THE COHOMOLOGY OF THE STEENROD ALGEBRA AND REPRESENTATIONS OF THE GENERAL LINEAR GROUPS NGUY^E~N H. V. HU_.NG Abstract.Let Trk be the algebraic transfer that maps from the coinvarian* *ts of certain GLk-representation to the cohomology of the Steenrod algebra.* * This transfer was defined by W. Singer as an algebraic version of the geometr* *ical transfer trk : iS*((BVk)+) ! iS*(S0). It has been shown that the algebra* *ic transfer is highly nontrivial, more precisely, that Trk is an isomorphis* *m for k = 1, 2, 3 and that Tr = kTrk is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from* * any degree d, and apply Sq0 repeatedly at most (k - 2) times, then we get in* *to the region, in which all the iterated squaring operations are isomorphis* *ms on the coinvariants of the GLk-representation. As a consequence, every fin* *ite Sq0-family in the coinvariants has at most (k - 2) non zero elements. Two applications are exploited. The first main theorem is that Trk is not an isomorphism for k 5. Furthermore, Trk is not an isomorphism in infinitely many degrees for ea* *ch k > 5. We also show that if Tr` detects a nonzero element in certain de- grees of Ker(Sq0), then it is not a monomorphism and further, Trk is not* * a monomorphism in infinitely many degrees for each k > `. The second main theorem is that the elements of any Sq0-family in the cohomology of the Steenrod algebra, except at most its first (k - 2) ele* *ments, are either all detected or all not detected by Trk, for every k. Applica* *tions of this study to the cases k = 4 and 5 show that Tr4 does not detect the th* *ree families g, D3, p0and Tr5 does not detect the family {hn+1gn| n 1}. 1.Introduction and statement of results There have been several efforts, implicit or explicit, to analyze the Steenro* *d alge- bra by using modular representations of the general linear groups. (See M`ui [2* *1, 22, 23], Madsen-Milgram [18], Adams-Gunawardena-Miller [3], Priddy-Wilkerson [26], Peterson [24], Wood [31], Singer [27], Priddy [25], Kuhn [14] and others.) In p* *ar- ticular, one of the most direct attempt in studying the cohomology of the Steen* *rod algebra by means of modular representations of the general linear groups was the surprising work [27] by W. Singer, which introduced a homomorphism, the so-call* *ed algebraic transfer, mapping from the coinvariants of certain representation of * *the general linear group to the cohomology of the Steenrod algebra. Let Vk denote a k-dimensional F2-vector space, and P H*(BVk) the primitive subspace consisting of all elements in H*(BVk), which are annihilated by every positive-degree operation in the mod 2 Steenrod algebra, A. Throughout the pape* *r, the homology is taken with coefficients in F2. The general linear group GLk := ____________ 1The work was supported in part by the National Research Program, Grant N014* *0801. 22000 Mathematics Subject Classification. Primary 55P47, 55Q45, 55S10, 55T15. 3Key words and phrases. Adams spectral sequences, Steenrod algebra, Modular * *representa- tions, Invariant theory. 1 2 NGUY^E~N H. V. HU_.NG GL(Vk) acts regularly on Vk and therefore on the homology and cohomology of BVk. Since the two actions of A and GLk upon H*(BVk) commute with each other, there are inherited actions of GLk on F2 H*(BVk) and P H*(BVk). In A [27], W. Singer defined the algebraic transfer T rk : F2 P Hd(BVk) ! Extk,k+dA(F2, F2) GLk as an algebraic version of the geometrical transfer trk : ßS*((BVk)+ ) ! ßS*(S0* *) to the stable homotopy groups of spheres. It has been proved that T rk is an isomorphism for k = 1, 2 by Singer [27] and for k = 3 by Boardman [4]. Among other things, these data together with the fact that T r = kT rk is an algebra homomorphism (see [27]) show that T rk is highly nontrivial. Therefore, the algebraic transfer is expected to be a useful tool i* *n the study of the mysterious cohomology of the Steenrod algebra, Ext*,*A(F2, F2). Direct calculating the value of T rk on any non-zero element is difficult (se* *e [27], [4], [11]). In this paper, our main idea is to exploit the relationship between* * the algebraic transfer and the squaring operation Sq0. It is well-known (see [17]) * *that there are squaring operations Sqi (i 0) acting on the cohomology of the Steen* *rod algebra, which share most of the properties with Sqi on the cohomology of space* *s. However, Sq0 is not the identity. On the other hand, there is an analogous squa* *ring operation Sq0, the Kameko one, acting on the domain of the algebraic transfer a* *nd commuting with the classical Sq0 on ExtkA(F2, F2) through the algebraic transfe* *r. We refer to Section 2 for its precise meaning. The key point is that the behaviors of the two squaring operations do not agr* *ee in infinitely many certain degrees, called k-spikes. A k-spike degree is a num* *ber that can be written as (2n1- 1) + . .+.(2nk - 1), but can not be written as a s* *um of less than k terms of the form (2n - 1). (See a discussion of this notion af* *ter Definition 3.1.) The following result is originally due to Kameko [12]: If m i* *s a k-spike, then fSq0: P H*(BVk)m-k_! P H*(BVk)m 2 0 is an isomorphism of GLk-modules, where fSqis certain GLk-homomorphism such 0 0 that Sq0 = 1 Sfq . (See Section 2 for an explanation of fSq.) GLk We recognize two phenomena on the universality and the stability of k-spikes: First, if we start from any degree d that can be written as (2n1-1)+. .+.(2nk-1* *), and apply the function ffik with ffik(d) = 2d + k repeatedly at most (k - 1) ti* *mes, then we get a k-spike; Secondly, k-spikes are mapped by ffik to k-spikes. So, w* *e have Theorem 1.1. Let d be an arbitrary non negative integer. Then 0 i-k+2 (fSq) : P H*(BVk)2k-2d+(2k-2-1)k! P H*(BVk)2id+(2i-1)k is an isomorphism of GLk-modules for every i k - 2. From the result of Carlisle and Wood [8] on the boundedness conjecture, one c* *an see that, for any degree d, there exists t such that 0 i-t (fSq) : P H*(BVk)2td+(2t-1)k! P H*(BVk)2id+(2i-1)k is an isomorphism of GLk-modules for every i t. However, this result does not confirm how large t should be. Theorem 1.1 shows that a rather small number STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 3 t = k - 2 commonly serves for every degree d. It will be pointed out in Remark * *6.5 that t = k - 2 is the minimum number for this purpose. An inductive property of k-spikes, which will also play a key role in the pap* *er, is that if m is a k-spike, then (2n - 1 + m) is a (k + 1)-spike for n big enoug* *h. Two applications of the study will be exploited in this paper. The first appl* *ica- tion is the following theorem, which is one of the paper's main results. Theorem 1.2. T rk is not an isomorphism for k 5. Furthermore, T rk is not an isomorphism in infinitely many degrees for each k > 5. In order to prove this theorem, using the notion of k-spike, we introduce the