathrm{k}\mathrm{e}\mathrm{r}D_{E}-\dim \mathrm{c}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}D_{E}\in \mathbb{Z}$(cf. [1]).
is the numerical index of
Now since the real part offollows from (2) that the equality$D_{E}$$(1-\xi_{p}^{-k})^{-1}$$\frac{1}{2\pi i}\log\det(D_{E},g)\equiv\frac{p-1}{2p}$Ind

is 1/2 for any $p\geq 2$ and any $1\leq k\leq p-1$ , it$(D_{E})- \frac{1}{p}\sum_{k=1}^{p-1}\frac{1}{1-\xi_{\overline{p}}^{k}}$holds if $g^{p}=1(p\geq 2)$ . Hence we can define$I(g)\in \mathbb{C}/\mathbb{Z}$Ind $(D_{E},g^{k})$ .

as follows.

Definition 1.1. Assume that $g\in G$ satisfies $g^{p}=1$ . Then(3)

if$g\neq 1$$(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$$I(g)\in \mathbb{C}/\mathbb{Z}$$I(g)= \frac{p-1}{2p}Ind(D_{E})-\frac{1}{p}\sum_{k=1}^{p-1}\frac{1}{1-\xi_{p}^{-k}}Ind(D_{E},g^{k})$is defined by$(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$. If $g=1$ , we define $I(g)=0$ .

Then we have(4)

$I(g) \equiv\frac{1}{2\pi i}\log\det(D_{E},g)$$(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$and hence $I(g)$ is independent of the choice of $p\geq 2$ such that $g^{p}=1$ .
Now since the equalities$\det(D_{E}, gh)=\det(D_{E},g)\det(D_{E}, h)$$\frac{1}{2\pi i}\log\det(D_{E}, g)^{N}\equiv N\frac{1}{2\pi i}\log\det(D_{E}, g)(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$hold, the next theorem follows ffom (4).

Thorem 1.2. Assume that there exists an action of $G$ on M. Then we have(a) $I(g)+I(h)-I(gh)=0$ for any , $h\in G$ ,(b) $NI(g)=0$ for any natural number $N$ and any $g\in G$ such that $\det(D_{E}, g)^{N}=1$ .
$g$

41
We can calculate
andand hence $I(h)$ by using the equivariant
index theorem. For example, the next proposition is proved by the same argument as in$\mathrm{I}\mathrm{n}\mathrm{d}(D_{E}, h)$$\mathrm{I}\mathrm{n}\mathrm{d}(D_{E})$[6].

Proposition 1.3. Assume that the fixed point set of consists of points , ,,$M$and the -action on
lifts to an action on a complex line bundle over M. Supposethat the eigenvalues of $h|T_{q_{j}}M$ arewith respect to an or iented
orthonormal basis of M. Then we have$h$$\mathrm{g}\mathrm{i}$$q_{2}$$\cdots$$q_{n}$$L$$\mathbb{Z}_{p}$$(\xi_{p}^{\tau_{j1}}, \xi_{p}^{-\tau_{j1}}, \cdots, \xi_{p}^{\tau_{jm}}, \xi_{p}^{-\tau_{jm}})$$T_{q_{j}}$,$Ind(D_{L})=e^{c_{1}(L)+c_{1}(\eta)}\hat{A}(TM)[M]$,
where
the -class and$c_{1}(\eta)\in H^{2}(M;\mathbb{Z})$$c_{1}(L)$$\hat{A}$$\lambda_{j}$$Ind(D_{L}, h)= \sum_{j=1}^{n}\frac{\xi_{p}^{\lambda_{j}}}{\Pi_{i=1}^{m}(1-\xi_{p}^{-\tau_{j}})}\dot{.}$are the first Chern classes of

$L$

and

$\eta$

respectively,$\hat{A}$isis an integer.

When $M$ has an almost complex structure, the next proposition follows from the
Riemann-Roch theorem (4.3) and the holomorphic Lefschetz theorem (4.6) in [1] (seealso Theorem 3.5.10 in [3] .
$)$Proposition 1.4. Let $M$ be an almost complex manifold with the natural $Spin^{c}- st$ ucture
ancl the action of a finite group $G,$ a complex -line bundle over $M$ and an elementof G. Assume that the action preserrves the almost complex ucture and that the fiedpoint set of consists only of points , ,, . Suppose that acts on the tangent
$G$

$L$

$h$

$G$

$st$

$h$

$\mathrm{g}\mathrm{i}$$q_{2}$$\cdots$$h$$q_{n}$

space
via multiplication by a diagonal matr $with$ diagonal entries
and acts on the fiber. Then we have
via multiplication by$T_{q_{j}}M$$7\dot{\mathrm{V}}X$$L|q_{j}$$(\xi_{p}^{\tau_{j1}}, \cdots, \xi_{p}^{\tau_{jm}})$$\xi_{p}^{\mu_{j}}$$Ind(D_{\ell})=Ch(L^{\ell})Td(TM)[M]$,$Ind$ (DE)$h)= \sum_{j=1}^{n}\frac{\xi_{p^{j^{\ell}}}^{\mu}}{\Pi_{i=1}^{m}(1-\xi_{p}^{-\tau_{j}})}\dot{.}$, $Ch$ is the Chern character, $Td$ is the Todd class and $[M]$ is thedenotesfundamental cycle of $M$ .
where$D_{\ell}$$\mathrm{D}\mathrm{L}\mathrm{e}$2. FINITE SUBGROUP OF THE MAPPING CLASS GROUP
Let $M$ be acompact Riemann surface of genus. In this section, we define an$M$action of afinite grouponas abiholomorphic action of with respect to somecomplex structure of $M$ . Then it is known that is not asubgroup of the mapping classgroup
if $M$ does not admit an action of (see [7]).
Assume that $M$ admits an action of the cyclic groupof order generated by and$Marrow
M/\mathbb{Z}_{p}$suppose that the quotient map :is abranched covering with branch
points ,,of order. For $1\leq i\leq b$ , set $r_{i}=p/n_{i}$ . Then the
Riemann-Hurwitz equation$\sigma\geq 2$

