H -spaces analogous to E8 mod 3 Dedicated to the memory of Masahiro Sugawara James P. Lin Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112, U.S.A. email:jimlin@euclid.ucsd.edu Abstract Let p be an odd prime. Let X0 be a finite, p-local, simply connected homotopy associative H-space. Suppose H* (X0; Zp) contains the subalgebra Zp [x0,_z0]_xp p(r0, P1 r0, Pp P1 r0, y0) 0, z0 satisfying z0 = Pp x0 = Q0Pp P1 r0, Pp P1 r0 = P1 y0 for r0 2 H3 (X0; Zp). The only known examples occur for p = 3 and involve the Lie group E8. In this note we prove that if X0 exists, then p must be 3. Mathematics Subject Classification (2000): 55N22, 55P35, 55P45, 55Q25, 55R05, 55S05, 55S10, 55T10 0 Introduction In this note we begin a study of the following questions. Question 1: For p an odd prime and X0 a finite H-space when is QHeven(X0; Zp) nontrivial? In the late 70's it was shown [21] that QHeven(X0; Zp) could be nontrivial only in degrees of the form 2(pk + pk-1 + ^pi+ . .+.1). On the other hand, there are examples only for 2p + 2 for any p, and 2p2 + 2 for p = 3. One of the major questions for finite H-space theory is to determine the possible degrees for QHeven(X0; Zp). If X0 admits the structure of a p-complete finite loop space, one can construct a generalized maximal torus and Weyl group [7], [8], [26]. Using 2 Dedicated to the memory of Masahiro Sugawara James P. Lin these ideas, one can classify the mod p cohomology algebras of finite p- complete loop spaces [1]. It is known [1] that QHeven(X0; Zp) can only be nontrivial in degrees 2p + 2 and 2p2 + 2 for p < 7 and QH2p2+2 (X0; Zp) 6= 0 only for p = 3. However if X0 does not admit a finite loop space structure very little progress has been made. Work of Milnor, Stasheff and Sugawara [24], [25], [28] shows there is a filtration H-spaces homotopy associative H-spaces An -spaces for n > 3 loop spaces. This filtration is related to the level of homotopy associativity of the H- space. There exist many p-local finite H-spaces that are not finite loop spaces. Among them are the examples due to Harper [10] whose mod p cohomology contains an even generator in degree 2p + 2. Also odd dimen- sional spheres localized at p are H-spaces and most of them are not finite loop spaces. The rank of a p-local finite H-space can be defined to be the number of odd generators in its mod p-cohomology. For H-spaces whose co- homology has no even degree generators, there is an infinite number of such H-spaces of a given rank [5], [6] that are not products of odd dimensional spheres. This is in contrast to p-local finite loop spaces for which there are only a finite number of a given rank. This suggests the second question. Question 2: What properties of finite H-spaces do not depend on the existence of a finite loop space structure? There is a curious lack of examples of finite Ap-spaces that do not admit p-local finite loop space structures. So one might suspect that many topolog- ical properties of finite H-spaces may be independent of the existence of a loop space structure. One recent example of this is that the mod 3 cohomol- ogy of finite H-spaces has been classified [20]. In particular for p = 3 this classification yields the information that QHeven(X0; Zp) is concentrated in degrees 8 and 20 for any finite H-space with associative homology ring H*(X0; Z3). In this note we study finite H-spaces whose mod p cohomology has two even degree algebra generators