Note on the β family in the homotopy of spheres

J Fac Educ Tottori Un (Nat Sci),45(1996)103-110

Note on the F family in the homOtoPy of Spheres

Katsumi SHIMo

RRA*

(R9cが υθ′ И″9クsι Br,r996)

§

1・ IntrOductiom

ln thc stablc homotopy groups of sphcrcs,thcrc arc _α,β

_and

_{γ familics, Thcsc}

ta■ lies arc now rccognized as υ_l―, υ

2 and

υ3‐periodic maps, rcspectivcly. In this

sensc, thcre secms to bc δ fanlily, or υ_{4 periodic maps, but so far wc havc no proof} of thc c

stence.Thc

α family is closcly related to thc rmデ for J:π

_*(SO)→

冗_*(S°),

and 11■ デ

is now wem undcrstood. Our next problc■ l is to understand Cokcr

デ.

Thcy consists of υ “

―periodic maps for η>1, and our flrst targct must bc υ2‐pcriodic

maps,say,thc

_β

_{family.Thc c}

_{stcnce of thc β}

_{family shows that the Cokcr J in}

thc homotopy groups of sphcrcs have nOn― zcro groups at a di14CnSiOn greater than

any largc dilncnsion. Besidcs, products of thcm add morc clcments in thc 4‐th linc of thc Adams‐ Novikov spectral scquencc. This shows the complcxity of thc groups. For cxamplc, wc havc not detcr江 ned the ring structure of thc sllbring of the stablc

homotopy groups of sphcres gcncratcd by

_β

_{family,Note that wc know he ring}

structurc of the subring gcncrated by α fan ly. O ginally,thcse fanlihcs arc obtained by constructing a suitablc spcctruna, but aftcr thcir ccrcbratcd paper[4], thcSC CIC‐

mcnts arc charactcrized by thc words of thc Adams‐Novikov spcctral scquence. This idca lcads to mOrc gcncral chromatic thcory in thc stable homotopy.

Hcrc we discuss how much vc havc known about thc

_β

_{ttmily,and so l make}

no dailns of originality.

§

2. Dennition of the′

family

First β―

ClCIncnts tt arc intrOduccd by Toda[25]for O<ι

<P in thc p―

pttmary

component of thc stablc homotOpy of spheres ior an odd prilne p. Thcn Lo Snlith [24]dcaned tt fOr anyす

>O by g

ing an essential sclμ

map

_β

:ガ2ク

227(1)→

_7(1)fOr

a prime p≧

3.Herc 7傲

_)for

η≧

-l arc the so―callcd Toda‐ Smith spcctrじ

m dcnncd

inductivcly by 7(-1)=S° and thc conbcr sequcnccs

(2.1)7(-1)47(-1)47(O14ガ 7(_1), andガ

222710147(0)二

7(1)ニ

ガ

2,17(0).

Herc P c Z tt π。(S°

)and

α

is an csscntial map known as Adams map.Using thcsc

notation,at thc primc p>3,β clcmCnts arc dcined by

(2.2)

虎

=″

lβ`'1'.

*Faculty Of Education,Tottori University, TOtto

Katsumi SHIMoMuRA

Wc also considcr thc

_β

_{fhmily in the track groups[7(0),7(0)]*:}

(23)

β

(r)=ブlβr'1.

Then

島

=′

_白っと

。

I It iS uscful to usc thc languagc of BP,the Brown―

Pctcrson spcctrum at a p

mc

1 4umbCr p・

Thc cocmcicnt ring of thc Brown―

Pcterson spcctrum】

P is

l B∴

=】桑

(S°

)=π *(BP)=Zl〕

[υl,υ2,…

・

]

i for gcncrators

υ

“￨=2p“ -2. Using thc ring structurc of the Brown― Petcrson spcc―

trunl wc can construct an Adams‐ typc spcctral scqucncc, which we call thc Adams― Novikov spcctral scqucncc:

E労

ど

_(ズ

_)=Ext≧

_´

_*ω

_⊃

_(】

P.,BP.(ズ

_))⇒

π

_*(ズ)

lor a p-local spcctrum ズ

. Hcrc thc Ext group is thc dcrivcd functor of Hom on

BP.(BP)‐COmOdules,where B=*(BP)iS a HOpf algcbroid associatcd to the ring spcc―

trum BP,By ttis,for a conbcr scqucnce/ム

r4z Ofρ

_10cal spcctra with漁

₌

B亀

(デ

)monOmorphic,wc havc a long cxact sequencc

・…→

Eら_(ズ)与

EL(7)各

EL(Z)4Eデ

1(ズ_)→

…

・

.

