SOME ACYCLIC RELATIONS IN THE LAMBDA ALGEBRA MIZUHO HIKIDA Abstract. We consider the relations !fl = 0 2 , and show that if !ff = 0 then ff = flfi for some fi. These relations give the acyclic chain complex -fl! -!! . We consider various cases, e.g. ! = ~n and fl = ~2n+1. Especially, we consider the case ! = wn = d~n for n = 2e+r+ 2e - 1, where fl = (he+r)r. 1. Introduction Consider the stable homotopy groups of the sphere ß*(S0) localized at prime 2. We have the 2-local Adams spectral sequence converging to ß*(S0) with E2-term Ext s,tA(Z=2, Z=2) = Hs,t( ) by [2]. Moreover, contains a subcomplex (n) whose cohomology is the E2-term of the unstable Adams spectral sequence converging to the 2-component of the unstable homotopy groups of Sn. There are corresponding p-local versions of algebra that we will not consider. The lambda algebra (at the prime p = 2) is a bigraded Z=2-algebra with generators ~n 2 1,n+1(n 0) and relations X `n - 1 - j ' (1) ~i~2i+1+n = ~i+n-j~2i+1+j (i, n 0) j 0 j with differential X `n - j ' (2) d~n = ~n-j~j-1 (n 0). j 1 j We refer to [9] for these relations and [2, 5 ] for that d is a well- defined endomorphism of . For a sequence I = (n1, n2, . .,.ns) of non-negative integers, a monomial ~I = ~n1~n2 . .~.nsis said to be ad- missible if 2ni ni+1 for 1 i s - 1. The admissible monomials form an additive basis of by [2, 5]. (n) is the subcomplex spanned by the admissible monomials with n1 < n (cf. [2, 9]). By [9], ____________ 2000 Mathematics Subject Classification: 55Q40 Key words and Phrases: Lambda algebra, Homotopy group of sphere, EHP sequence. 1 2 MIZUHO HIKIDA there is a unique differential algebra endomorphism ` : ! with `(~n) = ~2n+1. This ` is usually called Sq0. See [6] for a recent treat- ment of the lambda algebra. The Adem relation ~m ~2m+1 = 0 gives a chain complex of right modules, using left-multiplication by ~m and ~2m+1 . This complex, and an unstable analogue, are acyclic: Theorem 1.1. The following chain complexes are acyclic: ~2n+1^ ~n^ --- - ! --! , ~2n+1^ ~n^ (p + 2n + 3) --- - ! (p + 1) --! (p - n), for p 2n + 1. For p < 2n + 1, the composite (p + 1) -E! -~n^-! is injective. The unstable maps above are defined in Lemma 2.2. The unstable composition formulas in x2 (of Wang, Mahowald and Singer) are crucial to our proofs. Furthermore, in Theorem 1.3 below, we prove the following chain complex of right modules is acyclic: (~3)2^ (~1,~0)^ --- - ! --- - - ! . This implies that the following chain complex, defined by Proposition 2.3, is acyclic: (h2)2^ w4^ (15) --- - ! (7) --! (4), where hi = ~2i-1, wn = d~n (cf. Theorem 1.5). The unstable maps above are well-defined by Singer's result (Proposition 2.3 below which extends Wang's earlier result), which we use heavily. We have many other, more complicated, acyclic chain complexes, e.g. (cf. 1.4 and 1.6): (hi+2)2^ (hi+1,hi)^ --- - - ! --- - - - ! (hi+2)2^ i+3 w2i+2+2i-1^ i+2 i (2i+4 - 1) --- - - ! (2 - 1) --- - - - - - !