Bol. Soc. Mat. Mex. DOI 10.1007/s40590-014-0012-z ORIGINAL ARTICLE

A family of nonzero products in the cohomology of the odd primary Steenrod algebra Xiang Li · Xiugui Liu

Received: 18 November 2013 / Accepted: 14 March 2014 © Sociedad Matemática Mexicana 2014

Abstract Let p be a prime number greater than five and let A be the mod p Steenrod (F p , F p ) is algebra. In this paper, we prove that the product h n g0 δ˜s+4 ∈ Ext s+7,∗ A nontrivial, where n  5, 0  s < p − 4. As a corollary, we obtain that the products h n g0 , h n δ˜s+4 and g0 δ˜s+4 are all nontrivial. Keywords

Steenrod algebra · Cohomology · May spectral sequence

Mathematics Subject Classification (2010)

55Q45

1 Introduction and statement of results Let p be an arbitrary odd prime number and S denote the sphere spectrum localized at p. To determine the homotopy groups π∗ S of spheres S is one of the core problems in the stable homotopy theory. Let A denote the mod p Steenrod algebra. The Adams spectral sequence E 2s,t = Ext s,t A (F p , F p ) ⇒ πt−s S is a useful tool to study the stable homotopy groups of spheres, where the E 2 -term is the cohomology of A. So far, only a few families of homotopy elements in π∗ S have been detected.

Research was partially supported by the NSFC (No. 11171161) and SRF for ROCS, SEM. X. Li School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China e-mail: [email protected] X. Liu (B) School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China e-mail: [email protected]

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For example, Cohen [3] constructed a certain infinite family of elements denoted by ζk ∈ π2( p−1)( pk+1 +1)−3 S, k  1 and p > 2. Note that the family ζk is represented by 3,2( p−1)( p k+1 +1)

(F p , F p ) in the Adams spectral sequence. h 0 bk ∈ Ext A Throughout this paper, we will fix q = 2( p − 1). To compute the stable homotopy groups of spheres with the classical Adams spectral sequence, we should compute the E 2 -term of the Adams spectral sequence Ext∗,∗ A (F p , F p ). Two best methods of determining Ext ∗,∗ (F , F ) are the May spectral sequence and the lambda algebra. p p A ∗,∗ Some known results on Ext A (F p , F p ) are as follows. Ext 0,∗ A (F p , F p ) equals F p by definition. From [7], we know that Ext1,∗ (F , F ) has F -basis consisting of a0 ∈ p p p A 1, pi q

(F p , F p ) for all i  0. Ext 2,∗ Ext 1,1 A (F p , F p ) and h i ∈ Ext A A (F p , F p ) has F p -basis 2 consisting of α2 , a0 , a0 h i (i > 0), gi (i  0), ki (i  0), bi (i  0), and h i h j ( j  i +2, i  0) whose internal degrees are 2q+1, 2, pi q+1, pi+1 q+2 pi q, 2 pi+1 q+ pi q, pi+1 q and pi q + p j q, respectively. In 1980, Aikawa [2] determined Ext 3,∗ A (F p , F p ) by λ-algebra. Higher-dimensional cohomology of the mod p Steenrod algebra A is an interesting topic and is studied by several authors. For example, Liu and Zhao [6] proved the following theorem. Theorem 1.1 [6] For p  7 and 4  s < p, the product h 0 b0 δ˜s = 0 in the classical Adams spectral sequence, where δ˜s is given in [10]. In this paper, our main result can be stated as follows. Theorem 1.2 Let p  7, n  5 and 0  s < p − 4. Then in the cohomology s+7,t (s,n)+s (F p , F p ), the product h n g0 δ˜s+4 is of the mod p Steenrod algebra A, Ext A nontrivial, where t (s, n) = 2( p − 1)[(s + 3) + (s + 3) p + (s + 3) p 2 + (s + 4) p 3 + p n ]. The main method of proof is the (modified) May spectral sequence, so we will recall some knowledge of the May spectral sequence in Sect. 2. In Sect. 3, we will describe E 1s,t,u with particular degrees (s, t, u) and use it in the proof of Theorem 1.2. 2 The May spectral sequence In this paper, we will use the May spectral sequence to prove our main result. From [8], there is a May spectral sequence {Ers,t,∗ , dr } which converges to Ext s,t A (F p , F p ) with E 1 -term E 1∗,∗,∗ = ∧F p (h m,i |m > 0, i  0) ⊗ F p [bm,i |m > 0, i  0] ⊗ F p [an |n  0], (2.1) where ∧F p ( ) denotes the exterior algebra, F p [ ] denotes the polynomial algebra, and 1,2( p m −1) pi ,2m−1

h m,i ∈ E 1

123

2,2( p m −1) pi+1 , p(2m−1)