connected by a Steenrod operation. The only known examples occur for p = 3 and they contain H* (E8; Z3) as a factor, where E8 is the compact exceptional Lie group. We note that H* (E8; Z3) contains the subalgebra Z3 [x0,_z0]_x3 3(r0, P1 r0, P3 P1 r0, y0) 0, z0 where deg r0 = 3, P3 x0 = z0 = Q0P3 P1 r0 and P3 P1 r0 = P1 y0. The main result of2the present paper is that for homotopy associative finite H-spaces X0 QH2p +2 (X0; Zp) is nontrivial only for p = 3. We prove H-spaces analogous to E8 mod 3 3 Main Theorem Let X0 be a finite, simply connected, p-local, homotopy associative H-space. Suppose H* (X0; Zp) contains the subalgebra Zp [x0,_z0]_xp p(r0, P1 r0, Pp P1 r0, y0) 0, z0 with deg r0 = 3, Pp x0 = z0 = Q0Pp P1 r0 and Pp P1 r0 = P1 y0. Then p = 3. Thus there are no H-space analogues of E8 mod 3 for primes greater than 3. We remark that the Main Theorem provides partial information about Questions 1.2 and 1.5 posed in [19]. In [20] we prove if p = 3, then H* (X0; Z3) contains a copy of H* (E8; Z3) for each 20 dimensional generator. Let Y be a simply connected, p-local, homotopy associative finite H- space. We conjecture the following: Conjecture A For p > 3, the even degree algebra generators lie in degrees 2p + 2. Conjecture B Pp P1 H3 (Y ; Zp) imP1 The proof of the Main Theorem is based on a study of the restricted Lie algebra of homology primitives P H*(Y ; Zp). We briefly describe its struc- ture. Given ~s, ~t2 H*(Y ; Zp) the commutator [~s, ~t] is defined by [~s, ~t] = ~s~t- (-1)|~s| |~t|~t~swhere|~s| is the degree of ~s H* (Y ; Zp) and H*(Y ; Zp) are dual Hopf algebras over the Steenrod alge- bra A(p). The left A(p) action on H* (Y ; Zp) dualizes to a right action on H*(Y ; Zp). Let < , >: H*(Y ; Zp) H* (Y ; Zp) -! Zp be the dual pairing. If ~s2 H*(Y ; Zp), t 2 H* (Y ; Zp), ff 2 A(p), then < ~s, fft >= (-1)|~s| |ff|< ~sff, t > . If X0 satisfies the hypotheses of the Main Theorem, there are non-zero primitives ~z0, ~x0, ~y0, ~r02 P H*(X0; Zp) with < ~z0, z0 > = 1,~z0Pp = ~x0, ~z0Pp Q0P1 = ~r0, ~z0Q0P1 = ~y0 < ~x0, x0 > = 1 < ~r0, r0 > = 1 < ~y0, y0 > = 1 If z0 6= 0 it will follow that [~x0, [~x0, ~y0]] = 0 (Theorem 2). In the pr* *oof of the Main Theorem, we will show if p > 3, then [~x0, [~x0, ~y0]] 6= 0. This will prove there are no H-spaces X0 satisfying the hypotheses of the Main Theorem for p > 3. For all known homotopy associative finite H-spaces, all nonprimitive cohomology generators are dual to primitive commutators, so an under- standing of this Lie algebra is key to an understanding of the cohomology. 4 Dedicated to the memory of Masahiro Sugawara James P. Lin Kane [13], [15] originally studied the Lie algebra of homology primitives us- ing secondary operations, and Yagita [29] used Morava K-theory. Recently the adjoint action has been used by Hara, Hamanaka, Iwase, Kono, Kozima and Nishimura [9], [12], [16], [17]. In this note we use the projective spaces defined in [18] to study the Lie algebra. We make the following assumptions throughout this paper. All spaces will be connected and basepointed. All homotopies will respect basepoint. Unless otherwise specified, all cohomologies and homologies will have coef- ficients Zp, for p an odd prime. All cohomologies and homologies will be of finite type. We follow the notation of [25], [15] in discussing Hopf algebras and com- mutators. All Hopf algebras will be connected. If A is such a Hopf algebra, I(A) will