A n-Related Family of Homotopy Elements in the Stable Homotopy of Spheres?

Xiugui LIUJianming XIAODa ZHENG

Abstract To determine the stable homotopy groups of spheres π?(S)is one of the central problems in homotopy theory.Let p be a prime greater than 5.The authors make use of the May spectral sequence and the Adams spectral sequence to prove the existence of an-related family of homotopy elements,β1ωnγs,in the stable homotopy groups of spheres,where n>3,3≤s

Keywords Stable homotopy groups of spheres,Adams spectral sequence,May spectral sequence

1 Introduction

Let S be the sphere spectrum localized at an odd prime number p.To determine the stable homotopy groups of spheres π?(S)is one of the central problems in homotopy theory.One of the powerful tools to determine π?(S)is the classical Adams spectral sequence(see[1,7])based on the Eilenberg-MacLane spectrum KZ/p for the prime p,

Consider the Smith-Toda spectra V(k)given in[8],and we have the following four cofiber sequences:

Here α,β and γ are the v1-,v2-and v3-mappings,respectively.

In 1998,Wang and Zheng[9]defined the third Greek letter family elementin the ASS for p ≥ 7 andmod p,

In[5],Liu constructed a new nontrivial family of homotopy elements in the stable homotopy groups of spheres and proved the following theorems.

Theorem 1.1(cf.[5])Let p≥5,n≥3.Thenis a permanent cycle in the Adams spectral sequence and it converges to a non-trivial family of homotopy elementsin the stable homotopy of spheres

In this paper,we make use of the above result to consider the composite mapand prove its non-triviality under some conditions.The main result can be stated as follows.

Theorem 1.2Let p≥7,n>3,3≤s

The paper is arranged as follows.After recalling some knowledge on the May spectral sequence in Section 2,we compute some May Er-terms and Adams E2-terms which are used in the proof of Theorem 1.2 in Section 3.Section 4 is devoted to showing Theorem 1.2.

2 The May Spectral Sequence

From[7],there is a May spectral sequencewhich converges towith E1-term

where E()denotes the exterior algebra,P[]denotes the polynomial algebra,and

One has

In particular,the first May differential d1is given by

There also exists a graded commutativity in the May spectral sequence as

for x,y=hm,i,bm,ior an.

where i>1,j>0.

3 Some May Er-Terms and Two Adams E2-Terms

In this section,we first determine some May Er-terms(r≥1).Then we give two important theorems about Adams E2-term which will be used in the proof of Theorem 1.2.

Lemma 3.1Let p≥7,n>3,0≤s

Here t(s,n)=q[pn+(s+3)pn+(s+5)p+(s+2)]+s,G1is the Z/p-module generated by the unique elementand G2is generated by the element

ProofWhen r≥s+2,we can easily show that in the May spectral sequence

Thus in the rest of the proof,we assume that 1≤r

where ci,j=0 or 1,ei=1 if wi=aki,or ei=0.It follows that

and

We claim that s+1?r≥0.On the one hand,it is easy to get the following inequality

from the fact that ei=0 or 1.On the other hand,using 1≤r

Using 0≤s+3,s+1?r

Case 1λ3=0.

We claim that

If λ4=1,we would have the following equations

By induction on j,we have that

Then we have the following two cases.

Case 1.1If there is a factor h1,nin g,we have that up to signwith

When r≥2,we can make use of(2.5)to get that in this case the generator g is impossible to exist.

Case 1.2If there is a factor b1,n?1in g,then up to signwith

implying that the generator g is impossible to exist,either.

Case 2λ3=1.

If r≥3,we would have

It is easy to see that λ3is impossible to equal 1.Thus in the rest of this case,we always assume

By induction on j,

Thus(3.4)can turn into

Note that l≤s+7.Thus s≥p?7.By 0≤s

Case 2.1WhenIn this case,l may equal p or p+1.