$G$

$G$

$G$

$G$

$\Gamma_{\sigma}$$\mathbb{Z}_{p}$$y_{1}$$y_{b}\in M/\mathbb{Z}_{p}$$\cdots$

(5)

$p$

$g$

$b$

$\pi$

$(n_{1}, \cdots, n_{b})$$2 \sigma-2=p(2\overline{\sigma}-2)+\sum_{i=1}^{b}(p-r_{i})$holds where is the genus of. In this section, applying Theorem 1.2 and the
Riemann-Hurwitz equation, we examine whether $M$ admits actions of cyclic groups anddihedral groups.
Let be the tangent bundle of $M$ andthe -valued Spinc-Dirac operator on $M$ .
Under the notation above, we have the next theorem.
$\overline{\sigma}$$L$$M/\mathbb{Z}_{p}$$D_{\ell}$$L^{\ell}$42
Thorem 2.1. Assume that $M$ admits an action of $G=h$. Then for $1\leq i\leq b$there exists a natuml number $1\leq t_{:}\leq n:-1$ which is prime to :such that$=\langle g\rangle$$n$,$\varphi_{\ell,z}(t_{1}, \cdots, t_{b})\in \mathbb{Z}$for any$\ell$$N\psi_{\ell,z}(t_{1}, \cdots, t_{b})\in \mathbb{Z}$and for any $z(1\leq z<p)$ which is prime to

$p$

where

$\varphi_{\ell,z}(t_{1}, \cdots, t_{b})$$=(1-z) \frac{p-1}{2p}(1-\sigma)(2\ell+1)-\sum_{\dot{|}=1}^{b}\frac{1}{n_{\dot{l}}}\sum_{j=1}^{n.-1}\frac{1}{1-\xi_{\overline{n}}^{j}}.\cdot(\frac{\xi_{n}^{jzt.\ell}}{1-\xi_{\overline{n}}^{jzt}}.\cdot\dot{.}.\cdot-z\frac{\xi_{4}^{jt.\ell}}{1-\xi_{n}^{-jt}}.\dot{.}\dot{.})$$\psi_{\ell,z}(t_{1}, \cdots, t_{b})=\frac{p-1}{2p}(1-\sigma)(2\ell+1)$,$- \sum_{\dot{l}=1}^{b}\frac{1}{n_{\dot{l}}}\sum_{j=1}^{u-1}\frac{\xi_{n}^{jzt\dot{.}\ell}}{(1-\xi_{\overline{n}}^{j})(1-\xi_{n}^{-jzt})}.\cdot\cdot.\dot{.}\dot{.}$and $N$ is a natural number such that $\det(D_{\ell},g)^{N}=1$ .

Proof We have$\mathrm{C}\mathrm{h}(L^{\ell})=1+\ell x$

where

$x$

,

$\mathrm{T}\mathrm{d}(TM)$$= \frac{x}{1-e^{-x}}=1+\frac{1}{2}x$is the first Chern class $c_{1}(TM)$ of the tangent bundle $TM$ . Moreover since$x[M]=c_{1}(TM)[M]=2-2\sigma$ , it follows ffom Proposition 1.4 that

Ind(D\ell ) $=( \ell+\frac{1}{2})x[M]=(1-\sigma)(2\ell+1)$ .

Now letbe the fixed point set of $g^{k}(1\leq k\leq p-1)$ andThen we can see thatconsists of :points ,,,points ofand therefore it follows that$\Omega(k)$$\pi^{-1}(y:)$$r$$q_{\dot{l}}$$g\cdot q_{\dot{1}}$$\cdots$$q.\cdot$

any point in.
, which are fixed$\pi^{-1}(y:)$$g^{r\dot{.}-1}\cdot$$q_{\dot{l}}$$g^{\Gamma:}$$\pi^{-1}(y_{i})\subset\Omega(k)\Leftrightarrow\pi^{-1}(y_{\dot{1}})\cap\Omega(k)\neq\phi\Leftrightarrow k=r_{\dot{1}}j(j=1,2, \cdots, n:-1)$

Since

.

acts transitively on $\pi^{-1}(y:)$ ,
acts on the tangent space of each point in $\pi^{-1}(y:)$
via the same rotation and therefore we can suppose that :acts on the tangent spaceof each point in $\pi^{-1}(y:)$ via multiplication by:where $1\leq t_{:}\leq\eta$. –1 and is prime
to . Let be any integer with $1\leq z<p$ such that $gcd(z,p)=1$ andthe primitive-th root of unity. Then since the order ofis ,coincides withand:acts on the tangent space of each point in
via multiplication by, it
follows from Propotion 1.4 that$g^{T:}$$g$$g^{f}$$\xi_{p}^{r.t}.$$n_{i}$$t_{:}$$z$$\xi_{n}.\cdot$$g^{z}$$n_{i}$$(g^{z})^{r}$$p$$M/\langle g^{z}\rangle$$M/\langle g\rangle$$\pi^{-1}(y_{\dot{l}})$$\xi_{p}^{z\mathrm{r}.t}.:$$I(g^{z}) \equiv\frac{p-1}{2p}(1-\sigma)(2\ell+1)-\frac{1}{p}.\cdot\sum_{=1}^{b}r:\sum_{j=1}^{n.-1}\frac{\xi_{p}^{r.jzt.\ell}}{(1-\xi_{p}^{-r\dot{.}j})(1-\xi_{p}^{-\mathrm{r}.jzt})}..\dot{.}$.