denote the elements of positive degree. P (A) and Q(A) will denote the modules of primitives and indecomposables respectively._Given a Hopf algebra A, we define_the_iterated_reduced_coproduct 4_`_: I(A) ! I(A) `+1 inductively by 4 1 = 4 and 4 ` = (4 1 . . .1)(4 `-1 ). The Steenrod algebra A(p) is a Hopf algebra generated by the symbols Q0, P` for ` 1. Q0 denotes the Bockstein. We may define the Milnor primitive Qj inductively by j-1 pj-1 Q0 = Q0, Qj = Pp Qj-1 - Qj-1 P for j 1. There are relations [21, Eq(1.3.4)]: æ j 0 ifp n Pn Qj - QjPn = Q n-pj j j+1 P ifp < n (* *1) Q2j= 0, QjQk = -QkQj Many of the ideas for this paper were formulated while the author visited Yutaka and Keiko Hemmi in the Spring of 2002. It gives me great pleasure to thank them and Kochi University for their gracious hospitality. I am indebted to Yutaka Hemmi for showing me that BP operations can be used to control certain coproducts. I would also like to thank John Harper and the referee for several helpful suggestions. 1 Preliminaries We begin by citing some known results. Theorem 1. Let Z be a p-local, finite, simply connected H-space with H*(Z) an associative ring (a) [21, Thm. 5.2.1, Thm 2.4.1] There exists an even dimensional, primi- tively generated A(p) subHopf algebra B H* (Z) with the property that the inclusion induces an isomorphism QB ~= QHeven(Z). In degrees less than 2p2 + 2p + 2, QB ~= P B is concentrated in degrees 2p + 2, 2p2 + 2 H-spaces analogous to E8 mod 3 * * 5 and 2p2 + 2pP+ 2. Pp B2p+2 = B2p2+2 , and P1 B2p2+2p+2 = 0. QB2k = 0 unless k = aipi where ai is 0 or 1 and a0 = 1. i=0 ___ (b) [3] Let R = {u 2 H* (Z)|4 u 2 B H* (Z)}. R is an A(p) coalgebra. The inclusion R H* (Z) induces an isomorphism Rodd ~= QHodd (Z). Reven = B. (c) [21, Thm 5.4.1] Given ~s, ~t2 P Hodd (Z), ~s2= ~t2= [~s, ~t] = 0. Given ~s, ~t2 P Heven(Z), ~sp= ~tp= [~s, ~t] = 0. (d) [25] If A is a commutative Hopf algebra, there is an exact sequence 0 -! P (,A) -! P (A) -! Q(A) If A is a bicommutative Hopf algebra, there is an exact sequence 0 -! P (,A) -! P (A) -! Q(A) -! Q(~A) -! 0 (e) B2p2+2 = Q0Pp P1 R3 = Q0P1 R2p2-2p+3 = Pp Q0P1 R3, Q0R3 = 0 = Q0R2p2-2p+3 . Proof. (a)(b)(c)(d) are proved as cited. By [21] QH2p2+2 (Z) = Q0Pp P1 QH3 (Z) = Q0P1 QH2p2-2p+3 (Z). By (a) and (b), given b 2 B2p2+2 there exist elements a1 2 R3, a2 2 R2p2-2p+3 such that b = Q0Pp P1 a1+d12= Q0P1 a2+d22where the di are decomposable. By (a) and (b), di 2 B2p +2 ~=P B2p +2 . By (d) di = 0 for i = 1, 2. Hence b = Q0Pp P1 a1 = Q0P1 a2. There is an Adem relation Q0Pp = ff1Pp Q0 + ff2P1 Q0Pp-1 and Q0Pp-1 P1 a1 = 0. Hence b = ff1Pp Q0P1 a1 and b 2 Pp Q0P1 R3. By (a) and (b), Q0R3 B4 = 0 and Q0R2p2-2p+3 B2p2-2p+4 = 0. Definition 1. The symbol X0 will be used to denote a p-local, simply con- nected, finite homotopy associative H-space satisfying the following proper- ties: (a) H* (X0; Zp) contains the subalgebra Zp [x0,z0]_xp0,zp0 (r0, P1 r0, Pp P1* * r0, y0) where Pp x0 = z0 = Q0Pp P1 r0, x0 = Q0P1 r0, r0, y0 2 Rodd , x0, z0 2 B, deg r0 = 3. (b) Pp P1 r0 = P1 y0 We remark that if X0 satisfies the conditions of the Main Theorem, then by Theorem 1, we may choose r0, y0 2 Rodd , x0, z0 2 B to satisfy the Steenrod module relations described in part (a). By the Adem relations and part (b), we have xp0= P1+p x0 = P1 z0 = P1 Q0P1 y0 = c1P2 Q0y0 + c2Q0P1 Pp P1 r0 = 0. Given an element u 2 H* (X0), let < u > denote the vector space span of u. By Theorem 1, there exist complementary subspaces Bi, Ri such that 2+2 B2p = < z0 > B1 R3 = < r0 > R1 where B1 = Q0Pp P1 R1 (* *2) 2+1 p 1 p 1 R2p = < P P r0 > R2 where P P R1 R2, Q0R2 = B1 B2p+2 = < x0 > B2 where Q0P1 R1 = B2, Pp B2 = B1. 