* Smith-Toda Spectrum V (1) exists for all p > 5 Xueguang Zhou Department of Mathematics, Nankai University Tianjin 300071, China 1 The main result In [6], Milnor proved that the dual Steenrod algebra A*phas the following algeb* *ra structure A*p= E(ø0, . .,.øn, . .). P (,1, . .,.,n, . .). where E denotes the exterior algebra and P denotes the polynomial algebra and |* *øi| = 2pi- 1, |,i| = 2pi- 2, p > 3. In [8], Smith proved that there exists a spectrum V (n) for 0 6 n 6 2, p > 5* * such that as a comodule over the dual Steenrod algebra H*(V (n), Zp) = E(ø0, . .,.øn) In [9], Toda proved that there exists a spectrum V (n) for 0 6 n 6 3, p > 7 * *such that as a comodule over the dual Steenrod algebra H*(V (n), Zp) = E(ø0, . .,.øn) In recent years, the Smith-Toda spectrum V (n) plays an important role in ho* *motopy theory. It is natural to ask that whether V (n) exist for n > 4. After long yea* *rs of hard work on this subject, we finally proved that V (1) exists for p > 5. We first introduce some notations. For a graded module A, we use A(n)to deno* *te the submodule of A generated by all elements of degree 6 n. Since any connected spe* *ctrum ________________________________ *Supported by National Natural Science Foundation of China 1 X may be considered as a spectrum, we use Er(X) to denote the r-th space of X* *. The following is the main result of this paper. Main Theorem Let p > 5 be a fixed prime and m be a non-negative integer, then there exists a spectrum W (m) such that the following properties hold. (a) As a comodule over the dual Steenrod algebra, H*(W (m), Zp)(m+1)= E(ø0, . .,.øn, . .).(m+1) (b) ßn(W (m)) = 0 for n > m+ 1. (c) W (m) is a ring spectrum with unit. (d) W (m) is homotopy commutative. (e) W (m) is homotopy associative. (f) There exists a map æm : W (m) ! W (m- 1) such that æm is the (m- 1) stag* *e Post- nikov decomposition of W (m) and æm is also an H map for m > 1. (g) There exists a map fm : k(Zp, 1) ! E1(W (m)) such that E1(æ1) O . .O.E1(* *æm ) O fm = id: k(Zp, 1) ! k(Zp, 1), where k(Zp, 1) denotes the Eilenberg-Maclane spa* *ce and Er(æm ): Er(W (m)) ! Er(W (m- 1)) denotes the natural map induced by æm : W (m)* * ! W (m- 1). (h) fm : k(Zp, 1) ! E1(W (m)) is an H map with respect to the loop multiplic* *ation structure of E1(W (m)). In this paper, p is always assumed to be a fixed prime > 5. Let X be a local finite connected spectrum, then it can be easily seen that * *there exists a sequence of subspectrum X(n), n = 1, 2, . .,.such that (a) X(1) X(2) . . .X(n) . . .X. (b) X(n)contains no i-cell with i> n. (c) Every cell represents a cycle modp. (d) The injection X(n)! X induces an isomorphism from H*(X(n), Zp) to H*(X, * *Zp)(n) The above sequence of spectra {X(n)} are called the p standard CW-decomposit* *ion of X and X(n)is called the p standard n-skeleton of X. If there is no confusion, w* *e simply 2 call them the standard decomposition and standard n-skeleton of X. We have that H*(W (q(n))q(n), Zp) = E(ø0, . .,.øn) n+1-1 where q(n) = |ø0. .ø.n| = 2p_____p-1- (n+ 1). It is obvious that for p > 5, 3q(* *n) < |øn+1| = 2pn+1- 1. We also have that for all q(n) 6 m 6 2pn+1 - 2, H*(W (m)m , Zp) = E(ø0, . .,.øn) Therefore, we have Theorem 2. Let p > 5 be an odd prime, then for all non-negative integers n,* * V (n) exists. Since 3q(n) < (2pn+1- 1), it can be easily seen that the multiplication map * *of n+1-2 W (2pn+1 - 2)2p and the homotopy maps for the commutativity and associativi* *ty of n+1 + * * + W (2pn+1-2)2p respectively map V (n)^V (n), V (n)^V (n)^I , V (n)^V (n)^V (n* *)^I into V (n), so we have Theorem 3. Let p > 5 be an odd prime, then for all integers n > 0, the V (n)* * constructed above is a commutative, associative ring spectrum with unit and the natural inj* *ection V (n- 1) ! V (n) is an H-map. Now let V (1) = [06n61 V (n) or limW (n), then it is easy to see that H*(V (1), Zp) = E(ø0, . .,.øn . .). Since the injections V (m) ! V (m+ 1) and W (m) ! W (m- 1) are all H-maps, V (1* *) is also an ring spectrum with unit. Notice that V (1) is an infinite spectrum and * *ßm (V (1)) may be nonzero for infinite m's. So we can not prove that V (1) is a commutati* *ve or associative ring spectrum. Let E be an infinite ring spectrum. If homotopy commutativity and associativ* *ity hold on any finite subspectrum of E, we say that E is a Q.C.A ring spectrum. We have 3 Theorem 4. Let p > 5 be an odd prime, then V (1) exists and is a Q.C.A ring * *spectrum with unit. Since H*(K(Zp, 0), Zp) = A*p = E(ø0, . .,.øn, . .). P (,1, . .,.,n, . .). = H*(V (1), Zp) H*(BP, Zp) where BP denotes the Brown-Peterson spectrum and K(Zp, 0) denotes the Eilenberg- Maclane spectrum. So we have Theorem 5. Let p > 5 be an odd prime, then K(Zp, 0) = V (1) ^ BP. So K(Zp, 0) is decomposable. Since Theorem 2 to Theorem 5 are all deduced from the main theorem, we need * *only prove the main theorem. We shall prove it by induction. Suppose the main theo* *rem holds for m, we shall prove that it holds for m+ 1. We use the letter (a)m+1 , * *. .,.