A new family of filtration three in the stable homotopy of spheres

A new family of filtration three in the stable homotopy of spheres

Jinkun Lin

(Received October 2, 2000) (Revised May 24, 2001)

Abstract. This paper proves the existence of a new nontrivial family of filtration three in the stable homotopy group of spheresp2ðpⁿþpÞðp1Þ3Swhich is of order pand is represented by (b0hnþh1bn1Þ in the E₂^{3;2ðpnþpÞðp1Þ}-term of the Adams spectral sequence, where pb5 is a prime and nb3.

1. Introduction

Let A be the mod p Steenrod algebra and Sthe sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres pS is one of the central problems in homotopy theory. One of the main tools to reach it is the Adams spectral sequence (ASS) E₂^s;t¼Ext_A^s;tðZp;Z_pÞ )ptsS, where the E₂^s;t-term is the cohomology of A. If a family of generators xi in E₂^s; converges nontrivially in the ASS, then we get a family of nontrivial homotopy elements f_i in pS and we say that f_i is represented by x_iAE₂^s; and has filtration sin the ASS. So far, not so many families of homotopy elements in pS have been detected. For example, a family z_n1Ap_pⁿqþq3S for nb2 which has filtration 3 and is represented byh0bn1AExt_A^3;pnqþqðZp;ZpÞhas been detected in [2], where q¼2ðp1Þ.

From [6], Ext_A^1;ðZp;ZpÞ has Zp-base consisting of a0;hn ðnb0Þ whose internal degrees are 1, pⁿq respectively and Ext_A^2;ðZp;Z_pÞ has Z_p-base con- sisting of a²₀;a₂;a₀h_n ðn>0Þ, g_n;k_n;b_n ðnb0Þ and h_nh_m ðmbnþ2;nb0Þ whose internal degrees are 2; 2qþ1; pⁿqþ1, ðpⁿþ2pⁿ¹Þq, ð2pⁿþpⁿ¹Þq, p^nþ1q, pⁿqþp^mq respectively.

Let M be the Moore spectrum modulo a prime pb5 given by the co- fibration

S!^p S!ⁱ M!^j SS:

ð1:1Þ

2000 Mathematics Subjects Classification. 55Q45.

Key words and phrases. Stable homotopy of spheres, Adams spectral sequence, Toda-Smith spectrum.

Let a:S^qM!M be the Adams map and K be its cofibre given by the co- fibration

S^qM!^a M!ⁱ⁰ K!^j

0

S^qþ1M:

ð1:2Þ

The spectrum which we briefly write as K is known to be the Toda-Smith spectrum Vð1Þ and the elements in the stable homotopy of spheres pS are closely related to the elements in pM or pK. The main purpose of this paper is to detect a new family of ðb0hnþh1bn1Þ-element in pS as stated in the following Theorem A.

Theorem A. Let pb5, nb3, then,

(1) iðh1hnÞAExt_A^2;pnqþpqðHM;ZpÞ is a permanent cycle in the ASS and converges to a nontrivial element x_n Appⁿqþpq2M.

(2) For x_nAp_pⁿqþpq2M obtainedin (1), jx_nAp_pⁿqþpq3S is a nontrivial element of order p which is represented (up to nonzero scalar) byðb0hnþh1bn1Þ AExt_A^3;pnqþpqðZp;Z_pÞ in the ASS.

Theorem A(2) is an easy consequence of Theorem A(1) which will be proved by an argument processing in the Adams resolution of certain spectra related to M and using the known z_n1-map in [2] as a geometric input. The main step is to show that there exists a map xx~_nA½S^pnqþpq3M;S such that

aixx~_n ¼j⁰bh_n⁰i⁰ modulo higher filtration;

where bA½S^ðpþ1ÞqK;K is the known second periodicity element and h_n⁰i⁰iA ppⁿqþq2K is an h0hn-map induced by the known zn1Appⁿqþq3S so that the right hand side of the above equation has filtration 4 in the ASS.

The new family in pS obtained in Theorem A(2) actually is a third peri- odicity family represented by g_pn2=pⁿ²1þother termsAExt_BP^3;_BPðBP;BPÞ in the Adams-Novikov spectral sequence based on the Brown-Peterson spectrum BP. Roughly speaking, we have the relation that a1g_pⁿ²_=pⁿ²₁¼b₁b_pⁿ¹_=pⁿ¹₁ AExt^4;_BPBPðBP;BPÞ and after the Thom map F:Ext^;_BPBPðBP;BPÞ ! Ext_A^;ðZp;ZpÞ we have Fðg_pⁿ²_=pⁿ²₁Þ ¼b0hnþh1bn1AExt_A^3;ðZp;ZpÞ since Fðb₁b_pn1=pⁿ¹1Þ ¼b0h0hnAExt_A^4;ðZp;ZpÞ.