$= \frac{p-1}{2p}(1-\sigma)(2\ell+1)-\sum_{\dot{l}=1}^{b}\frac{1}{n_{\dot{1}}}\sum_{j=1}^{n.-1}\frac{\xi_{n}^{jzt.\ell}}{(1-\xi_{\overline{n}}^{j})(1-\xi_{\overline{n}}^{jzt})}.\cdot\cdot.\cdot.\cdot.\cdot(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$Therefore it follows from Theorem 1.2 (a) that
$0=I(g^{z})-zI(g)\equiv\varphi_{\ell,z}(t_{1}, \cdots, t_{b})$$(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$and it follows ffom Theorem 1.2 (b) that
$0=NI(g^{z})\equiv N\psi\ell_{z},(t_{1}, \cdots, t_{b})$$(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$.

43
Example 2.2. Let $M$ be a compact Riemann surface of genus . Then the necessar$ry$
and sufficient condition for $M$ to admit-action is given in Theorem 4in [5] (seealso Proposition 2.2 in [4] . In this example, we consider one hundred cases that $2\leq$, $p\leq 11$ . Then if$\sigma$$a\mathbb{Z}_{p}$$)$$\sigma$(6)$(\sigma,p)=(2, 7)$, $(2, 11)$ , $(3, 11)$ , $(4, 11)$ , $(5, 7)$ , $(7, 11)$ , $(8, 11)$ , $(9, 11)$ ,the Riemann-Hurwitz equation is not satisfied for any , ,and hence $M$ does not
admit-action. Moreover using Thoerem 4in [5], we can see that $M$ does not admit
an action ofis contained in (6) or
if and only if$\overline{\sigma}$$b$$r_{i}$$a\mathbb{Z}_{p}$$(\sigma,p)$$\mathbb{Z}_{p}$(7)$(\sigma,p)=(2, 9)$, $(3, 5)$ , $(3, 10)$ , $(4, 7)$ , $(5, 9)$ , $(6, 11)$ , $(1, 7)$ .

In this example, using the Riemann-Huritz equation and Theorem 2.1, we prove that $M$
does not admit
action forin (7).
Now using the Riemann-Hurwitz equation, we can see that$(\sigma,p)$$a\mathbb{Z}_{p}$$(\sigma,p)=(2, 9)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(3, \{3,3, 9\})$$(\sigma,p)=(3, 5)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(1, \{5\})$$(\sigma,p)=(3, 10)$\Rightarrow (b,$\{n_{1}$,$\cdots$,$n_{b}\}$) $=(3, \{5,5, 5\})$ , (4, {2, 2, 2, 10})$(\sigma,p)=(4, 7)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(1, \{7\})$$(\sigma,p)=(5, 9)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(4, \{3,3,3, 9\})$( , ll)\Rightarrow (b,, (1, {9})) $=(1, \{11\})$$(\sigma,p)=(11, 7)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(1, \{7\})$ .
$(\sigma,p)=$$6$$\{n_{1}$,$\cdots$,$n_{b}\}$

When $(\sigma,p)=(2, 9)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(3, \{3,3, 9\})$ , direct computation using a computer shows that
, $1<\varphi_{1,2}(2,1, 1)=\varphi_{1,2}(1,2, 1)<2,0<\varphi_{1,2}(2,2, 1)<1$ ,$2<\varphi_{1,2}(1, 1, 2)<3$ , $1<\varphi_{1,2}(2,1, 2)=\varphi_{1,2}(1,2, 2)<2$ , $0<\varphi_{1,2}(2,2, 2)<1$ ,$1<\varphi_{1,2}(1, 1, 1)<2$, $1<\varphi_{1,2}(2,1, 4)=\varphi_{1,2}(1,2, 4)<2$ , $1<\varphi_{1,2}(2,2, 4)<2$ ,$1<\varphi_{1,2}(1, 1, 5)<2$ , $1<\varphi_{1,2}(2,1, 5)=\varphi_{1,2}(1,2, 5)<2$ , $0<\varphi_{1,2}(2,2, 5)<1$ ,$2<\varphi_{1,2}(1, 1, 7)<3$ , $1<\varphi_{1,2}(2,1, 7)=\varphi_{1,2}(1,2, 7)<2$ , $0<\varphi_{1,2}(2,2, 7)<1$ ,$2<\varphi_{1,2}(1, 1, 8)<3$ , $1<\varphi_{1,2}(2,1, 8)=\varphi_{1,2}(1,2, 8)<2$ , $1<\varphi_{1,2}(2,2, 8)<2$ ,$2<\varphi_{1,2}(1, 1, 4)<3$

and therefore none ofis an integer. Hence it follows from Theorem 2.1 that
the Riemann surface of genus 2does not admit an action of .
hen $(\sigma,p)=(3, 5)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(1, \{5\})$ , direct computation shows that$\varphi_{1,2}(t_{1}, t_{2}, t_{3})$$\mathbb{Z}_{9}$$2<\varphi_{1,2}(1)$,$\varphi_{1,2}(2)$,$\varphi_{1,2}(3)$, $\varphi_{1,2}(4)<3$ .

Hence the Riemann surface of genus 3does not admit an action of . Hence it is clearthat the Riemann surface of genus 3does not admit an action of.
When $(\sigma,p)=(4, 7)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(1, \{7\})$ , direct computation shows that$\mathbb{Z}_{5}$$\mathbb{Z}_{10}$$3<\varphi_{1,2}(1)$,$\varphi_{1,2}(4)$, $\varphi_{1,2}(5)<4<\varphi_{1,2}(2)$ ,$\varphi_{1,2}(3)$, $\varphi_{1,2}(6)<5$ .

