LAMBDA ALGEBRA UNSTABLE COMPOSITION PRODUCTS AND THE EHP SEQUENCE WILLIAM RICHTER Abstract. Simple combinatorial proofs are given of various Lambda algebra results, mostly due to the MIT school [B-C-K& , Cu1, Pr], but also the unstable composition formulas of Wang, Mahowald and Singer, which imply the folklore EHP sequence. 1. Introduction Mahowald [Ma1 , Ma2 ] initiated a öl w-techä pproach to the unstable Adams spectral sequence, using a purely algebraic treatment of the Lambda algebra , and ad-hoc tower constructions. However, full details have not yet appeared. A few such details, combinatorial proofs, are given here. The power of Mahowald's approach is shown by his [Ma1 ] EHP se- quence chain-level map P : (2n + 1) ! (n), defined by composing with d(~n) 2 2,n+1(n). But the geometric analogue P : 2S2n+1 ! Sn is only composition with the Whitehead product ['n, 'n] under the double suspen- sion. His computation [Ma1 , Prop. 3.1] of the Hopf invariant of P is the the analogue of the author's result [Ri1 ]. Mahowald's P uses Ä dams- filtration betterü nstable compositions, due to Wang [Wa ], and codified (without proof) by Singer [Si]. Singer's formulas are proved here, first: Proposition 1.1 (Singer). Composition in restricts to an unstable com- position pairing, written as a cup product: s,t(n) (n + t)! (n), ff fiØ//_ff ^ fi. Singer's result follows by easy induction from the special case s = 1 of Mahowald [Ma2 , Lem. 3.5], or Wang's [Wa , Lem. 1.8.1] special case involving 1,*(n + t). Wang deduced [Wa , Thm. 1.8.4] the MIT school's result [B-C-K& ], that (n) is a subcomplex, and his proof showed the folklore (see Remark 3.3) result that H : (n + 1) ! (2n + 1) is a chain map. Curtis [Cu2 ] first stated without proof the EHP sequence, which any careful reader could've deduced from these papers [Wa , Ma2 , Si]: ____________ 1991 Mathematics Subject Classification. 55T15, 55Q40, 55Q25. Thanks to Paul Burchard for the diagram package, which uses XY-pic arrows. 1 2 WILLIAM RICHTER Theorem 1.2. There's an exact sequence of complexes and a chain map P (n) //___//_E (n + 1) _____////_H (2n + 1), (2n + 1) _____//_P (n), where H and P are defined by H(~nff) = ff and P (ff) = d(~n) ^ ff, for ff 2 (2a + 1), and H( (n)) = 0. P induces the cohomology boundary. Bousfield and Kan [B-K ] construct unstable cohomology compositions1, which they proved are compatible with the geometric compositions: (1) Hs,t-s (n) H* (n + t - s) ! H* (n), ßt-s+nSn ß*Sn+t-s ! ßt-s+nSn. Since the differential d of preserves the t-degree, Proposition 1.1 im- mediately implies an Ä dams-filtration better" improvement of (1): Corollary 1.3 (Singer). Unstable composition induces the pairing Hs,t (n) H* (n + t) ! H* (n). An EHPss approach to [Wa ] yields the 3-lines and relations on the 4- lines for H*( (n)). That's basically how Wang (who never mentions H) proves (cf. [Ko ]) the Adams differential d(hn) = h0h2n-1for n > 3. This systematization of Wang's work will appear later, as part of the author's work with Mahowald on 3-cell Poincare complexes and Unell's theorem. The admissible monomial basis of MIT school [B-C-K& , Pr] is proved in section 5, verifying Mahowald's conjecture (cf. [Ma1 , p. 78]) that a com- binatorial proof exists. Also proved (section 4) is the MIT school's related result, that d is well-defined (i.e. preserves Adem relations). In section 3, we prove Theorem 1.2, and in section 6, we prove the Mahowald-Singer Hopf invariant formula, and explain how [Ma1 , Prop. 3.1] motivates [Ri1 ]. In section 7, we reprove Wang's result on the equivalence of the admissible and symmetric Adem relations, by Tangora's recursion relation. This paper is part of an investigation of geometric applications of [Ma3 ] with Mark Mahowald, who I'd like to thank, especially for his guidance on the basis. Thanks to Paul Goerss for many helpful tutorials about and the uAss. Thanks to Pete