, bm,i ∈ E 1

1,2 p n −1,2n+1

, an ∈ E 1

.

A family of nonzero products in the cohomology

One has dr : Ers,t,u → Ers+1,t,u−r

(2.2)



and if x ∈ Ers,t,∗ and y ∈ Ers ,t ,∗ , then dr (x · y) = dr (x) · y + (−1)s x · dr (y).

(2.3)

In particular, the first May differential d1 is given by d1 (h i, j ) =





h i−k,k+ j h k, j , d1 (ai ) =

h i−k,k ak , d1 (bi, j ) = 0.

(2.4)

0k
0
There also exists a graded commutativity in the May spectral sequence: x · y = (−1)ss +tt y · x for x, y = h m,i , bm,i or an . For each element x ∈ E 1s,t,u , we define hdim x = s, intdim x = t, May(x) = u. Then we have that ⎧ hdim h i, j = hdim ai = 1, ⎪ ⎪ ⎪ ⎪ ⎪ hdim bi, j = 2, ⎪ ⎪ ⎪ ⎪ ⎪ intdim h i, j = q( pi+ j−1 + · · · + p j ), ⎪ ⎪ ⎪ ⎨ intdim b = q( pi+ j + · · · + p j+1 ), i, j (2.5) ⎪ intdim ai = q( pi−1 + · · · + 1) + 1, ⎪ ⎪ ⎪ ⎪ ⎪ intdim a0 = 1, ⎪ ⎪ ⎪ ⎪ ⎪ May(h i, j ) = May(ai−1 ) = 2i − 1, ⎪ ⎪ ⎩ May(bi, j ) = (2i − 1) p, where i  1, j  0. 3 Proof of the main theorem In this section, we give three lemmas which are needed in the proof of Theorem 1.2. The first lemma is on the representative of δ˜s+4 in the May spectral sequence. Lemma 3.1 [6, Lemma 3.1] For p  7 and 0  s < p − 4, the fourth Greek letter s+4,t (s)+s (F p , F p ) is represented by element δ˜s+4 ∈ Ext A 1 s+4,t1 (s)+s,∗

a4s h 4,0 h 3,1 h 2,2 h 1,3 ∈ E 1

(4) in the E 1 -term of the May spectral sequence, where δ˜s+4 is actually α˜ s+4 described 2 3 in [9] and t1 (s) = [(s + 1) + (s + 2) p + (s + 3) p + (s + 4) p ]q. 

Then we claim that, in the proof of Theorem 1.2, h n g0 δ˜s+4 is a permanent cycle in the Adams spectral sequence. To prove it, we give the following two lemmas.