Hence the Riemann surface of genus 4does not admit an action of .
When $(\sigma,p)=(5, 9)$ , ( , {ni,) $=(4, \{3,3,3, 9\})$ , direct computation shows thatnone ofis an integer for $1\leq t_{1}\leq t_{2}\leq t_{3}\leq 2$ , $1\leq t_{4}\leq 8$ , $t_{4}\neq 3,6$ .
$(\sigma,p)=(5,9)$
Moreover if, $(b, \{n_{1}, \cdots, n_{b}\})=(1, \{9\})$ , direct computation also show$\mathbb{Z}_{7}$$b$$\varphi_{1,2}(t_{1}, t_{2}, t_{3}, t_{4})$$\cdots,$$n_{b}\}$

44
is an integer for $1\leq t_{1}\leq 8$ , $t_{1}\neq 3,6$ . Hence the Riemann surface
that none ofof genus 5does not admit an action of .
) $=(1, \{11\})$ , direct computation shows that
, ( , {ni,Whennone ofis an integer for $1\leq t_{1}\leq 10$ . Hence the Riemann surface of genus 6.
does not admit an action of) $=(1, \{7\})$ , direct computation shows that none
, ( , {ni,Whenis an integer for $1\leq t_{1}\leq 6$ . Hence the Riemann surface of genus 11 does notofadmit an action of .
$\varphi_{1,2}(t_{1})$$\mathbb{Z}_{\mathrm{O}}$$(\mathrm{a},\mathrm{p})=(6,11)$$b$$n_{b}\}$$\cdots,$$\varphi_{1,2}(t_{1})$$\mathbb{Z}_{11}$$(\mathrm{a},\mathrm{p})=(1,7)$$b$$\cdots,$$n_{b}\}$$\varphi_{1,2}(t_{1})$$\mathbb{Z}_{7}$Example 2.3. Let $M$ be a compact Riemann surface of genus $\sigma(2\leq\sigma\leq 11)$ which
becauseadmits an action of $h$ $(3\leq p\leq 11)$ . Note that $M$ always admits an action ofwith respect to the -rotation around x-aocis.
we can embed $M$ symmetrically into
In this example, applying Theorem 2.1, we examine whether $M$ admits an action of thedihedral group $D(2p)$ generated by , with the relation$\mathbb{Z}_{2}$$\mathbb{R}^{3}$

$g$

$\pi$

$h$

$g^{p}=h^{2}=1$ ,(8)$h^{-1}gh=g^{-1}$

Note that $M$ clearly admits an action of the dihedral group $D(2p)$ if
with respect to the $2\pi/p$ -rotation aroundbecause we can embed $M$ symmetrically intoz-axis.
If $M$ admits an action of $D(2p)$ , the relation (8) implies that$\sigma\equiv 0,1(\mathrm{m}\mathrm{o}\mathrm{d} p)$$\mathbb{R}^{3}$$\det(D_{\ell}, g)=\det(D_{\ell}, h^{-1}gh)=\det(D\ell,g^{-1})=\det(D\ell,g)^{-1}\Leftrightarrow\det(D\ell,g)^{2}=1$

Since we have$\det(D_{\ell}, g)^{p}=\det(D_{\ell}, g^{\mathrm{p}})=1$, it follows from Theorem 2.1 that$\det(D_{\ell},g)^{2}=1\Rightarrow 2\psi_{\ell,z}(t_{1}, \cdots,t_{b})\in \mathbb{Z}$$\det(D_{\ell},g)=1\Rightarrow\psi_{\ell,z}(t_{1}, \cdots, t_{b})\in \mathbb{Z}$for any.

when is even,when is odd
$p$

$p$

and any $z(1\leq z<p)$ which is prime to .
Now it follows from the Riemann-Hurwitz equation and Thoerem 4in [5] that$\ell$$p$$(\sigma,p)=(2, 5)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(3, \{5,5, 5\})$$(\sigma,p)=(7, 5)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(3, \{5,5,5\})$$(\sigma,p)=(3, 9)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(3, \{3,9,9\})$$(\sigma,p)=(4,9)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(3, \{9,9,9\})$$(\mathrm{a},\mathrm{p})=(1, 9)\Rightarrow$( , {ni,$b$$\cdots$,$n_{b}\}$) $=(5, \{3, 9, 9, 9, 9\})$$(\sigma,p)=(2,10)\Rightarrow(b, \{n_{1}, \cdots, n_{b}\})=(3, \{2, 5, 10\})$\Rightarrow (b,$\{n_{1},$( , ll)\Rightarrow (b,$\{n_{1},$$(\sigma,p)=(7, 10)$$(\sigma,p)=$$5$$\cdots,$$\cdots,$$n_{b}\}$$n_{b}\}$) $=(4, \{2,10,10,10\})$ , (5, {2, 2, 2, 5, 10})) $=(3, \{11,11,11\})$ .

When $(\sigma,p)=(2, 5)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(3, \{5,5,5\})$ , direct computation shows that$-2<\psi_{1,1}(t_{1}, t_{2}, t_{3})<-1$

is an integer. Hence theand therefore none of
Riemann surface of genus 2does not admit an action of $D(10)$ .
When $(\sigma,p)=(7, 5)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(3, \{5, 5,5\})$ , direct computation shows thatfor any$1\leq t_{1}\leq t_{2}\leq t_{3}\leq 4$$\psi_{1,1}(t_{1}, t_{2}, t_{3})$$-8<\psi_{1,1}(t_{1}, t_{2}, t_{3})<-7$

45
for any

and therefore none ofis an integer. Hence the
Riemann surface of genus 7does not admit an action of $D(10)$ .
TAlhen $(\sigma,p)=(3,$9), (b,$\{n_{1},$\cdots ,$n_{b}\})=(3,${3,9,$9\})$ , direct computation shows that
$1\leq t_{1}\leq t_{2}\leq t_{3}\leq 4$$\psi_{1,1}(t_{1}, t_{2}, t_{3})$, (1, 1, 5), (1, 1, 7), (1, 5, 5), (1, 5, 7), (1, 7, 7),(2, 1, 1), (2, 1, 2), (2, 1, 4), (2, 1, 5), (2, 1, 7), (2, 1, 8),$(t_{1}, t_{2}, t_{3})=(1, 1, 1)$(2, 2, 5), (2, 2, 7), (2, 4, 5), (2, 4, 7), (2, 5, 5), (2, 5, 7),(2, 5, 8), (2, 7, 7), (2, 7, 8)$\Rightarrow-3<\psi_{1,1}(t_{1}, t_{2}, t_{3})<-2$