(2 + 2 - 1) (~2,~0)^ -~5~3^---! --- - - ! (20) -~5~3^---! (10) -w6^-! (6) Now we collect some acyclic chain complexes systematically. For integers n1 > . .>.nr 0, we denote fl(n1, . .,.nr) = `(~n1) . .`.r(~nr). Theorem 1.2. If ~nifl(n1, . .,.nr) = 0 for 1 i r, then the follow- ing chain complexes are acyclic: fl(n1,...,nr)^ (~n1,...,~nr)^r --- - - - - ! --- - - - - - !i=1 , SOME ACYCLIC RELATIONS IN THE LAMBDA ALGEBRA 3 fl(n1,...,nr)^ (~n1,...,~nr)^r (p + 1 + tr) --- - - - - ! (p + 1) --- - - - - - !i=1 (p - ni), P r for p 2n1 + 1, where tr = i=12i(ni+ 1). In the case ni = 2e+r-i-1 (e 0, 1 i r), we have fl(n1, . .,.nr) = (he+r)r and the assumption of Theorem above is satisfied. Theorem 1.3. The following chain complexes are acyclic: (he+r)r^ (he+r-1,...,he)^r --- - - ! --- - - - - - - !i=1 , (he+r)r^ (he+r-1,...,he)^r e+r-i (p + 1 + r2e+r) --- - - ! (p + 1) --- - - - - - - !i=1 (p - 2 + 1), for p 2e+r - 1. In the case ni = 2e+r-i+1 - 2e - 1 (e 0, 1 i r), we have fl(n1, . .,.nr) = ~2e+r+1-2e+1-1. .~.2e+r+1-2e+i-1. .~.2e+r+1-2e+r-1. We denote this element by ke,r. By Lemma 3.5, dke,r= 0 and the assumption of Theorem 1.2 is satisfied. Theorem 1.4. The following chain complexes are acyclic: ke,r^ (~n1,...,~nr)^r --- ! --- - - - - - !i=1 , for ni = 2e+r-i+1 - 2e - 1 (e 0, 1 i r), ke,r^ (~n1,...,~nr)^r (p + 1 + tr) --- ! (p + 1) --- - - - - - !i=1 (p - ni), for p 2e+r+1 - 2e+1 - 1, where tr = (r - 1)2e+r+1 + 2e+1. Using these acyclic chain complexes, we get the main theorems in this paper. Theorem 1.5. For n = 2e+r + 2e - 1 (e 0, r 1), the following is an acyclic chain complex. (he+r)r^ e+r-1 wn^ e (2n + 1 + u) --- - - ! (2n + 1 - 2 ) --- ! (n - 2 + 1), where u = (r - 1)2e+r + 2e+r-1. Theorem 1.6. For n = 2e+r+1 - 2e - 1 (e 0, r 1), the following is an acyclic chain complex. ke,r^ e+r e wn^ e (2n + 1 + u) --- ! (2n + 1 - 2 + 2 ) --- ! (n - 2 + 1), where u = (r - 2)2e+r+1 + 2e+r + 2e+1 + 2e. 4 MIZUHO HIKIDA Note that if r = 1 then Theorems 1.5 and 1.6 gives the same com- plexes, because 2e+1 + 2e - 1 = 2e+1+1 - 2e - 1, he+1 = ke,1, 2e+1-1 = 2e+1 - 2e. P n-j Since we can calculate wn = j j ~n-j~j-1 for n = 2e+r 2e - 1 explicitly, we can conclude Theorems 1.5-6. In fact, if wnff = 0 and ff is low-dimensional, then we get ~j-1ff = 0 for each j with n-jj = 1, and we can apply Theorems 1.2-3. In the case n 6= 2e+r 2e - 1, we can calculate wn partially, and get only a äp rtial acyclicity" result, which is too technical to state in this paper. Before closing the introduction we compare with the possible acyclic relations in the Steenrod algebra A (cf. [5]). The sequence of left A- modules Sq2n-1 Sqn A --- - !A --! A is exact for n = 1 and 2 (as is well-known from A-module resolutions of the spectra KZ and bo), but not exact for any odd n > 1, as Sq1 is in the homology. The sequence of right A-modules Sqn Sq2n-1 A --! A --- - !A is not exact for n = 3, because Sq4Sq2Sq1 is in the homology: Sq5Sq4Sq2Sq1 = Sq7Sq2Sq2Sq1 = Sq7Sq3Sq1Sq1 = 0, but Sq3Sq4 = Sq7, and Sq3Sq3Sq1 = Sq5Sq1Sq1 = 0. Adams