数理解析研究所講究録 1290 巻 2002 年 48-53

48
Congruences for the Burnside module
和歌山高専

・藤田

一般科目

亮介 ( Ryousuke Fujita )General Education, Wakayama National College of Technology・・.

・.

Notation 1.
: finite group$S(G)$ : the set of all subgroups ofand -set by conjugation$\Phi(G)$ : the conjugacy class set of: partially ordered set and acts on it preserving the partially order$\rho:\Piarrow S(G)$ : an order preserving G-map
$G$

$G$

$G$

$G$

$G$

$\Pi$

Notation 2.
For any $ax\in\Pi$ ,.

.

$\Pi_{\alpha}:=\{\beta\in\Pi|\beta\geq\alpha\}$$G_{\alpha}:=\{g\in G|g\alpha=\alpha\}$Defnition 3.
is called aApair$(\Pi, \rho)$$G$-poset if it is satisfying the following condition: for any$\rho(\alpha)\triangleleft G_{\alpha}$

Note thatand$5(\mathrm{G})\mathrm{p}(\mathrm{a})=S(\rho(\alpha))\subset S(G_{\alpha})$$\rho:\Pi_{\alpha}arrow S(G)_{\rho(\alpha)}$and$G_{\alpha}\subset$$\alpha\in\Pi$,

is injective.

$\mathrm{p}(\mathrm{a})=NG(\rho(\alpha))$, the normalizer of$\rho(\alpha)$

in

$G$.

$\underline{\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}4.}$A$G$-poset$(\Pi, \rho)$

is called complete if$p:\Pi_{\alpha}arrow S(G)_{\rho(\alpha)}$is bijective for all $at\in\Pi$ .

Definition 5.
Afinite G-GW-complex $X$ with the base point is called a -complex if it is equipped with aof $X$ , satisfying the following four conditions:of subcomplexes
specified set(i),(ii)
for $g\in G$ ,$q$$\{X_{\alpha}|\alpha\in\Pi\}$$X_{\alpha}$$X_{\alpha}\ni q$$gX_{\alpha}=X_{g\alpha}$$\alpha\in\Pi$

$\Pi$

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(iii)
(iv) for any$X_{\alpha}\subseteqq X_{\beta}$

if$\alpha\leqq\beta$$H\in \mathrm{S}(\mathrm{G})$

in

$\Pi$

, and,$X^{H}=\vee X_{\alpha}\rho(\alpha)=H$(the wedge sum of$X_{\alpha}$’$\mathrm{s}$).

Example 6.

Let$\alpha\in\Pi$. The G-GW-complex$(G/ \rho(\alpha))^{+}(=G/\rho(\alpha)\prod\{*\})$$(G/ \rho(\alpha))_{\beta}^{+}=\{g\rho(\alpha)|.q\alpha\leq\beta\}\prod\{*\}$$\Rightarrow(G/\rho(\alpha))^{+}$

is a -complex;$\Pi$for any$\beta\in\Pi$.

is aIl-complex.

Definition 7. ([7])
Let $Z$ and $W$ be II-complexes.
$Z\sim W\Leftrightarrow\chi(Z_{\alpha})=\chi(W_{\alpha})$for all$\alpha\in\Pi$

The set$\Omega(G, \Pi)=${ $[Z]|Z$ is aIl-complex}

is called Burnside module .
$[Z]+[W]:=[Z\vee W]$

Remark that$\Rightarrow\Omega(G,\mathcal{F})$is afinitely generated free abelian group$(arrow \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}2)$

Notation 8.

.

$S((G), \alpha):=\{K\in S(G)|(K/\rho(\alpha))\in\Phi(G_{\alpha}/\rho(\alpha))$.

and$\overline{\chi}(X)=\chi(X)-1$ for any space $X$$K/\rho(\alpha)$

Theorem A.
Let at be an element ofThen we have$\phi(|K/\rho(\alpha)|)$$\ll \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}$

Let

$\Pi$

.

$\sum_{K\in S((G),\alpha)}\frac{|G_{\alpha}/\rho(\alpha)|}{|N_{G_{\alpha}/\rho(\alpha)}(K/\rho(\alpha))|}\cdot\phi(|K/\rho(\alpha)|)\cdot\overline{\chi}(X_{\alpha}^{K})\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} |G_{\alpha}/\rho(\alpha)|$is the number of generators of the cyclic groupof Theorem
be a -poset and$(\Pi, \rho)$

is cyclic}$K/\rho(\alpha)$, where.