After giving some preliminaries on low dimensional Ext groups in O2, the proof of Theorem A will be given in O3.

2. Some preliminaries on low dimensional Ext groups

In this section, we prove some results on low dimensional Ext groups which will be used in the proof of the main theorem.

Proposition 2.1. Let pb5, nb3, hnAExt_A^1;pnqðZp;ZpÞ, a0AExt_A^1;1

ðZp;ZpÞ and a2;bn be generators in Ext_A^2;ðZp;ZpÞ with internal degrees 2qþ1;

p^nþ1q respectively. Then we have the following.

(1) The product a₂b₀h_n00AExt_A^{5;pnqþðpþ2Þqþ1}ðZp;Z_pÞ.

(2) Ext_A^4;pnqþpqþ1ðZp;ZpÞGZpfa0b0hn;a0bn1h1g, Ext_A^{4;pnqþðpþ1Þq}ðZp;Z_pÞGZ_pfb0h₀h_ng.

(3) Ext_A^{4;pnqþðpþtÞqþr}ðZp;ZpÞ ¼0 for t¼1;2 andr¼1;2;3.

(4) Ext_A^{4;ðpþ2Þqþr}ðZp;ZpÞ ¼0 for r¼2;3.

Proof. From [8], p. 82, Theorem 3.2.5, there is a May spectral sequence (MSS) fE_r^s;t;;d_rg which converges to Ext_A^s;tðZp;Z_pÞ with E₁-term

E^;;₁ ¼Eðhm;ijm>0;ib0ÞnPðbm;ijm>0;ib0ÞnPðanjnb0Þ;

where E is the exterior algebra, P is the polynomial algebra, and h_m;iA E₁^{1;2ðpm1Þpi;2m1},b_m;iAE₁^{2;2ðpm1Þpiþm;pð2m1Þ},a_nAE₁^1;2pn1;2nþ1. Observe the sec- ond degree of the following generators ðmodpⁿqÞ for 2ai<n, nb3:

jh1;ij ¼pⁱq ðmodpⁿqÞ;

jb1;i1j ¼pⁱq ðmodpⁿqÞ;

jhs;i1j ¼ ðp^sþi2þ þpⁱ¹Þq ðmodpⁿqÞ;iasþi2<n;

jbs;i1j ¼ ðp^sþi1þ þpⁱÞq ðmodpⁿqÞ;iasþi2<n;

jaiþ1j ¼ ðpⁱþ þ1Þqþ1 ðmodpⁿqÞ:

At degree t¼pⁿqþ ðpþrÞqþk for r¼0;1;2 and k¼0;1;2;3, E₁^4;t; has no generator which has factors consisting of the above elements, because such generators will have internal degree ðcn1pⁿ¹þ þc₁pþc₀Þqþd ðmodpⁿqÞ with some ci00, 2aian1, where 0acs<p, s¼0;. . .;n1; 0ada4.

Exclude the degree>pⁿq, then we know that E₁^4;t; for t¼pⁿqþ ðpþrÞqþk with r¼0;1;2, k¼0;1;2;3 has elements of the form h1;nx or b1;n1x for some xAEðh1;0;h_1;1;h_2;0ÞnPða0;a₁;a₂;b_1;0Þ. So we have

E₁^{4;pnqþðpþ1Þq;}¼Zpfh1;nb1;0h1;0;b1;n1h1;1h1;0g;

E^4;pnqþðpþ2Þqþ1;

1 ¼Zpfh1;nh1;1h1;0a1;h1;nh2;0h1;0a0;b1;n1h1;0a2;b1;n1h2;0a1g; E^4;pnqþðpþ1Þqþ1;

1 ¼Z_pfb1;n1h_1;1a₁;b1;n1h_2;0a₀;h_1;nh_1;1h_1;0a₀g;

E^4;pnqþðpþ2Þqþ2;

1 ¼Zpfb1;n1a2a1;h1;nh2;0a1a0;h1;nh1;0a2a0;h1;nh1;1a₁²g;

E^4;pnqþðpþ1Þqþ2;

1 ¼Zpfb1;n1a2a0;h1;nh1;0a1a0;h1;nh2;0a²₀g;

E^4;pnqþðpþ2Þqþ3;

1 ¼Zpfh1;na2a1a0g; E^4;pnqþðpþ1Þqþ3;

1 ¼Zpfh1;na2a²₀g;

E₁^{4;pnqþpqþ1;}¼Zpfb1;n1h1;1a0;h1;nb1;0a0g;

E₁^{3;pnqþpqþ1;}¼Z_pfh1;nh_1;1a₀g; E₁^{3;pnqþðpþ1Þq;}¼Z_pfb1;n1h_2;0;h_1;nh_1;1h_1;0g;

E₁^3;pnqþpq;¼Z_pfh1;nb_1;0; b_1;n1h_1;1g:

From [8], p. 82, Theorem 3.2.5, the formulas for the di¤erential d1 are d₁ðh1;nÞ ¼0, d₁ðb1;n1Þ ¼0, d₁ða0Þ ¼0, d₁ða1Þ ¼ a0h_1;0, d₁ðh2;0Þ ¼ h1;0h_1;1, d1ða2Þ ¼ a0h2;0a1h1;1 and drðxyÞ ¼drðxÞyþ ð1Þ^sxdrðyÞ for xAE_r^s;t;, yA E_r^s0;t0; ðrb1Þ, xy¼ ð1Þ^ss0þtt0yx for x;y¼h_m;i;b_m;i or a_n. Thus, we have

d1ðh1;nh1;1h1;0a1Þ ¼0¼d1ðh1;nh2;0h1;0a0Þ;

d₁ðb1;n1h_1;0a₂Þ ¼b_1;n1h_1;0a₀h_2;0þb_1;n1h_1;0a₁h_1;100; ðÞ d₁ðb1;n1h_2;0a₁Þ ¼b1;n1h_2;0a₀h_1;0b1;n1h_1;0h_1;1a₁00; ðÞ and the last two elements are linearly independent. Therefore, E₂^{4;pnqþðpþ2Þqþ1;}GZ_pfh1;nh_1;1h_1;0a₁;h_1;nh_2;0h_1;0a₀g and these two generators are permanent cycles in the MSS since it is known that h1;n;b1;n;h2;0h1;0; h1;0a1;a0AE₁^;; are permanent cycles which converge in the MSS for all nb0 to h_n;b_n;g₀;a₂, a₀AExt_A^;ðZp;Z_pÞ respectively. Then, h_1;0a₁b_1;0h_1;n A E₁^{5;pnqþðpþ2Þqþ1;} cannot be hitted by di¤erential and it converges in the MSS nontrivially to a₂b₀h_nAExt_A^{5;pnqþðpþ2Þqþ1}ðZp;Z_pÞ, and so (1) is proved.

Note that h1;nh1;1h1;0a1, h1;nh1;0h2;0a0 converge in the MSS to hnh1a2¼0, h_ng₀a₀¼0 (Note: a₂h₁¼0, g₀a₀¼0 by [1], Table 8.2) in Ext respectively.

Combining with the linearly independent equations (*), (**), this shows that Ext_A^{4;pnqþðpþ2Þqþ1}ðZp;ZpÞ ¼0. Look at the following:

d1ðb1;n1h1;1a1Þ ¼b1;n1h1;1a0h1;0¼ d1ðb1;n1h2;0a0Þ

and b1;n1h1;1a1þb1;n1h2;0a0¼d1ðb1;n1a2Þ. Moreover, h1;nh1;1h1;0a0AE₁^4;;

converges in the MSS to h_nh₁h₀a₀¼0 in Ext, then we have Ext_A^{4;pnqþðpþ1Þqþ1}ðZp;ZpÞ ¼0. By a straightforward calculation we have

d₁ðb1;n1a₂a₁Þ ¼ b1;n1a₀h_2;0a₁b_1;n1a₁h_1;1a₁þb1;n1a₂a₀h_1;000;

d1ðh1;nh2;0a1a0Þ ¼h1;nh1;0h1;1a1a0h1;nh2;0a0h1;0a000;

d1ðh1;nh1;0a2a0Þ ¼ h1;nh1;0a1h1;1a0h1;nh1;0a0h2;0a000;

d1ðh1;nh1;1a₁²Þ ¼ h1;nh1;1a0h1;0a1þh1;nh1;1a1a0h1;0 ¼2h1;nh1;1a0a1h1;000;

where the first three elements are linearly independent and d1ðh1;nðh2;0a1a0þh1;0a2a0þh1;1a²₁ÞÞ ¼0:

However, h1;nðh2;0a1a0þh1;0a2a0þh1;1a₁²Þ ¼ d1ðh1;na2a1Þ, which shows that all the generators in E^4;pnqþðpþ2Þqþ2;

1 die, i.e. E₂^{4;pnqþðpþ2Þqþ2;}¼0, so Ext_A^{4;pnqþðpþ2Þqþ2}ðZp;ZpÞ ¼0 as desired. By a straightforward calculation we have

d1ðb1;n1a2a0Þ ¼ b1;n1a0h2;0a0b1;n1a1h1;1a000;

d1ðh1;nh2;0a²₀Þ ¼h1;nh1;0h1;1a²₀00

which are linearly independent. In addition, h1;nh1;0a1a0 is a permanent cycle which converges in the MSS to hna2a0¼0 in Ext (Note: a0a2¼0 by [1], Table 8.2). This shows thatExt_A^{4;pnqþðpþ1Þqþ2}ðZp;Z_pÞ ¼0. Moreover, we have

d₁ðh1;na₂a₁a₀Þ ¼h_1;na₀h_2;0a₁a₀þh_1;na₁h_1;1a₁a₀þh_1;na₂a₀h_1;0a₀00;

d₁ðh1;na₂a₀²Þ ¼ h1;nh_2;0a₀³h_1;nh_1;1a₁a₀²00:

This shows that Ext_A^{4;pnqþðpþrÞqþ3}ðZp;ZpÞ ¼0 forr¼1;2 and finishes the proof of (3).