Therefore,the conber scqucnccs of(2.1)yiCld the connccting homomorphisms

l

δ

:Eら_(7(0))→Eす

1(7(-1))=】

す

1(S°

) and

δ

:EL(7(1))→ Eゴ1(7(0)).

Composing these, we obtain a map

η

=δ

δ

:Eら(7(1))→ Eす2(sO).

H,Miller,D.Ravenel and S.Wilson[4]dcancd

_β

family in thc E2 tCrm of thc

Adams―

Novikov spcctral sequencc by

(2.4)

島

=η

(υ

ケ

)∈ E二 (S°

) and plサ

)=δ

(υ

ち

)∈ E】(7(0)).

Hcrc notc thc fonowing

LEMMA 2.5.T力 9】

θttθηι_υち 'Sα

じοじノC'9,ι力αヶ,s,テι,s,乃 E9(7(1)).

PROOF,Notc that BPx(7(1))=B4/1p,υ

_{l)and rCCali thc Landwcbcr's formula}

ηR(υ

2)=υ 2mOd(p,υ l).BCSidcs,x∈

ExtBP*伊

D(BPx,BP.AP,υ

l))if and Only if ηR(χ

)=

l

χ mod(p,υ

_{l),by the dcanition of thc Ext groups.Thus thc lcmma fonows, q.c.d.}

Silnilarly, wc obtain thc followlngi

l LEMMA 2.6.([4])И

_{ι ヶ力ι} prヵη9猾 "附bθ

r p>2,ι

力θ ぞ′θ阿ぞηιχ

,,sα

じοCノC'9Q′

ExtBPtlBD(BP.,BP.xp;υ

_f))√ p'11デ ≦ α “ _:+1沃ノ 猾 ≧

O

αη冴 ら ブ

>0.

Note on the F Family in the homotopy oF sphcrcs

Here the integcr α

“

is dcincd by

α。

=l and

α打

=p"十

p打

1+l for

η

>0,

and thc clcmcnt

_{χ. satisncs}

χ

.=υ

ダ

mod(P,υ

l), which is dcaned inductivcly by

XO=υ

_{2' Xl=υ}

_{】― υ}

:υ

テ

lυ 3;

χ

_2=χ

_{f υ}

:2_lυ

】

2P+1_υ

:2+コ

ー

lυ3'

χ

.=Xμ

l-2υ

?

υ

ダ

P「

1+l for

η≧

3

With仇 =(p+1)(p“

1-1)fOr

η>1. At thc primc 2,wc havc anothcr

LEMMA 2,7.([16])И

ヶι力θ _prヵηゼη傷胞

b9r 2,

ι力ι】ι “ ιηιχ_{, 'd α Cο じノ}Cケι _q′ ExtBP*(BP.,BP.X2を υ_f))ア p:11ブ _{≦ αヵ}_,キ

1カ

rη ≧

0,η

冴 らブ

>0,Fク

rιルrttοr9,χ

,ぬ

αε09Cι_{θ げ ExtBP半 伊}

_D(B亀

,BP./(グ+2,ノ″))√ 2協 ≦ α刀

_,1)/η

>テ

+2

αη冴 附,テ >0。 IIcre

ノo=υ

l'

ノ1=υ ♀

-4υ

_『

lυ

_{2' and}

ノ

:=ノ

盈

l fOr J>1.

Consider thc short exact sequenccs:

0→

】

P.4】

Px→

BPx/(pり

→

0,

0→

_{】亀 Д}pり

4BP./(pり

→

BttXPを

_υf)→

O fOrブ =れ

p'l Withブ

≦ α “

_,+1,and

O→

BPx/(グ+2)雪BPx/(グ+2)→

B4/(グ

+2,ガ

_)→ O fOr ttpι ≦ α “ _ヶ_l with η

>,キ

2.

We dcnotc thc connccting homomorphisms assOciatcd to this short exact scqucnccs by

δど:Extら_P*ωD(】Px,】 P./(pり_)→

Ext既

_伊

_D(BPx,】

P.),

ェ,I ExtらP.lBD(B亀,BPx/(ptt υl))→

Ext釜

_§

D(BPx,】

P./(pり

), and

δ岳,1:Ext≧P*伊

D(B4,BP.Xグ

+2,ノr))→Ex娼

=lω

D(】P.,】 PxX2'+2)). Using thcsc rcsultsぅ _{thcy dcnncd thc gcncral}

_β‐

CICmcntsI

(2.8) APれ

ガ,i=δ:牡,(χ ,Pれ)∈Extζ

P.lBD(BP.,B4), and

ん れsP坊,,+2=δ,+2弘,,(χ,P流)∈Extζ P*ωD(】

4,BR∂

・

§3. E

stence of the F family

l For thc

β

clements denncd in thc E2 tCrm of thc Adams‐

No

kov spcctral se―

l qucncc, wc haVe not yct dcterimined which detects an essential homotopy elcmcnt.