$\mathrm{A}\gg$$G$$G_{\alpha}$

the isotropy subgroup at$\alpha$. Given a -complex $X$ , we see$\Pi$50-complex
is equipped with a -complex structure as following:
for all. By our definition of the Il-complex, it can be shown thatfor all
Let $\chi(X)$ be the Euler characteristic, and $\overline{\chi}(X)=\chi(X)-1$ Note that amap :satisfies aBurnside relation. By Burnside’s lemma [6, Lemma 4.1], we havethe desired result.

the$X^{\rho(\alpha)}$$G_{\alpha}/\rho(\alpha)- CW$$X_{\alpha}^{\rho(\alpha)}$$\Pi$$(X^{\rho(\alpha)})_{\alpha}=$$X_{\alpha}^{\rho(\alpha)}=X_{\alpha}$$\alpha\in\Pi$$\alpha$$f$$\mathrm{o}\mathrm{f}X$$\mathcal{F}_{c}(G_{\alpha}/\rho(\alpha))-$$\mathbb{Z};I\mathrm{f}/\rho(\alpha)\mapsto\overline{\chi}(X_{\alpha}^{K})$Lem
Suppose that a -poset
is complete. Let cx be an elemnet of. For a -complex $X$ , it holds that
$G$$(\Pi, \rho)$$\sum$$\overline{\chi}(X_{\alpha}^{K})=$$\beta\in\Pi$

Theorem B.
If a -poset

with$\overline{\chi}(X_{\beta})$.

$\rho(\beta)=Kfl<\alpha$is complete, one h&s$(\Pi, \rho)$$Im(\overline{\chi} :and $K$ asubgroup with$\Pi$

$K\supset\rho(\alpha)$

$G$

$\Pi$

\Omega(G, \Pi)arrow\oplus_{\alpha\in A}\mathbb{Z})$,$= \{(x_{\alpha})\in\oplus_{\alpha\in A}\mathbb{Z}|\sum_{K\in S((G),\alpha)}\frac{|G_{\alpha}/\rho(\alpha)|}{|N_{G_{\alpha}/\rho(\alpha)}(I\mathrm{f}/\rho(\alpha))|}\cdot\phi(|K/\rho(\alpha)|)\cdot x_{\alpha,(K)}\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} |G_{\alpha}/\rho(\alpha)|\}$

where$x_{\alpha,(K)}$is some integer such that$x_{\alpha,(K)}=\{$$x_{\alpha}$$(K=\rho(\alpha))$$\sum_{\beta}x_{\beta}$($I\mathrm{f}\neq p(\alpha)$,$\beta$

is some element of$p(\beta)=K,\beta<\alpha)$

$\Pi$

with.

of ProoQ)
First we use for the right side, andfor the left side in the equation of Theorem B. Let. By [4, Lemma 1.80], we can arrange elements of $A$ such that$\ll \mathrm{O}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}$${\rm Im}$$\mathrm{S}$$A=\{\alpha_{1}, \cdots, \alpha_{m}\}$$\alpha_{i}\leq\alpha_{j}\Rightarrow i\leq j$

Define amap

$P_{\leq k}$::by$\oplus \mathbb{Z}_{\alpha}m:arrow\oplus^{k}\mathbb{Z}_{\alpha}$$\dot{l}=1$$k$.

coordinate maps $n$. :$\oplus \mathbb{Z}_{\alpha}m:arrow \mathbb{Z}_{\alpha_{l}}$such that$i=1$$.\cdot=1$$P_{\leq k}(x)=(p_{1}(x), \cdots,p_{k}(x))$

where each Za. is acopy of Z. Note that$\mathrm{S}\subset\bigoplus_{i=1}^{m}\mathbb{Z}_{\alpha:}$,. It will now suffice to prove that

$P_{\leq m}(\mathrm{S})=P\leq m({\rm Im})$.

$[$

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We proceed by induction on .
the case of $k=1$
By Theorem Aand the previous Lemma, we have that.
(Irn)
the case of $k=m$
Remark that. (previous Lemma) Suppose that
We prove.

$k$.

$=P_{\leq 1}(\mathrm{S})$$P_{\leq 1}$$P_{\leq m}({\rm Im})\subset P_{\leq m}(\mathrm{S})$$P_{\leq k-1}(\mathrm{S})=P_{\leq k-1}({\rm Im})$$arrow \mathrm{t}\mathrm{h}\mathrm{e}$$P_{\leq k}(\mathrm{S})\subset P_{\leq k}({\rm Im})$

Suppose that(Irn). Letelement of S. By assumption, there exists an element$P_{\leq k-1}(\mathrm{S})=P_{\leq k-1}$$y=(y_{\alpha_{1}}, \mathrm{y}\mathrm{a}\mathrm{i}, \cdots, y_{\alpha_{k-1}}, \mathrm{y}\mathrm{a}\mathrm{i}, \tau/\alpha_{k+1}’\ldots, y_{\alpha_{m}}.)$

be an$x=(x_{\alpha_{1}},x_{\alpha_{2}}, \cdots,x_{\alpha_{k-1}}.,x_{\alpha_{k}}, x_{\alpha_{k+1}}., \cdots,x_{\alpha_{m}})\in{\rm Im}$

such that$x_{\alpha_{1}}=2/\mathrm{a}2$,$x_{\alpha_{2}}=\mathrm{y}\mathrm{a}\mathrm{i}$$z=y-x=$Here we let$\alpha=\alpha_{k}$,$\cdots$,$x_{\alpha_{k-1}}=\mathrm{t}/\alpha_{k-1}$. Then we have$(0, 0, \cdots, 0,y_{\alpha_{k}}-x_{\alpha_{k}},y_{\alpha_{k+1}}-x_{\alpha_{k+1}}, \cdots,y_{\alpha_{m}}-x_{\alpha_{m}})\in \mathrm{S}$$z_{\alpha}.\cdot=y_{*}-x_{\alpha}:$’and$n_{\alpha,K}= \frac{|G_{\alpha}/\rho(\alpha)|}{|N_{G_{a}/\rho(\alpha)}(K/\rho(\alpha))|}\cdot\phi(|K/\rho(\alpha)|)$.