6 Dedicated to the memory of Masahiro Sugawara James P. Lin By Theorem 1 (a), B4 = 0. Hence the integral Bockstein vanishes on H3 (X0). Therefore, all elements of R1 can be lifted to integral classes. Sup- mQ pose dim R1 = m. Let K0 = K(Z, 3) and define f0 : X0 ! K0 to be a s=1 map such that f0*: H3 (K0; Zp) ! H3 (X0; Zp) maps H3 (K0; Zp) isomorphically onto R1. Since X0 is simply connected, H3 (X0; Z(p)) is primitive. The A3 obstruction of f0 lies in H3 (X0 ^ X0 ^ X0; Z) = 0. Hence, f0 is an A3-map. Let X be the fibre of f0. We have a Hopf fibration K0 # j X (3) # q X0 -f0! K0 where X is a homotopy associative H-space. The Serre spectral sequence associated to this fibration is a spectral sequence of Hopf algebras. Let A(p)R1 be the Steenrod module generated by R1. A calculation using [4, Thm.5.8] shows ffiffi * H* (X0) A(p)R1 is a sub Hopf algebra of H (X) (4) The map q* is the composition ffiffi * H* (X0) -ß! H* (X0) A(p)R1 - ! H (X). (5) By [22], there is an algebra isomorphism ffiffi H* (X) ~= H* (X0) A(p)R1 Zp[u1, . .,.u`] (t1, . .,.tn ) (6) where deg ui = 2pki and deg ti 1 mod 2p. Equations (2) and (6) imply 2+2 2p+2 3 dim QH2p (X) = dim QH (X) = dim QH (X) = 1 (7) with non-zero generators z = q*(z0), x = q*(x0), and r = q*(r0). Let y = q*(y0). By Definition 1 and (5), H* (X) contains the subalgebra Zp _[x,_z]_xp, zp (r, P1 r, Pp P1 r, y) (8) with z = Q0Pp P1 r, Pp P1 r = P1 y and x = Q0P1 r. Definition 2. Let ~z 2 P H2p2+2 (X) satisfy < ~z, z > = 1, ~x= ~zPp , ~r= ~xQ0P1 = -~xQ1, y~= ~zQ0P1 = -~zQ1. Let ~x0= q*(~x) and ~r0= q*(~r). Then < ~x, x > = 1 = < ~r, r >. Theorem 2. (a) [~z, ~x] = [~x, ~r] = [~x0, ~r0] = 0 H-spaces analogous to E8 mod 3 * * 7 (b) ~zp= 0 (c) [~x, [~x, ~y]] = 0 (d) ~zP1 = 0 = ~zQ0P2 Proof. Theorem 1 (a), (5) and (6) imply 2+2p+4 2p2+2p 2p2-2p+4 QH2p (X) = 0, QH (X) = 0, and QH (X) = 0. Therefore [~z, ~x] = 0, ~zp= 0 and ~zP1 = 0. By Definition 2 and (1), ~zQ2 = -~r. Q2 acts as a derivation on H*(X0). Therefore 0 = [~z, ~x]Q2 = [~x, ~r] and 0 = q*[~x, ~r] = [~x0, ~r0] By (1), ~zQ0P2 = ~zP2 Q0 - ~zP1 Q1 = 0. This proves (a), (b) and (d). Using Definition 2, we have 0 = [~z, ~x]Q1 = [-~y, ~x] + [~z, -~r]. Therefore [~x, ~y] = [~z, ~r]. The Jacobi identity and (a) implies [~x, [~x, ~y]] = [~x, [~z, ~r]] = -[~r, [~x, ~z]] - [~z, [~r, ~x* *]] = 0. We now sketch the strategy for the proof of the Main Theorem. Step 1 Equation (36) There exists an element W 2 IH* (X) p satisfying 2-p+1 ___p-1 Q0Pp W = -z . . .z + 4 (u) for some u 2 H* (X). ___2 Step 2_(Theorem_4) If Pp P1 r = P1 y then 4 y = 0. In particular for p > 3, 4 p-3y = 0. This is shown in Section 2 using BP theory. Step 3 Let p > 3. For 1 i p - 2. Let ___2 4i = 1IH*(X)xi-1 4 1IH*(X) p-i-2 (Theorem 6) For 1 i p - 2 there exist elements Vi, Ui such that W = Q2Vi + 4i(Ui). Step 4 (Section 4) For p > 3, [~x[~x, ~y]] 6= 0. This contradicts Theorem 2 and proves the Main Theorem. 