It is easily seen thatdrðE₁^{3;pnqþpqþ1;}Þ ¼0 for allrb1 and d1ðE₁^{3;pnqþðpþ1Þq;}Þ

¼Z_pfb1;n1h_1;0h_1;1g. Therefore,E₂^{4;pnqþðpþ1Þq;}GZ_pfh1;nb_1;0h_1;0g,E₂^{4;pnqþpqþ1;}

GZpfb1;n1h1;1a0;h1;nb1;0a0g and drðEr^{3;pnqþðpþ1Þq;}Þ ¼0 for all rb2, which proves (2). The result in (4) follows from E₁^{4;ðpþ2Þqþ3;}¼Zpfh1;1a₁²a0g, E₁^{4;ðpþ2Þqþ2;}¼Z_pfb1;0a₁²;h_1;1h_1;0a₁a₀g and d₁ðh1;1a²₁a₀Þ00, d₁ðb1;0a²₁Þ00, 2h1;1h1;0a1a0¼d1ðh1;1a₁²Þ by a straghtforward calculation. Q.E.D.

Proposition 2.2. Let pb5, nb3, then (1) Ext_A^{4;pnqþðpþ2Þqþ2}ðHK;HMÞ ¼0.

(2) Ext_A^r;pnqþ2qþtðHK;ZpÞ ¼0 for r¼3;t¼0;1 or r¼4;t¼1;2.

Proof. (1) Consider the exact sequence (k¼pⁿqþ ðpþ2Þq;r¼2;3) Ext_A^4;kþrðHM;ZpÞ !ⁱ

0

Ext_A^4;kþrðHK;ZpÞ !^j

0

Ext_A^4;kþrq1ðHM;ZpÞ !^a induced by (1.2), where a is the connecting homomorphism associated with the short exact sequence in Zp-cohomology induced by (1.2). The first and the third groups are zero by Proposition 2.1(3) except for the third group in caser¼2 which has unique generatoraiðb0hnÞby Proposition 2.1(2),(3), since h₀b₀h_n¼jaiðb0h_nÞAExt_A^{4;pnqþðpþ1Þq}ðZp;Z_pÞ (cf. Remark below). However, aðaiÞðb0hnÞ00AExt_A^{5;pnqþðpþ2Þqþ2}ðHM;ZpÞ by the fact that jaðaiÞðb0hnÞ

¼a₂b₀h_n00 (cf. Proposition 2.1(1)). This shows that the above a is monic and im j⁰¼0. So the middle group is zero for r¼2;3 and the result follows by the exact sequence (k¼pⁿqþ ðpþ2Þq)

0¼Ext_A^4;kþ3ðHK;Z_pÞ !^j

Ext_A^4;kþ2ðHK;HMÞ !ⁱ

Ext_A^4;kþ2ðHK;Z_pÞ ¼0 induced by (1.1).

(2) Look at the exact sequence Ext_A^r;pnqþ2qþtðHM;Z_pÞ !ⁱ

0

Ext_A^r;pnqþ2qþtðHK;Z_pÞ !^j

0

Ext_A^r;pnqþqþt1ðHM;Z_pÞ induced by (1.2). The right group is zero for ðr;tÞ ¼ ð3;0Þ;ð4;1Þ;ð4;2Þ by [1], Table 8.1 or [4], Proposition 2.1 and has unique generator iðh0b_n1Þ for ðr;tÞ ¼ ð3;1Þ which satisfies aiðh0bn1Þ ¼ia₂bn100AExt_A^4;pnqþ2qþ1ðHM;Z_pÞ by [4], Proposition 2.1 and Ext_A^3;pnqþ2qðZp;Z_pÞ ¼0 by [1], Table 8.1. The left group is zero for ðr;tÞ ¼ ð3;0Þ and has unique generator iða2hnÞ, iða2bn1Þ, aðaiÞðbn1Þ for ðr;tÞ ¼ ð3;1Þ, ð4;1Þ, ð4;2Þ respectively (cf. [1], Table 8.1 and [4], Proposition 2.1(2)). However, i⁰iða2hnÞ ¼0, i⁰iða2bn1Þ ¼0 and i⁰aðaiÞðbn1Þ ¼0 by i⁰ija²i¼0Ap2q1K, then the result follows. Q.E.D.