IIere 、

vc writc down somc kno、 vn rcsults. In ordcr to explain these by modcrn

Katsumi SmMoMuRA

languagc,wc introducc thc lorava K― thcory. For cach non― negativc intcger η_,thcrc

is a hOmology thcory K(■)*(―)Witt COemcicnt ttng K(η

)*=K(D*(S° )=Z/PEυ

.,υ_「 1]

for

η

>O and K(0)*=2,WC Call a p…

local a

tc spCCtrumズ

a typc

η

spcctrum

if

η

is the smallest integcr such that K(づ _*(ズ)≠

0.Ifズ

iS Contractiblc,thcn/has

typc∞

.Thc Toda―

Smith spcctrum 7(η_{)is a typical example of typc乃}

+l spcctrum

if it exists. 7(0)=′

r is thc mOd p

foorc spcctrum,and 7(1)iS a cOnbcr of thc

gcncrator α

of[7(0),7(0)]27-2 fOr an odd primc p.Thcn in[24],Smih gavc an

esscntial self―

map

_β

:Σ2″2_27(1)→

7(1)fOr p>3.A conbcr of β

iS dCnOtcd by 7(2)。

So far,it is known that thc Toda― Smith spcctrum 7(η _{)cXiStS if and only if 2η}

<p

for η

<4([28],[14]).On type

η spcctra,wc havc thc following

THEOREM 3.1.(HOpkins and Smith[1])L防

ズ bθ α _p‐Jοcαケヶ_ノpι _{η ヵη}

'ι θ・Spιc― ιr"陶

.Tル

η サル

r"s,scJ/―

れ

,P/:ガ '/→

χ S,c力 す力αι K(め_*(デ)'S,η 'Sο ttOrpttdれ αη冴 K(“)*(デ)'S ιr′υ 'α`沃

/陶

>η

.河

θrι 冴

=0リ

カぞ乃 η

=Oα

乃冴 冴 ね α 胞レJヶ_{げ θげ} 2p“

-2.

Thc mapデ

of thiS thcorcm is callcd

υ

_刀

―

map.Thc maps pク α

_{and β}

abOVC arc

υ

O‐,91‐

and υ

2 maps,respect

cly.Using thc υ

2 map β,WC Can dcanc β

family(2.2)

in thc homotopy groups of

π

_*(S°

_{) Lct}

(3.2)

χ′→ χ → ズ〃

be a conbcr scqucncc, and lct

∂

:π

よズ

〃

_)→

π

_{,_1(ズ}

′

₎

bc thc associated “

geometric" boundary homomorphisna, induccd from

力:】イー→,ど勇ζ′.

Suppose that BP.(力

_{)=0.ThCn(3.2)induCCS a short exact scqucncc of】}

P.―

homology

and thcn thc boundary homomorphism

δ:Ext≧_{P.lBD(BPx,BP.(ズ}〃_))→

Ext講

_ゎ

_{D(B亀 ,B民}

_(ズ′_)),

which is

ln fact, thc E2‐ tC■

n of the

Ext≧_P.lBD(BP.,β

Px(〃

))。

THEOREM 3.3.(Gcomctic Boundary Theorem [2])丁

ア ∈_{Eら (/〃)S,rυ} 'υ 9Sケο

χ∈冗

_{*∈Y″ ),す}

力

9乃

δ

_(⊃

∈

Eす1(ズ

′

_)sゲ

υ

'υ gSケ

ο∂

(χ)∈

π

*(χ

′

).

By virtue of this thcorcnl, we havc thc following

THEOREM 3.4.L9ヶ

_{P>3. T力}

9 9′ι “ 9肪

s

_{島 ∈}E二(S°)σ 'υ ιη _{加 (2.4)s"rυ} 'υ ι

S

ιο ん ∈π*(S°)'力 (2.2).

Wc want to provc similar thcorcl■ for the clc■

lcnts of(2.8),but WC knOw thc

following noncxistence thcorcms:

THEOREM 3.5,([13])と

す_{P bι σ′θα}ttθrヶ力α冷

2. T力

ιη 才んι 】θ翻?ηお ィ_{ЪvPi'η (2.8)}

冴ο ηοtt s,7υ

'υ

9 ιθ 力Ottοヶοpノ _じ修ユηιηιs.