Bousfield for 2 very interesting and encouraging discussions. Thanks to Halvard Fausk, who listened to an early version of the paper and encouraged me to write it up. Thanks to Charles Rezk, for explaining that [Pr] is a purely algebraic treatment, using nothing of the sim- plicial Lie algebras of [B-C-K& ]. Thank to Stewart Priddy, who explained that genealogy [C-M ] strongly indicates that unstable Lambda composition should be in the same order as composition in unstable homotopy groups. ____________ 1Actually somewhat less, due to "fringing" problems, which Bousfield says we* *re later overcome. Bousfield and Kan work for all spaces, not just spheres, and not actu* *ally use , but a description of H* (n) as an Extgroup in a category of unstable A-modules. LAMBDA ALGEBRA EHP SEQUENCE 3 2. unstable Lambda algebra composition products The Lambda algebra is generated by {~i : i 0}, and has relations the admissible Adem relations X ` n - k - 1 ' (2) ~p~2p+1+n + ~p+n-k ~2p+1+k, p, n 0. k 0 k A monomial ~(a1, . .,.as) is admissible iff ai 2ai-1 for 1 < i s. Adem relations reduce the right-lexicographical order, while fixing s and the t-degree a1 + . .+.as + s, so by induction the admissible monomials span. The MIT school [B-C-K& ] showed the harder fact that has a basis of admissible monomials, and also that (n) is a subcomplex, where (n) has the basis of admissible monomials ~I(a1, . .,.as) with a1 < n. To motivate Proposition 1.1, let's ask how we could construct Bousfield and Kan's unstable compositions (1)in . Their geometric compatibility (same order e.g.) shows us we need ~(a1, . .,.as) to belong to (n), for any sequence (a1, . .,.as) satisfying the inequalities a1 < n a2 < n + a1 . . .as < n + a1 + . .+.as-1. And it's not hard to see this is true, because Adem relations preserve these inequalities, and then we'd have (n) . (n + t - s) (n). But we note: (1) Left multiplication by ~-1 is more or less d, and this [B-K ] type unstable composition isn't enough to prove (n) is a subcomplex. (2) Performing an Adem relation improves the above inequalities. This leads us to stronger inequalities: Definition 2.1. APmonomial ~(a1, . .,.as) is called n-pseudo-admissible if ai < n + i - 1 + j0 k Then d(~n)J 2 (n) by Proposition 1.1: 2,n+1(n) . (2n + 1) (n) Now we develop the EHP sequence of the MIT school [Cu1 ]. Note that this follows from our proof of Corollary 3.1, but not the statement itself. Clearly the inclusion E : (n) ! (n + 1) is a chain map, so we have some EHP sequence, but we want to a better grip on the quotient complex (n + 1)= (n). Recall that the Hopf invariant H : (n + 1) ! (2n + 1) is defined to annihilate E, and H(~nJ) = J, for admissible monomials J 2 (2n + 1). Now we have Corollary 3.2. The linear map H : (n + 1) ! (2n + 1) is a chain map. Proof. It suffices to show dH = Hd holds for an element ~nJ, for an ad- missible monomial J 2 (2n + 1), since d (n) (n), by Corollary 3.1. But replicating the proof of Corollary 3.1, we have Hd(~nJ) = H(d(~n)J + ~nd(J)) = 0 + d(J) = dH(~nJ) since d(~n)J 2 (n), and d(J) 2 (2n + 1). 6 WILLIAM RICHTER Mahowald's description [Ma2 ] of P is now immediate, and we've proved Proposition 1.2 of the introduction. Remark 3.3. Ravenel and Kochman [Ra , Ko ] implausibly assert that Corol- laries 3.2 and 3.1 follow immediately from Formulas (2)and (3). Curtis and Mahowald [C-M , p. 128] implausibly offer no proof or citation for these 2 results. Curtis [Cu1 , sec. 11] fails to prove that H is a chain map, first by merely citing [Cu2 ], and second by an error in his proof that H is in- duced by the geometric Hopf invariant. Curtis claims that a sum of maps induce an isomorphism on E1 terms, but clearly each map induces zero, as they're all Whitehead product, with positive Adams filtration. I give Curtis credit for his bold attempt, and I think a version of his argument works with Mahowald's [Ma1 ] äm pping