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Lemma 3.2 Let p  7, n  5 and 0  s < p − 4. Then we have the May E 1 -term  G n > 5 and s = p − 5, s+6,t (s,n)+s,∗ E1 = 0 otherwise. Here, t (s, n) = 2( p − 1)[(s + 3) + (s + 3) p + (s + 3) p 2 + (s + 4) p 3 + p n ], G is the F p -module generated by the following n − 4 elements: ⎧ p−5 ⎪ d = an h 1,0 h 5,0 h n,0 h n−1,1 h n−3,3 h n−4,4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f = anp−5 h 2,0 h 5,0 h n,0 h n−2,2 h n−3,3 h n−4,4 , ⎪ ⎪ ⎪ p−5 ⎪ ⎪ ⎪ ⎨ g1 = an h 5,0 h 6,0 h n,0 h n−3,3 h n−4,4 h n−6,6 , ... ⎪ ⎪ ⎪ ⎪ gk = anp−5 h 5,0 h k+5,0 h n,0 h n−3,3 h n−4,4 h n−(k+5),k+5 , ⎪ ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎪ ⎩ gn−6 = anp−5 h 5,0 h n−1,0 h n,0 h n−3,3 h n−4,4 h 1,n−1 . Before proving Lemma 3.2, we first introduce some notations. For convenience, we denote ai , h i, j and bi, j by x, y and z, respectively. By the graded commutativity of E 1∗,∗,∗ , we can suppose a generator g is of the form (x1 · · · xb )(y1 · · · yv )(z 1 · · · zl ) ∈ E 1s,t+b,∗ , where t = (c¯0 + c¯1 + · · · + c¯n p n )q with 0  c¯i < p (0  i < n), 0 < c¯n < p, s < q and 0  b < q. By (2.5), the intdims of xi , yi and z i can be uniquely expressed as: ⎧ n ⎪ ⎨ intdim xi = (xi,0 + xi,1 p + · · · + xi,n p )q + 1, intdim yi = (yi,0 + yi,1 p + · · · + yi,n p n )q, (3.1) ⎪ ⎩ n intdim z i = (0 + z i,1 p + · · · + z i,n p )q, and ⎧ ⎪ ⎨ (a) (xi,0 , xi,1 , · · · , xi,n ) is of the form (1, · · · , 1, 0, · · · , 0), (b) (yi,0 , yi,1 , · · · , yi,n ) is of the form (0, · · · , 0, 1, · · · , 1, 0, · · · , 0), (3.2) ⎪ ⎩ (c) (0, z i,1 , · · · , z i,n ) is of the form (0, · · · , 0, 1, · · · , 1, 0, · · · , 0). By the graded commutativity of E 1∗,∗,∗ , g = (x1 · · · xb )(y1 · · · yv )(z 1 · · · zl ) ∈ E 1s,t+b,∗ can be arranged in the following way: ⎧ (i) If i > j, we put ai on the left side of a j , ⎪ ⎪ ⎪ ⎨ (ii) If j < k, we put h i, j on the left side of h w,k , ⎪ (iii) If i > w, we put h i, j on the left side of h w, j , ⎪ ⎪ ⎩ (iv) Apply the rules (ii) and (iii) to bi, j .

123

(3.3)

A family of nonzero products in the cohomology

Now we give the proof of the above lemma. Proof From here on the argument to prove the lemma splits into two cases: n  6 and n = 5. s+6,t (s,n)+s,∗ First, we assume n  6. Consider g ∈ E 1 , where t (s, n) = [(s + 3) + 2 3 n (s + 3) p + (s + 3) p + (s + 4) p + p ]q with (c¯0 , c¯1 , · · · , c¯n ) = (s + 3, s + 3, s + 3, s + 4, 0, · · · , 0, 1). Since s < p, according to hdim g, we obtain that the number of xi ’s in g must be s. By the reason of dimension, all the possible monomials of g can be listed as x1 · · · xs z 1 z 2 z 3 , x1 · · · xs y1 y2 z 1 z 2 , x1 · · · xs y1 y2 y3 y4 y5 y6 , x1 · · · xs y1 y2 y3 y4 z 1 . Case1 g = x1 x2 · · · xs z 1 z 2 z 3 . Note that z i,0 = 0 for 1  i  3, one can easily get s xi,0 + z 1,0 + z 2,0 + z 3,0  s < s + 3 = c¯0 . Then the equation about intdim that i=1 has no solution. It follows that such g is impossible to exist. Case 2 g = x1 x2 · · · xs y1 y2 z 1 z 2 . By an argument similar to that used in Case 1, we can obtain that such g is impossible to exist either. Case 3 g = x1 x2 · · · xs y1 y2 y3 y4 y5 y6 . Subcase 3.1 If there exists some xi or y j with intdim xi or intdim y j being not less than qp n , we obtain that y6 =h 1,n by (3.3). Note that c¯3 = s + 4, we have the following cases: 5 Subcase 3.1.1 x1 =x2 = · · · = xs =a4 and i=1 yk,3 = 4. One has y1,0 + y2,0 + y3,0 + y4,0 + y5,0 = 3. Thus there exists at least two h 4,0 ’s. It follows that such nontrivial g is impossible to exist by h 24,0 = 0. Subcase 3.1.2 x2 =· · · =xs =a4 and