concept of critical element in ExtkA(F2, F2) in such a way that if d is the ste* *m of a critical element, then T rk is not an isomorphism either in degree d or in de* *gree 2d + k. Further, we show that if x is critical, then so is hnx for n big enough* *. Our inductive procedure starts with the initial critical element P h2 for k = 5. Combining Theorem 1.2 and the results by Singer [27], Boardman [4] and Bruner- H`a-Hu_.ng [7], we get Corollary 1.3. (i)T rk is an isomorphism for k = 1, 2 and 3. (ii)T rk is not an isomorphism for k 4. (iii)T rk is not an isomorphism in infinitely many degrees for k = 4 and k > * *5. Remarkably, we do not know whether the algebraic transfer fails to be a monom* *or- phism or fails to be an epimorphism for k > 5. Therefore, Singer's conjecture is still open. Conjecture 1.4. ([27]) T rk is a monomorphism for every k. The following theorem is related to this conjecture. Theorem 1.5. If T r` detects a critical element, then it is not a monomorphism and further, T rk is not a monomorphism in infinitely many degrees for each k >* * `. A family {ai| i 0} of elements in ExtkA(F2, F2) (or in F2 P H*(BVk)) is GLk called a Sq0-family if ai= (Sq0)i(a0) for every i 0. The root degree of a0 is* * the maximum non negative integer r such that Stem(a0) can be written in the form Stem(a0) = 2rd + (2r - 1)k, for some non negative integer d. The second application of our study is the following theorem, which is also o* *ne of the paper's main results. Theorem 1.6. Let {ai| i 0} be a Sq0-family in ExtkA(F2, F2) and r the root degree of a0. If T rk detects an for some n max{k - r - 2, 0}, then it detect* *s ai for every i n and detects aj modulo Ker(Sq0)n-j for max {k - r - 2, 0} j < * *n. A Sq0-family is called finite if it has only finitely many non zero elements. Corollary 1.7. (i)Every finite Sq0-family in F2 P H*(BVk) has at most GLk (k - 2) non zero elements. (ii)If T rk is a monomorphism, then it does not detect any element of a fini* *te Sq0-family in ExtkA(F2, F2) with at least (k - 1) non zero elements. The following is an application of Theorem 1.6 into the investigation of T r4. Proposition 1.8. Let {bi| i 0} and {ci| i 0} be the Sq0-families in Ext4A(F* *2, F2) with b0 one of the usual five elements d0, e0, p0, D3(0), p00, and c0 one of th* *e usual two elements f0, g1. 4 NGUY^E~N H. V. HU_.NG (i)If T r4 detects bn for some n 1, then it detects bi for every i 1. (ii)If T r4 detects cn for some n 0, then it detects ci for every i 0. Based on this event, we prove the following theorem by showing that T r4 does not detect g1, D3(0), D3(1), p00, p01. Theorem 1.9. T r4 does not detect any element in the three Sq0-families {gi| i 1}, {D3(i)| i 0} and {p0i| i 0}. That T r4 does not detect the family {gi| i 1} is due to Bruner-H`a-Hu_.ng * *[7]. Recently, T. N. Nam privately informed to prove that T r4 does not detect D3(0). Conjecture 1.10. T r4 is a monomorphism that detects all elements in Ext4A(F2, * *F2) except the ones in the three Sq0-families {gi| i 1}, {D3(i)| i 0} and {p0i|* * i 0}. The following theorem would complete our knowledge in Corollary 1.3 on whether T r5 is not an isomorphism in infinitely many degrees. Theorem 1.11. If hn+1gn is non zero, then it is not detected by T r5. It has been claimed by Lin [15] that hn+1gn is non zero for every n 1. The paper is divided into nine sections and organized as followed. Section 2 is a recollection of the Kameko squaring operation. In Section 3, we explain t* *he notion of k-spike and then study the Kameko squaring and its iterated operations in k-spike degrees. Section 4 deals with an inductive way of producing k-spike* *s, which plays a key role in the proofs of Theorems 1.1, 1.2, 1.5 and 1.6. In Sect* *ion 5, based on the concept of critical element, we prove Theorems 1.2 and 1.5. Sectio* *n 6 is devoted to the proofs of Theorems 1.1 and 1.6. Sections 7 and 8 are applicat* *ions to the study of the fourth and the fifth algebraic transfers. Final remarks and conjectures are given in Section 9. Acknowledgment: The research was in progress during my visit to Wayne State University, Detroit (Michigan) in the academic year 2002-2003. I would like to express my warmest thank to Lowell Hansen and all colleagues at the Department of Mathematics for their hospitality and for the wonderful working atmosphere. In particular, I am grateful to Robert Bruner, Daniel Frohardt, Kay Magaard and Sergey Shpectorov for fruitful conversations on the Ext groups and Modular Representations. 2.Preliminary on the squaring operation To make the paper self-contained, this section is a recollection of the Kameko squaring operation Sq0 on F2 P H*(BVk). The most important property of the GLk Kameko Sq0 is that it commutes with the classical Sq0 on Ext*A(F2, F2) (defined* * in [17]) through the algebraic transfer (see [4], [20]). This squaring operation is constructed as follows. As well known, H*(BVk) is the polynomial algebra, Pk := F2[x1, ..., xk], on k generators x1, ..., xk, each of degree 1. By dualizing, H*(BVk) = (a1, . .,.ak) is the divided power algebra generated by a1, . .,.ak, each of degree 1, where * *ai is dual to xi 2 H1(BVk). Here the duality is taken with respect to the basis of H*(BVk) consisting of all monomials in x1, . .,.xk. STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 5 In [12] Kameko defined a homomorphism fSq0: H*(BVk) ! H*(BVk), a(i1)1.a.(.ik)k7!a(2i1+1)1.a.(.2ik+1)k, where a(i1)1.a.(.ik)kis dual to xi11. .x.ikk. The following lemma is well known. 0 Lemma 2.1. fSq is a homomorphism of GLk-modules. See e. g. [7] for a proof. Further, there are two well known relations 0 2t 0 0 t Sq2t+1*fSq= 0, Sq* fSq= fSqSq*. 0 See [10] for an explicit proof. Therefore, fSq maps P H*(BVk) to itself. The Kameko Sq0 is defined by 0 Sq0 = 1 Sfq : F2 P H*(BVk) ! F2 P H*(BVk). GLk GLk GLk 0 0 The dual homomorphism fSq*: Pk ! Pk of fSq is obviously given by ( j1-1_ jk-1_ fSq0*(xj11. .x.jkk) = x12 . .x.2k , j1, ..., jk odd, 0, otherwise. Hence 0 _____ _____ Ker(fSq*) = Even, where Even denotes the vector subspace of Pk spanned by all monomials xi11. .x.* *ikk with at least one exponent it even. The following lemma is more or less obvious. Lemma 2.2. ([7]) Let k and d be positive integers. Suppose that each monomial xi11. .x.ikkof Pk in degree 2d + k with at least one exponent it even is hit. T* *hen fSq0*: (F2 Pk)2d+k! (F2 Pk)d A A is an isomorphism of GLk-modules. Here, as usual, a polynomial is called hit if it is A-decomposable in Pk. A proof of this lemma is sketched as follows. 