and $that-4<\psi_{1,1}(t_{1}, t_{2}, t_{3})<-3$ for other $1\leq t_{1}\leq 2$ , $1\leq t_{2}\leq t_{3}\leq 8$ , , $t_{3}\neq 3,6$ .
Hence the Riemann surface of genus 3does not admit an action of $D(18)$ .
When $(\sigma,p)=(4, 9)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(3, \{9,9, 9\})$ , direct computation shows thatnone ofis an integer for $1\leq t_{1}\leq t_{2}\leq t_{3}\leq 8$ , , , $t_{3}\neq 3,6$ . Hence the
Riemann surface of genus 4does not admit an action of $D(18)$ .
When $(\sigma,p)=(11, 9)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(5, \{3,9,9,9, 9\})$ , direct computation showsthat none ofis an integer for $1^{\cdot}\leq t_{1}\leq 2,1\leq t_{2}\leq t_{3}\leq t_{4}\leq t_{5}\leq$$t_{5}\neq
3,6$, , , ,. Hence the Riemann surface of genus 11 does not admit an action$t_{2}$$\psi_{1,1}(t_{1}, t_{2}, t_{3})$$t_{1}$$t_{2}$$\psi_{1,1}(t_{1}, t_{2}, t_{3}, t_{4}, t_{5})$$8$$t_{2}$$t_{3}$$t_{4}$

of $D(18)$ .

When $(\sigma,p)=(2, 10)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(3, \{2,5, 10\})$ , direct computation shows thatnone ofis an integer for $t_{1}=1,1\leq t_{2}\leq 4,1\leq t_{3}\leq 9$ , $t_{3}\neq 2,4,5,6,8$ .
Hence the Riemann surface of genus 2does not admit an action of $D(20)$ .
When $(\sigma,p)=(7, 10)$ , $(b, \{n_{1}, \cdots)n_{b}\})=(4, \{2,10,10, 10\})$ , direct computation showsthat none ofis an integer for $t_{1}=1,1\leq t_{2}\leq t_{3}\leq t_{4}\leq 9$ , , ,, 4, 5, 6, 8. When $(\sigma,p)=(7, 10)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(5, \{2,2,2,5, 10\})$ , direct computation also shows that none ofis an integer for $t_{1}=t_{2}--t_{3}=1$ ,$1\leq t_{4}\leq 4,1\leq t_{5}\leq 9$ , $t_{5}\neq 2,4,5,6,8$ . Hence the Riemannsurface of genus 7does
not admit an action of $D(20)$ .
When $(\sigma,p)=(5, 11)$ , $(b, \{n_{1}, \cdots, n_{b}\})=(3, \{11,11, 11\})$ , direct computation showsthat$2\psi_{1,1}(t_{1}, t_{2}, t_{3})$$2\psi_{1,1}(t_{1}, t_{2}, t_{3}, t_{4})$$t_{2}$$t_{3}$$t_{4}\neq$$2$$2\psi_{1,1}(t_{1}, t_{2}, t_{3}, t_{4}, t_{5})$$\{(t_{1}, t_{2}, t_{3})|\psi_{1,1}(t_{1}, t_{2}, t_{3})\in \mathbb{Z}\}\cap\{(t_{1}, t_{2}, t_{3})|\psi_{2,1}(t_{1}, t_{2}, t_{3})\in \mathbb{Z}\}=\phi$.

Hence the Riemann surface of genus 5does not admit an action of $D(22)$ .

Theorem 2.1 is useful in determining the rotation angles around the fixed points of theaction of an element of the mapping class group.
Example 2.4. Assume that a Riemann surface M of genus $\sigma(2\leq\sigma\leq 11)$ admits anaction ofgenerated by and let ,, $q_{b}\in M$ be the fixed points of . Note that$b=0$ if acts freely on M. In this example, we use Theorem 2.1 to deter mine therotation angle, where we can assume that $2\geq t_{1}\geq t_{2}\geq\cdots\geq t_{b}\geq 1$ .
ofvia rotation of , thenacts on $T_{qb}M$ via rotation of . Hence
If acts on
it suffices to determine , ,,under the condition that $t_{b}=1$ , which eve assume$\mathbb{Z}_{3}$$g$$\mathrm{q}\mathrm{i}$$\cdots$$g$$g$$\frac{2\pi t}{3}$$g$$g|T_{q}\dot{.}M$$T_{q_{b}}M$$\frac{4\pi}{3}$$t_{1}$$t_{2}$$\cdots$$t_{b}$$g^{2}$$\frac{2\pi}{3}$46
Now it follows from the Riemann-Hu rwitz equation and Theorem 4in [5] that a Riemann surface $M$ of genus $\sigma(2\leq\sigma\leq 11)$ admits an action of
if and only if$\mathbb{Z}_{3}$, $(3, 2)$ , $(3, 5)$ , $(4, 0)$ , $(4, 3)$ , $(4, 6)$ , $(5, 4)$ , $(5, 7)$ , $(6, 2)$ , $(6, 5)$ , $(6, 8)$ ,$(7, 0)$ , $(7, 3)$ , $(7, 6)$ , $(7, 9)$ , $(8, 4)$ , $(8, 7)$ , $(8, 10)$ , $(9, 2)$ , $(9, 5)$ , $(9, 8)$ , $(9, 11)$ ,$(10, 0)$ , $(10, 3)$ , $(10, 6)$ , $(10, 9)$ , $(10, 12)$ , $(11,4)$ , $(11, 7)$ , $(11, 10)$ , $(11, 13)$ .