8 Dedicated to the memory of Masahiro Sugawara James P. Lin 2 Primitivity of y By [21], H* ( X0; Z(p)) is torsion free. We use BP operations to derive information about the coproduct of y. Recall that BP operations are related to Steenrod operations via the Thom map, T : BP *( X0; Z(p)) -! H* ( X0; Zp). There is a commutative diagram rE BP *( X0; Z(p)) _______________- BP *( X0; Z(p)) | | T || ||T |? |? H* ( X0; Zp) _______________- H* ( X0; Zp) Ø(PE ) We list some of the properties from [14, Sec 47.2] (a) T is onto and ker T = (p, v1, v2, . .).. (9) (b) T rp = -Pp T (c) For 0 < j < p, T rj = cjPjT for cj 6= 0 in Zp (d) prp = rp-2 r2 mod (p2, v1, v2, . .).. Let oe* : QH* (X0) -! P H*-1 ( X0) be the mod p cohomology suspension map. By [4, Thm. 5.14], oe* : QH2k+1 (X0) -! P H2k ( X0) is monic and is epic if 2k 6 -2 mod 2p (10) Theorem 3. Let p be greater than 3. There exists an element u0 2 Rodd with Pp y0 = Pp-3 u0 Proof. If Pp y0 = 0, let u0 = 0. If Pp y0 6= 0, Pp y0 2 Rodd ~= QHodd (X0) by Theorem 1 (b). Hence by (10), oe*(Pp y0) = Pp oe*(y0) 6= 0. (11) By Definition 1, Pp P1 r0 = P1 y0, so P2 y0 = 1_2P1 P1 y0 = 0. Therefore, P2 oe*(y0) = 0. (12) By (9) (a), there exists Y 2 BP *( X0; Z(p)) with T (Y ) = oe*(y0). (13) H-spaces analogous to E8 mod 3 9 Equations (9) (c) and (13) imply T (r2Y ) = c2P2 T (Y ) = c2P2 oe*(y0) = 0. (14) By (9) (a) r2Y = pZ0 + v1Z1 mod (p2, v21, v2, . .).. (15) Applying rp-2 , rp-2 r2Y = prp-2 Z0 + rp-2 (v1Z1) mod (p2, pv1, v21, v2, . .). = prp-2 Z0 + prp-3 (Z1) mod (p2, v1, v2, . .). by [14, p.448] = prpY mod (p2, v1, v2, . .). by (9) (d) . Since BP *( X0; Z(p)) (v1, . .).~= H* ( X0; Z(p)) is torsion free, rpY = rp-2 Z0 + rp-3 (Z1) mod (p, v1, v2, . .).. (16) Applying T , we obtain from (9) (b) and (c), T (rpY ) = -Pp oe*(y0) = cp-2 Pp-2 T (Z0) + cp-3 Pp-3 T (Z1). Therefore since Pp-3 P1 = cPp-2 for some c 2 Zp, there is an element w 2 H* ( X0) such that Pp oe*(y0) = Pp-3 (w). (17) and deg w = 2p2 + 4p - 4. We have degree Pp oe*(y0) 6 0 mod 2p. By Theorem 1 (d), w is indecompos- able. Theorem 1 (b) (d) and (10) imply w = oe*(u0) + d for u0 2 Rodd and d decomposable . (18) Then Pp oe*(y0) - Pp-3 oe*(u0) = Pp-3 d is primitive decomposable . (19) By Theorem 1 (d), Pp-3 d = 0. Therefore Pp oe*(y0) = Pp-3 oe*(u0). (20) Equation (10) implies Pp y0-Pp-3 u0 2 Rodd \ker oe* = 0. Therefore, Pp y0 = Pp-3 u0. ___2 Theorem 4. Let p be greater than 3. Then 4 y = 0 and Q2y = 0. 10 Dedicated to the memory of Masahiro Sugawara James P. Lin ___ Proof. By Theorem 1 (b), (2) and (7), 4 y has the form ___ p-1X s * odd 4 y = x æ1 + x æs foræi 2 q (R ). (21) s=2 ___ Let u = q*(u0). By Theorem 1 (b), (2) and (7), 4 u has the form ___ p-1X s 0 4 u = z i + x is + zx i (22) s=1 for i, is, i0 2 q*(Rodd ), deg i = 4p - 5, deg i0 = 2p - 7. Theorem 1 (a) and (7) imply x, z 2 ker P1 and z 2 ker Pp . Hence the Cartan formula implies ___p p p-1X s-1 p-1X s p 4 P y = z æ1 + x P æ1 + szx æs + x P æs (23) s=2 s=2 ___p-3 p-3 p-1X s p-3 p-3 0 4 P u = z P i + x P is since P i = 0 (24) s=1 ___ ___ 4 Pp y = 4 Pp-3 u implies Pp-3 i = æ1, and æs = 0 for s 2. (25) Therefore, by (21) ___ 4 y = x Pp-3 i. (26) If i is not primitive, by Theorem 1 (b), it must satisfy ___ * 2p-7 p-3 4 i 2 x q (R ) ker P . ___ ___2 Hence 4 y 2 P H* (X) 2 and 4 y = 0. By Theorem 1 (a), Q2x = 0. By (1), ___ p-3 p-3 4 Q2y = x Q2P i = x P Q2i. (27) We have 2 Q2i 2 q*(Q2R4p-5 ) q*(B2p +4p-6 ). ___ 2 Therefore 4 Q2i 2 x Q2q*(R2p-7 ) 2 x P H2p +2p-8 (X). By (6), P H2p2+2p-8 (X) = 0. Hence 2+4p-6 Q2i 2 P H2p (X) = 0. Therefore 2 Q2y 2 P H4p -2p+2 (X). (28) Theorem 1 (a), (d) and (6) imply 2-2p+2 4p2-2p+2 * 4p2-2p+2 P H4p (X) ~= QH (X) = q QH (X0) = 0. Hence Q2y = 0. H-spaces analogous to E8 mod 3 11 3 Projective