0 Let s : Pk ! Pk be a left inverse of fSq*defined by s(xi11. .x.ikk) = x2i1+11.x.2.ik+1k. It should be noted that s does not commute with the doubling map on A, that is, in general _Sq2ts_6= sSqt. However, Im(Sq2ts - sSqt) Even. Let A+ denote the ideal of A consisting of all positive degree operations. Un* *der the hypothesis of the lemma, we have _____ + (A+ Pk + Even)2d+k (A Pk)2d+k. Therefore, the map _s: (F 2 Pk)d! (F2 Pk)2d+k A_ A s[X] = [sX] 6 NGUY^E~N H. V. HU_.NG is a well-defined linear map. Further, it is the inverse of Sfq0*: (F2 Pk)2d+k! (F2 Pk)d. A A 0 So, fSq*is an isomorphism in degree 2d + k. 3. The iterated squaring operations in k-spike degrees The following notion, which is due back to Kraines [13], formulates some spec* *ial degrees that we will mainly be interested in. Definition 3.1. A natural number m is called a k-spike if (a) m = (2n1- 1) + . .+.(2nk - 1) with n1, ..., nk > 0, and (b) m can not be written as a sum of less than k terms of the form (2n - 1). Note that k-spike is our terminology. Other authors write ~(m) = k to say m is a k-spike. (See e. g. Wood [32, Definition 4.4].) One easily checks e. g. that 20 is a 4-spike, 27 is a 5-spike and 58 is a 6-s* *pike. Let ff(m) denote the number of ones in the dyadic expansion of m. The followi* *ng two lemmas are more or less obvious, but useful later. Lemma 3.2. Condition (a) in Definition 3.1 is equivalent to ff(m + k) k m, m k (mod 2). Proof.Suppose m = (2n1- 1) + . .+.(2nk - 1) with n1, ..., nk > 0. Then m k = (21 - 1) + . .+.(21 - 1) (k terms). In addition, from m + k = 2n1+ . .+.2nk with n1, ..., nk > 0, it implies ff(m + k) k and m k (mod 2). The equality ff(m + k) = k occurs if and only if n1, ..., nk are different each* * other. Conversely, suppose that ff(m + k) k m and m k (mod 2). Let i = ff(m + k). Then we have m + k = 2m1 + . .+.2mi, where m1, ..., mi> 0, as m + k is even. If at least one exponent mj > 1, then we write (m + k) as a sum of (i + 1) te* *rms of 2-powers as follows m + k = 2m1 + . .2.mj-1+ 2mj-1 + . .+.2mi. This procedure can be continued if at least one of the exponents m1, ..., mj-1,* * mj- 1, ..., miis bigger than 1. After each step, the number of terms in the sum inc* *reases by 1. The procedure stops only in the case when the sum becomes m+k = 2+. .+.2 with the number of term is (m + k)=2 2k=2 = k. In particular, we reached at some step a sum of exactly k terms m + k = 2n1+ . .+.2nk with n1, ..., nk > 0, or equivalently m = (2n1- 1) + . .+.(2nk - 1). The lemma is proved. The following lemma helps to recognize k-spikes. STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 7 Lemma 3.3. A natural number m is a k-spike if and only if (i)ff(m + k) k m, m k (mod 2), and (ii)ff(m + i) > i for 1 i < k. Proof.From Lemma 3.2, if m satisfies (i), then m = (2n1- 1) + . .+.(2nk- 1) with n1, ..., nk > 0. Also by Lemma 3.2, if m satisfies (ii), then it can not be wri* *tten as a sum of less than k terms of the form (2n - 1). So, if m satisfies (i) and (ii), then it is a k-spike. Conversely, suppose m is a k-spike, then (i) holds by Lemma 3.2. It suffices to show (ii). Suppose the contrary ff(m + i) i for some i with 1 i < k. We then have ff(m + i) i < k m. Let us consider the two cases. Case 1: m i ( mod 2). Then, by Lemma 3.2, we get m = (2n1-1)+. .+.(2ni-1) with n1, ..., ni> 0. This contradicts to the definition of k-spike. Case 2: m i - 1 (mod 2). It implies i > 1. Indeed, if i = 1, combining the hypothesis ff(m + 1) 1 with the fact m + 1 is odd, we get m + 1 = 1. This contradicts to the hypothesis that m is a natural number. By Lemma 4.3 below, we have ff(m + (i - 1)) = ff(m + i) - 1 i - 1. As m i - 1 (mod 2), we apply Lemma 3.2 again to see that m can be written as a sum of (i - 1) terms of the form (2n - 1). This is also a contradiction. Combining the two cases, we see that if m is a k-spike, then (i) and (ii) hol* *d. The lemma follows. The following proposition is originally due to Kameko [12]. We give a proof of it to make the paper self-contained. Proposition 3.4. If m is a k-spike, then Sfq0*: (F2 Pk)m ! (F2 Pk)m-k_ A A 2 is an isomorphism of GLk-modules. Proof.By using Lemma 2.2, it suffices to show that any monomial R of Pk in degr* *ee m with at least one even exponent is hit. Such a monomial R can be written, up to a permutation of variables, in the form R = x1. .x.iQ2, with 0 i < k, where Q is a monomial in degree (m - i)=2. Ifmi = 0, then R = Q2mis simply in the image of Sq1. (It is also in the image* * of Sq __2, as R = Q2 = Sq __2Q.) So, it suffices to consider the case 0 < i < k. Let Ø be the anti-homomorphism in the Steenrod algebra. The so-called Ø-trick, which is known to Brown and Peterson in the mid-sixties, states that uSqn(v) Ø(Sqn)(u)v mod A+ M, for u, v in any A-algebra M. (See also Wood [31].) In our case, it claims that m-i_ R = x1. .x.iQ2 = x1. .x.iSq 2 Q is hit if and only if Ø(Sq m-i_2)(x1. .x.i)Q is. We will show Ø(Sq m-i_2)(x1. .* *x.i) = 0. 8 NGUY^E~N H. V. HU_.NG As A is a commutative coalgebra, Ø is a homomorphism of coalgebras (see [19, Proposition 8.6]). Then we have the Cartan formula X Ø(Sqn)(uv) = Ø(Sqi)(u)Ø(Sqj)(v). i+j=n Furthermore, it is shown by Brown and Peterson in [5] that æ 2q q Ø(Sqn)(xj) = xj0,,ifno=t2he-r1wforisome,qse, for xj in degree 1. So, in order to prove Ø(Sq m-i_2)(x1. .x.i) = 0 we need only to show that m-i* *_2 can not be written in the form m_-_i_= (2`1- 1) + . .+.(2`i- 1) 2 with `1, ..., `i 0. This equation is equivalent to m = (2`1+1- 1) + . .+.(2`i+1- 1). Since 0 < i < k, this equality contradicts to the hypothesis that m is a k-spik* *e. The proposition is completely proved. The following lemma is the base for an iterated application of Proposition 3.* *4. Lemma 3.5. If m is a k-spike, then so is (2m + k). Proof.(a) From the definition of k-spike, m = (2n1- 1) + . .+.(2nk - 1), for n1, ..., nk > 0. It implies that 2m + k = (2n1+1- 1) + . .+.(2nk+1 - 1). So, 2m + k satisfies the first condition in the definition of k-spike. (b) Also by this definition, we have ff(m + k - j) > k - j, for 1 j < k. Hence ff(2m + k + (k - 2j))= ff(2(m + k - j)) = ff(m + k - j) > k - j > k - 2j, ff(2m + k + (k - 2j + 1))= ff(2(m + k - j) + 1) = ff(2(m + k - j)) + 1 (by Lemma 4.3) = ff(m + k - j) + 1 > (k - j) + 1 > k - 2j + 1. Note that each i satisfying 1 i < k can be written either in the form i = k -* * 2j (for 1 j k-1_2) or in the form i = k - 2j + 1 (for 1 j k_2). So, the ab* *ove two inequalities show that ff(2m + k + i) > i, for 1 i < k. Thus, 2m + k satisfies the second condition in Definition 3.1. Combining parts (a) and (b), we see that 2m + k is a k-spike. STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 9 Remark 3.6. The converse of Lemma 3.5 is false. For instance, 27 is a 5-spike, whereas 11 = (27 - 5)=2 is not. Proposition 3.7. If m is a k-spike, then 0 i+1 (fSq) : P H*(BVk)m-k_2! P H*(BVk)2im+(2i-1)k is an isomorphism of GLk-modules for every i 0. Proof.If m is a k-spike, then by the dual of Proposition 3.4, we have an isomor* *phism of GLk-modules fSq0: P H*(BVk)m-k_! P H*(BVk)m . 