BROWN-PETERSON COHOMOLOGY OF $BPU(p)$(Cohomology Theory of Finite Groups and Related Topics)

103

BROWN-PETERSON

COHOMOLOGY OF $BPU(p)$

MASAKI KAMEKO (亀子正喜 富山国際大学地域学科)

NOBUAKI YAGITA(柳田伸顕 茨城大学教育学部)

Let$p$be afixedoddprimearid denoteby$BP^{*}(X)$ ( resp.

$P(m)^{*}(X)$) the

Brovvn-Peterson cohomology ofa space $X$ with the coefficient ring $BP^{*}=\mathbb{Z}(p)[v_{1}, v\mathrm{z}, \cdots]$

(resp. $P(m)^{*}=\mathbb{Z}/p[v_{m},$ $v_{m+1}$, $\cdots$]) where $\deg v_{k}=-2p^{k}+2$

.

We denote by PU(n) the projective unitary group which is the quotient of the unitary

group

$U(n)$ by its center $S^{1}$. Recall that the cohomologies of PU(p) and exceptional

Lie

groups

$F_{4}$,$E_{6)}E_{7}$,$E_{8}$ have odd torsion elements. In this

paper,

we compute

the Brown-Peterson cohomologies of classifying spaces $BG$ of these Lie groups $G$

as $BP^{*}$-modules using the Adams spectral sequence. Let us write $H^{*}(X;\mathbb{Z}/p)$ by

simply $H^{*}(X)$ and let $A$be the mod$p$

Steenrod

algebra. Our main result is as follows:

Theorem 0.1. Let $(G, p)$ _be

one

of

eases

$(G=PU(p),p)$

for

an arbitrary odd

prime $p$ and $G=F_{4}$,$E_{7}$

for

$p=3$, and $G=$

Es

for

$p=5$. Then the E2-te

rm$

of

the Adams spectral sequences abutting to $BP^{*}(BG)$ and$P(m)^{*}(BG)$

for

$m\geq 1$

$\mathrm{E}\mathrm{x}\mathrm{t}_{A}^{s}’{}^{t}(H^{*}(BP), H^{*}(BG))$, $\mathrm{E}\mathrm{x}\mathrm{t}_{A}^{s}’{}^{t}(H^{*}(P(m)),H^{*}(BG))$

have

no

odd degree elements.

As an immediate consequence is as follows:

Corollary 0,2. For $(G, p)$ _{in Theorem 1.1,} the Adams spectral sequence abutting

to $BP_{\backslash }^{*(}BG$) and$P(m)^{*}(BG)$ in theprevious theorem collapse at the

$E_{2}$-levet. In

particular BP $(BG)=P(m)^{odd}(BG)=0$

.

Recall If$(m)^{*}(X)\cong K(m)^{*}\otimes_{P(m)}\cdot$ $P(m)^{*}(X)$ is the Morava if-theory. From

above theorem and corollary, we see $K(m)^{odd}(BPU(p))=0$

.

Then we have the

following corollary ([Ko-Ya],[Ra-Wi-Ya])

Corollary 0.3. For (G, p) in Theorem 1.1 the followingholds:

$(’\mathit{1})BP^{*}(BG)$ is $BP^{*}$

-flat

for

$BP^{*}(BP)$-modules, $\iota.e.$,

$BP*$ $(BG\}< X)\cong BP*(BG)\otimes_{BP}\cdot BP^{*}(X)$

for

all

finite

complexes $X$

(2) If$(n)^{*}$ $(BG)\cong I_{\dot{\mathrm{L}}}^{r}4(n)^{*}\otimes_{BP}\cdot BP^{*}(BG)$.

(3) $P(n)^{*}(BG)\cong P(n)^{*}\otimes_{BP^{*}}BP^{*}(BG)$

.

We give the $BP^{*}$

-module structure

of$BP^{*}(BPU(p))$

more

explicitly, in this

talk.

Theorem 0.4. There is a $BP^{*}$-algebro isomorphism

$0arrow BP^{*}\otimes M\wedgearrow grBP^{*}(BPU(p))arrow BP^{*}\otimes IN/\wedge(f_{0}, f_{1})arrow 0$

where

1. $M$;$\mathbb{Z}_{(p)}[x_{4}, x_{6}, \cdots)x_{2p}]a\mathit{8}\mathbb{Z}_{(p)^{-}}modules$ (but not$\mathbb{Z}_{(p)}-$ algebras).

2, $IN\cong \mathbb{Z}_{(p)}[x_{2p+2}, x_{2p(p-1)}]\{x_{2p+2}\}\mathrm{i}$ theprincipal ideal

of

$\mathbb{Z}[x_{2_{P}+2}, x_{2p\langle p-1)}]$

generated

by$x_{2p+}\mathrm{z}$

.