and Margolis [1] proved there are exact sequences of right A-modules Pst Pst A -! A -! A s for 0 s < t, where Pts2 A is the Milnor-basis dual of ,2t, but their proof are quite different from ours. I conjecture that the sequences of left A-modules Sq2n+1-1 Sq2n A --- - - !A -- ! A are exact. I wish to thank Mark Mahowald for verifying the case n = 2 of my conjecture, and pointing out that a proof follows from his paper with Gorbounov [4]. I wish to thank the referee for many useful comments, and explained how Singer's results streamline my proofs. 2. The lambda algebra EHP sequences P P By [8, Lemma 2.6], if a = ai2i, b = bi2i (0 ai, bi < 2), then ` ' ` ' b Y bi (3) (mod 2). a ai SOME ACYCLIC RELATIONS IN THE LAMBDA ALGEBRA 5 By this formula, ` ' ` ' ` ' ` ' ` ' n 2n + 1 2n + 1 2n 2n (4) , 0. m 2m + 1 2m 2m 2m + 1 Consider a map ` :Z ! Z by taking `(n) = 2n + 1 = 2(n + 1) - 1. Then `e(n) = 2e(n + 1) - 1 = n2e + 2e - 1 and `(n)-2j2j n-jj , `(n)-2j-1 2j+1 0. For n 0, let F (n) = {j : n-jj = 1, 0 j n_2}. It is well- known that hr = ~2r-1 is a cycle for r 0. This is equivalent to F (2r - 1) = {0} by Equation (3). By Equations (3) and (4), we have F (2r) = {0} q {2a : 0 a < r}, F (2r - 2) = {2a - 1 : 0 a < r} and (5) F (`e(2r))= {0} q {2e+a : 0 a < r} (6) F (`e(2r - 2))= {2e+a - 2e : 0 a < r}. They are used to get acyclic chain complexes for wn, where n = `e(b) for b = 2r, 2r - 2. By [9], there is a unique differential algebra endomorphism ` : ! , s,t(n) ! s,2t(2n) with `(~i) = ~2i+1. This ` is usually called Sq0, and it commutes with Adem relations. Lemma 2.1 ([9, Proposition 1.7.3]). (i) ` is injective. (ii) If d(`(x)) = 0 then d(x) = 0. Now we explain the lambda algebra EHP sequence. We refer to [6] for recent proofs. Lemma 2.2 ([3, Lemma 3.5]). ~m (n + m + 1) (n) for m < n. In [3], this is proved by a double induction argument and it is similar to the proof of the dual result [9, Proposition 1.8.1]: s,t(n)~k (n) fork < n + t. By Lemma 2.2 (or Wang's dual) and induction on s, we have the fol- lowing proposition which is due to Singer. Proposition 2.3 ([7, Proposition 5.1]). s,t(n) (n + t) (n). This proposition and d~n 2 2,n+1(n) give Wang's result: Lemma 2.4 ([9, Proposition 1.8.3]). (d~n)x 2 (n) for x 2 (2n+1). Following Wang [9], we see that this lemma implies the result of [2]: Proposition 2.5 ([9, Proposition 1.8.4]). (n) is a subcomplex of the chain complex , i.e. d (n) (n), d : s,t(n) ! s+1,t(n). 6 MIZUHO HIKIDA Now we define a map (Hopf invariant) H : s,t(n + 1) ! s-1,t-n-1(2n + 1) by H(~n~I) = ~I, H(~i~I) = 0 for the admissible sequences (n, I), (i, I) with i < n. Lemma 2.4 also implies the following. Proposition 2.6. H : (n + 1) ! (2n + 1) is a chain map. Corollary 2.7 ([9, Theorem 1.8.5]). If dff = 0 then dH(ff) = 0. We define unstable composition product ff ^ fi = fffi 2 (n) for ff 2 s,t(n), fi 2 (n + t). Then we can define a chain map (Whitehead product) P : s,t(2n + 1) ! s+2,t+n+1(n) by P (ff) = wn ^ ff, where wn = d~n 2 2,n+1(n). Moreover, we have a chain map (suspension) E : s,t(n) ! s,t(n + 1) which is inclusion. Then we have short exact sequences 0 ! s,t(n) -E! s,t(n + 1) -H! s-1,t-n-1(2n + 1) ! 