2 On the other hand, from Lemma 3.5, if m is a k-spike, then so is 2im + (2i- 1* *)k for every i 0. Hence, applying repeatedly the dual of Proposition 3.4, we get* * an isomorphism of GLk-modules 0 i+1 (fSq) : P H*(BVk)m-k_2! P H*(BVk)2im+(2i-1)k. The proposition is proved. Corollary 3.8. If m is a k-spike, then (Sq0)i+1: (F2 P H*(BVk))m-k_! (F2 P H*(BVk))2im+(2i-1)k GLk 2 GLk is an isomorphism for every i 0. 4.Recognition of k-spikes In this section, we introduce an inductive way of producing k-spikes, which w* *ill play a key role in the proofs of Theorems 1.1, 1.2, 1.5 and 1.6 in the next two sections. Lemma 4.1. If m is a k-spike, then (2n - 1 + m) is a (k + 1)-spike for every n with 2n m + k - 1. To prove this lemma, we need the following two lemmas. Lemma 4.2. If 2n a, then ff(2n - 1 + a) ff(a). Proof.The proof proceeds by induction on ff(a). If ff(a) = 1, then a is a power* * of 2, say a = 2p 2n. We have 2n - 1 + 2p = 2n + (2p - 1) = 2n + (2p-1 + . .+.20). Thus ff(2n - 1 + 2p) = 1 + p 1 = ff(a). Suppose inductively that the lemma is valid for ff(a) = t. We now consider the case ff(a) = t + 1 > 1. That is a = 2nt+1+ 2nt+ . .+.2n1 with nt+1> nt> . .>.n1. Set b = 2nt+ . .+.2n1 < 2nt+1, then a = 2nt+1+ b, and ff(b) = t. From 2n a, it implies 2n > 2nt+1. So, we get ff(2n - 1 + a)= ff(2n + 2nt+1- 1 + b) = 1 + ff(2nt+1- 1 + b) 1 + ff(b) (by the inductive hypothesis) = 1 + t = ff(a). 10 NGUY^E~N H. V. HU_.NG The lemma is proved. The following lemma is an obvious observation. Lemma 4.3. If e is an even number, then ff(e + 1) = ff(e) + 1. Proof.Let ffi(a) denote the coefficient of 2iin the dyadic expansion of a. Then* *, as e is even, we obviously have ff0(e + 1) = 1, ff0(e) = 0, ff0(1) = 1, ffi(e + 1) = ffi(e), for i > 0. Hence, ff(e + 1) = ff(e) + ff(1) = ff(e) + 1. The lemma is proved. Proof of Lemma 4.1.(a) Since m = (2n1- 1) + . .+.(2nk - 1), we get (2n - 1) + m = (2n - 1) + (2n1- 1) + . .+.(2nk - 1). So the first condition in Definition 3.1 holds for (2n - 1 + m). (b) If 1 i < k, then 2n m + k - 1 m + i. By Lemma 4.2, we have ff(2n - 1 + m + i) ff(m + i) > i. The last inequality comes from the hypothesis that m is a k-spike. Finally, we need to show ff(2n - 1 + m + k) > k. Recall that, as m is a k-spi* *ke, then m k (mod 2). Hence, e = (2n - 1) + m + (k - 1) is even. By Lemma 4.3, we have ff(2n - 1 + m + k)= ff(2n - 1 + m + (k - 1) + 1) = ff(2n - 1 + m + (k - 1)) + 1. Now, applying Lemma 4.2 to the case 2n m + k - 1, we get ff(2n - 1 + m + (k - 1)) + 1 ff(m + (k - 1)) + 1 > (k - 1) + 1 = k. The last inequality comes from the fact that m is a k-spike. In summary, the second condition in Definition 3.1 holds for (2n - 1 + m). Combining parts (a) and (b), we see that (2n - 1 + m) is a (k + 1)-spike. The lemma is proved. Remark 4.4. Lemma 4.1 can not be improved in the meaning that the hypothesis 2n+1 m + k - 1 does not imply (2n - 1 + m) to be a (k + 1)-spike. This is the case of k = 5, m = 27 and 2n = 16, because 15 + 27 = 42 is not a 6-spike. The following corollary is a key point in the proof of Lemma 6.3 and therefore in the proofs of Theorems 1.1 and 1.6 . Corollary 4.5. 2k - k is a k-spike for every k > 0. Proof.We prove this by induction on k. The corollary holds trivially for k = 1. Suppose inductively that 2k - k is a k-spike. Then, as 2k > (2k - k) + k - 1, applying Lemma 4.2 to the case n = k and m = 2k - k, we have 2k+1 - (k + 1) = (2k - 1) + (2k - k) to be a (k + 1)-spike. The corollary follows. STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 11 5.The algebraic transfer is not an isomorphism for k 4 We first recall briefly the definition of the algebraic transfer. Let Pb1be * *the submodule of F2[x1, x-11] spanned by all powers xi1with i -1. The usual A- action on P1 = F2[x1] is canonically extended to an A-action on F2[x1, x-11] (s* *ee Adams [2], Wilkerson [30]). bP1is an A-submodule of F2[x1, x-11]. The inclusi* *on P1 bP1gives rise to a short exact sequence of A-modules: 0 ! P1 ! bP1! -1F2 ! 0 . Let e1 be the corresponding element in Ext1A( -1F2, P1). Singer set ek = e1 . . .e1 2 ExtkA( -kF2, Pk) (k times). Then, he defined T r*k: TorAk(F2, -kF2) ! TorA0(F2, Pk) = F2 Pk by T r*k(z) = ek\z. Its image is a submodule of (F2 Pk)* *GLk. A A The k-th algebraic transfer is defined to be the dual of T r*k. We will need to apply the following theoremnby D. Davis [9]. Let hn be the nonzero element in Ext1,2A(F2, F2). Theorem 5.1. ([9]) If x is a nonzero element in Extk,k+dA(F2, F2) with 4 d * *2j, then hnx 6= 0 for every n 2j + 1. The following concept plays a key role in this section. Definition 5.2. An nonzero element x 2 ExtkA(F2, F2) is called critical if (a) Sq0(x) = 0, and (b) 2Stem (x) + k is a k-spike. Note that, by Lemma 3.5, if Stem(x) is a k-spike, then so is 2Stem (x) + k. Lemma 5.3. If x 2 ExtkA(F2, F2) is critical, then so is hnx for every n with 2n max{4d2, d + k}, where d = Stem(x). Proof.First, we show that if x is critical, then Stem(x) > 0. Indeed, suppose t* *he contrary Stem(x) = 0, then x = hk0. As x is critical, Sq0(x) = Sq0(hk0) = hk1= * *0. This implies that k 4, as h1, h21, h31all are non zero, whereas h41= 0. Howev* *er, 2Stem (x) + k = k is not a k-spike for k 4, because it can be written as a sum k = 3 + 1 + . .+.1 of (k - 2) terms of the form (2n - 1). This contradicts to t* *he definition of critical element. Now we have Stem(x) > 0. Combining the facts that Sq0 is a monomorphism in positive stems of ExtkA(F2, F2) for k 4, and that x is critical, we get k > 4* *. As x is a non zero element of positive stem in ExtkA(F2, F2) with k > 4, by the vani* *shing line theorem (see [1]), we have Stem(x) > 7. So, x satisfies the hypothesis of Theorem 5.1 that d = Stem(x) 4. Let j be the smallest positive integer such that 2j d. Then, the smallest positive integer i with 2i d2 should be either 2j or 2j - 1. From the hypothes* *is 2n 4d2, it implies that 2n-2 d2. Hence, we get n - 2 i 2j - 1, or equivalently, n 2j + 1. Therefore, by Theorem 5.1, hnx 6= 0 if 2n 4d2. As Sq0 is a homomorphism of algebras, we have Sq0(hnx) = Sq0(hn)Sq0(x) = Sq0(hn) . 