. Consider the case of. Then we haveI$n_{\alpha_{k},K}.\cdot z_{\alpha_{k},(K)}\equiv 0$$\mathrm{m}\mathrm{o}\mathrm{d} |G_{\alpha_{k}}/\rho(\alpha_{k})|$.

$K\in S((G),\alpha_{k})$Observe that the coefficientwith $\rho(\beta)=K$,$\beta<\alpha_{k}$$z_{\alpha_{k},(K)}(K\neq\rho(\alpha_{k}))$is equal to, where$\beta$is some element of. Thus the above equation implies$n_{\alpha_{k\prime}\rho(\alpha_{k})}.\cdot z_{\alpha_{k},(\rho(\alpha_{k}))}\equiv 0$

Note that$\sum_{\beta}z_{\beta}$$\mathrm{m}\mathrm{o}\mathrm{d} |G_{\alpha_{k}}/\rho(\alpha_{k})|$.
. that is,$n_{\alpha_{k},\rho(\alpha_{k})}= \frac{|G_{\alpha_{k}}/\rho(\alpha_{k})|}{|N_{G_{\alpha_{k}}/\rho(\alpha_{k})}(\rho(\alpha_{k})/\rho(\alpha_{k}))|}\cdot\phi(|\rho(\alpha_{k})/\rho(\alpha_{k})|)=1$$z_{\alpha_{k}}\equiv 0$$\mathrm{m}\mathrm{o}\mathrm{d} |G_{\alpha_{k}}/\rho(\alpha_{k})|$On the other hand, we have$\alpha\in A\oplus\overline{\chi}([(G/\rho(\alpha_{k}))^{+}])=(\overline{\chi}_{\alpha}([(G/\rho(\alpha_{k}))^{+}])_{\alpha\in A}=$

Hence there exists an integer$a\in \mathbb{Z}$such that$y-x-a(\overline{\chi}_{\alpha}((G/\rho(\alpha_{k}))^{+}))=$

.

$\Pi$

52

y

$=x+a(\overline{\chi}_{\alpha}((G/\rho(\alpha_{k}))^{+}))+$

By induction, we see immediately that
$P_{\leq k}(y)=P_{\leq k}(x+a(\overline{\chi}_{\alpha}((G/\rho(\alpha_{k}))^{+}))\in P_{\leq k}({\rm Im})$.

Example 9
Let be aprime number. We set $G=C_{p}$ (a cyclic group of order ). Since $S(G)=\{\{e\}, G\}(e$is the unit element of ), and the -action on $S(G)$ is trivial, aBurnside module $\Omega(G, S(G))$ isafree abelian group generated by $[(G/\{e\})^{+}]$ , $[(G/G)^{+}]$ . Clearly $\Phi(G)=\{\{e\}, G\}$ .
First, consider the case of $\alpha=\{e\}$ . Since $S((G), \alpha)=\{\{e\}, G\}$ , we get
$p$

$p$

$G$

$G$

$\frac{|G|}{|G|}\cdot 1\cdot x_{\{e\rangle,(\{e\})}+\frac{|G|}{|G|}\cdot$$(p-1)\cdot x\{e\},(G)\equiv 0$$\mathrm{m}\mathrm{o}\mathrm{d} p$.

That is,.

$\mathrm{m}\mathrm{o}\mathrm{d} p$$x_{\{e\},(\{e\rangle)}\equiv x_{\{e\},(G)}$By Theorem 1.7, there exists a -complex $X$ such that
Thus we have$\Pi$$\overline{\chi}(X_{\{e\}})\equiv\overline{\chi}(X_{G})$$\overline{\chi}(X\{e\})=x\{e\},(\{e\})$$\mathrm{m}\mathrm{o}\mathrm{d} p$and$\overline{\chi}(X_{G})=x\{e\},(G)$.

In particular, if $X$ has a -complex structure as follows:
$\Pi$

$X$

$(\alpha=\{e\})$$X^{G}$$(\alpha=G)$$X_{\alpha}=\{$,

the previous expression implies$\chi.(X)\equiv\chi(X^{G})$$\mathrm{m}\mathrm{o}\mathrm{d} p$.

Next for $\alpha=G$ , since $S((G), \alpha)=\{G\}$ , we obtain$\frac{1}{1}\cdot 1\cdot \mathrm{X}\mathrm{G},(\mathrm{G})\equiv 0$$\mathrm{m}\mathrm{o}\mathrm{d} 1$.

Immediately,$x_{G,(G)}\equiv 0$$\mathrm{m}\mathrm{o}\mathrm{d} 1$.

This equation is true for any integer, and so there is no relation for Il-complexes.

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