cone" construction for an unstable Adams resolution over the fiber of a map, in this case E :Sn ! Sn+1 , although I think we'd have to abandon the Lower Central Series filtration. Bousfield and Curtis [B-C , Rem. 5.3] construct a long exact cohomology EHP sequence, using unstable A-modules, but I believe that one cannot glean a proof of Theorem 1.2 from their argument, but instead, that they use [B-C , Lem. 3.5] Corollary 3.2. Singer [Si, top p. 380] reconstructs the long exact cohomology EHP sequence, and it's clear that his proof that h : Ls(Sn) ! Ls-1(S2n-1) is a chain map uses Corollary 3.2, which of course he could've proved himself. Wang [Wa , Prop. 1.8.3] "immediately" deduced that d(~n) . (2n + 1) (n), and therefore Corollary 3.1, from his special case s,t(n) . ~m (n) for m < n + t of Proposition 1.1. I contend that Wang's leap shows the importance of stating Singer's result, from which his result does follows immediately. Wang could easily have deduced Corollary 3.2 from his Prop. 1.8.3, and he point out its obvious corollary, that H(x) 2 (2n + 1) is a cycle if x 2 (n + 1) is a cycle. 4. d preserves the Adem relations Before proving the admissible monomial basis, we'll prove an easier result of the MIT school [B-C-K& , Pr]: Proposition 4.1. The differential d : ! is well-defined. Proof. is a tensor algebra modulo the 2-sided ideal generated by the Adem relations. The Leibniz rule defines d on the tensor algebra, but we must show that d sends Adem relations to the 2-sided ideal. To prove this, we'll expand the tensor algebra to include ~-1, well-known to be related to d, and use what Pete Bousfield calls "pension operators", i.e. selfmaps of tensor powers which preserve Adem relations. Let V be the Z=2 vector space with basis {~p : p -1}. Let e be the selfmap of V given by e(~p) = ~p+1. Define the selfmap of V 2 by LAMBDA ALGEBRA EHP SEQUENCE 7 D = e 1 + 1 e. As Mahowald recommends (cf. [Ma1 , p. 78]), we'll use the original [B-C-K& ] symmetric Adem relations, for p -1, n 0: X ` n' (4) [p, n] := Dn(~p ~2p+1) = ~p+i ~2p+1+j 2 V 2. i+j=n i The (original [B-C-K& ] symmetric) differential on comes from p = -1: (5) d(~n) = [-1, n + 1] + ~-1~n + ~n~-1 2 V 2, n 0. Now define the selfmap C = e e2 of V 2. Then C preserves Adem relations as well, and we have C([p, n]) = [p + 1, n], D([p, n]) = [p, n + 1]. It's well known that all the Adem relations are obtained from [-1, -1] by applying powers of C and D. Call I2 = 1 1 the identity selfmap of V 2. Now we'll define selfmaps of V 3, and we'll apply them to ~p ~2p+1 ~4p+3 = [p, 0] ~4p+3 = ~p [2p + 1, 0]. We'll also call D the selfmap D = e I2 + 1 e 1 + I2 e of V 3, so D = D 1 + I2 e = 1 D + e I2. We venture into new territory with the selfmap E of V 3 defined by E = e e2 1 + e 1 e2 + 1 e e2 = C 1 + D e2 = 1 C + e D2. We've written both E and D as the sum of 2 commuting operators on V 3, in 2 different ways, so the binomial theorem computes powers of Dm and En, just as with [p, n] above. Let's define, for p -1, n, m 0, elements [p, n, m] := Dm En([p, 0] ~4p+3) = Dm En(~p [2p + 1, 0]) 2 V 3. By the binomial theorem, [p, n, m] has 2 expressions. Equating them gives X ` n' ` m ' ` [p + i, j + s] ~4p+3+2j+t' (6) = 0. i+j=n,s+t=m i s + ~p+j+t [2p + 1 + i, 2j + s] Now we specialize to p = -1, and assume n > 0, and project this equation onto the positive part of V 3. I.e. we throw out the terms containing ~-1. This will be our equation for why d preserves Adem relations. The terms in Equation (6)containing ~-1 come from either j = t = 0 or i = 0, and add up to X ` m ' ` [-1, n + s] ~2n+t-1' ~-1 [n-1, m]+[n-1, m] ~-1+ . s+t=m s + ~n+t-1 [-1, 2n + s] 8 WILLIAM RICHTER By formula (5), the Leibniz rule, and switching s and t in the second part, the positive projection of this expression is X ` m ' ` d(~n+s-1) ~2n+t-1' = d([n - 1, m]), s+t=m s + ~n+t-1 d(~2n+s-1) So the positive projection of Equation (6)shows, for n > 0, m 0, that d([n - 1, m]) is the sum of the positive Adem relations X ` n' `m ' ` [i - 1, j + s] ~2j+t-1' (7) i+j=n, s+t=m i s +~j+t-1 [i - 1, 2j + s] i>0, (j,t)6=(0,0) 5.the admissible monomial basis Let W V be the subvectorspace with basis {~p : p 0}, and let R W 2be the subvectorspace Z=2{[p, n] : p, n 0}. Then with I the 2-sided ideal generated by R, we have = T (W )=I, I = T (W ) . R . T (W ) We now prove the MIT school's result [B-C-K& , Pr] Proposition 5.1. has a basis of the admissible monomials. First we prove an analogue of Proposition 4.1: Lemma 5.2. For p, n, m 0, we can rewrite ~p [2p + 1 + n, m] as a sum X X ~p [2p + 1 + n, m] = ~xi [pi, ni] + [qj, mj] ~yj 2 W 3 i j where for each i, the triple (xi, pi, 2pi+ 1 + ni) has lower right-lex order than (p, 2p + 1 + n, 4p + 3 + 2n + m). Proof. Equation (6)simplifies to X `n '` m ' (8) ~p+j+t [2p + 1 + i, 2j + s] 2 R W W 3. i+j=n,s+t=m i s The (i, s)-term produces the triple (p+j +t, 2p+1+i, 4p+3+2n+s), and the maximum right-lex order occurs uniquely at s = m and i = n, which corresponds to the term ~p [2p + 1 + n, m]. Remark 5.3. We proved what we will use below, but here's a more straight- forward analogue of Proposition 4.1. Define the excess of (a, b) to be b - 2a - 1. Then the excess of (p + j + t, 2p + 1 + i) is i - 2j - 2t n, and the maximum n is achieved only for j = t = 0. So Formula (8)rewrites ~p [2p+1+n, m] as an element of R W plus a sum of elements ~a [b, c] LAMBDA ALGEBRA EHP SEQUENCE 9 with b - 2a - 1 < n. By induction ~p [2p + 1 + n, m] is an element of R W plus a sum of elements ~a [b, c] with each (a, b) is admissible. Proof of Proposition 5.1.The problem is that 2-sided ideal I is öt o big". We first define a sub-vectorspace J of I so that T (W )=J has a basis of the admissible monomials. J will be the sub-vectorspace I that's defined by the algorithm of performing an Adem relation on the left-most inadmissible pair of a monomial. Formally, let K I be the subvectorspace with basis {~(a1, ..., as)[p, n] : p, n, s 0, ~(a1, ..., as) admissible, p 2as if}s,> 0 and define J = K . T (W ). It's obvious that T (W )=J has a basis of the admissible monomials. We'll use to Lemma 5.2 to show I = J. I is spanned by spanning elements ff = ~(a1, ..., as)[p, n]OE, ai, p, n, s 0, OE 2 T (W ). By abuse of notation, let's call s the Adams filtration of ff. We'll say that f* *f is an admissible spanning element if (a1, . .,.as, p) is admissible. Of course, ff 2 J if ff is admissible. If ff is inadmissible, we'll perform reductions until ff is a sum of admissible spanning elements, and then ff 2 J. We need an ordering on the spanning element, derived from the orderings Priddy [Pr] and Mahowald [Pr, Ma1 , Prop. 5.5] used in their cohomological proofs of this basis result. We order the spanning elements ff of a given word-length N = s + 2 + r and a given stem degree a1 + . .+.as + p + (2p + 1 + n) + b1 + . .+.br first by the Adams filtration s and then by right lexicographical order on the N-tuple (a1, . .,.as, p, 2p + 1 + n, b1, . .,.br). We can now induct because there are only a finite number of elements with lower filtration than ff. We're going to perform a sequence of reductions until ff is a sum of admissible spanning elements, and then ff 2 J. Our two reduction moves are: (1) Apply a symmetric Adem relations [q, n] to any inadmissible pair in ~(a1, . .,.as) (2) Apply a higher Adem relations Dm En(q 2q + 1 4q + 3) to as[p, n], if (as, p) is inadmissible. We'll see that both moves strictly lower the filtration order. It will be ob- vious that both moves preserve the word-length and the stem degree.Then I = J by the same inductive argument that proves why admissibles span : keep applying moves in any order until (by finiteness) we have a sum of admissible spanning elements. 