Remark. Let us intepret why the connecting homomorphism p: Ext_A^s;tðZp;Z_pÞ !Ext_A^sþ1;tþ1ðZp;Z_pÞ is a multiplication by a₀. Let W !^f X !^g Y !^h SW be a cofibration such that h induces the zero homomorphism in Z_p- cohomology. From [8], p. 63–64, Theorem 2.3.4, the connecting homo- morphism h:Ext_A^s;tðHY;Z_pÞ !Ext_A^sþ1;tðHW;Z_pÞ can be described as the Yoneda product with the element of Ext_A¹ðHW;HYÞ corresponding to the short exact sequence 0!HY !^g

HX !^f

HW!0 inZp-cohomology. Ap- plying this result to the cofibration (1.1), for the connecting homomorphism p:Ext_A^s;tðZp;ZpÞ !Ext_A^sþ1;tþ1ðZp;ZpÞ,pðxÞis the Yoneda product ofxAExt_A^s;t ðZp;ZpÞ with a0AExt_A^1;1ðZp;ZpÞ corresponding to the short exact sequence 0!HS!^j

HM!ⁱ

HS!0. (Note: The latter also follows from the fact that the degree pmap S!Sis represented by a0AExt_A^1;1ðZp;ZpÞ in the ASS.) This shows that pðxÞ ¼a0x and pðxÞ x⁰¼pðxx⁰Þ. Similarly we have

pðxÞ ¼a₀x and jaiðxÞ ¼h₀x.

Let L be the cofibre of a1¼jai:S^q1S!S and K_r⁰ be the cofibre of a^ri:S^rqS!M given by the following cofibrations:

S^q1S^a!¹ S ⁱ!⁰⁰ L^j!

00

S^qS;

ð2:3Þ

S^rqS^a!^ri M !^vr K_r^0y!^r S^rqþ1S:

ð2:4Þ

Then K₂⁰ also is the cofibre of vj⁰:S¹K!S^qK⁰ given by the cofibration S¹K ^vj!

0

S^qK⁰ !^c K₂⁰ !^r K;

ð2:5Þ

where we briefly write K₁⁰ as K⁰ etc., which can be seen by the following

commutative diagram of 33 lemma in stable homotopy category (cf. [9], p.

292–293):

!

!

!

!

S¹K ^vj!

0

S^qK^{0 y}! S^2qþ1S

j⁰

v c

y2

S^qM K₂⁰

ai a

v2

r

S^2qS ^a2!ⁱ M ⁱ⁰! K;

ð2:6Þ

and we have the following relations:

y¼y₂c; cv¼v₂a; rv₂¼i⁰; j⁰r¼ aiy2: ð2:7Þ

Let m_M :SM!M5M be the injection and m_M :M5M!M be the mul- tiplication of M satisfying mMði51MÞ ¼1M, ðj51MÞmM ¼ ð1M5jÞmM ¼ 1_M and ði51_MÞmMþm_Mðj51_MÞ ¼1_M5M. By the commutative diagram of 33 lemma

!

!

!

!

SM ^v ! SK^{0 1K05p}! SK⁰

ðv51MÞmM

1_K05j y

z

K⁰5M S^qþ2S

1_K05i p

jj⁰ ai

K^{0 x} ! K ^j !

0a⁰

S²M;

ð2:8Þ

we have two cofibrations

K⁰!^x K ^jj!

0

S^qþ2S!^z SK⁰; ð2:9Þ

S¹K^j !

0a⁰

SM ^v51Mm!^M K⁰5M ^p !K;

ð2:10Þ

where ½K;S^qþ2SGZ_pfjj⁰g, since ½M;S^qþ2S ¼0,½M;SSGZ_pfjg andaijj⁰¼ ða151MÞj⁰¼j⁰a⁰ with a⁰¼a151K. In addition, ðvi51MÞmMðai51MÞ þ ðv51_MÞmMðjai51_MÞ ¼ ðv51_MÞðai51_MÞ ¼0, which shows that

ðvi51_MÞa¼ ðv51_MÞmMða151_MÞ;

ð2:11Þ

since mMðai51MÞ ¼a¼mMði51MÞa.

By the commutative diagram of 33-lemma in the stable homotopy category

M ! L5K ^j !

0051K

S^qK

i⁰ i⁰⁰51K r p

K S^qK⁰5M

j⁰ e

S^q1K ^j !

0a⁰

S^qþ1M ^a ! SM;

ð2:12Þ

a⁰ ðv51MÞmM

we have a cofibration

!

!

!

!

M ^ði !