Spaces `X Let p be a prime greater than 3. We summarize some results from [18]. Let ~ : X x X ! X be the multiplication for X. Consider the map ` : S1 ^ X ^ X ! S1 ^ X defined by X X X ` = - ß1 + ~ - ß2. By [2], we may assume `(t, x1, x2) = (t, x1x2). (29) The induced homology map 2 `* : IH*(X) ! IH*(X) is the multiplication and the induced cohomology map 2 `* : IH* (X) ! IH* (X) ___ is the reduced coproduct 4 . For ` 2 we define ``-1 : S1 ^ X` ! S1 ^ X inductively by `1 = ` ``-1 = `(``-1 ^ 1). Let `X be the cofibre of ``-1 . There exist cofibration sequences ` ``-1 1 i` ~` 1 1 ^` S1 ^ X^ - ! S ^ X - ! `X - ! S ^ S ^ X that induce exact sequences [18, p.143] ` 1 __4`-1 * 1 * IH* (S1) IH* (X) - IH (S ) IH (X) (30) -i*` H* ( ~*` * 1 2 * ` `X) - IH (S ) IH (X) . ___2 By Theorem 4, 4 y = 0. By (30), there exists ^y2 H* ( pX) with i*p(^y) = s y where s 2 H1 (S1) is a generator . (31) Theorem 1 (a), (e) and (8) imply i*p(Q1y^) = -s Q1y = -s (P1 Q0y - Q0P1 y) = s z. By [18, Prop 1.2], 2+1 * Q0Pp Q1y^= ~p(-s s z . . .z). (32) By [15, Eq(3.1)] and (1), 2+1 p2-p+1 p2-p+1 Q0Pp Q1y^= -Q2Q0P ^y= Q0P Q2y^ (33) By Theorem 4, i*p(Q2y^) = -s Q2y = 0. (34) 12 Dedicated to the memory of Masahiro Sugawara James P. Lin By (30), p Q2y^= ~*p(s s W ) for some W 2 IH* (X) . (35) Equations (32), (33) and (35) imply 2-p+1 * ~*p(s s Q0Pp W ) = ~p(-s s z . . .z). Therefore by (30), 2-p+1 ___p-1 * Q0Pp W = -z . . .z + 4 (u) for some u 2 H (X). (36) Recall by Definition 2, ~z2 P H2p2+2 (X) satisfies < ~z, z > = 1. Let A* H*(X) be the A(p) sub Hopf algebra generated by z~. By Definition 2, P A* contains ~z, ~y, ~x, ~r. Dualizing, there is an epimorphism * *of Hopf algebras ß : H* (X) -! A* (37) and A* contains the subalgebra Zp _[ß(x),_ß(z)]_ß(x)p, ß(z)p (ß(r), P1 ß(r), Pp P1 ß(r), ß(y)).(38) Let < ß(x) i-1 ß(y) ß(x) p-i > be the vector space span of ß(x) i-1 ß(y) ß(x) p-i in (A*) p . Then there exists a complementary vector space C such that p Mp i-1 p-i (A*) = < ß(x) ß(y) ß(x) > C (39) i=1 where for 1 i p, and C annihilates ~x i-1 ~y ~x p-i 2 (A*) p Pp j-1 p-j Theorem 5. ß p (W ) = ffjß(x) ß(y) ß(x) +c where ffj 2 Zp j=1 pP satisfy ffj = -1 and c 2 C. j=1 Proof. -1 = -1+ < ~zp, u > by Theorem 2 (b) ___p-1 = < ~z . . .~z, -z . . .z + 4 (u) > 2-p+1 = < ~z . . .~z, Q0Pp W > by (36) 2-p+1 = < (~z . . .~z)Q0Pp , W > Xp i-1 p-i = < ~x ~y ~x , W > by Theorem 2 (d) and Definition 2 i=1 Xp i-1 p-i p = < ~x ~y ~x , ß (W ) > i=1 X p = ffj. i=1 H-spaces analogous to E8 mod 3 13 By (39), ß p (W ) has the desired form. Definition 3. Let Ti : X^i-1 ^ S1 - ! S1 ^pX^i-1-2be the twistpmap. For 1 i p - 2, define 4i : IH* (X) -! IH* (X) to be ___2 4i = 1IH*(X) i-1 4 1IH*(X) p-i-2 . Lemma 1. For 2 i p - 2 `p-1 (Ti ^ 1) ' `p-2 (Ti ^ 1)(1 ^ `2 ^ 1). Proof. The proof is by induction on i. For i = 2, by (29) `p-1 (T2 ^ 1)(x1, t, x2, . .x.p)= `p-1 (t, x1, . .x.p) = t, (((x1x2)x3)x4 . .).xp `p-3 (T2 ^ 1)(1 ^ `2 ^ 1)(x1, t, x2 . .x.p)= `p-3 (T2 ^ 1) (x1, t, (x2x3)x4, . .,.xp) = t, ((x1[(x2x3)x4])x5 . .).xp. Since X is homotopy associative, there exist paths ((x1x2)x3)x4 (x1(x2x3))x4 x1((x2x3)x4) _________________________________________________ooo The homotopies defined by these paths prove the Lemma for i = 2. Now assume by induction that for 2 i < p - 2 `p-1 (Ti ^ 1) ' `p-3 (Ti ^ 1)(1 ^ `2 ^ 1). Consider `p-3 (Ti+1 ^ 1)(1 ^ `2 ^ 1)(x1, . .,.xi, t, xi+1 , . .,.xp) = `p-3 (t, x1, . .,.xi, (xi+1 xi+2 )xi+3 , . .,.xp) = t, ((x1x2) . .x.i-1)xi)[(xi+1 , xi+2 )xi+3 ]) . .).xp. There are paths connecting the following elements ((x1x2) . .).xp' (((x1x2) . .x.i-1)[(xixi+1 )xi+2 ]xi+3 ) . .x.p by the induction asumption ' (((x1x2) . .x.i-1)[xi(xi+1 xi+2 )])xi+3 ) . .x.p by homotopy associativity ' (((x1x2) . .x.i-1)xi)(xi+1 xi+2 ))xi+3 ) . .x.p by homotopy associativity ' (((x1x2) . .x.i-1)xi)[(xi+1 xi+2 )xi+3 ]) . .).xp by homotopy associativity . 