0 = 0. Since x is critical, m := 2d + k is a k-spike. We need to show that 2Stem (hn* *x) + (k + 1) is a (k + 1)-spike. We have Stem(hnx) = 2n - 1 + Stem(x) = 2n - 1 + d. 12 NGUY^E~N H. V. HU_.NG A routine calculation shows 2Stem (hnx) + (k + 1)= 2(2n - 1 + d) + (k + 1) = 2n+1 - 2 + (2d + k) + 1 = 2n+1 - 1 + m. By Lemma 4.1, this number is a (k + 1)-spike for every n with 2n+1 m + k - 1 = 2(d + k) - 1, or equivalently 2n d + k In summary, hnx is critical for every n with 2n max{4d2, d + k}. The lemma is proved. Remark 5.4. (a) Suppose hnx 6= 0 although 2n < 4(Stem (x))2. If x is critical and 2n Stem(x) + k, then hnx is also critical. (b) There is no critical element for k 4, as Sq0 is a monomorphism in posi* *tive stems of ExtkA(F2, F2) for k 4. Proposition 5.5. (i)For k = 5, there is at least one number, which is the stem of a critical element. (ii)For each k > 5, there are infinitely many numbers, which are stems of critical elements. Proof.For k = 5, P h2 2 Ext5,16A(F2, F2) is critical. Indeed, it is well known * *(see e. g. Tangora [29]) that Ext5,32A(F2, F2) = 0, so we get Sq0(P h2) = 0. Further, by Lemma 3.3, 2Stem (P h2) + 5 = 27 is a 5-spike. We can start the inductive argument of Lemma 5.3 with the initial critical el* *e- ment P h2. The proposition follows. The following theorem is also numbered as Theorem 1.2 in the introduction. Theorem 5.6. T rk is not an isomorphism for k 5. Furthermore, T rk is not an isomorphism in infinitely many degrees for each k > 5. Proof.In order to prove the theorem, by means of Proposition 5.5, it suffices to show that T rk is not an isomorphism either in degree d or in degree 2d + k, wh* *ere d denotes the stem of a critical element x 2 ExtkA(F2, F2). We consider the following two cases. Case 1: x is not in the image of T rk. Then, T rk is not an epimorphism in degree d. Case 2: x = T rk(y) for some y 2 F2 P H*(BVk). GLk From x 6= 0, it implies y 6= 0. We have a commutative diagram (F2 P H*(BVk))d _________-TrkExtk,k+d(F2, F2) GLk A | | | 0 | 0 |Sq |Sq | | |? |? (F2 P H*(BVk))2d+k _________-TrkExtk,2(k+d)(F2, F2) , GLk A STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 13 where the left vertical arrow is the Kameko Sq0 and the right vertical one is t* *he classical squaring operation. As m = 2d+k is a k-spike, by Corollary 3.8, the Kameko Sq0 is an isomorphism. So, from y 6= 0, we have z = Sq0(y) 6= 0. Now, by the commutativity of the diagram, we get T rk(z) = T rk(Sq0(y)) = Sq0(T rk(y)) = Sq0(x) = 0. This means that T rk is not a monomorphism in degree 2d + k. The theorem is completely proved. Remark 5.7. (a) We can show that F2 P H*(BV5)11 = 0. It implies that GL5 P h2 is not detected by T r5. (b) By Lemma 5.3, hnP h2 is critical for every n 9, as Stem(P h2) + 5 < 4(Stem (P h2))2 = 4 . 112 = 484 < 29 = 512. Also, by Remark 5.4, hnP h2 * *is critical for n = 4, 5, 6, as it is non zero (see [6]) and 24 Stem(P h2* *) + 5 = 16. R. Bruner privately claimed h7P h2 6= 0. It seems likely that h8P h2* * 6= 0. If so, by the same argument, these two elements are also critical. The following corollary is also numbered as Corollary 1.3 in the introduction. Corollary 5.8. (i)T rk is an isomorphism for k = 1, 2 and 3. (ii)T rk is not an isomorphism for k 4. (iii)T rk is not an isomorphism in infinitely many degrees for k = 4 and k > * *5. This result is due to Singer [27] for k = 1, 2, to Boardman [4] for k = 3, an* *d to Bruner-H`a-Hu_.ng [7] for k = 4. The fact that T r5 is not an isomorphism in de* *gree 9 is also due to Singer [27]. The remaining part is shown by Theorem 5.6. Our knowledge's gap on whether T r5 is not an isomorphism in infinitely many degrees will be studied in Section 8. The following theorem is also numbered as Theorem 1.5 in the introduction. Theorem 5.9. If T r` detects a critical element, then it is not a monomorphism and further, T rk is not a monomorphism in infinitely many degrees for each k >* * `. Proof.The proof proceeds by induction on k `. For k = `, suppose T r` detects a critical element x` 2 Ext`A(F2, F2). Then, by Case 2 in the proof of Theorem 5.6, T r` is not a monomorphism in degree 2Stem (x`) + `. By means of this argument, it suffices to show that if T rk detects a critical element xk, then T rk+1 detects infinitely many critical elements, whose stems * *are different each other. From the hypothesis, xk = T rk(yk) for some yk 2 F2 P H*(BVk). With ambi- GLk guity of notation, let hn also denote the element in F2 P H*(BV1), whose image GL1 under T r1 is the usual hn 2 Ext1A(F2, F2). As T r = kT rk is a homomorphism of algebras (see [27]), we have T rk+1(hnyk) = T r1(hn)T rk(yk) = hnxk. By Lemma 5.3, the element hnxk is critical for every n with 2n max{4d2, d + k* *}. 14 NGUY^E~N H. V. HU_.NG By the first part of the theorem, since T rk+1 detects the critical element h* *nxk, it is not a monomorphism in degree 2Stem (hnxk) + (k + 1) for every n with 2n max{4d2, d + k}. Thus, T rk+1 is not a monomorphism in infinitely many degrees. The theorem follows. 6. The stability of the iterated squaring operations The following theorem, which is also numbered as Theorem 1.1 in the introduc- tion, shows that Sq0 is eventually isomorphic on F2 P H*(BVk). More precisely, GLk it claims that if we start from any degree d of this module, and apply Sq0 repe* *at- edly at most (k - 2) times, then we get into the region, in which all the itera* *ted squaring operations are isomorphisms. Theorem 6.1. Let d be an arbitrary non negative integer. Then 0 i-k+2 (fSq) : P H*(BVk)2k-2d+(2k-2-1)k! P H*(BVk)2id+(2i-1)k is an isomorphism of GLk-modules for every i k - 2. In the theorem, for k = 1 we take the convention that 21-2d + (21-2 - 1)k = d. Let us denote 0 Sq0 (Sq0)-1(F2 P H*(BVk))d = lim{. .S.q-!(F2 P H*(BVk))2id+(2i-1)k-!. .}.. GLk -!i GLk The following corollary is an immediate consequence of Theorem 6.1. Corollary 6.2. Let d be an arbitrary non negative integer. Then, (i)the following iterated operation is an isomorphism for every i k - 2: (Sq0)i-k+2: F2 P H*(BVk)2k-2d+(2k-2-1)k! F2 P H*(BVk)2id+(2i-1)k; GLk GLk (ii) (Sq0)-1(F2 P H*(BVk))d ~=(F2 P H*(BVk))2k-2d+(2k-2-1)k; GLk GLk (iii)If d = 2k-2d0+ (2k-2 - 1)k for some non negative integer d0, then (Sq0)-1(F2 P H*(BVk))d ~=(F2 P H*(BVk))d. GLk GLk In order to prove Theorem 6.1, we need the following lemma. Let ffik denote the function given by ffik(d) = 2d + k. Lemma 6.3. If d is a non negative integer with ff(d + k) k, then ffik-1k(d) = 2k-1d + (2k-1 - 1)k is a k-spike. Proof.The lemma holds trivially for k = 1. Indeed, from the hypothesis ff(d+1) 1 it implies that d = 2n - 1 for some n. Then ffi01(d) = d = 2n - 1 is an 1-spi* *ke. We now consider the case of k 2. First, we observe that k 2k-1d + (2k-1 - 1)k k (mod 2) and ff(2k-1d + (2k-1 - 1)k + k) = ff(2k-1(d + k)) = ff(d + k) k. By Lemma 3.2, ffik-1k(d) = 2k-1d+(2k-1-1)k satisfies condition (a) of Definitio* *n 3.1. So, in order to prove the lemma, it suffices to show that ff(2k-1d + (2k-1 - 1)k + i) > i for 1 i <.k STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 15 We now work modulo 2k-1. First, we have 2k-1d + (2k-1 - 1)k (2k-1 - 1)k (mod 2k-1). Let k = 2nt+ . .+.2n1 be the dyadic expansion of k with nt> . .>.n1. We get (2k-1 - 1)k = 2k-1(2nt+ . .+.2n2) + (2k-1+n1- (2nt+ . .+.2n1)). Thus (2k-1 - 1)k 2k-1+n1- (2nt+ . .+.2n1) (mod 2k-1) 2k-1 - (2nt+ . .