0. Proposition 2.8 ([7, Proposition 5.3]). EH(ff ^ fi) = EH(ff) ^ fi + `(ff) ^ EH(fi) 2 (2n + 2) for ff 2 s,t(n + 1), fi 2 (n + t + 1). Since E is injective, E(ff ^ fi) = Eff ^ Efi and `(Eff) = E2`(ff). We have two special cases and the second case is [7, Proposition 5.2]: Corollary 2.9. Let ff 2 s,t(n), fi 2 (n + t + 1). Then H(E(ff) ^ fi) = E`(ff) ^ H(fi) 2 (2n + 1). Also, if ff 2 s,t(n + 1), fi 2 (n + t), then H(ff ^ E(fi)) = H(ff) ^ fi 2 (2n + 1). Singer gave proofs of Propositions 2.3, 2.6 and 2.8 in the preprint version of his paper [7], but unfortunately omitted them from the pub- lished version. Proposition 2.8 is proved by generalizing the proof of [3, Lemma 3.1] which is the case of ff = d(~2n) and fi 2 (4n + 1). This is essentially Singer's preprint proof. We prove Proposition 2.3 by double induction. Note that our proof does not use Lemma 2.2. Proof of Proposition 2.3.We shall show that s1,t(n) s2,*(n + t) (n) by double induction on s = s1 + s2 and n. Consider ff = ~m x, for m < n and x 2 s1-1,t-m-1(2m+1), and fi 2 s2,*(n+t). Since m < n, x 2 s1-1,t-m-1(n + m + 1), and so we have fl = xfi 2 s-1,*(n + m + 1) by induction on s. We shall show that ~m fl 2 (n). SOME ACYCLIC RELATIONS IN THE LAMBDA ALGEBRA 7 If m = n-1 then this is trivial since (2n) = (2n-1)+~2n-1 (4n- 1). If m < n - 1 then we take the admissible form fl = ~n+m x + y with x 2 (2n + 2m + 1) and y 2 (n + m). By induction on n, ~m y 2 (n - 1). We have an Adem relation ~m ~n+m = ~n-1~2m+1 + z with z 2 2,2m+n+2(n - 1). By induction on s, ~2m+1 x 2 (2n - 1) and_ zx 2 (n - 1). Thus ~m fl 2 (n). |__| The case s = 1 for the first part of Corollary 2.9 is proved by a similar argument, and induction proves the case s > 1. The second part of Corollary 2.9 follows easily by Proposition 2.3. Proposition 2.8 requires in addition some tricky cancellation, which we leave to the reader, since we do not use Proposition 2.8, but only Corollary 2.9. 3. Some relations on the lambda algebra Consider elements ff, ffi 2 . We define ff ^ : ! and (ff1, . .,.ffr) ^ : ! ri=1 by taking ff ^ (x) = ffx, (ff1, . .,.ffr) ^ (x) = (ff1x, . .,.ffrx). If fffi = 0 then we have a chain complex fi^ ff^ (7) --! --! . If ffifi = 0 for 1 i r then we have a chain complex fi^ (ff1,...,ffr)^r (8) --! --- - - - - !i=1 . For ff 2 s,t(n) and m n + t, we define the map ff ^ : (m) ! (n) by Proposition 2.3. Sometimes we will suspend alpha without mention- ing it to give a larger n, but this is clear from context. For instance, in Theorem 1.3, we use (he+r)r ^ : (p + 1 + r2e+r) ! (p + 1), where e+r e+r e+r (he+r)r 2 r,r2 (2 ), and 2 p + 1. So we suspended to think of e+r (he+r)r 2 r,r2 (p + 1). Proof of Theorem 1.1. By the Adem relation, (~n ^) O (~2n+1 ^) = 0. Consider an element ff 2 s,t(p+1) with ~n ^ ff = 0. For p < 2n+1, ~nff is admissible, and so ff = 0. For p = 2n + 1, ff = ~2n+1x + y 2 (2n + 2), where x = H(ff) 2 (4n + 1) and y 2 (2n + 1). So ~n ^ ff = ~ny, and so y = 0 by the case p < 2n + 1 above. Thus ff = ~2n+1 ^ H(ff). 