δ

:EL(ズ

〃

_)→Eす1(ズ

′

).

Note on the β family in the homotopy or sphCrcs

This thcorcnl is closely fclatcd to thc Kervairc invariant one problcm. On this

problc■1, wC haVC

THEOREM 3.6.([23])Lθ

ι

/b￠

ヶ_{力?7-dた 冴防οη げ ヶん9 Ar,力 οり},′冴spじcιrク胞 ズ (1), αЮ′L2ケル

B"已

β】冴 ケοじαJ'Zαヶ,οヵy♭乃cι

orり

,ヶ力rθsp9じ才ιου51】

P,T力

θ猾 "ち v2i'η (2.8) s夕′υ,υιs ι

O

π*(L2/).

Wc also havc nonexistcncc thcorenl:

THEOREM 3.7.([22]) T力

99'ι_{れιηヶ島} ,Ю _(2.4)αι才力θ _pr,れじ3冴οθs ηοι s,rυ,υθι0

α 力οttοιοpノ 】ιttθηιア ι三

4,7,8 mod 9.

Corrcsponding to this thcorcnl,

THEOREM 3.8.([12]) T力

99'ιttθηι島,η _(2.4)αιと力ι _prヵηゼ3s夕rυ,υθsケοα力Ottθιοpノ 9,9翻θttι _{√ ヶ}

=1,2,3,5

αη冴6.

This thcorcm scems to hold for

ヶ三三

1, 2, 3, 5 and 6 mod 9, but so far, the

author has not got any information,

Hopkins and SInith thcorcm 3.l ccrtincs thc cxistence:

THEOREM 3.9.Lθ

ケ

p>2.T力

9乃 ybr 9,cヵ ,,ブ リ

'オ

カ p.11ブ,オカ99'9陶ιttι APx万_,,テη (2.8)s,rυ

力θ

S

ι

ο

α力

ο

ttο

ヶ

ο

pノ

θ

ι

ι

駒

?η

ι派ノ α′

α

rσ9 s≫ 0.

In fact,thcir theorcm suggcsts thc existcncc of a

υ

_{2 mapデ}

On a typc 2 spcctrum

χ(らブ

)WhOSC BPx― homology is

βPx/(ptt υ

F)fOr an odd p

mc p.Furthcrmore,

B鳥

(デ

)=υ

yれ for a largc s.

At a p

me>3,S.Oka constructcd many

β―

clemcntsi

THEOREM 3.10。

_{([5],[7],[8])L9ι}

s>O

α

Ю

冴

0<r≦

p.T力

ι

Ю

_AP/r力 (2.8)s,′

‐

υ

テ

υ

θ

sケOα

力

0“

ο

ι

ο

pノ

冴ι

加例ι√′<p,ο″√

d>l

α

η

tt r=p.Bcs,′ ι

s,βs,2/ク,2'Ю (2.8)

sクrttυ

ι

s

ι

O

α力

0“Oι

ο

pノ

ι

′

ぞ

れじ

所 ュ

/s>1.

THEOREM 3.11.([9],[10])A″

m万 加 (2.8)s,rυテυ9SケOα 力Ottο ιοpノ θJθ脇ゼηヶ力

rs>0

α

_Ю冴デ≦

2・

lp力

rι,C・

L

ο

tt pr,れ

θ

p.

For more β―

clCmcnts,wc havc Sadofsky's rcsults:

THEOREM 3.12. FOrブ

, た ≧

1,S≧

2, αηtt η>log2(ブp2た+1_2), ιヵι′θ 'S αη ιル躍 乃ケ Q′ πコ θιθCιθ冴 肋 ιtt Иttαtts―Nου,たOυ E2 ιCrtt bノ

β

sp2た

・

11+漁

灯

P2た+11,2た+1+p2た(d夕

“

[り

rβ's,■

υ

ο

Jυ,ヵ_,sttα

′

ん

r pοwθrsモ

_リ

r

υ

2)・

§

4.Ring structure of the subring generated by the′ family

Lct B dcnote the subring of thc stablc homotopy groups

π_*(S°

)Of Sphcrcs

gcncratcd by thc

_β

_{ttmily.In this scction,wc statc some relations on this ring,} Consider thc subalgebra Bl of B gcncratcd by島 's fOrヶ

>0. At a primc p>3,

Katsumi SHIMoNIuRA

THEOREM 4.1.([29])Lι

ι ′

,s,猾

冴 ι うゼ ,所9σじ

rs >0.T力

ιヵ rS島

_島

_+s

_ι

=す

_(r+S―

ι

_)舟

_A.