104

MASAKI KAMEKO ($\mathrm{f}\mathrm{l}\mp \mathrm{i}\mathrm{E}\mathrm{B}$ $\mathrm{g}\mathfrak{U}\mathrm{E}\ovalbox{\tt\small REJECT}\lambda \mathrm{r}*\mathrm{f}\mathrm{f}\mathrm{l}\mathfrak{B}\mu\neq \mathrm{f}41$ NOBUAKI YAGITA$(\Phi \mathrm{E}l*\mathit{5}\Xi \mathrm{f}\mathrm{f}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{l}\lambda\infty\neq \mathrm{a}\mathrm{e}\Leftrightarrow\infty\neq\oplus)$

3. relations$f_{0}$,$f_{1}$ are given with modulo (p,$v_{1}, v_{2}, \cdots)^{2}$

$f_{0}\equiv v_{0}-v_{2}x_{2p+2}^{\mathrm{p}-1}+\cdots$ , _{$f_{1}\equiv v_{1}-v_{2}x_{2p(p-1)}+\cdots$}

.

Remark 0.5. In the above theorem,

_suffix

$\mathrm{i}$

of

$x_{i}$ means its degree. $BP^{*}(BPU(p))$ does not contain the subalgebra $BP^{*}\otimes \mathbb{Z}(p)[x_{4}\wedge$

,

.

. .

,

$x_{2p}]_{\mathit{1}}$ but contains a subalgebra

which is isomorphic as $BP^{*}$-modules to the above BP-subalgebra.

For an algeb raic group $G$ over $\mathbb{C}$, Totaro defines its Chow ring [To]

and con-jectures that $BP^{*}(BG)\otimes_{BP^{\alpha}}\mathbb{Z}_{(p)}\cong GH^{*}(BG)_{(p)}$

.

Recall that $PGL(p, \mathbb{C})$ is the

algebraic group over $\mathbb{C}$ corresponding the Lie group PU(p).

Theorem 0.6. There is the isomorphism

$BP^{*}(BPU(p))$ $\otimes_{BP}\cdot$ $\mathbb{Z}_{\langle p)}\cong GH^{*}(BGL(p, \mathbb{C}))_{(p)}$.

Hence there is the additive isomorphism

$CH^{*}(BGL(p, \mathbb{C}))_{(p)}\cong \mathbb{Z}_{\{p)}[x_{4}, x_{6}, \cdots, x_{2p}]\oplus \mathrm{F}_{p}[x_{2p+2}, x_{2p(p-1\rangle}]\{x_{2p+2}\}$.

Remark. Recently Vistoli [Vi] also determined the additive structure of the Chowring andintegral cohomology ofBPGL$(p, \mathrm{F}_{p})$ by using stratified methods of

Vessozi. Moreover he shows that for $G=PGL(p, \mathbb{C})$

$H^{*}(G;\mathbb{Z})arrow H^{*}(BT;\mathbb{Z})^{W_{G}(T)}$

1s epic.

Let $MGL^{**},(X)$ bethemotiviccobordismringdefinedbyV.Voevodsky [Vo] and

$MGL^{2*,*}(X)=\oplus_{i}MGL^{2i,i}(X)$.

Corollary 0.7. $MGL^{2*_{7}*}(BPGL(p, \mathbb{C}))_{(p)}\cong MU^{*}(B$PU$(p))_{(p)}$

.

We prove Theorem 1.1 using the Adams spectral sequence converging to the Browen-Petersoncohomology. The $E_{1}$-termofthe spectral sequence could be given

by

$\mathrm{F}_{p}[v_{0}, v_{1}, \cdots]\otimes H^{*}(X)\wedge$ with $d_{1}x= \sum_{k=0}^{\infty}v_{k}Q_{k}x$

where Qk’s are Milnor’s operations. By the change-of-rings isomorphism, the $E_{2^{-}}$

term is

$\mathrm{E}\mathrm{x}\mathrm{t}_{A}$($H$”(BP),$H^{*}(X)$) $\cong$ $\mathrm{E}\mathrm{x}\mathrm{t}_{\mathcal{E}}(\mathrm{F}_{p}{}_{:}H^{*}(X))$

where $\mathcal{E}=\Lambda’(Q_{0}, Q_{1}, \cdots)$

.

The $E_{\infty}$-term is given by $grBP^{*}(X)$

.

To state the cohomology $H^{*}(BPU(p))$, we recall the Dickson algebra. Let $A_{n}$

be an elementary abelian $p$-group ofrank $n$

,

and

$H^{*}(BA_{n})\cong \mathrm{F}_{p}[t_{1,\}}\ldots t_{n}]\otimes\Lambda(dt_{1}, \ldots, dt_{n})$ with $\beta(dt_{i})=t_{i}$

.