$(\sigma, b)=(2,4)$(9)

If $(\sigma, b)=(4,3)$ , then the direct computation shows that,$\varphi_{\ell,z}(t_{1}, t_{2}, t_{3})\in \mathbb{Z}$$3\psi_{\ell,z}(t_{1}, t_{2},t_{3})\in \mathbb{Z}$$(or=(2, 2, 2))$ . Hence it, $z\leq 2$ if and only if
for any,,
via the rotation
on
acts
that
Theorem
2.1
follows fromrespectively. On the other hand, if $(\sigma, b)=(4,6)$ , the direct computation shows that$(t_{1}, t_{2}, t_{3})=(1, 1,1)$$1\leq\ell$$T_{q_{1}}M,$$g$,$\varphi_{\ell,z}(t_{1}, t_{2,3}t, t_{4}, t_{5}, t_{6})\in \mathbb{Z}$for any$1\leq\ell$$T_{q_{2}}M$$T_{q\mathrm{s}}M$$\frac{2\pi}{3},$$\frac{2\pi}{3}$$\frac{2\pi}{3}$$3\psi_{\ell,z}(t_{1,2,3}tt, t_{4}, t_{5}, t_{6})\in \mathbb{Z}$, $z\leq 2$ if and only if $(t_{1}, t_{2},t_{3}, t_{4}, t_{5},t_{6})=(1,1,1,1,1,1)$ or (2, 2, 2, 1, 1, 1).

this result does not imply that there are too types of rotation angles because Theorem 2.1
gives only a necessary condition. But this result implies that there does not exist anothertype of rotation angles. Further computation leads to the next result.
(10)$b=2\Rightarrow(t_{1}, t_{2})=(2,1)$,$b=3\Rightarrow(t_{1}, t_{2}, t_{3})=(1,1,1)$$b=4\Rightarrow(t_{1}, t_{2}, \cdots, t_{4})=(2,2,1,1)$,,$b=5\Rightarrow(t_{1}, t_{2}, \cdots, t_{5})=(2,1,1,1,1)$$b=6\Rightarrow$ $(t_{1}, t_{2}, \cdots, t_{6})=(1,1,1,1,1, 1)$,or (2, 2, 2, 1, 1, 1)$b=7\Rightarrow$ $(t_{1}, t_{2}, \cdots, t_{7})=(2,2,1, 1,1,1, 1)$,or (2, 2, 2, 2, 1, 1, 1, 1)$b=9\Rightarrow(t_{1}, t_{2}, \cdots, t_{9})=(1,1,1, 1,1,1,1,1,1)$ or (2, 2, 2, 1, 1, 1, 1, 1, 1)$b=10\Rightarrow$ $(t_{1}, t_{2}, \cdots, t_{10})=(2,2,1,1,1,1,1,1,1,1)$ or (2, 2, 2, 2, 2, 1, 1, 1, 1, 1)$b=11\Rightarrow(t_{1}, t_{2}, \cdots, t_{11})=(2,1,1,1,1,1,1,1,1,1,1)$ or (2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1)$b=12\Rightarrow$ $(t_{1}, t_{2}, \cdots, t_{12})=(1, 1,1,1,1,1,1,1,1,1, 1, 1)$ or (2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1)
or (2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1)$b=8\Rightarrow$$(t_{1}, t_{2}, \cdots, t_{8})=(2,1,1,1,1,1,1,1)$$b=13\Rightarrow$ $(t_{1}, t_{2}, \cdots, t_{13})=(2,2,1, 1,1,1,1,1,1, 1, 1, 1, 1)$or (2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1).
3. ALMOST FREE ACT1ON
In this section we call the action of afinite group on $M$ is almost free if the fixed
point set of any $G\ni g\neq 1$ is empty or consists only of points. Note that $M$ does not
admit an almost free action of the cyclic group % if $M$ does not admit an almost ffee
action of $h$ .
Now let $h$ be the cyclic group of prime order generated by and acomplex $h$-line
bundle over $M$ . Then since the fixed point set of is independent of , the number ofthe fixed points of is independent of and the action of $h$ is almost free if and only
if the fixed point set of is empty or consists only of points. In this section, applying
act almost ffeely on $M$ .
Theorem 1.2, we examine whether $h$
$G$

$p$

$g$

$g^{k}$$k$$g^{k}$$g$$\mathrm{c}\mathrm{n}$

$L$

$k$

$n$

47
First we have the next theorem for $p=2$ .

Thorem 3.1. Assume that $M$ admits an almost free action of . Then we have thefollowing results.
(1) If the almost free action oflifts to an action on a complex line bundle over $M$
and $Ind(D_{L})$ is an odd number, then we have $n\geq 2^{m}$ .
preserves theucture and the almost free action of(2) If $M$ has an almost complex
almost complex structure then we have $n=0$ or $n\geq 2^{m}$ .
$\mathbb{Z}_{2}$$L$$\mathbb{Z}_{2}$$st$$\mathbb{Z}_{2}$Proof (1) It follows from Proposition 1.3 that
$2I(g) \equiv\frac{1}{2}(\mathrm{I}\mathrm{n}\mathrm{d}(D_{L})-\frac{1}{2^{m}}\sum_{j=1}^{n}(-1)^{\lambda_{j}})$$(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$.

The right-hand side of the equality above is not an integer if $n<2^{m}$ because
an odd number. Hence it follows from Theorem 1.2 (b) that $n\geq 2^{m}$ .
(2) It follows from Proposition 1.4 that$\mathrm{I}\mathrm{n}\mathrm{d}(D_{L})$$(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$$2I(g) \equiv\frac{2-1}{2}\mathrm{I}\mathrm{n}\mathrm{d}(D)-\frac{1}{1-(-1)}\sum_{j=1}^{n}\frac{1}{(1-(-1))^{m}}=\frac{1}{2}(\mathrm{I}\mathrm{n}\mathrm{d}(D)-\frac{n}{2^{m}})$is.

The right-hand side of the equality above is not an integer if $0<n<2^{m}$ . Hence it
follows from Theorem 1.2 (b) that $n=0$ or $n\geq 2^{m}$ .
$\square$Remark 3.2. Let L be the trivial complex line bundle over M. Then any action of $h$
lifts to the trivial action on L.