+.2n1) (mod 2k-1) 2k-1 - k (mod 2k-1), where 2k-1 - k 0 because of k 2. As a consequence, we get 2k-1d + (2k-1 - 1)k + i 2k-1 - k + i (mod 2k-1) for 1 i < k. Since k 2 and d 0 we have 2k-1d + (2k-1 - 1)k + i (2k-1 - 1)2 + 1 > 2k-1. From this inequality it implies that, in the dyadic expansion of 2k-1d + (2k-1 - 1)k + i, there is at least one nonzero term 2n with n k - 1. On the other han* *d, as 2k-1 - k + i < 2k-1 for 1 i < k, the dyadic expansion of 2k-1 - k + i is j* *ust a combination of the 2-powers 20, 21, ..., 2k-2. Therefore, in order to prove ff(2k-1d + (2k-1 - 1)k + i) > i for 1 i < k, we need only to show that ff(2k-1 - k + i) i. From Corollary 4.5, 2k-1 - (k - 1) is a (k - 1)-spike. Then we have ff(2k-1 - (k - 1) + j) > j for 1 j < k - 1. Set i = j + 1, we get ff(2k-1 - k + i) i for 2 i < k. In addition, it is obvious that ff(2k-1 - k + 1) 1. In summary, we have shown that ff(2k-1 - k + i) i for 1 i < k. The lemma is proved. Remark 6.4. (a) Lemma 6.3 can not be improved in the meaning that the number ffik-2k(d) = 2k-2d + (2k-2 - 1)k is not a k-spike in general. Indeed, taking d = 2t+ 1 - k with t big enough so that d 0, we have ff(2k-2d + (2k-2 - 1)k + (k - 1)) = ff(2t+k-2+ (2k-2 - 1)) = k - 1. By Lemma 3.3, 2k-2d + (2k-2 - 1)k is not a k-spike. 16 NGUY^E~N H. V. HU_.NG (b) However, a number could be a k-spike although it is not of the form ffik* *-1k(d) for any non negative integer d. For instance, this is the case of the fo* *llowing numbers with k = 4: Stem(e2) = 80, Stem (f1) = 40, Stem(p2) = 144, Stem(D3(2)) = 256,Stem (p02) = 288, where e2, f1, p2, D3(2), p02are the usual elements in Ext4A(F2, F2). Th* *is observation will be helpful in the proof of Proposition 7.2 below. Proof of Theorem 6.1.According to Wood's theorem [31] (it was originally Peter- son's conjecture), the primitive part P H*(BVk) is concentrated in the degrees * *d's with ff(d + k) k. This fact together with the equality ff(ffiik(d) + k) = ff(2i(d + k)) = ff(d + k) show that, if ff(d + k) > k, then the domain and the target of the homomorphism in the theorem both are zero. If ff(d + k) k, then the theorem is an immediate consequence of Lemma 6.3 and Proposition 3.7. The theorem is proved. Remark 6.5. Let k = 5 and d = 0. As ffi5-25(0) = 35, Theorem 6.1 claims that 0 i-3 (fSq) : P H*(BV5)35! P H*(BV5)5(2i-1) is an isomorphism of GL5-modules for i 3. In the final section we will see th* *at Sq0 : F2 P H*(BV5)15! F2 P H*(BV5)35 GL5 GL5 is not a monomorphism. This shows that Theorem 6.1 can not be improved in the meaning that (k - 2) is, in general, the minimum times that we must repeatedly apply Sq0 to get into "the isomorphism regionö f the iterated squaring operati* *ons. A family {ai| i 0} of elements in ExtkA(F2, F2) is called a Sq0-family if a* *i = (Sq0)i(a0) for every i 0. Sq0-family in F2 P H*(BVk) is similarly defined. GLk Definition 6.6. Let a0 2 ExtkA(F2, F2). The root degree of a0 is the maximum non negative integer r such that Stem(a0) can be written in the form Stem (a0) = ffirk(d) = 2rd + (2r - 1)k, for some non negative integer d. The following theorem is also numbered as Theorem 1.6 in the introduction. Theorem 6.7. Let {ai| i 0} be a Sq0-family in ExtkA(F2, F2) and r the root degree of a0. If T rk detects an for some n max{k - r - 2, 0}, then it detect* *s ai for every i n and detects aj modulo Ker(Sq0)n-j for max {k - r - 2, 0} j < * *n. Proof.It is easy to see that ff(Stem (ai) + k) = ff(2i(Stem (a0) + k)) = ff(Stem (a0) + k). Suppose ff(Stem (a0) + k) > k, then we have ff(Stem (ai) + k) > k for every i * * 0. By Wood's theorem [31] (it was originally Peterson's conjecture), P H*(BVk)t= 0 in any degree t with ff(t + k) > k. So, all elements of the family {ai| i 0} * *are not detected by T rk. STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 17 Now we consider the case where ff(Stem (a0) + k) k. We observe that ff(Stem (a0) + k) = ff(2r(d + k)) = ff(d + k) k. Set q = max{k - r - 2, 0}, and we have Stem (aq+1) = ffiq+1k(Stem (a0)) = ffiq+r+1k(d). Note that q + r + 1 = max{k - r - 2, 0} + r + 1 (k - r - 2) + r + 1 = k - 1. So, by Lemmas 6.3 and 3.5, Stem(aq+1) is a k-spike. According to Theorem 6.1, if c = Stem(aq), then 0 i-q (fSq) : P H*(BVk)c ! P H*(BVk)2i-qc+(2i-q-1)k is an isomorphism of GLk-modules for every i q. Suppose T rk detects an with n q, that is an = T rk(ean) for some eanin F2 P H*(BVk). If i n, then we set eai= (Sq0)i-n(ean). As the squaring GLk operations commute with each other through the algebraic transfer, we have ai = (Sq0)i-n(an) = (Sq0)i-nT rk(ean) = T rk(Sq0)i-n(ean) = T rk(eai). Thus, ai is detected by T rk for every i n. Next we consider j with max{k - r - 2, 0} j < n. Then we set eaj= [(Sq0)n-j]-1(ean). This makes sense, as it is shown above that (Sq0)n-j is isomorphic in degree of* * eaj. Again, as the squaring operations commute with each other through the algebraic transfer, we have (Sq0)n-jT rk(eaj)= T rk(Sq0)n-j(eaj) = T rk(ean) = an = (Sq0)n-j(aj). As a consequence, we get T rk(eaj) = aj (mod Ker (Sq0)n-j). This means that T rk detects aj modulo Ker(Sq0)n-j. The theorem is proved. Remark 6.8. (a) Under the hypothesis of Theorem 6.7, let a0i= T rk(Sq0)i-n(ean) for every i max {k - r - 2, 0} no matter whether i n or i < n. Then we get a new Sq0-family {a0i| i max{k - r - 2, 0}}, whose every element is detected by T rk and æ a0i= ai,a 0 n-ifii n, i(mod Ker (Sq ) i),f i < n. The new Sq0-family is called the adjustment of the original one. (b) Theorem 6.7 is still valid and can be shown by the same proof if we repl* *ace max{k - r - 2, 0} by any number q such that Stem(aq+1) is a k-spike. This remark will be useful in the proof of Proposition 7.2 for the case k = 4. 18 NGUY^E~N H. V. HU_.NG Corollary 6.9. Let {ai| i 0} be a Sq0-family in ExtkA(F2, F2) and r the root degree of a0. Suppose the classical Sq0 is a monomorphism in the stems of the elements {ai| i max{k-r-2, 0}}. If T rk detects an for some n max{k-r-2, 0} then it detects ai for every i max{k - r - 2, 0}. A Sq0-family is called finite if it has only finitely many non zero elements,* * infinite if all of its elements are non zero. The following is also numbered as Corollar* *y 1.7 in the introduction. Corollary 6.10. (i)Every finite Sq0-family in F2 P H*(BVk) has at most GLk (k - 2) non zero elements. (ii)If T rk is a monomorphism, then it does not detect any element of a fini* *te Sq0-family in ExtkA(F2, F2) with at least (k - 1) non zero elements. Proof.(i) Suppose that {eai| i 0} is a Sq0-family in F2 P H*(BVk) with at l* *east GLk (k-1) non zero elements. Then ea0, ea1, ..., eak-2are its first (k-1) non zero * *elements. Set d = deg(ea0), then deg(eak-2) = 2k-2d + (2k-2 - 1)k. So, by Corollary 6.2, (Sq0)i-k+2: F2 P H*(BVk)2k-2d+(2k-2-1)k! F2 P H*(BVk)2id+(2i-1)k GLk GLk is an isomorphism for every i k - 2. Therefore, from eak-26= 0 it implies that eai= (Sq0)i-k+2(eak-2) is non zero for every i k - 2. Thus, the Sq0-family is infinite. (ii) Let a0, a1, ..., ak-2 be the last (k - 1) non zero elements of the given f* *inite Sq0- family in ExtkA(F2, F2). As ak-2 is the last non zero element in the Sq0-family* *, we have Sq0(ak-2) = 0. Set d = Stem(a0), then by Lemma 6.3, 2Stem (ak-2) + k = 2k-1d + (2k-1 - 1)k ia a k-spike. So, ak-2 is critical. Suppose the contrary that T rk detects some (non zero) element in the Sq0-fam* *ily. Then, as the squaring operations commute with each other through the algebraic transfer, T rk also detects the critical element ak-2. According to Theorem 5.9* *, this contradicts to the hypothesis that T rk is a monomorphism. The corollary is proved. 