The Dickson algebra is

$D_{n}=]\mathrm{F}_{p}[t_{1}, \ldots,t_{n}]^{GL(n,\mathrm{F}_{p}\}}\cong \mathrm{F}_{p}[c_{n,0}, \ldots, c_{n,n-1}]$

with $|c_{n_{J}i}|=2(p^{n}-p^{j})$. The

invariant

ring under $SL(n,\mathrm{F}_{P})$ is also given

$SD_{n}=\mathrm{F}_{p}[t1, \ldots, t_{n}]^{SL(n,\mathrm{F}_{p})}\cong D_{n}\{1, e_{n}, \ldots, e_{n}^{p-2}\}$ with $e_{n}^{p-1}=\mathrm{c}_{n,0}$

.

We also recall the Mui’s ([Mu]) result by using $Q_{i}$ by [Ka-Mi]

$grH^{*}(BA)^{SL_{n}(\mathrm{N}_{p}^{i})}\cong SD_{n}/(e_{n})\oplus SD_{n}\otimes\Lambda(Q_{0}, \ldots, Q_{n-1})\{u_{n}\}$

105

BROWN-PETERSON COHOMOLOGY OF $BPU(p)$

Theorem 0.8. There is the short exact sequence

$\mathrm{O}arrow M/parrow H$’ BPU(p) $arrow \mathrm{N}arrow \mathrm{O}$

wftere

$M/p$ is the trivial$\mathcal{E}$-module

given in Theorem 1,4 and

$N=SD_{2}$ $(\otimes \mathrm{A}(\mathrm{Q}0, \mathrm{Q}\mathrm{i})\{\mathrm{u}2\}\cong \mathrm{F}_{p}[x_{2p+2}, x_{2(p^{2}-p)}]\otimes\Lambda(Q_{0}, Q_{1})\{u_{2}\}$

identifying $x_{2_{P}+2}=e_{2}$ and $x_{2(p^{2}-p)}=c_{2,1}$

.

This theorem is proved by using the following facts. The group $G=PU(p)$

has just two conjugacy classes ofmaximal elementary abelian $p$-subgroups, one of

whichis toral and the other is non-toral$A$of_{$rank_{p}=2$}

.

The cohomology $H^{*}(BG)$

is detected by thistwo subgroups. Therestriction image to the non-toral subgroup is $\mathrm{i}_{A}^{*}(H^{*}PU(p))\cong H^{*}(BA)^{SL(2,\mathrm{F}_{p})}$

.

Similar (but not same) facts also hold for the

exceptional Lie groups given in Theorem 1.1. Algebraic mainresult in this talk is as follows: Theorem 0.9. Form $\geq 0$,

define

$f_{0}$,

\ldots ,$f_{n-1}$ in $P(m)^{*}\otimes SD_{n}\wedge$ by

$d_{1}u_{n}= \sum_{h\geq m}v_{k}Q_{k}(u_{n})=f_{0}Q_{0}u_{n}+\cdot$ . .

$+f_{n-1}Q_{n-1}u_{n}$

.

Then the sequence $f_{0t}\ldots$ , $f_{n-1}$ is a regufar sequence in $P(m)^{*}\otimes SD_{n}\wedge$

.

With the notation in this theorem, we prove that the complex

$C=(P(m)^{*}\otimes SD_{n}\otimes\Lambda(Q_{0}, Q_{1}\wedge, \cdots, Q_{n-1})\{u_{n}\}, d_{1})$

with the differential $d_{1}u_{n}= \sum_{i=0}^{n-1}f_{\mathrm{i}}Q_{i}u_{n}$ is a Koszul complex. This means that

$H_{i}(C, d_{1})=\{$

$P(m)^{*}\otimes SD_{n}\{e_{n}\}\wedge/(f_{0}, \cdots, f_{\mathrm{n}-1})$

for

$i=0$ 0

_for

$i\geq 1$

.

Thus Theorem 1.1 follows from the above theorem.

Remark about the convergence

of

the Adams spectral sequence. By Theorem

15.6 in

Boardman’s

paper [Bo2], since $H^{*}(BP)$ is offinite type, the above Adams

spectral sequence is conditionally convergent. Moreover, since we prove the above Adams spectral sequence coilapses at the $E_{2}$-level, by the remark after Theorem

7.1

in [Bol], the above Adams spectral sequence is strongly convergent

so

that we know the Brown-Peterson cohomology up to group extension.

$\mathrm{R}:3\mathrm{F}\mathrm{E}\mathrm{R}\mathrm{E}\mathrm{N}\mathrm{C}\mathrm{E}\mathrm{S}$

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108

MASAKI KAMBKO (亀子正喜 富山国際大学地域学科) NOBUAKI YAGITA(柳田伸顕 茨城大学教育学部)

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Faculty of Regional science, Toyama University of Internal Studies, Toyama,

Japan

Faculty ofEducation, Ibaraki University, Mito, Ibaraki, Japan