0051KÞi⁰

L5K ^r !S^qK⁰5M ^e !SM ð2:13Þ

with the relation

eðv51MÞmM ¼a; eð1K⁰5iÞvj⁰¼ 2j⁰a⁰A½S²K;M: ð2:14Þ

Note that eð1K⁰5iÞvA½S^q1M;MGZpfija;aijg so that eð1K⁰5iÞv¼l1ijaþ l₂aij, l₁ijaaiþl₂aijai¼0 and l₂¼ 2l1 by 2aija¼ija²þa²ij. By applying d on eð1K⁰5iÞv¼eð1K⁰5ijÞðv51MÞmM we have l1¼1 and so eð1K⁰5iÞvj⁰¼ 2j⁰a⁰. In addition, by the 33-lemma in the stable homotopy category one can easily check that there is a cofibration

M!^vi K⁰!^k SL ! SS ð2:15Þ

with the relation that kv¼i⁰⁰j.

From [7], p. 434, there are DA½S^q1L5K;K and DD~A½S¹K;L5K satisfyingDði⁰⁰51KÞ ¼ ðj⁰⁰51KÞDD~¼i⁰j⁰A½S^q1K;K and jj⁰D¼0. Then, by (2.9), there is D_K⁰A½S^q1L5K;K⁰ such that

D_K⁰ði⁰⁰51_KÞ ¼vj⁰A½S^q1K;K⁰; Dði⁰⁰51_KÞ ¼ ðj⁰⁰51_KÞDD~¼i⁰j⁰: ð2:16Þ

Proposition 2.17. Let pb5;nb3, then (1) Ext_A^3;pnqþtqþ1ðHK;HMÞ ¼0 for t¼0;1.

(2) Ext_A^3;pnqþpqðZp;HMÞ ¼0 andExt_A^{3;pnqþðpþ2Þqþ2}ðHK⁰5M;Z_pÞ has unique generator ðv51MmMÞiðhng0Þ, where hng0AExt_A^{3;pnqþðpþ2Þqþ1} ðHM;HMÞ satisfies jiðhng0Þ ¼hng0, the unique generator of Ext_A^{3;pnqþðpþ2Þq}ðZp;Z_pÞ statedin [1], Table 8.1.

(3) ðv51MmMÞ:Ext_A^{5;pnqþðpþ2Þqþ1}ðHM;HMÞ !Ext_A^{5;pnqþðpþ2Þqþ2}ðHK⁰5 M;HMÞ is monic.

Proof. (1) Consider the exact sequence (k¼pⁿqþtqþ1 with t¼0;1) Ext_A^3;kðHM;HMÞ^ði!

0Þ

Ext_A^3;kðHK;HMÞ^ðj!

0Þ

Ext_A^3;kq1ðHM;HMÞ

induced by (1.2). The left group is zero for t¼0 and has unique generator aðbb~_n1Þfort¼1 (cf. [4], Proposition 2.3(1) and 2.4(1)), and hence im ði⁰Þ¼0.

The right group is zero for t¼0;1, since Ext_A^{3;pnqþðt1Þqþr}ðZp;ZpÞ ¼0 for t¼0;1 and r¼ 1;0 (cf. [1], Table 8.1). Thus im ðj⁰Þ¼0 and the result follows.

(2) By [1], Table 8.1, Ext_A^3;pnqþpqþ1ðZp;Z_pÞ ¼0 and Ext_A^3;pnqþpqðZp;Z_pÞ has two generators hnb0;h1bn1 such that a0hnb0;a0h1bn1 are linearly inde- pendent inExt_A^4;ðZp;ZpÞ(cf. Proposition 2.1(2)). Then the first result follows.

For the second half, look at the exact sequence

Ext_A^{3;pnqþðpþ2Þqþ2}ðHM5M;Z_pÞ^ðv51M!^Þ Ext_A^{3;pnqþðpþ2Þqþ2}ðHK⁰5M;Z_pÞ !

ðy51MÞ

Ext_A^{3;pnqþðpþ1Þqþ1}ðHM;ZpÞ

induced by (2.4). The right group is zero by Ext_A^{3;pnqþðpþ1Þqþr}ðZp;Z_pÞ ¼0 for r¼0;1;2 and the left group has unique generator ðmMÞiðhng0Þ by Ext_A^{3;pnqþðpþ2Þqþr}ðZp;Z_pÞ ¼0 forr¼1;2 andExt_A^{3;pnqþðpþ2Þq}ðZp;Z_pÞGZ_pfhng₀g (cf. [1], Table 8.1). Thus the result follows.

(3) Consider the exact sequence induced by (2.10) Ext_A^{4;pnqþðpþ2Þqþ2}ðHK;HMÞ^ðj !

0a⁰Þ

Ext_A^{5;pnqþðpþ2Þqþ1}ðHM;HMÞ !

ðv51MmMÞ

Ext_A^{5;pnqþðpþ2Þqþ2}ðHK⁰5M;HMÞ:

The left group is zero by Proposition 2.2(1) and so the result follows. Q.E.D.