14 Dedicated to the memory of Masahiro Sugawara James P. Lin This shows `p-1 (Ti+1 ^ 1) ' `p-3 (Ti+1 ^ 1)(1 ^ `2 ^ 1). By induction the Lemma is proved. Theorem 6. p Let p > 3. Thenpfor-each2i satisfying 1 i p - 2, there ex* *ist Vi 2 IH* (X) , Ui 2 IH* (X) with W = Q2Vi + 4i(Ui). Proof. By Lemma 1 for 2 i p - 2, we obtain the following homotopy commutative cofibration ladder: _________Ti_^_1_____________- 1 ^p X^i-1 ^ S1 ^ X^3 ^ X^p-i-2 S ^ X | | * *p-1 1 ^ `2 ^ 1 | | ` |? |? _Ti_^_1___- 1 ^p-2 ___`p-3___- 1 X^i-1 ^ S1 ^ X ^ X^p-i-2 S ^ X S ^ X 1 ^ i3 ^ 1 || || * *ip |? ffi |? ______________i_____________- X^i-1 ^ 3X ^ X^p-i-2 pX 1 ^ ~3 ^ 1 || || * *~p |? (T |? ____________i_^_1)__________- * * p X^i-1 ^ S1 ^ S1 ^ X^p-i+1 S1 ^ S1 ^* * X^ * *(40) Applying cohomology, we obtain a ladder of exact sequences Ti* 1 * 1 * * * p IH* (X) i-1 IH* (S1) IH* (X) p-i+1 __________oe IH (S ) IH * *(X) ___2 |6 |6 * * ___p-1 1 4 1 | | 1 * * 4 | (Ti* 1)(1 __4p-3) | IH* (X) i-1 IH* (S1) IH* (X) p-i-1 __________________oeIH* (S1) I* *H* (X) 1 i*3 1 ||6 ||6 * *i*p | ffi*i | IH* (X) i-1 H* ( 3X) IH* (X) p-i-2 __________oe H* ( pX) 1 ~*3 1 ||6 ||6 * *~*p | (Ti* 1) | IH* (X) i-1 IH* (S1) 2 IH* (X) p-i+1 __________oe IH* (S1) 2 IH* ** (X) p * *(41) H-spaces analogous to E8 mod 3 15 Because p > 3, ___p-3 * * ___p-3 (Ti* 1)(1 4 )ip(^y)= (Ti 1)(s 4 (y)) = 0 by Theorem 4 = (1 i*3 1)(ffi*i(^y)) by (41) By exactness of the left column of (41), ffi*i(^y) = (1 ~*3 1)(Vi) for some Vi. Applying Q2 we have ffi*i(Q2y^)=(1 ~*3 1)(Q2Vi) (42) = ffi*i(~*p(s s W )) by (35) = (1 ~*3 1)( Ti* 1)(s s W ) by (41) (42) implies ( Ti* 1)(s s W ) - Q2Vi 2 ker 1 ~*3 1. Note Ti* 1 does not change sign since two H* (S1) factors are twisted. Suppressing H* (S1) factors, ker 1 ~*3 1 = im4i. Therefore there exist Ui such that W = Q2Vi + 4iUi for 2 i p - 2. For i = 1 we note `p-1 = `p-3 (`2 ^ 1). The proof follows using a diagram similar to (40) without the twist maps. 4 Proof of the Main Theorem Let p be a prime greater than 3. pP Lemma 2. If ffj = -1, then there exists 1 i p - 2 such that j=1 ffi - 2ffi+1 + ffi+2 6= 0. Proof. Suppose for all i such that 1 i p - 2, ffi - 2ffi+1 + ffi+2 = 0. Then p-3X 0 = (1 + . .+.i)(ffi - 2ffi+1 + ffi+2 ) i=1 = (ff1 + . .+.ffp-3 ) + (p_-_3)(p_-_2)_2ffp-1 = ff1 + . .+.ffp-2 + 3ffp-1 since (p_-_3)(p_-_2)_2 3 mod p = (ff1 + . .+.ffp-3 + 3ffp-1 ) + (ffp-2 - 2ffp-1 + ffp) Xp = ffj j=1 Xp This contradicts ffj = -1. j=1 16 Dedicated to the memory of Masahiro Sugawara James P. Lin Let Ad2(~x)(~y) = ~x (~x ~y- ~y ~x) - (~x ~y- ~y ~x) ~x. We note if ~ : X x X ! X is the multiplication on X, then ___2 (~(~ x 1))* = 4 and (43) ~*(~* 1)(Ad2(~x)(~y)) = [~x, [~x, ~y]] = 0 by Theorem 2(c) . Let i satisfy the conditions of Lemma 2. Then for p > 3, 0 6= ffi - 2ffi+1 + ffi+2 i-1 2 p-i-2 Xp j-1 p-j = < ~x Ad (~x)(~y) ~x , ffjß(x) ß(y) ß(x) + c > j=1 i-1 2 p-i-2 p = < ~x Ad (~x)(~y) ~x , ß (W ) > by Theorem 5 i-1 2 p-i-2 p p-2 = < ~x Ad (~x)(~y) ~x , Q2ß (Vi) + 4iß (Ui) > by Theorem 6 i-1 2 p-i-2 p-2 = < ~x Ad (~x)(~y) ~x , 4iß (Ui) > since ~x, ~y2 ker Q2 i-1 p-i-2 p-2 = < ~x [~x, [~x, ~y]] ~x , ß (Ui) > by Definition 3 and (43)* * . Therefore [~x, [~x, ~y]] 6= 0. We have proved if X0 exists for p > * * 3 then [~x, [~x, ~y]] 6= 0. This contradicts Theorem 2. Hence p = 3. This completes the proof of the Main Theorem. References 1. K. K. Anderson, J. Grodal, J. Moller and A. Viruel, The classification of p-compact groups for p odd, (to appear). 