7.On behavior of the fourth algebraic transfer This section is an application of the previous section into the study of T r4* *. We refer to [29], [6], [16] for an explanation of the generators of Ext4A(F2, F2). It has been known (see [16]) that the graded module Ext4A(F2, F2) is generated by hihjh`hm , hicj, di, ei, fi, gi+1, pi, D3(i), p0iand subject to the relation* *s: hihi+1= 0, hih2i+2= 0, h3i= h2i-1hi+1, h2ih2i+3= 0,hicj = 0 fori = j - 1, j, j + 2, j + 3. The following is also numbered as Conjecture 1.10 in the introduction. Conjecture 7.1. T r4 is a monomorphism that detects all elements in Ext4A(F2, F* *2) except the ones in the three Sq0-families {gi| i 1}, {D3(i)| i 0} and {p0i|* * i 0}. That T r4 does not detect the family {gi| i 1} is due to Bruner-H`a-Hu_.ng * *[7]. Recently, T. N. Nam privately informed to prove that T r4 does not detect the element D3(0). The following proposition, which is also numbered as Proposition 1.8 in the introduction, is an attempt to prepare for a proof of Conjecture 7.1. STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 19 Proposition 7.2. Let {bi| i 0} and {ci| i 0} be the Sq0-families in Ext4A(F* *2, F2) with b0 one of the usual five elements d0, e0, p0, D3(0), p00, and c0 one of th* *e usual two elements f0, g1. (i)If T r4 detects bn for some n 1, then it detects bi for every i 1. (ii)If T r4 detects cn for some n 0, then it detects ci for every i 0. Proof.Although the stems of b2 and c1 can not be written as ffi34(d) for some n* *on negative integer d (except for b2 = d2 and c1 = g2), it is easy to check by usi* *ng Lemma 3.3 that they all are 4-spikes. Following part (b) of Remark 6.8, we can show this proposition by the same argument as given in the proof of Theorem 6.7. Furthermore, as Sq0 is a monomor- phism in positive stems of Ext4A(F2, F2) (see e. g. [16]), the proposition has* * the strong formulation liked Corollary 6.9. The proposition is proved. By means of Proposition 7.2, to prove Conjecture 7.1 it suffices to show that (1) T r4 detects d0, d1, e0, e1, f0, p0, p1; (2) T r4 does not detect g1, D3(0), D3(1), p00, p01; and (3) T r4 is a monomorphism. The following theorem is also numbered as Theorem 1.9 in the introduction. Theorem 7.3. T r4 does not detect any element in the three Sq0-families {gi| i 1}, {D3(i)| i 0} and {p0i| i 0}. Outline of proof.First, we show that F2 P H*(BV4) is zero in degree 20. So, T* * r4 GL4 does not detect g1 of stem 20 and therefore, by Proposition 7.2, does not detec* *t any element in the Sq0-family {gi| i 1}. (Notice again that this part of the theo* *rem is due to Bruner-H`a-Hu_.ng [7].) Secondly, we show that F2 P H*(BV4) is zero in degrees 61 and 69 and has GL4 dimension 1 in degrees 126 = 2 . 61 + 4 and 142 = 2 . 69 + 4. Note that, as T r1 detects the family {hn| n 0} (see [27]), the homomorphis* *m of algebras T r = kT rk detects the subalgebra generated by the family {hn| n 0* *}. So, T r4 definitely sends the two generators of its domain in degrees 126 and 1* *42 to the nonzero elements h20h26and h20h4h7 respectively. Therefore, the four elemen* *ts D3(0), p00, D3(1), p01of respectively stems 61, 69, 126, 142 are not detected b* *y T r4. The theorem is proved by combining this fact and Proposition 7.2. 8. An observation on the fifth algebraic transfer From Corollary 5.8, the following conjecture naturally comes up. Conjecture 8.1. T r5 is not an isomorphism in infinitely many degrees. The facts that gn is not detected by T r4 and that T r = kT rk is a homomor- phism of algebras do not imply that hign is not detected by T r5. For instance, h0g1 = h2e0 and h1g1 = h2f0 are presumably detected by T r5, as e0 and f0 are expectedly detected by T r4. The purpose of this section is to prove the following, which is also numbered* * as Theorem 1.11 in the introduction. Theorem 8.2. If hn+1gn is non zero, then it is not detected by T r5. 20 NGUY^E~N H. V. HU_.NG Outline of proof.We first observe that, as Sq0 is a homomorphism of algebras, {hn+1gn| n 1} is a Sq0-family, that is (Sq0)n-1(h2g1) = hn+1gn, for every n 1. Next, using Lemma 3.3 we easily show that Stem(h2g1) = 23 is not a 5-spike, but ffi5(23) = 2 . 23 + 5 = 51 is. So, by Proposition 3.7, 0 i (fSq) : P H*(BV5)23! P H*(BVk)2i.23+(2i-1)5 is an isomorphism of GL5-modules for every i 0. In addition, a routine computation shows that F2 P H*(BV5)23= 0. GL5 As a consequence, we get F2 P H*(BV5)2i.23+(2i-1)5= 0, GL5 for every i 0. So, the domain of T r5 is zero in the degree that equals to Stem (hn+1gn) = 2n-1 . 23 + (2n-1 - 1)5, for every n 1. Therefore, if hn+1gn is non zero, then it is not detected by T r5. The theorem is proved. Corollary 8.3. If hn+1gn is non zero for every n 1, then T r5 is not an epimo* *r- phism in infinitely many degrees. The corollary's hypothesis is claimed to be true by Lin [15]. So, Conjecture * *8.1 is established. Remark 8.4. As h3g2 = h5g1 (see [29]) and Sq0 is a homomorphism of algebras, Theorem 8.2 also shows that if hn+4gn is non zero, then it is not detected by T* * r5. Which elements in Ext5A(F2, F2) are detected by T r5? This question can partially be answered by using the fact that T r = kT rk is an algebra homomorphism and the information on elements detected by T rk for k 4. For instance, h3D3(0) = h0d2 (see [6]) is presumably detected by T r5, a* *s h0 is detected by T r1 and d2 is expectedly detected by T r4 (see Conjecture 7.1). Based on Theorem 6.7 and concrete calculations, the following conjecture pres* *ents some "new" families, which are expectedly detected by T r5. Conjecture 8.5. T r5 detects every element in the Sq0-families initiated by the classes n, x, h0g2, D1, H1, h1D3(0), h2D3(0), Q3, h4D3(0), h6g1, h0g3 of stems * *31, 37, 44, 52, 62, 62, 64, 67, 76, 83, 92 respectively. Conjectures 8.5 and 7.1 together with the fact that T r = kT rk is an algebra homomorphism predict that T r5 detects all Sq0-families initiated by the classe* *s of stems < 125, except possibly the three families, which are respectively initiat* *ed by P h1, P h2 and h0p0. Since Sq0(P h1) = h2g1, every element of the Sq0-family initiated by P h1 is not detected by T r5 (see [27] for P h1 and Theorem 8.2 for hn+1gn). It has been known that T r5 does not detect the Sq0-family of exactly * *one non zero element {P h2} (see Remark 5.7). We have no prediction on whether the Sq0-family initiated by h0p0of stem 69 is detected or not. STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 21 9.Final Remarks Remark 9.1. We still do not know whether T rk fails to be a monomorphism or fails to be an epimorphism for k > 5. If Singer's Conjecture 1.4 that T rk is a monomorphism for every k is true, then