Proposition 2.18. Let pb5;nb3, then ð1Þ ½S¹L5K;M ¼0:

ð2Þ Ext_A^{1;pnqþðpþ2Þq}ðHL5K;HMÞ ¼0:

Proof. (1) By (2.3), it su‰ces to prove ½S¹K;M ¼0¼ ½S^q1K;M. Consider the exact sequence (t¼0;1)

½S^ðtþ1ÞqM;M^ðj!

0Þ

½S^tq1K;M^ði!

0Þ

½S^tq1M;M^a!

induced by (1.2). The left group has unique generator a;a² for t¼0;1 re- spectively, and hence im ðj⁰Þ¼0. The right group has unique generator ij fort¼0 and two generatorsija;aij fort¼1. However, ija00,ðl1ijaþl₂aijÞa

¼0 implies l1¼l2¼0, and hence a is monic. Thus im ði⁰Þ ¼0 and the result follows.

(2) Since the top cell of L5K has degree 2qþ2, the result follows from the fact that Ext_A^1;ðZp;Z_pÞ has Z_p-base consisting of h_n for all nb0 with internal degree pⁿq.

3. Proof of the main theorem

We first prove Theorem A(1) which will be done by an argument pro- cessing in the Adams resolution of some spectra related to K. Let

^a2! S²E2 ^a1! S¹E1 ^a0! E0 ¼S

??

?y^b2

??

?y^b1

??

?y^b0 S²KG2 S¹KG1 KG0¼KZp

ð3:1Þ

be the minimal Adams resolution of S satisfying the following.

(1) Es!^bs KGs!^cs Esþ1!^as SEs are cofibrations for all sb0 which induce short exact sequences in Zp-cohomology.

(2) KG_s is a wedge sum of suspensions of Eilenberg-MacLane spectra of type KZp.

(3) p_tKG_s are the E₁^s;t-terms, ðbscs1Þ:p_tKGs1!p_tKG_s are the d₁^s1;t- di¤erentials of the ASS and ptKGsGExt_A^s;tðZp;ZpÞ (cf. [3], p. 180).

Then an Adams resolution of an arbitrary finite spectrum V can be obtained by smashing V on (3.1). We first prove the following lemmas.

Lemma 3.2. (1) Let pb3;nb3 andh_nAExt_A^1;pnqðZp;Z_pÞ, b0;a₂;g₀ be the generators in Ext_A^2;ðZp;ZpÞ with internal degrees pq;2qþ1;ðpþ2Þq re- spectively, then d₂ðhng₀Þ ¼a₂h_nb₀00AExt_A^{5;pnqþðpþ2Þqþ1}ðZp;Z_pÞ (up to non- zero scalar), where d2:Ext_A^3;ðZp;ZpÞ !Ext_A^5;þ1ðZp;ZpÞ is the di¤erential of the ASS.

(2) The di¤erential satisfies d2ðg₀~hhnÞ ¼aaðbb~0~hhnÞAExt_A^{5;pnqþðpþ2Þqþ2} ðHM;HMÞ up to nonzero scalar, where g₀AExt_A^{3;ðpþ2Þqþ1}ðHM;HMÞ, b~

b0AExt_A^2;pqðHM;HMÞ and ~hhnAExt_A^1;pnqðHM;HMÞ satisfy ijg₀¼g0, iðbb~₀Þ ¼iðb0Þ andið~hh_nÞ ¼iðhnÞ respectively.

Proof. (1) Let bA½S^ðpþ1ÞqK;K be the second periodicity element (cf.

[7], p. 426). It is known that b₁¼jj⁰bi⁰iAppq2S is represented by b0 A Ext_A^2;pqðZp;ZpÞ in the ASS and a2b₁¼ja²ijj⁰bi⁰i¼0, a2b000AExt_A^4;pqþ2qþ1 ðZp;Z_pÞ. Thena₂b₀ must be hitted by the di¤erential and the only possibility isd2ðg0Þ ¼a2b0 up to nonzero scalar. From [8], p. 11, Theorem 1.2.14, d2ðhnÞ

¼a₀bn1AExt_A^3;pnqþ1ðZp;Z_pÞ, then d₂ðhng₀Þ ¼d₂ðhnÞg0þhnd₂ðg0Þ ¼a₂h_nb₀ up to nonzero scalar. (Note: g0a0¼0 by [1], Table 8.2).

(2) Since pðg0Þ ¼a₀g₀¼0, we have g₀AjExt_A^{3;ðpþ2Þqþ1}ðHM;Z_pÞ.

Moreover, pExt_A^{3;ðpþ2Þqþ1}ðHM;Z_pÞHiExt_A^{4;ðpþ2Þqþ2}ðZp;Z_pÞ ¼0 (cf. Prop- osition 2.1(4)), then there is g₀AExt_A^{3;ðpþ2Þqþ1}ðHM;HMÞ such that ijg₀¼ g₀. By (1), the di¤erential satisfies d₂ðijg₀Þ ¼a₂b₀¼ijaaðbb~₀Þ and so d2ðig₀Þ ¼iaaðbb~0Þmodulo iExt_A^{4;ðpþ2Þqþ2}ðZp;ZpÞ ¼0 (cf. Proposition 2.1(4)).