5

i=1 yk,3

= 5. One has

intdim x1 y1 y2 y3 y4 y5 = (5 p 3 + 4 p 2 + 4 p + 4)q + 1. By an argument similar to that used in Subcase 3.1.1, we can obtain that such nontrivial g is impossible to exist. Subcase 3.2 If there exists no xi or y j with intdim xi or intdim y j being not less than qp n , by(3.3) we obtain that y6 =h 1,n . we have the following cases: Subcase 3.2.1 If s  p − 7, then intdimx1 x2 · · · xs y1 y2 · · · y6  s +

n−1 i=0

(s + 6)qpi

< q[(s +3)+(s +3) p+(s +3) p 2 +(s +4) p 3 + p n ]+s = intdimx1 x2 · · · xs y1 y2 · · · y6 . This is a contradiction. Then in this case we obtain g is impossible to exist.

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s 6 Subcase 3.2.2 If s = p − 6, then i=1 xi,n−1 + i=1 yi,n−1 = s + 5. It is easy to 6 yi,n−1  5 by xi,n−1 = 0 or 1. From c¯0 = s + 3, we deduce that there know that i=1 exists two h n,0 ’s among yk . Such nontrivial g is impossible to exist by h 2n,0 = 0. s 6 Subcase 3.2.3 If s = p − 5, then i=1 xi,n−1 + i=1 yi,n−1 = s + 4. If xs =a0 , by an argument similar to that used in Subcase 3.2.2, we obtain that such nontrivial g is impossible to exist. If xs = a0 , there exists no a0 among xk ’s. Note that c¯0 = s + 3, 6 y = 3. If g exists and is nontrivial, there must exist at most one i,0 i=1 s h n,0 among xi,n−1 + yk ’s, that is to say, there are at least two y j ’s whose y j,n−1 =0. By i=1 6 y = s + 4, we obtain that x =x =· · · = x =a and y =h . It follows 1 2 s 4 1 n,0 i=1 i,n−1 intdim y2 y3 y4 y5 y6 = (3 p n−1 + · · · + 3 p 5 + 4 p 4 + 3 p 3 + 2 p 2 + 2)q.

(3.4)

We can obtain that, up to sign, g can equal the following: ⎧ p−5 d = an h 1,0 h 5,0 h n,0 h n−1,1 h n−3,3 h n−4,4 , ⎪ ⎪ ⎪ ⎪ p−5 ⎪ ⎪ f = an h 2,0 h 5,0 h n,0 h n−2,2 h n−3,3 h n−4,4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ g1 = anp−5 h 5,0 h 6,0 h n,0 h n−3,3 h n−4,4 h n−6,6 , ⎨ ... ⎪ ⎪ ⎪ p−5 ⎪ gk = an h 5,0 h k+5,0 h n,0 h n−3,3 h n−4,4 h n−(k+5),k+5 , ⎪ ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎩ p−5 gn−6 = an h 5,0 h n−1,0 h n,0 h n−3,3 h n−4,4 h 1,n−1 , Case 4 g = x1 x2 · · · xs y1 y2 y3 y4 z 1 . Similar to Case 3, we easily obtain that such g is impossible to exist. From Cases 1–4, we obtain that the generator g only exists when n > 5 and p+1,t ( p−5,n)+ p−5,∗ has n − 4 generators s = p − 5. In this case, the May E 1 -term E 1 d, f, g1 ,· · · , gn−6 . Now we turn to the case of n = 5. The only difference between the case of n = 5 and the case of n > 5 is Eq. (3.4). When n = 5, Eq. (3.4) becomes intdim y2 y3 y4 y5 y6 = (4 p 4 + 3 p 3 + 2 p 2 + 2)q. 6 yi,4  3, by (3.2) we obtain that such g is impossible to exist. From i=2 The desired result follows. Lemma 3.3



Let p  7, n  5 and 0  s < p − 4. Then we have the May Er -term s+6,t (s,n)+s,(2n+1)( p−5)+8n−10

Er

=0

for r  2. Proof By Lemma 3.2, we may assume that n > 5 and s = p − 5, in which case p+1,t ( p−5,n),∗ s+6,t (s,n)+s,∗ = E1 = F p {d, f, g1 , g2 , · · · , gn−6 }. By (2.4), we have E1 that up to sign