COROLLARY 4.2.И

_{prο冴,cι 舟 1舟 2 ・ 舟 た(た} ≧

2),szゼ

rο _√ rl r2…_生

_{=O mod P,}

99"αtts ιο_{β: 1舟}_κ

+1√

r≠ た

-l mOd P,α

乃冴 99傷 α,s,θ _{β1 2β}

_{2AP l√ r=た -1+sp}

クpォο sο 翻ゼ 附 ク′_{ι″ 修 げ α 傷れた。 π}

9r9r=Σ

_kl吟.

By this, in order to dctc■

4ine Bl, wc havc to know thc rctation on

_β

_{4. For}

this, H. TOda also gave thc following

THEOREM 4.3.([26],[27])

_β

_f=O

α猾冴_βF≠

O

αすす力θ _{pr,胞 ゼ}

_{p=3 ,η}

_{】1, αη冴}

β:2+1=0,ヶ α

pr,附

ι

p>5.

S.Oka showcd

THEOREM 4.4.([6]]

β

?1=0,ι

ケカ9 pr,用 θ

p=5.

Some til■e later, Ravencl dctcrn

ncd the homotopy groups of sphcrcs at thc

prilnc 5 up to diinension l,000 Thus wc rcad olf the fonowingi

THEOREM 4.5。

_([14])

β:7≠

0,.冴

β

!8=Oα

ヶ ι力じpr,“θ5,

Recendy,at a p

me >5, C―N. Lcc and Do Ravcnel showcd

THEOREM 4.6.([3])

Иιιカゼ_Pr,“θ

p>5,β

_{:2_P_1≠} 0.

Furtherl■orc, wc have relations in 】_1:

THEOREM 4,7.([19]) Lぞ

チ ιうθ

,Pο

s,河υ9'ηι99ゼr pr力η9ιο

_{p. Tん}

ゼ■,

β

l舟

≠

0

√ ι

=た

p.―

_{(p'1-l1/1p-1)-1折}

/Sο

“

θ

'>O

α

η

'た

前ι

_{カフィた+1.}

In thc grccn book [14], Ravenel also showcd

THEOREM 4,8.И

ιι力θ _pr,翻θ

_p〓

5, Aん

=0

_{√ αηtt οηル √ 51sι} αヶ Jο乃ク αS

s+ι

≦ フ

2_p+1.

Now turn to the product AttP万

_{リカカブ≦}

p.

THEOREM 4.9。

_{([11],[17],[18],[20])AttP/i=0√ ブ}

<p

ο

r piSι,α

η

′

AttP/P≠

0

√フイ

Sι

(S-1),ο

r√

s=rp+lα

猾

冴

pィ

ヶ

"(ク

+1)沃

/例 =(r+ι

)/p“

.

By this,wc havc dctcrmincd whcthcr or not the product of thc forln Attptt iS

trivial except for the casc whcrc

ブ

=p,S=rp+1,p才

ケ

and pИ

+l r tt ι+pμ

for someヵ

≧0。

Note on the F family in the homotopy oF spheres

THEOREM 4.10.([21])Lθ

ォs αηtt ι bι pοs,ι ,υθ,猾ιゼσιrs, T力θれ ,ヵ ι力9カθttοヶοpノ

σ

rο

"ps

π

*(L2S°

),AttP/p=0√ α

η

tt

ο

rtlノ

√

Oη

θげ す

力

9yガ ′

。

り

,η

σ

cο

η

ttι,ο

η力

ο

ι

tts:

1) pISι οr

2)s=rp+lα

力冴p“+11r+ι +p“ _ヵr sOη

9'所

9♂ιrs r'η冴 猾≧0.

Usc the relation A島

_ク2/P,2=鬼+`(P2_2)島

P/P giVCn in[11]in the E2 tCrm of the

Adams― Nowikow spcctral scqucncc, and wc havc thc non― trivianty thcOrcnl:

THEOREM 4.11. A島

_ク2/P,2≠

0√

pィSι

(S-1)ο

″ √

S=rp+1,■

'pィ

ォ "(夕

+1)力

r

,=(r+ι

)/P・

力

r dοttθ

猾

・

For thc product of the form AP/れ

_フ

_ガ

,WC alS° Scc that

THEOREM 4,12.([17],E20])

'+デ

≦

p, '+ブ

=p■

1,pイ

S+ι

,

,+J=p+2,

p+2<,十

_{デ≦2p,pls+ι}

,α

η

冴

p2/t(d tt

ι

+p)

力

r Sοttι

χ≠

0. References

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