Hence we have d2ðg₀Þ ¼aaðbb~0Þ modulo jExt_A^{4;ðpþ2Þqþ3}ðHM;ZpÞ ¼0 by Proposition 2.1(4). Since d2ð~hhnÞAExt_A^3;pnqþ1ðHM;HMÞ which is zero by Ext_A^3;pnqþrðHM;ZpÞ ¼0 for r¼1;2 (cf. [4], Proposition 2.3(1)), we have d₂ðg₀~hh_nÞ ¼d₂ðg₀Þ~hh_n¼aaðbb~₀Þ ~hh_n¼aaðbb~₀~hh_nÞ (cf. Remark of Proposition 2.2) as desired. Q.E.D.

Lemma 3.3. Let pb5;nb3, then there exists h_n;2⁰ A½S^pnqþqK;E25K such thatðb251KÞh_n;⁰ ₂¼h0hn51KA½S^pnqþqK;KG25Kand ð1E25a⁰Þh_n;⁰ ₂¼0, where h₀h_nAp_pⁿqþqKG₂GExt_A^2;pnqþqðZp;Z_pÞ and a⁰¼jai51_KA½S^q1K;K.

Proof. From [5], Proposition 3.4, there is a d₁-cycle ðh0h_nÞ⁰⁰A

½S^pnqþq1K;KG25K such that ð1KG25j⁰Þðh0hnÞ⁰⁰¼ ð1KG25ijj⁰Þðh0hn51KÞ and ðc251_KÞðh0h_nÞ⁰⁰¼0. It follows that ðc251_L5KÞð1KG25DDÞðh~ 0h_nÞ⁰⁰¼0 and there exists ff~₁A½S^pnqþq2K;E25L5K such that ðb251L5KÞff~₁¼ ð1KG25DDÞðh~ 0hnÞ⁰⁰A½S^pnqþq2K;KG25L5K. Then, by (2.16) we have

ðb251KÞð1E25j⁰⁰51KÞff~₁¼ ð1KG25ðj⁰⁰51KÞDDÞðh~ 0hnÞ⁰⁰

¼ ð1KG25i⁰j⁰Þðh0h_nÞ⁰⁰¼ ð1KG25i⁰ijj⁰Þðh0h_n51_KÞ and so

ðb251KÞð1E25j⁰⁰51KÞð1E251L5mÞðff~₁51KÞn

¼ ð1KG25mÞð1KG25i⁰ijj⁰51_KÞðh0h_n51_K51_KÞn¼h₀h_n51_K; where m:K5K!K is the multiplication ofK satisfyingmði⁰i51KÞ ¼1K and n:S^qþ2K!K5K is the injection such that ðjj⁰51_KÞn¼1_K (cf. [7], p. 433).

This shows that h_n;2⁰ ¼ ð1E25j⁰⁰51KÞð1E251L5mÞðff~₁51KÞn is our desired map. Q.E.D.

Proof of Theorem A(1). For the map h_n;2⁰ in Lemma 3.3, we have

½ðb251_KÞh_n;2⁰ i⁰ ¼ ½ðh0h_n51_KÞ⁰i⁰ ¼ ða151_KÞ½ðhn51_KÞi⁰, then ½ðb251_L5KÞ ð1E25i⁰⁰51KÞh_n;2⁰ i⁰ ¼0AExt_A^2;pnqþqðHL5K;HMÞ. By (2.3) and Proposi- tion 2.17(1) we have Ext_A^3;pnqþqþ1ðHL5K;HMÞ ¼0, then ða0a151L5KÞ ð1E25i⁰⁰51_KÞh_n;2⁰ i⁰A½S^pnqþq2M;L5Khas filtrationb4. Moreover, the sec- ond periodicity element bA½S^ðpþ1ÞqK;K has filtration one, then ða0a15 1L5KÞð1E251L5bÞð1E25i⁰⁰51KÞh_n;⁰ ₂i⁰¼ ða0a1a2a3a451L5KÞf2 which has fil- trationb5 with f₂A½S^{pnqþðpþ2Þqþ3}M;E₅5L5K. It follows that ð1E25i⁰⁰ 51KÞð1E25bÞh_n;2⁰ i⁰¼ ða2a3a451L5KÞf2þ ðc151L5KÞg and the d1-cycle gA

½S^{pnqþðpþ2Þq}M;KG₁5L5K is zero by Proposition 2.18(2). That is, we have ð1E25ði⁰⁰51KÞbÞh_n;⁰ ₂i⁰¼ ða2a3a451L5KÞf2

ð3:4Þ

for some f₂A½S^{pnqþðpþ2Þqþ3}M;E₅5L5K.