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A family of nonzero products in the cohomology p−5

d1 (d) = d1 (an =

h 1,0 h 5,0 h n,0 h n−1,1 h n−3,3 h n−4,4 ) p−5 an h 1,0 h 5,0 h n,0 h n−1,1 h n−3,3 d1 (h n−4,4 ) + · · · p−5 an h 1,0 h 5,0 h n,0 h n−1,1 h n−3,3 h 1,4 h n−5,5 + · · · d

= = 0,

p−5

d1 (f) = d1 (an =

h 2,0 h 5,0 h n,0 h n−2,2 h n−3,3 h n−4,4 ) p−5 an h 2,0 h 5,0 h n,0 h n−2,2 h n−3,3 d1 (h n−4,4 ) p−5 an h 2,0 h 5,0 h n,0 h n−2,2 h n−3,3 h 1,4 h n−5,5 f

= = 0,

+ ··· + ···

p−5

d1 (g1 ) = d1 (an = =

h 5,0 h 6,0 h n,0 h n−3,3 h n−4,4 h n−6,6 ) p−5 an h 5,0 h 6,0 d1 (h n,0 )h n−3,3 h n−4,4 h n−6,6 + · · · p−5 an h 1,0 h 5,0 h 6,0 h n−1,1 h n−3,3 h n−4,4 h n−6,6 + · · · g1

= 0, .. . p−5

d1 (gk ) = d1 (an = =

h 5,0 h k+5,0 h n,0 h n−3,3 h n−4,4 h n−(k+5),k+5 ) p−5 an h 5,0 h k+5,0 d1 (h n,0 )h n−3,3 h n−4,4 h n−(k+5),k+5 + · · · p−5 an h 1,0 h 5,0 h k+5,0 h n−1,1 h n−3,3 h n−4,4 h n−(k+5),k+5 + · · · gk

= 0, .. . p−5

d1 (gn−6 ) = d1 (an = =

h 5,0 h n−1,0 h n,0 h n−3,3 h n−4,4 h 1,n−1 ) p−5 an h 5,0 h n−1,0 d1 (h n,0 )h n−3,3 h n−4,4 h 1,n−1 + · · · p−5 an h 1,0 h 5,0 h n−1,0 h n−1,1 h n−3,3 h n−4,4 h 1,n−1 + ··· g n−6

= 0. p−5

We easily obtain that an

h 1,0 h 5,0 h n,0 h n−1,1 h n−3,3 h 1,4 h n−5,5 is not in d1 (f) or p−5

d1 (gi ) (1  i  n − 6). Similarly, an

d

h 2,0 h 5,0 h n,0 h n−2,2 h n−3,3 h 1,4 h n−5,5 is not in f

p−5

d1 (d) or d1 (gi ) (1  i  n −6); an h 1,0 h 5,0 h k+5,0 h n−1,1 h n−3,3 h n−4,4 h n−(k+5),k+5 gk is not in d1 (d), d1 (f) or d1 (gi ) (i = k). Thus we obtain that d1 (d), d1 (f), d1 (g1 ), · · · , d1 (gn−6 ) are linearly independent. Thus we have s+6,t (s,n)+s,(2n+1)( p−5)+8n−10

Er

=0

for r  2. The lemma follows.



Now we begin to prove the main theorem of this paper. Proof of Theorem 1.2 It is known that h 1,n , h 2,0 h 1,0 ∈ E 1∗,∗,∗ are all permanent cycles in the May spectral sequence and represent h n , g0 ∈ Ext ∗,∗ A (F p , F p ), respec-

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tively. Meanwhile, we also have that δ˜s+4 is represented by a4s h 4,0 h 3,1 h 2,2 h 1,3 ∈ s+4,t (s)+s,∗ in the May spectral sequence by Lemma 3.1. Thus we obtain that E1 1 a4s h 4,0 h 2,0 h 1,0 h 3,1 h 2,2 h 1,3 h 1,n ∈ E 1s+7,t (s,n)+s,9s+21 is a permanent cycle in the May spectral sequence and represents s+7,t (s,n)+s (F p , F p ). h n g0 δ˜s+4 ∈ Ext A

Case 1 n > 5 and s = p − 5. After direct computations, from (2.5) we obtain that May(d) = May(f) = May(g1 ) = May(g2 ) = · · · = May(gn−6 ) = (2n + 1)( p − 5) + 8n − 10 and p−5

May(a4

h 4,0 h 2,0 h 1,0 h 3,1 h 2,2 h 1,3 h 1,n ) = 9( p − 5) + 21.