2. M. Arkowitz and P. Silberbush, Some properties of Hopf type construction* *s, Math. Proc. Cambridge Philos. Soc., 117(1995) no.2, 287-301. 3. P. Baum, W. Browder, The Cohomology of quotients of classical groups, Topology, 3 (1965), 305-336. 4. W. Browder, On differential Hopf Algebras, Trans. Amer. Math. Soc., 107 (1963), 153-178. 5. F. Cohen and J. Neisendorfer, A construction of p-local H-spaces, Algebr* *aic Topology Lec. Notes in Math. v. 1051 (1982) Springer, 351-359. 6. G. Cooke, J. Harper and A. Zabrodsky, Torsion free mod p H-spaces of low rank, Topology 18(1979), 349-359. 7. W. Dwyer, and C. Wilkerson, Homotopy fixed point methods for Lie groups and finite loop spaces, Ann. of Math. 139(1994) 395-442. 8. W. Dwyer, and C. Wilkerson, Homotopical uniqueness of classifying spaces, Topology 31 (1992), 29-45. 9. H. Hamanaka, and S. Hara, The mod 3 homology of the space of loops on the exceptional Lie groups and the adjoint action, J. Math. Kyoto, 37-3 (199* *7) 441-453. 10. J. Harper, H-spaces with Torsion, Mem. Amer. Math. Soc., no.223(1979). 11. Y. Hemmi, Maps inducing iterated reduced coproduct, preprint. 12. N. Iwase, Adjoint action of a finite loop space, Proc. Amer. Math. Soc., 125(1997), 9, pp.2753-2757. H-spaces analogous to E8 mod 3 17 13. R. Kane, The Homology Algebra of finite H-spaces, J. P.A.A., 41(1986), 213-232. 14. R. Kane, The Homology of Hopf Spaces, North Holland, (1988). 15. R. Kane, Torsion in Homotopy Associative H-spaces, Ill. J. Math* *., 20(1976), 476-485. 16. A. Kono and K. Kozima, The adjoint action of Lie groups on the space of loops, J. Math. Soc. Japan, 45 No. 3 (1993), 445-510. 17. A. Kono, J. Lin and O. Nishimura, Characterization of the mod 3 cohomol- ogy of E7, Proceedings of the American Math. Soc., 131(2003), 3289-3295. 18. J. Lin, Commutators in the Homology of H-spaces, Japanese American Math. Institute Proceedings, AMS Contemporary Math., 293(2002), 141- 152. 19. J. Lin, H-spaces with Finiteness Conditions, Handbook of Algebraic Topol- ogy, chapter 22 (1995), 1095-1141. 20. J. Lin, Mod 3 cohomology algebras of finite H-spaces, Math. Z., 240(2002* *), 389-403. 21. J. Lin, Torsion in H-spaces II, Ann. Math. 107(1978), 41-88. 22. J. Lin, Indecomposables of Multiplicative fibrations, Proc. AMS, (to app* *ear). 23. J. McCleary, User's Guide to Spectral Sequences, Mathematical Lecture Series 12, Publish or Perish Inc. 1985. 24. J. Milnor, Construction of Universal Bundles, I, II, Ann. Math., 63(1956* *), 272-284, 430-436. 25. J. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. Math., 81(1965), 211-264. 26. J. Moller, Homotopy Lie Groups, Bull. AMS, 32(1995), 413-428. 27. J. Stasheff, Homotopy Associativity of H-spaces, I, II, Trans. Amer. Mat* *h. Soc., 108(1963), 2275-292, 293-312. 28. M. Sugawara, A condition that a space is group-like, Math. J. of Okayama Univ., 7(1957), 123-149. 29. N. Yagita, Homotopy nilpotency for simply connected Lie groups, Bulletin London Math. Soc., 25(1993), 481-486.