8 MIZUHO HIKIDA For p > 2n + 1, we have a commutative diagram by Corollary 2.9: ~2n+1^ ~n^ (p + 2n + 3) -- - - ! (p + 1) -- - ! (p - n) ? ? ? ? ? ? yH yH yH ~4n+3^ ~2n+1^ (2p + 4n + 5) -- - - ! (2p + 1) -- - - ! (2p - 2n - 1) Then 0 = H(~n ^ ff) = ~2n+1 ^ H(ff). By induction on s, H(ff) = ~4n+3 ^ fl for some fl 2 (2p + 4n + 5). Since H is surjective, we have an element f 2 (p + 2n + 3) with H(f) = fl. Then H(~2n+1 ^ f) = ~4n+3 ^ H(f) = ~4n+3 ^ fl = H(ff). Hence ff0 = ff + ~2n+1 ^ f 2 (p + 1) has H(ff0) = 0, and so ff0 2 (p) and ~n ^ ff0 = 0. By induction on p, ff0 = ~2n+1 ^ fi0 for some fi0 2 (p + 2n + 2). Thus __ ff = ~2n+1 ^ fi for fi = f + fi0 2 (p + 2n + 3). |__| Lemma 3.1. For integers n1 > . .>.nr 0, if s < r then a composite (~n1,...,~nr)^r s,t(p + 1) -E! --- - - - - - !i=1 is injective. Proof. Consider ff 2 s,t(p + 1) with ~ni ^ ff = 0 (1 i r). We prove this lemma by induction on r, s, p. For r = 1 or p = 0 or s = 0, this is trivial. If p < 2n1 + 1 then ff = 0 by ~n1 ^ ff = 0 and Theorem 1.1. If p = 2n1 + 1 then ff = ~p ^ H(ff) by the proof of Theorem 1.1 for the case p = 2n + 1. Now 0 = H(~ni ^ ff) = `(~ni) ^ H(ff) for 2 i r, and so H(ff) = 0 by induction on r and ff = ~p ^ H(ff) = 0. If p > 2n1 + 1 then 0 = H(~ni ^ ff) = `(~ni) ^ H(ff) for 1 i r, and so H(ff) = 0 by induction on s, and ff 2 (p). By induction on p, __ ff = 0. |__| P For integers n1 > . . .> ni > . . .> nr 0, we denote tr = r i i=12 (ni+ 1) and (9) fl(n1, . .,.nr) = `(~n1) . .`.i(~ni) . .`.r(~nr) 2 r,tr(2n1 + 2). The proof of Theorem 1.2 is very similar to the proof of Theorem 1.1, which is the case r = 1. Proof of Theorem 1.2. By the assumption, ((~n1, . .,.~nr) ^) O (fl(n1, . .,.nr) ^) = 0. Consider an element ff 2 s,t(p + 1) with ~ni ^ ff = 0 for 1 i r. If r = 1 then this is Theorem 1.1. If s = 0 then ff = 0, because the generator is the identity element * 2 0,0(p + 1) = Z=2 where * is the monomial of length 0. But * is not in the kernel since ~n ^ * = ~n 6= 0. SOME ACYCLIC RELATIONS IN THE LAMBDA ALGEBRA 9 For p = 2n1 + 1, ff = `(~n1) ^ H(ff) by the proof of Theorem 1.1 for the case p = 2n + 1. Now 0 = H(~ni ^ ff) = `(~ni) ^ H(ff) for 2 i r, and 0 = H(~ni ^ fl(n1, . .,.nr)) = `(~ni) ^ H(fl(n1, . .,.nr)) = `(~ni) ^ `(fl(n2, . .,.nr)). By inductionPon r, H(ff) = `(fl(n2, . .,.nr)) ^ fi, where fi 2 (2p + 1 + ri=22i-1(2ni+ 1 + 1)) = (p + tr). Then ff = fl(n1, . .,.nr) ^ fi. For p > 2n1 + 1, we have a commutative diagram by Corollary 2.9: fl^ (~n1,...,~nr)^ r (p + 1 + tr) -- - ! (p + 1) --- - - - - - ! i=1 (p - ni) ? ? ? ? ? ? yH y H y H `(fl)^ (`(~n1),...,`(~nr))^r (2p + 1 + 2tr) -- - ! (2p + 1) --- - - - - - - - - !i=1 (2p - 2ni- 1), where fl = fl(n1, . .,.nr) 2 r,tr(2n1+2) r,tr(p). Then 0 = H(~ni ^ ff) = `(~ni) ^ H(ff) for 1 i r. By induction on s, H(ff) = `(fl) ^ fi0 for some fi0 