By assumption that n  5, it follows (2n + 1)( p − 5) + 8n − 10 − [9( p − 5) + 21] = (2n + 1 − 9)( p − 5) + (8n − 31) > 1. p−5

From (2.2), we easily obtain that the permanent cycle a4 h 4,0 h 2,0 h 1,0 h 3,1 h 2,2 h 1,3 p+1,t ( p−5,n)+ p−5,(2n+1)( p−5)+8n−10 h 1,n is not in d1 (E 1 ), i.e., p−5

a4



p+1,t ( p−5,n)+ p−5,(2n+1)( p−5)+8n−10 . h 4,0 h 2,0 h 1,0 h 3,1 h 2,2 h 1,3 h 1,n ∈ d1 E 1

Meanwhile, from Lemma 3.3 we also have p−5

a4



p+1,t ( p−5,n)+ p−5,(2n+1)( p−5)+8n−10 h 4,0 h 2,0 h 1,0 h 3,1 h 2,2 h 1,3 h 1,n ∈ dr Er p−5

for r  2. Thus the permanent cycle a4 by any May differential, meaning that

h 4,0 h 2,0 h 1,0 h 3,1 h 2,2 h 1,3 h 1,n cannot be hit

p+2,t ( p−5,n)+ p−5 

h n g0 δ˜ p−1 = 0 ∈ Ext A

Fp, Fp .

Case 2 n > 5 and 0  s < p − 5, or n = 5 and 0  s < p − 5. From Lemma 3.2, we can easily obtain that the May E 1 -term E 1s+6,t (s,n)+s,∗ = 0, which implies Ers+6,t (s,n)+s,∗ = 0

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A family of nonzero products in the cohomology

for r ≥ 1. Consequently, the permanent cycle a4s h 4,0 h 2,0 h 1,0 h 3,1 h 2,2 h 1,3 h 1,n cannot be hit by any May differential. Thus in this case we have that h n g0 δ˜s+4 = 0. From the discussion above, Theorem 1.2 follows. 

As an immediate consequence of the main theorem above, we have the following Corollary 3.4 Let p  7, n  5 and 0  s < p − 4. Then the products h n g0 , 

h n δ˜s+4 and g0 δ˜s+4 are all nontrivial. References 1. Adams, J.F.: Stable homotopy and generalised homology. Univ. Chicago Press, Chicago (1974) 2. Aikawa, T.: 3-dimensional cohomology of the mod p Steenrod algebra. Math. Scand. 47(1), 91–115 (1980) 3. Cohen, R.: Odd primary infinite families in stable homotopy theory. Mem. Amer. Math. Soc. 30(242), viii+92 (1981) 4. Liu, X.: A nontrivial product in the stable homotopy groups of spheres. Sci. China Ser. A 47(6), 831–841 (2004) 5. Liu, X., Wang, H.: On the cohomology of the mod p Steenrod algebra. Proc. Japan Acad. Ser. A Math. Sci. 85(9), 143–148 (2009) 6. Liu, X., Zhao, H.: On a product in the classical Adams spectral sequence. Proc. Amer. Math. Soc. 137(7), 2489–2496 (2009) 7. Liulevicius, A.: The factorizations of cyclic reduced powers by secondary cohomology operations. Memo. Amer. Math. Soc. 42, 112 pp (1962) 8. Ravenel, D.C.: Complex cobordism and stable homotopy groups of spheres. Academic Press, Orlando (1986) 9. Toda, H.: On spectra realizing exterior parts of Steenord algebra. Topology 10, 55–65 (1971) (n) 10. Wang, X., Zheng, Q.: The convergence of α˜ s+4 h 0 h k . Sci. China Ser. A 41(6), 622–628 (1998)

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