10 WILLIAM RICHTER Let's illustrate the type (1) move for s = 2. If (a1, a2) = (q, 2q + 1 + m) is inadmissible, then X `m ' ff = [q, m]([p, n]OE) + ~(q + i, 2q + 1 + j)[p, n]OE, i+j=m, j 1 follows by easy induction by the strictly associativity of the formula. First some obvious properties of unstable composition, involving asso- ciativity, suspension naturality, Sq0 and admissible concatenation: 0,t0 0 Lemma 6.2. If ff 2 s,t(n), fi 2 s (n + t), and fl 2 (n + t + t ), then ff ^ (fi ^ fl)= (ff ^ fi) ^ fl 2 (n), E(ff ^ fi)= E(ff) ^ E(fi) 2 (n + 1), 0,2(t+t0) Sq0(ff ^ fi)= Sq0(ff) ^ Sq0(fi) 2 s+s (2n). For fi 2 (2n + 1), we have ~nfi = ~n ^ E(fi) 2 (n + 1). Proof. We must only check that all of the unstable compositions are defined, since unstable composition is just the multiplication desuspended to the appropriate sub-vectorspace (i) . Proof of Proposition 6.1.We'll prove Equation (9) by induction on s, the Adams filtration of the first argument ff. First we'll do s = 1, and be very pedantic about unstable products. So ff = ~a, with 0 a n. For fi 2 (n + a + 2), we need (10) EH(~a ^ fi) = ffia,nfi + ~2a+1 ^ EH(fi) 2 (2n + 2) Let's write m = n + a + 1, so fi 2 (m + 1). 12 WILLIAM RICHTER Assume a < n. Write ~a 2 1,a+1(n), and ~2a+1 2 1,2a+2(2n + 1). Then ~2a+1 ^ H(fi) 2 (2n + 1), since (2n + 1) + (2a + 2) = 2m + 1, and H(fi) 2 (2m + 1). So Equation (10)desuspends to H(E(~a) ^ fi) = ~2a+1 ^ H(fi) 2 (2n + 1). Let's write fi = ~m ^ E(x) + E(y) in admissible form, for x 2 (2m + 1), and y 2 (m). Now let's write the Adem relation for ~a~m as E(~a) ^ ~m = ~n~2a+1+E(Ra,m) 2 (n+1), for Ra,m 2 2,m+a+2(n). Then Ra,m ^ x 2 (n), since n+m+a+2 = 2m+1, and ~a ^ y 2 (n), since n + a + 1 = m. Then we have E(~a) ^ fi= ~n ^ E(~2a+1 ^ x) + E(Ra,m ^ x + ~a ^ y), so H(E(~a) ^ fi) = ~2a+1 ^ x = ~2a+1 ^ H(fi) 2 (2m + 1). This finishes the case a < n. Now assume a = n. Then write fi 2 (2n + 2) in admissible form as fi = ~2n+1 ^ EH(fi) + E(y), for y 2 (2n + 1). Since ~n~2n+1 = 0, we have ~n ^ fi = ~ny, and the case s = 1 is concluded by EH(~n ^ fi) = E(y) = fi + ~2n+1 ^ EH(fi) 2 (2n + 2). The induction step follows from the strict associativity of the RHS. Take 0,t0 0 ff fi fl 2 s,t(n + 1) s (n + t + 1) (n + t + t + 1). Assuming the result for s and s0, the Adams filtrations of ff and fi, we'll show it's true for ff ^ fi in the first argument. Using Lemma 6.2, we have EH((ff ^ fi) ^ fl) = EH(ff ^ (fi ^ fl)) = EH(ff) ^ fi ^ fl + Sq0(ff) ^ EH(fi ^ fl) 0 = EH(ff) ^ fi ^ fl + Sq0(ff) ^ EH(fi) ^ fl + Sq (fi) ^ EH(fl) 0 0 0 = EH(ff) ^ fi + Sq (ff) ^ EH(fi) ^ fl + Sq (ff) ^ Sq (fi) ^ EH(fl) = EH(ff ^ fi) ^ fl + Sq0(ff ^ fi) ^ EH(fl). So Equation (9)is true with ff ^ fi in the first argument. This completes our induction, since every ffP2 s,t(n + 1) is a sum of such products. Just write ff admissibly as ff = ni=0~i^ E(xi), for xi 2 s-1,t-i-1(2i + 1), and we've proved the result for Adams filtration 1 and s - 1. There are two important special cases when Proposition 6.1 desuspends. First, when the second argument fi desuspends, we have [Si, Prop. 5.2] Corollary 6.3 (Singer). For ff 2 s,t(n + 1) and fi 2 (n + t), we have H(ff ^ E(fi)) = H(ff) ^ fi 2 (2n + 1). LAMBDA ALGEBRA EHP SEQUENCE 13 That is, letting m = n + t, the diagram commutes: H id s,t(n + 1) (m) ___________________//s-1,t-n-1(2n_+ 1) (m) | | id E | |^ fflffl| fflffl| ^ s,t(n + 1) (m + 1) ____//(n_+ 1) __________//_(2nH+ 1) We only need observe that both sides actually desuspend. Proposition 6.1 also implies the desuspension when the first argument ff desuspends: Corollary 6.4. For ff 2 s,t(n) and fi 2 (n + t + 1), we have H(E(ff) ^ fi) = E(Sq0(ff)) ^ H(fi) 2 (2n + 1). As Singer explains [Si], we can now perform analogues of various geometric EHP construction that Toda, Barratt and others used. Consider Toda's calculation [To ] of ßs7= Z=16, generated by oe 2 ß15S8. The problem is to construct his elements oe0, oe00, oe000which are born on S7, S6, and S5 respectively, and are stably 2oe, 4oe, and 8oe respectively, with