Remark 3.3. Professor Akio Hattori has pointed out to the author that (2) of the theOrem above is also deduced from the equivar iant index theorem by using the fact that theequivariant index of any involution is an integer.
”
be the $m$ -dimensional complex projective space with the
Example 3.4. Let
SpinO-structure determined by the condition that $c_{1}(\eta)=(m+1+2s)x$ where is an. Assume that $M$ admits an
integer and is the positive generator ofand let be any element of G. Then $g^{*}x=\pm x$ ,almost free action of a finite group$(m+1+2s)g^{*}x=(m+1+2s)x$ and $(g^{*}x)^{m}=x^{m}$ imply that $g^{*}x=x$ . Hence it follows
fied points. For example, iffrom the Lefschetz fied point theorem that has $m+1$
on $M$ defined by$m<p$ , the fied point set of the action of$M=\mathbb{C}\mathrm{I}\mathrm{P}$$s$$H^{2}(M;\mathbb{Z})\cong \mathbb{Z}$

$x$

$G$

$g$

$g$

$\mathbb{Z}_{p}=\langle g\rangle$$g\cdot[z_{0} :z_{1}\mathrm{c}z_{2} :\ldots :z_{m}]arrow[z_{0}$:$\xi_{p}z_{1}$:$\xi_{p}^{2}z_{2}$:. . . :$\xi_{p}^{m}z_{m}]$consists of $m+1$ points and hence the action is almost free. Moreover it follows from

48
Proposition 1.3 that
$Ind(D)=e^{\frac{(m+1+2s)x}{2}}$$=x^{m}$\^A(M)[M]-coefficient of$e^{sx}( \frac{x}{1-e^{-x}})^{m+1}=\frac{1}{2\pi i}\oint_{C(z)}\frac{e^{(m+s)z}}{(e^{z}-1)^{m+1}}e^{z}dz$(where $C(z)$ is a sufficiently small counterclockwise loop around the origin)$= \frac{1}{2\pi i}\oint_{C(u)}\frac{(u+1)^{m+s}}{u^{m+1}}$

du(via the substitution $u=e^{z}$ , where $C(u)$ is a counterclockise loop around the origin)$=u^{m}$-coefficient of$(u+1)^{m+s}=(\begin{array}{l}m+sm\end{array})$

Now we assume that $m\geq 2$ , which implies that $m+1<2^{m}$ . Then it follows from
Theorem 3.1 (1) that $M$ does not admit an almost free involution which preserves the
SpinO-structure of $M$ if the number$(\begin{array}{l}m+sm\end{array})\iota\dot{s}$odd.

Example 3.5. Let $M=S^{6}$ be the 6-dimentional sphere with any almost complex stmcture. Note that any orientation-preser ving free involution has two fied points. Thendoes not admit any almost freesince $2<2^{m}=8$ , it follows from Theorem 3.1 (2) thatclearly
involution which preserves the almost complex structure. On the other hand,$or^{1}ientation$
admits an-poeseruing almost free involution defined by$S^{6}$$S^{6}$$\mathbb{R}^{7}\supset S^{6}\ni$$(x_{1}, \cdots, x_{6}, x_{7})arrow(-x_{1}, \cdots, -x_{6}, x_{7})$,which preserrves the unique SpinO-structure of . Note that the involution above has two
fixed points and that $Ind(D)$ is equal to 0because Ind(D)=\^A(TM)[M] is a Pontrjagin
number of .
$S^{6}$$S^{6}$

For p $=3,$ 5, we have the next theorem.
$where$is an odd
Thorem 3.6. Assume that $M$ admits an almost free action ofover $M$ .
prime number and that the action lifts to an action on a complex line bundle.
toLet be the distance fromdefined by, we have
Then for any real number such that$p$$\mathbb{Z}_{p}$

$L$

$d$

$\epsilon_{\frac{-1}{2}Ind(D_{L})}$$\gamma$$d= \min_{s\in}\mathrm{z}|sp-\mathrm{L}^{\underline{1}}Ind(2D_{L})|$$p\mathbb{Z}$$0\leq\gamma\leq d$$n \geq\frac{\gamma}{3(p-1)}(2\sin\frac{\pi}{p})^{m+1}$

Moreover if $\det(D_{L},g)=1$ , then we have$(\begin{array}{l}2\mathrm{s}\mathrm{i}\mathrm{n}\underline{\pi}p\end{array})m+1$$n \geq\frac{\gamma}{p-1}$

Proof Set
$K_{1}= \sum_{k=1}^{p-1}\frac{1}{1-\xi_{p}^{-k}}\{\mathrm{I}\mathrm{n}\mathrm{d}(D_{L},g^{2k})-2\mathrm{I}\mathrm{n}\mathrm{d}(D_{L},g^{k})\}$,$K_{2}= \sum_{k=1}^{p-1}\frac{1}{1-\xi_{p}^{-k}}$Ind$(D_{L},g^{k})$.

49Then since
Proposition 1.3 that$|1-\xi_{p}^{t}|\geq|1-\xi_{p}|$for any integer which is not amultiple of , it follows from
$t$

$p$

$|K_{1}| \leq\sum_{k=1}^{p-1}\sum_{j=1}^{n}\frac{1}{|1-\xi_{p}^{-k}|}\{\frac{1}{\prod_{i=1}^{m}|1-\xi_{p}^{-2k\tau_{j}}|}\dot{.}+2\frac{1}{\prod_{i=1}^{m}|1-\xi_{p}^{-k\tau_{j}}|}\dot{.}\}$$\leq\frac{3n(p-1)}{|1-\xi_{p}|^{m+1}}=\frac{3n(p-1)}{(2\sin\frac{\pi}{p})^{m+1}}$.