2 (2p + 1 + 2tr). Since H is surjective, we have an element f 2 (p + 1 + tr) with H(f) = fi0. Then H(fl ^ f) = `(fl) ^ H(f) = `(fl) ^ fi0 = H(ff). Hence ff0 = ff + fl ^ f 2 (p + 1) has H(ff0) = 0. So ff0 2 (p), and ~ni ^ ff0 = 0 for 1 i r by the assumption. By induction on p, ff0= fl ^ fi00for some fi002 (p + tr). __ Thus ff = fl ^ fi for fi = f + fi002 (p + 1 + tr). |__| Lemma 3.2. If ~nifl(n1, . .,.ni) = 0 then ~nifl(n1, . .,.nr) = 0. Two examples where the hypotheses of Theorem 1.2 are satisfied are given in Lemmas 3.3 and 3.5 below. By [9], hi(hi+r)r = 0 for hi = ~2i-1, and so we have the following. Lemma 3.3. Let ni = 2e+r-i- 1 (e 0, 1 i r) be integers. Then fl(n1, . .,.nr) = (~2e+r-1)r = (he+r)r and fl(nj, . .,.ni) = (he+r-j+1)i-j+1. Moreover ~nifl(nj, . .,.ni) = 0 for 1 j i r. This lemma and Theorem 1.2 imply Theorem 1.3. Our next example leads to Theorem 1.4, and the proof is similar to Wang's calculation hi(hi+r)r = 0, so let's recall Wang's proof. It suffices by ` to prove that h0hrr= 0. An Adem relation writes h0hr as a sum of terms ~mihi, and by induction, hihr-1r= 0. Next we consider integers na = 2a - 2. Then we shall show that fl(nb, . .,.na) satisfies the conditions in Theorem 1.2. We write fi(b, a) = fl(nb, . .,.na) for b a. 10 MIZUHO HIKIDA Lemma 3.4. (i) ~nxfi(a + r, a) = 0 for a + r x a. (ii) d(fi(r, 1)) = 0. Proof. (i) Because fi(a + r, a) = fi(a + r, x)`a+r-x+1 (fi(x - 1, a)) for x > a, it suffices to prove that ~nafi(a + r, a) = 0. For r = 0, this is the Adem relation. We assume r > 0 and induction on r. Then fi(a + r, a) = `(~na+r)`(fi(a + r - 1, a)). The Adem relations imply X ~p~`(p)+2en= ~p+2e(n-k)~`(p)+2ek. k2F(n-1) Now F (2r - 2) = {2b - 1 : 0 b < r} by (6), and `(na+r) = `(na) + 2(na+r - na) = `(na) + 2a+1(2r - 1). By substituting b for r, we have `(na) + 2a+1(2b- 1) = `(na+b). Hence r-1X ~na`(~na+r) = ~m(a,r,b)`(~na+b) b=0 for some m(a, r, b) we are not concerned with. This implies r-1X ~nafi(a + r, a) = ~m(a,r,b)`(~na+bfi(a + r - 1, a)) = 0 b=0 by induction on r. (ii) For r > 1, fi(r, 1) = `(~nr)`(fi(r - 1, 1)), so it by induction, it suffices to show that d(~nr)fi(r - 1, 1) = 0. Then X r-1X d(~nr) = ~nr-k~k-1 = ~nr-2b+1~nb 0Pji since 2n + 1 - 3ji ji+ 1 > 0 by n - ji ji. Then wn = d~n = ri=1~n-ji~ji-1+ w0 2 n+1(n-j1+1), where w02 (n-jr). The lemma above and Theorem 1.2 imply the following. Lemma 4.2. If wnff = 0 for ff 2 (2n + 1 - jr), then ~ji-1ff = 0 for 1 i r. Moreover, if ~ji-1fl(jr - 1, . .,.j1 - 1) = 0 for 1 i r then ff = fl(jrP- 1, . .,.j1 - 1)fi for some fi 2 (2n + 1 - jr + tr), where tr = ri=12ijr-i+1. 