Hopf invariants j, j2 & j3 respectively. oe000is a Toda bracket < , 8', >, used in the constructing the Adams selfmap [Ad ]. But oe0 and oe00are more mysterious, not expressed as Toda brackets. In , the oe, oe0, oe00, oe000story is easy. Starting with the cycle ~7 2 (8* *), with H(~7) = * 2 (15), Proposition 1.1 and Corollary 6.4 imply ~0~7 2 (7), H(~0~7) = ~1, ~20~72 (6), H(~20~7) = ~21, ~30~72 (5), H(~30~7) = ~31. Of course, these equations are trivial to verify by hand. Note that ~30~7 therefore is a cycle with leading term 4111. Compare [Ra , Ex. 3.3.11], where 4111 is completed to a cycle by the Curtis algorithm. ~30~7 brings up an obvious corollary of Proposition 1.1 and Corollary 3.1: Corollary 6.5. For ff 2 s,t(n) and fi 2 (n + t + 1), we have d(ff ^ fi) = d(ff) ^ fi + ff ^ d(fi) 2 (n). Now take ff = d~n 2 2,n+1(n) and note H(ff) = (n - 1)~0. Propo- sition 6.1 and Theorem 1.2 immediately imply Mahowald's result [Ma1 , Prop. 3.1]: The composition (2n + 1) -P! (n) -H! (2n - 1) -E! (2n) sends fi to (n-1)~0 ^ fi +Sq0(d~n) ^ H(fi). Then d~2n+1 = ESq0(d~n), and specializing to n even, Mahowald observed that the composition 2 (4n + 1) -P! (2n) -H! (4n - 1) -E! (4n + 1) 14 WILLIAM RICHTER sends fi to ~0 ^ fi + d(~4n+1) ^ H(fi). Recall the Hilton-Hopf expan- sion [B-S , Wh ] James used for his 2-primary exponent [Ja, Co , B-C-G& ]: (11) 2' . ff = ff . 2' + ['n, 'n] . H(ff), for ff 2 ß*(Sn). It's well-known that d(~n) corresponds to ['n, 'n], and ~0 corresponds to 2'. Assuming this, Bousfield and Kan's (1), leads us to expect that left/right composition with ~0 corresponds to left/right geometric composition by 2'. Mahowald then observed the following result: Proposition 6.6 (Mahowald). The composition 2 (4n + 1) -P! (2n) -H! (4n - 1) -E! (4n + 1) induces a selfmap of H* (4n + 1), which is E2 . H . P (fi) = fi~0. Proof. We only need to prove the analogue of Equation (11). Singer [Si, Thm. 4.1] proves the full analogue of the Barratt-Toda commutation for- mula [To ]: for f 2 ßm+a Sa and g 2 ßn+aSb we have f . g - (-1)nm g . f = [1a+b-1, 1a+b-1] . H(f) ^ H(g) 2 ßm+n+a+b Sa+b-1. We'll only prove a special case. For a cycle f 2 (p + 1), we'll show ~0 ^ f + f ^ ~0 = d(~p+1) ^ H(f) 2 H* (p + 1) To prove this, write f admissibly as f = ~pA + B, for B 2 (p) and A 2 (2p + 1). Since f is a cycle, A must be a cycle, as Wang (who didn't use H) observed [Wa , Thm. 1.8.4]: dA = dH(f) = Hd(f) = H(0) = 0. By Equation (5), commutation with ~-1 is the boundary map d: df = [f, ~-1] 2 T (V ). Now we'll extend our operator D to T (V ), so D satisfies the Leibniz rule, and D(~p) = ~p+1. Writing D(ff) = ff0, we have (df)0= [f, ~-1]0= [f0, ~-1] + [f, ~0] = d(f0) + [f, ~0] 2 T (W ). We now pass to , since D, as an operator on T (W ), preserves Adem rela- tions, i.e. D([a, m]) = [a, m + 1]. Since f is a cycle, i.e. df = 0, that's d(f0) = [f, ~0] = ~0f + f~0 2 . We need to show d(f0) is cohomologous to d(~p+1) ^ H(f) 2 (p + 1). Differentiate the defining equation for f and apply the boundary map d: f = ~pA + B f0 = ~p+1A + ~pA0+ B0 (12) ~0f + f~0 = d(f0) = d(~p+1)A + d(~pA0+ B0) 2 , since d(A) = 0. We'll show that (~pA0+ B0) 2 (p + 1), because (13) C 2 (k) =) C0 2 (k + 1) LAMBDA ALGEBRA EHP SEQUENCE 15 To see this, take an admissible monomial C = ~(a1, . .,.as) 2 (k). Then C0 is a sum of s terms, each of which is either admissible or zero, since ~a~2a+1 = 0. So the first term ~(a1 + 1, a2, . .,.as) 2 (k + 1), and the re- maining terms of the sum C0 belong to (k). This proves implication (13). Thus B0 2 (p + 1) and A02 (2p + 2), so ~pA0= ~p ^ A02 (p + 1), by Proposition 1.1. So Equation (12)now reads ~0f + f~0 d(~p+1)A 2 (p + 1). Since H(f) = A, we've proved our formula: For any cycle f 2 (p + 1), ~0 ^ f + f ^ ~0 = d(~p+1) ^ H(f) 2 H* (p + 1). Now recall E2 . H . P (fi) = ~0 ^ fi + d(~4n+1) ^ H(fi) 2 (4n + 1). Mahowald then conjectured the geometric analogue of Proposition 6.6: (P) 2n H 4n-1 E2 3 4n+1 the composite 3S4n+1 --- ! S -! S -! S is homotopic to the H-space squaring map on 3. The author [Ri1 ] proved this conjec- ture, which implies the following infinite statement in homotopy groups: 4n-1 (14) 2ßkS4n+1 E2 ßk-2S , for k 3. [B-C-G& ] shows that (14) is not due to James or Selick [Ja, Se ], even though (14)does not improve on the James-Selick 2-primary exponent. 