the algebraic transfer does not detect t* *he kernel of Sq0 in k-spike degrees. This leads us to the study of the kernel of Sq0 in F2 P H*(BVk). The map GLk fSq0: P H*(BVk) ! P H*(BVk) is obviously injective. Taking this event together with Corollary 3.8 into acco* *unt, one would expect that the Kameko map 0 Sq0 = 1 Sfq : F2 P H*(BVk) ! F2 P H*(BVk) GLk GLk GLk is also a monomorphism. However, this is false. Indeed, P H*(BV5) has dimen- sion 432 and 1117 in degrees 15 and 35 respectively, while F2 P H*(BV5) has GL5 dimension 2 and 1 in degrees 15 and 35 respectively. Combining these data with the fact that Ext5,5+15A(F2, F2) = Span{h40h4, h1d0} and the technique in the proof of Theorem 5.6, we claim Remark 9.2. (a) There is an element t5 2 F2 P H*(BV5) in degree 15 such GL5 that Sq0(t5) = 0 and T r5(t5) 6= 0. (b) If tk 2 F2 P H*(BVk) is a positive degree element with Sq0(tk) = 0 GLk and T rk(tk) 6= 0, then Sq0(hntk) = 0 and T rk(hntk) 6= 0 for every n wi* *th 2n 4(Stem (tk))2. As an immediate consequence, we have Corollary 9.3. (i)Ker(Sq0) \ (F2 P H*(BVk)) is nonzero for k = 5 and GLk has an infinite dimension for k > 5. (ii)T rk detects a non zero element in the kernel of Sq0 for k = 5 and infin* *itely many elements in this kernel for each k > 5. It has been known (see [27], [4]) that Sq0 is injective on F2 P H*(BVk) for GLk k 3. Conjecture 9.4. Sq0 is a monomorphism in positive degrees of F2 P H*(BV4). GL4 In other words, Sq0 is a monomorphism in positive degrees of F2 P H*(BVk) if GLk and only if k 4. The following is an analogue of Corollary 6.2 and is related to Corollary 6.1* *0. Conjecture 9.5. (Sq0 is eventually isomorphic on the Ext groups.) Let Im(Sq0)i denote the image of (Sq0)i on ExtkA(F2, F2). There is a number t depending on k such that (Sq0)i-t: Im(Sq0)t! Im(Sq0)i is an isomorphism for every i > t. 22 NGUY^E~N H. V. HU_.NG In other words, Ker(Sq0)i = Ker(Sq0)t on ExtkA(F2, F2) for every i > t. As a consequence, any finite Sq0-family in ExtkA(F2, F2) has at most t non zero elem* *ents. Is the conjecture true for t = k - 2? An observation on the known generators of the Ext groups supports the above conjecture with t much smaller than k - 2. It also leads us to the question on whether Sq0 is an isomorphism on Im(Sq0)t F2 P H*(BVk) for some t < k - 2. (This question has an affirmative answer GLk given by Corollary 6.2 for t = k - 2.) References [1]J. F. Adams, A periodicity theorem in homological algebra, Proc. Cambridge * *Philos. Soc. 62 (1966), 365-377. [2]J. F. Adams, Operations of the nth kind in K-theory, and what we don't know* * about RP1 , New Developments in Topology, G. Segal (ed.), London Math. Soc. Lect. Note S* *eries 11 (1974), 1-9. [3]J. F. Adams, J. H. Gunawardena and H. Miller The Segal conjecture for eleme* *ntary Abelian p-groups, Topology 24 (1985), 435-460. [4]J. M. Boardman, Modular representations on the homology of powers of real p* *rojective space, Algebraic Topology: Oaxtepec 1991, M. C. Tangora (ed.), Contemp. Math. 146 (* *1993), 49- 70. [5]E. Brown and F. P. Peterson, H*(MO) as an algebra over the Steenrod algebra* *, Notas Mat. Simpos. 1 (1975), 11-21. [6]R. R. Bruner, The cohomology of the mod 2 Steenrod algebra: A computer calc* *ulation, WSU Research Report 37 (1997), 217 pages. [7]R. R. Bruner, L^e M. H`a and Nguy^e~n H. V. Hu_.ng, On behavior of the alge* *braic transfer, Submitted. [8]D. P. Carlisle and R. M. W. Wood, The boundedness conjecture for the action* * of the Steenrod algebra on polynomials, Adams Memorial Symposium on Algebraic Topology 2, N.* * Ray and G. Walker (ed.) London Math. Soc. Lect. Note Series 176 (1992), 203-216. [9]D. M. Davis, An infinite family in the cohomology of the Steenrod algebra, * *J. Pure Appl. Algebra 21 (1981), 145-150. [10]Nguy^e~n H. V. Hu_.ng, Spherical classes and the algebraic transfer, Trans.* * Amer. Math. Soc. 349 (1997), 3893-3910. [11]Nguy^e~n H. V. Hu_.ng, The weak conjecture on spherical classes, Math. Zeit* *. 231 (1999), 727-743. [12]M. Kameko, Products of projective spaces as Steenrod modules, Thesis, Johns* * Hopkins Uni- versity 1990. [13]D. Kraines, On excess in the Milnor basis, Bull. London Math. Soc. 3 (1971)* *, 363-365. [14]N. J. Kuhn, Generic representations of the finite general linear groups and* * the Steenrod algebra, Amer. Jour. Math. 116 (1994), 327-360. [15]W. H. Lin, Private communication, December 2002. [16]W. H. Lin and M. Mahowald, The Adams spectral sequence for Minami's theorem* *, Contemp. Math. 220 (1998), 143-177. [17]A. Liulevicius, The factorization of cyclic reduced powers by secondary coh* *omology opera- tions, Mem. Amer. Math. Soc. 42 (1962). [18]I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordi* *sm of manifolds, Ann of Math. Studies, No. 92, Princeton Univ. Press, 1979. [19]J. Milnor and J. Moore, On the structure of Hopf algebras, Ann. of Math. 81* * (1965), 211-264. [20]N. Minami, The iterated transfer analogue of the new doomsday conjecture, T* *rans. Amer. Math. Soc. 351 (1999), 2325-2351. [21]Hu`ynh M`ui, Modular invariant theory and cohomology algebras of symmetric * *groups, Jour. Fac. Sci. Univ. Tokyo, 22 (1975), 310-369. [22]Hu`ynh M`ui, Dickson invariants and Milnor basis of the Steenrod algebra, T* *opology, theory and application, Coll. Math. Soc. Janos Bolyai 41, North Holland (1985), 345* *-355. STEENROD ALGEBRA AND REPRESENTATIONS OF GENERAL LINEAR GROUPS 23 [23]Hu`ynh M`ui, Cohomology operations derived from modular invariants, Math. Z* *eit. 193 (1986), 151-163. [24]F. P. Peterson, Generators of H*(RP1 ^ RP1 ) as a module over the Steenrod * *algebra, Abstracts Amer. Math. Soc., No 833, April 1987. [25]S. Priddy, On characterizing summands in the classifying space of a group, * *I, Amer. Jour. Math. 112 (1990), 737-748. [26]S. Priddy and C. Wilkerson, Hilbert's theorem 90 and the Segal conjecture f* *or elementary abelian p-groups, Amer. Jour. Math. 107 (1985), 775-785. [27]W. M. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), * *493-523. [28]W. M. Singer, On the action of Steenrod squares on polynomial algebras, Pro* *c. Amer. Math. Soc. 111 (1991), 577-583. [29]M. C. Tangora, On the cohomology of the Steenrod algebra, Math. Zeit. 116 (* *1970), 18-64. [30]C. Wilkerson, Classifying spaces, Steenrod operations and algebraic closure* *, Topology 16 (1977), 227-237. [31]R. M. W. Wood, Steenrod squares of polynomials and the Peterson conjecture,* * Math. Proc. Cambridge Phil. Soc. 105 (1989), 307-309. [32]R. M. W. Wood, Problems in the Steenrod algebra, Bull. London Math. Soc. 30* * (1998), 449-517. Current Address: Department of Mathematics, Wayne State University 656 W. Kirby Street, Detroit, MI 48202 (USA) E-mail address: nhvhung@math.wayne.edu Permanent Address: Department of Mathematics, Vietnam National University, Hanoi 334 Nguy^e~n Tr~ai Street, Hanoi, Vietnam E-mail address: nhvhung@vnu.edu.vn