Case 2.1.1l=p.From the following two equations:

we have that up to sigh the generator g must be of the form

In this case r must equal 1,then we have that up to sign

Case 2.1.2l=p+1.From the following two equations:

we have that up to sign the generator g must be of the following form:

In this case r must equal 1,then we have that up to sign

Case 2.2When

Case 2.2.1l=p.From the following two equations:

we have that up to sign the generator g must be of the form

If r=1,we have that up to sign

If r=2,we have that up to sign

Case 2.2.2l=p+1.From the following two equations:

we have that up to sign the generator g must be of the form

From Cases 1 and 2,the lemma follows.

We need the following theorem about the γ-element.

Theorem 3.1(cf.[4])Let p≥7,0≤s

detects the second Greek letter elementin the May spectral sequence,where r ≥ 1,t=(s+3)p2q+(s+2)pq+(s+1)q+s anddetects the γ-element

in the Adams spectral sequence.

Now we consider some results on the product

Lemma 3.2(1)The productis represented by

in the May spectral sequence,where

In particular,

Proof(1)Since it is known thatand aare all permanent cocycles in the May spectral sequence and converge nontrivially tofor 0≤s

is a permanent cocycle in the May spectral sequence and converges to

(2)It is easy to get the desired results.

By Lemmas 3.1–3.2,we have the following corollary.

Corollary 3.1For the May E1-module G1in Lemma 3.1,we have

where

For the May E1-module G2in Lemma 3.1,we have

where

To show the non-triviality of the productwe need to show the following two lemmas.

Lemma 3.3The May Er-modulefor r ≥ 2.

ProofFrom Corollary 3.1,

By use of(2.2)–(2.3),we have that up to sign

showing

Then it follows that

for r≥2.The proof of Lemma 3.3 is completed.

Lemma 3.4The May Er-modulefor r ≥ 2.

ProofFrom Corollary 3.1,

By use of(2.2)–(2.3),we have that up to sign

showing

Then it follows that

for r≥2.The proof of this lemma is completed.

By use of Lemmas 3.3–3.4,we can prove the non-triviality of the productas follows.

Theorem 3.2Let p≥7,n>3,0≤s

ProofFrom Lemma 3.2(1),the productis represented byin the May spectral sequence.Now we show that nothing hitsunder the May differential drfor r≥1.

We divide the proof into the following three cases.

Case 1When 0≤s

Then we have that in the May spectral sequence

Case 2When s=p?7,from Lemma 3.1 and Corollary 3.1,we have

By Corollary 3.2[2],we have

By direct computations,we have

Thus by the reason of May filtration,we have

Moreover,by Lemma 3.4 one has

From the above discussion,the permanent cocyclecannot be hit by any differential in the May spectral sequence.Consequently,converges nontrivially toin the May spectral sequence.It follows that

Case 3When s=p?6,from Lemma 3.1 and Corollary 3.1,we have

By Lemma 3.2,we have

By direct computations,we have

Thus by the reason of May filtration,we have

Moreover,using Lemma 3.3,one has

From the above discussion,the permanent cocyclecannot be hit by any differential in the May spectral sequence.Thus,converges nontrivially toin the May spectral sequence.Consequently,

From Cases 1–3,the desired result follows.

Theorem 3.3Let p≥7,n>3,0≤s

ProofFrom Lemma 3.1,in this case

By the May spectral sequence,the desired result follows.

4 Proof of the Main Result

We are now in a position to prove the main theorem in this paper.It is easy to see that to prove Theorem 1.2 is equivalent to proving the following theorem.

Theorem 4.1Let p≥7,n>3,0≤s

is a permanent cycle in the Adams spectral sequence,and converges nontrivially to the composite map

of order p,where t(s,n)=q[pn+(s+3)p2+(s+5)p+(s+2)]+s.

ProofWe know that β1,ωnand γs+3are represented in the Adams spectral sequence by b0,k0hnandrespectively.Thus,the composite map

is represented by

up to nonzero scalar in the Adams spectral sequence.

By Theorem 3.1 and the knowledge of Yoneda products,we know that the composite

is a multiplication up to nonzero scalar by

It follows that the composite map β1ωnγs+3is represented by

up to nonzero scalar in the Adams spectral sequence.

AcknowledgementThe authors would like to express their deep thanks to the referee.He read carefully the manuscript of this paper and gave the authors many helpful comments and suggestions.