7. symmetric and admissible Adem relations We'll prove Wang's result [Wa , Thm. 1.6.1] that the admissible Adem relations (2)are equivalent to the original [B-C-K& ] symmetric Adem rela- tions (4). First We'll prove the MIT school's result [B-C-K& ] that d2 = 0. Lemma 7.1. d2(~n) = 0 2 , for n 0. Proof. We'll use the symmetric Adem relations, and show even more, that d2(~a) vanishes in the tensor algebra. By formula (5), for n 0, we have X ` n + 1' d(~n) = ~i-1~j-1. i+j=n+1, ij>0 i Then we instantly derive d2(~n) = 0. Using the Leibniz rule d(~i-1~j-1) = d(~i-1)~j-1 + ~i-1d(~j-1), d2(~n) is the sum of two terms, the first of which is X `n + 1 ' ~s-1~t-1~j-1, s+t+j=n+1, stj>0 s, t, j 16 WILLIAM RICHTER n+1 i n+1 as we see by using the binomial identity i s = s,t,j, where as usual, n+1 s,t,j= (n + 1)!=(s! t! j!). But the other term, arising from ~i-1d(~j-1), is equal, so the sum d2(~n) is zero. Remark 7.2. En arose in a way showing the power of the symmetric formu- las: Suppose (p, b) is inadmissible. ThenP~p~b~2b+1 = 0 since ~b~2b+1 = 0, but perform the Adem relation ~p~b = ~x~y first. Each pair (y, 2b + 1) is inadmissible, so perform an Adem relation on each one. The basis requires this sum to vanish in , but why? Using the admissible formulas, this isn't at all clear.PBut using the symmetric formulas, it's easy to rewrite this sum as a sum [q, r]s, using identities like ni 2i2a= in,a,b. So we avoided a relation, and the calculation basically hands us the operator En. Wang [Wa , Thm. 1.6.1] used formal power series to ä dmissify" the symmetric formulas. We'll use a simple recursion formula due to Tan- gora [Ta2 , Ta1]. Define Ca,k2 Z=2, for a 0, k 2 Z, recursively by (15) C0,k= 0, C1,k= ffik,0, and for a 2, Ca,k= Ca-1,k+ Ca-2,k-1. Then for p -1, and a 0, let's define X (16) (p, a) := ~p ~2p+1+a + Ca,k~p+a-k ~2p+1+k 2 V 2. k P By easy induction on a, we see that k Ca,k~p+a-k ~2p+1+k is a finite sum of admissibles: Ca,k= 0 for either k < 0 or 2k + 1 > a. This justifies calling Formula (16)the admissible Adem formulas. Now we obtain relations between the symmetric and admissible Adem relations, by the usual procedure of applying D to formula (16): Lemma 7.3. Assume p -1. Then (p, a) = [p, a] for a = 0, 1, 2, and (17) (p, a + 1) = D(p, a) + (p + 1, a - 2) 2 V 2, for a 2. Proof. The case a = 0, 1, 2 is obvious, and we'll deduce formula (17) by induction on a 2. First, by replacing k by k - 2, we have X (p + 1, a - 2) = ~p+1~2p+1+a + Ca-2,k-2~p+1+a-k ~2p+1+k. k Then D(p, a) + (p + 1, a - 2) equals ~p~2p+1+a+1 plus the sum X (Ca,k+ Ca,k-1+ Ca-2,k-2) ~p+1+a-k ~2p+1+k k X = (Ca,k+ Ca-1,k-1) ~p+1+a-k ~2p+1+k, by (15)for k - 1 k X = Ca+1,k~p+(a+1)-k~2p+1+k, by (15)for k k LAMBDA ALGEBRA EHP SEQUENCE 17 So by this double application of the Tangora recursion formula (15), we have D(p, a) + (p + 1, a - 2) = (p, a + 1). Now we have Lemma 7.4. For all p, a 0, (p, a) is the admissible Adem relation (2). a-k-1 Proof. We must only show that Ca,k= k , for k 0 and 2k + 1 a. Again this follows from induction: ` ' ` ' ` ' a - k - 1 a - k - 1 a - k Ca+1,k= Ca,k+ Ca-1,k-1= + = k k - 1 k by Pascal's triangle. Now we'll show that the admissible and symmetric Adem relations imply each other. Let A W 2be the admissible analogue of R, so A has basis {(p, a) : p, a 0}. Then we have an immediate corollary of Lemma 7.3: Lemma 7.5. A = R, so can be defined either by the admissible Adem relations (2)or the symmetric Adem relations (4). Proof. Since D(R) R, Lemma 7.3 