12 MIZUHO HIKIDA Proof of Theorem 1.5. Let n = 2e+r+2e-1 = `e(2r). Then ji = 2e+i-1 for 1 i r by Equation (5), and so fl(jr - 1, . .,.j1 - 1) = (he+r)r 2 r,tr(2e+r) and tr = r2e+r. P r Hence wn(he+r)r = i=1~n-ji~ji-1(he+r)r = 0 by Lemma 3.3. If wn ^ ff = 0 for ff 2 (2n + 1 - 2e+r-1) then ff = (he+r)r ^ fi for_ some fi 2 (2n + 1 + (r - 1)2e+r + 2e+r-1) by Lemma 3.3 and 4.2. |__| Proof of Theorem 1.6. Let n = 2e+r+1 - 2e - 1 = `e(2r+1 - 2). Then ji = 2e+i- 2e for 1 i r by Equation (6), and so fl(jr - 1, . .,.j1 - 1) = ke,rand tr = (r - 1)2e+r+1 + 2e+1. P r Hence wnke,r= i=1~n-ji~ji-1ke,r= 0 by Lemma 3.5. If wn ^ ff = 0 for ff 2 (2n + 1 - 2e+r + 2e) then ff = ke,r^ fi for some fi 2 (2n + 1 + (r - 2)2e+r+1 + 2e+r + 2e+1 + 2e) by Lemma 3.5 __ and 4.2. |__| For a general n, we do not get chain complexes. That is, our methods produce necessary but not sufficient conditions. If wnff = 0, we can conclude that ff = flfi for some fi, but it's not generally true that wnfl = 0. Consider n = 10, 12: By F (10) = {0, 1, 3, 4, 5}, F (12) = {0, 1, 2, 5, 6}, w10 = ~9~0 + ~7~2 + ~6~3 + ~5~4 = w0+ ~9~0 + ~7~2, fl(4, 3, 2, 0)= ~9~15~23~15, w12 = ~11~0 + ~10~1 + ~7~4 + ~6~5 = w00+ ~11~0 + ~10~1, fl(5, 4, 1, 0)= ~11~19(~15)2, in which w02 2,11(7), w002 2,13(10). Now ~ji-1fl(jk - 1, . .,.ji- 1) = 0 except for ~0fl(3, 2, 0)= ~0~7~11~7 = ~4(~7)3, ~0fl(4, 3, 2,=0)~0~9~15~23~15 = ~8~9(~15)3, ~1fl(4, 1)= ~1~9~7 = (~5)2~7, ~1fl(5, 4, 1)= ~1~11~19~15 = ~9(~11)2~15, ~0fl(5, 4, 1,=0)~0~11~19(~15)2 = ~8(~11)2(~15)2. Moreover ~1fl(4, 1, 0) = (~5)2(~7)2. Hence fl(2, 0) and fl(1, 0) satisfy the condition of Theorem 1.2, but the other fl(jr - 1, . .,.j1 - 1) don't satisfy this condition. So we apply Lemma 4.2 to fl(2, 0) = ~5~3 and fl(1, 0) = (h2)2 as follows: If ff 2 (18) and w10 ^ ff = 0 then ff = fl(2, 0)fi for some fi 2 (28). If ff 2 (23) and w12 ^ ff = 0 then ff = fl(1, 0)fi for some fi 2 (31). SOME ACYCLIC RELATIONS IN THE LAMBDA ALGEBRA 13 However, we don't have chain complexes fl(2,0)^ w10^ (28) --- - ! (18) --- ! (10), fl(1,0)^ w12^ (31) --- - ! (23) --- ! (12) because w10fl(2, 0)= w10~5~3 = ~6~3~5~3 + ~5~4~5~3, w12fl(1, 0)= w12(~3)2 = ~7~4(~3)2 + ~6~5(~3)2. References 1.J. F. Adams and H. R. Margolis: Modules over the Steenrod algebra, Topology 10 (1971), 271-282. 2.A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector and J.* * W. Schlesinger: The mod-p lower central series and the Adams spectral sequence, Topology 5 (1966), 331-342. 3.M. Mahowald: On the double suspension homomorphism, Trans. Amer. Math. Soc. 214 (1975), 169-178. 4.M. Mahowald and V. Gorbounov: Some homotopy of the cobordism spec- trum MO<8>, Homotopy theory and its applications (Cocoyoc, 1993), Contemp. Math. 188, Amer. Math. Soc., 1995, 105-119. 5.S. Priddy: Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39-60. 6.W. Richter: Lambda algebra unstable composition products and the EHP se- quence, preprint. 7.W. M. Singer: The algebraic EHP sequence, Trans. Amer. Math. Soc. 201 (1975), 367-382. 8.N. E. Steenrod and D. B. A. Epstein: Cohomology operation, Annals of Math. Studies No. 50, Princeton Univ. Press, Princeton, (1962). 9.J. S. P. Wang: On the cohomology of the mod-2 Steenrod algebra and non- existence of elements of Hopf invariant one, Illinois J. Math. 11 (1967), 48* *0-490. Hiroshima Prefectural University, Shobara-shi, 727-0017, Japan E-mail address: hikida@bus.hiroshima-pu.ac.jp