On the other hand, it follows from Theorem 1.2 (a) that
$2I(g)-I(g^{2})= \frac{p-1}{2p}\mathrm{I}\mathrm{n}\mathrm{d}(D_{L})+\frac{1}{p}K_{1}\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})$$\Leftrightarrow\frac{p-1}{2}\mathrm{I}\mathrm{n}\mathrm{d}(D_{L})+K_{1}\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d} p)$

Hence we have$|K_{1}|\geq\gamma$.

and therefore it follows that$m+1$$(\begin{array}{l}2\mathrm{s}\mathrm{i}\mathrm{n}\underline{\pi}p\end{array})$$\frac{3n(p-1)}{(2\sin\frac{\pi}{p})^{m+1}}\geq\gamma\Leftrightarrow n\geq\frac{\gamma}{3(p-1)}$

If $\det(D_{L}, g)=1$ , then we have$I(g)= \frac{p-1}{2p}\mathrm{I}\mathrm{n}\mathrm{d}(D_{L})-\frac{1}{p}K_{2}\equiv 0$

which implies that$|K_{2}|\geq\gamma$$( \mathrm{m}\mathrm{o}\mathrm{d} \mathbb{Z})\Leftrightarrow\frac{p-1}{2}\mathrm{I}\mathrm{n}\mathrm{d}(D_{L})-K_{2}\equiv 0$$(\mathrm{m}\mathrm{o}\mathrm{d} p)$,. Hence it follows from the same argument as above that$\gamma\leq|K_{2}|\leq\frac{n(p-1)}{(2\sin\frac{\pi}{p})^{m+1}}\Rightarrow n\geq\frac{\gamma}{p-1}(2\mathrm{s}.\mathrm{n}\frac{\pi}{p})^{m+1}$

口

Remark 3.7. Note that if M admits a free action of
p and hence $\gamma=0$ .

$\mathbb{Z}_{p}$, then $Ind(D_{L})$ is a multiple of

Example 3.8. Let
be the $m$ -dimensional complex projective space with theSpin-structure determined by the condition that $c_{1}(\eta)=(m+1+2s)x$ . As was seen in$M=\mathbb{C}\mathrm{P}^{m}$

Example 3.4, we have$Ind(D)=(\begin{array}{l}m+sm\end{array})$

and hence we can set $\gamma=1$ unless$(\begin{array}{l}m+sm\end{array})$

is a multiple of . Therefore it follows from Theorem 3.6 that$p$$3(m+1)(p-1) \geq(2\sin\frac{\pi}{p})^{m+1}$

if

$p$

is an odd

$pr\dot{v}me$

number and$(\begin{array}{l}m+sm\end{array})$

is not a multiple of . This inequality implies$p$that $M$ does not admit any almost free actions ofMoreover if $p=5$ andfollows that$M$$(\begin{array}{l}m+sm\end{array})\equiv 1$, 4$(\mathrm{m}\mathrm{o}\mathrm{d} 5)$$\mathbb{Z}_{3}$,$\mathbb{Z}_{5}$if $m\geq 6$ , $m\geq 37$ respectively.

, then we can set $\gamma=2$ and hence itdoes not admit any almost free actions of$\mathbb{Z}_{5}$if $m\geq 32$ .

50
be the $m$ -dimensional complex projective space with the
Example 3.9. Let
SpinO-structure detemined by the condition that $c_{1}(\eta)=(m+1+2s)x$ , an odd prime
number and $D(2p)$ the dihedral group generated by , with the relation in (8).
Then there eists an action of $D(2p)$ on $M$ defined by$M=\mathbb{C}\mathrm{P}^{m}$

$p$

$h$

$g$

$g$

$h$

if

:

$[z_{0}$::$[z_{0}$:$z_{1}$$z_{1}$:. .. : $z_{m}]arrow[z_{0}$ ::... : $z_{m}]arrow[z_{0}$ :$\xi_{p}z_{1}$$z_{m}$:... ::. .. ::$\xi^{\frac{m}{p^{2}}}z_{\frac{m}{2}}$$z_{\frac{m}{2}+1}$:$\xi_{p}^{p-\frac{m}{2}}z_{\frac{m}{2}+1}$$z_{\frac{m}{2}}$:. .. ::.. . :$\xi_{p}^{p-1}*]$,$z_{1}]$

is even, and

$m$

$g$

$h$

:

$[z_{0}$::$[z_{0}$:$z_{1}$$z_{1}$:. . . :$z_{m}]arrow[\xi_{p}z_{0}$:$\xi_{p}^{2}z_{1}$:. . . : $z_{m}]arrow[z_{m}$ :. . . ::. . . :$\xi^{\frac{m+1}{p2}}z_{\frac{m-1}{2}}$$z_{\frac{m-\backslash 1}{2}+1}$:$z_{\frac{m-1}{2}}$:$\xi_{p}^{p-\frac{m+1}{2}}z_{\frac{m-1}{2}+1}$:. . . :$z_{1}$::. . . :$\xi_{p}^{p-1}*]$,$z_{0}]$

is odd. Note that the action of $h$defined above is almost free if $m<p$ .
On the other hand, the same argument as in Example 2.3 shows that $\det(D_{L}, g)=1$
for any action of $D(2p)$ on M. Therefore as in the previous example, it follows from
Theorem 3.6 that the inequality

if

$m$

$=\langle g\rangle$$(m+1)(p-1) \geq\gamma(2\sin\frac{\pi}{p})^{m+1}$

holds if $M$ admits an almost free action of $h$$=\langle g\rangle$. If$(\begin{array}{l}m+sm\end{array})$

as not a multiple of ,$p$then we can set $\gamma=1$ and the inequality above implies that $M$ does not admit any actionis almost free if $p=3$ , $m\geq 3$ orof $D(2p)=\langle g, h\rangle$ such that the action of$\mathbb{Z}_{p}=\langle g\rangle$$p=5$ , $m\geq 29$ . Moreover

if $p=5$ and$(\begin{array}{l}m+sm\end{array})\equiv 1$, 4$(\mathrm{m}\mathrm{o}\mathrm{d} 5)$, then we can set $\gamma=2$

and the inequality above implies that $M$ does not admit any action of $D(10)=\langle g, h\rangle$ suchis almost free if $m\geq 23$ .
that the action of$\mathbb{Z}_{5}=\langle g\rangle$

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