implies that (p, a) 2 R by induction on a. So A R. But Lemma 7.3 also implies that D(p, a) 2 A. Thus D(A) A. Since [p, 0] = (p, 0), and D[p, n] = [p, n + 1], we have R A. Hence A = R. Lemma 7.6. The differential d : ! can be defined either by the admis- sible Adem relations (3)or the symmetric Adem relations (5). Proof. For p -1 and a 0, we can measure the difference between (p, a) and [p, a] as follows. Let Xp,a= [p, a] + (p, a) 2 V 2. We can restate Lemma 7.3 as Xp,a= 0 for a = 0, 1, 2, and (18) Xp,a+1= DXp,a+ (p + 1, a - 2) 2 V 2, for a 2. Let [p^, a]and (p^, a)be [p, a] and (p, a) plus ~p ~2p+1+a + ~p+a ~2p+1. Then clearly Xp,a= [p^,+a]^(p,.a)Now [-^1, a]and (-1^, a)are the formulas in W 2for the symmetric and admissible Adem relations. So specializing Equation (18)to p = -1 shows by induction on a that [-^1, a]+ (-1^, a)= X-1,a2 R. The Tangora recursion relations (15)are a theoretical improvement over the usual recursion formula (which arose in the proof of Lemma 7.3) (19) Ca+1,k= Ca,k+ Ca,k-1+ Ca-2,k-2, because it was clear that we had a sum of admissibles, and it was easy to see that Ca,k= a-k-1k. 18 WILLIAM RICHTER In doing hand calculations, the Tangora recursion relations also give a big improvement over the usual recursion scheme, because Tangora's involves 2 terms instead of 3, and the calculation stays on the same äp ge." For instance, we quickly and independently obtain on the ~0 and ~1 pages: ~0~1 = 0 ~1~3 = 0 ~0~2 = ~1~1 ~1~4 = ~2~3 ~0~3 = ~2~1 ~1~5 = ~3~3 ~0~4 = ~3~1 + ~2~2 ~1~6 = ~4~3 + ~3~4 ~0~5 = ~4~1 ~1~7 = ~5~3 ~0~6 = ~5~1 + ~4~2 + ~3~3 ~1~8 = ~6~3 + ~5~4 + ~4~5 ~0~7 = ~6~1 + ~4~3 ~1~9 = ~7~3 + ~5~5 ~0~8 = ~7~1 + ~6~2 + ~4~4 ~1~10 = ~8~3 + ~7~4 + ~5~6 ~0~9 = ~8~1 ~1~11 = ~9~3 In usual recursive scheme, based on (19), one applies D to each equation to get the next one. In the Steenrod algebra [M-T ], this works OK. To compute Sq3Sq4 = Sq7 on the Sq4 page, we need the Sqi page for i = 1, 2, 3, and this presents no hardship. But in , the ~0 page requires part of the ~1 page, which requires part of the ~2 page, etc. For instance, to compute ~0~9, we apply D to the equations for ~0~8, ~1~7 and ~2~6 to obtain ~0~9 = (~8~1 + ~6~3 + ~5~4 + ~4~5) + (~6~3 + ~5~4) + ~4~5 = ~8~1. References [Ad] Adams, J. F.: On the groups J(X). IV. Topology 5, 21-71 (1966). Correc* *tion Topology 7 (1968), p. 331 [B-C] Bousfield, A. K., Curtis, E. B.: A spectral sequence for the homotopy * *of nice spaces. Trans. Amer. Math. Soc. 151, 457-479 (1970) [B-C-G&] Barratt, M. G., Cohen, F., Gray, B., Mahowald, M., Richter, W.: Two re* *sults on the 2-local EHP spectral sequence. Proc. Amer. Math. Soc. 123, 1257-12* *61 (1995) [B-C-K&] Bousfield, A. K., Curtis, E. B., Kan, D. M., Quillen, D. G., Rector, D* *. L., Schlesinger, J. 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Soc. 123, 3889-3900 (1995) [Ri2] Richter, W.: A homotopy theoretic proof of Williams's Poincar'e embedd* *ing theorem. Duke Math. J. 88, 435-447 (1997) [Se] Selick, P.: 2-primary exponents for the homotopy groups of spheres. To* *pology 23, 97-99 (1984) [Si] Singer, W.: The algebraic EHP sequence. Trans. Amer. Math. Soc. 201, 3* *67- 382 (1975) [Ta1] Tangora, M.: Generating Curtis tables. In: Algebraic topology (Proc. C* *onf., Univ. British Columbia, Vancouver, B.C.), pp. 243-253. Springer 1978 [Ta2] Tangora, M.: Some remarks on the lambda algebras. In: Geometric applic* *a- tions of homotopy theory II, pp. 476-487. Springer 1978 [To] Toda, H.: Composition Methods in the Homotopy Groups of Spheres. Princ. Univ. Press 1962 [Wa] Wang, J.: On the cohomology of the mod -2 Steenrod algebra and the non- existence of elements of Hopf invariant one. Ill. J. Math. 11, 480-490* * (1967) [Wh] Whitehead, G. W.: Elements of Homotopy Theory. (GTM, Vol. 61). Springer 1980 William Richter, Mathematics Department, Northwestern University, Evanston IL 60208 E-mail address: richter@math.nwu.edu