(e)m+1 respectively to denote (a), . .,.(e) hold for m+ 1. In the literatures, V (n) is constructed from K(Zp, 0) by killing all non øi* *0. .ø.interms. The difficulty is that some øi0. .ø.inmay not be killed in this process. It sh* *ould be noticed that all øi, i> 0 exist in k(Zp, 1) and all the cohomology operationss * *Qi are non- trivial at the fundamental cohomology class of k(Zp, 1). So, we reduce the prob* *lem of the existence of V (1) to the problem of lifting the natural map from the stable ho* *motopy P -1 type ~k(Zp, 1) to K(Zp, 0) to the Postnikov decomposition W (m) of V (1). I* *f W (m) is an associative and commutative ring spectrum with unit, then E(øi0. .ø.in. .* *).is a natural subalgebra of H*(W (m), Zp). So V (1) exists. This is the main idea of * *the proof of (a) to (e) which is also the first part of our paper. If W (m) is a commutative and associativbe ring spectrum with multiplicative* * map Mm , then it can be easily seen that there exists multiplicative map Mm+1 on W * *(m+ 1) 4 such that W (m+ 1) ! W (m) is an H-map. However, it can not be easily seen that* * Mm+1 is commutative and associative. Notice that the difference of the different mul* *tiplicative maps is a cohomology class of H*(W (m)^ W (m)), so by studying the relations be* *tween D(Mm+1 , ~Mm+1) we prove the existence of a new associative and commutative mul* *tiplica- tive map Mm+1 . Since the computation needs a coefficient 1_3, the result hold* *s only for p> 5. The above computation is the proof of (f) to (g) and is the second part o* *f our paper. By the theory of maps from the stable homotopy type of CW-complex to spectra* *, we P -1 reduced the problem of lifting the map from ~k(Zp, 1) to W (m) to the probl* *em of lifting the identity map of k(Zp, 1) to a map from k(Zp, 1) to E1(W (m)). The m* *ain toool is to use Milnor's construction Br(G) (r = 1, 2) for a topological group G. To * *study the relation between Br(k(Zp, 1)) and Br(E1(W (m))), we introduce the minus product* *, plus product and semi-mixed product of CW-complexes. Using this construction, we get* * the proof of (h). This is the third part of our paper. The next section introduces some preliminaries and notations used in our pro* *of. We use here twice the notion of Ext group of an algebra. Firstly, we use it to pr* *ove that k(Zp, 1) ! E1(W (n)) is an H-map. Secondly, we use it to prove that W (m) is ho* *motopy associative. So we introduce the basic properties of Ext group of an algebra. On page 289 of [7], D.C. Ravenal claimed in Theorem 7.5.1 that V (3) does no* *t exist for p = 5. His proof depends on the Toda's result ff1fip1= 0. In the appendix, * *we will show that all the proofs of the statement ff1fip1= 0 are incorrect. So, the prooof o* *f Ravenal's result is also incorrect. The author wishes here to express his gratitude to J.K. Lin and Q.B. Zheng a* *nd X.J. Wang for their helps in our work. The author also wishes to express his gratitu* *de to D. Ravenal and J.P. Meyer for their pointing out some errors in our unpublished pr* *evious works [11] and [12] about this subject. 2 Notations about spectra Since any connected spectrum is equivalent to a spectrum, any connected spect* *rum X can be expressed in the following form {Er(X), r: Er(X) ! Er+1(X), r> 1} such* * that 5 (a) Er(X) is the space of Er+1(X) for r> 1. We call Er(X) r-th space of X. (b) r: Er(X) ! Er+1(X), r> 1 is the adjoint of (Er+1(X)). Let Y be a CW-complex. We use ~Yto denote the stable type] { Y, 2Y, . .,. n* *Y, . .}., then there is a natural isomorphism between ß[ rY, Er(X)] and ß[Y~, X], r> 1. L* *et f: Y ! Er(X) be a map, we use -rf~: -rY~ ! X to denote the map of spectra determined* * by f. Let X1, X2 be two connected spectra, f: X1 ! X2 be a map. For r> 1, we use Er(f): Er(X1) ! Er(X2) to denote the map of space determined by f. 3 Properties of homology groups of algebras Let M be a commutative and associative graded algebra over Zp with unit. We use* * M~ to denote the kernel of the augmentation ö f M . We denote Cn(M) = M~__._._.~M-z_____" n copies We define @n,i: Cn(M) ! Cn-1(M) by that for x1, . .,.xn 2 M~, @n,i(x1 . . .xn) = x1 . . .xi-1 xixi+1 . . .xn and define @n: Cn(M) ! Cn-1(M) by @n = n-1i=1(-1)i-1@n,ifor n> 2 and @1 = 0, t* *hen C*(M) = {Cn(M), @n, n> 1} is a chain complex. The dual C*(M) = {C*n(M), @*n, n>* * 1} is a cochain complex. We call H*,*(C*(M), @*) and H*,*(C*(M), @*) respectively * *the Tor and Ext group of M and simply denote them by H*,*(M) and H*,*(M). Notice that the tensor product from C*m(M) C*n(M) ! C*m+n(M) makes C*(M) a DGA and thus H*,*(M) is an algebra over Zp and H*,*(M) is a coalgebra over Zp. For two commutative associative graded algebrs M and N over Zp with unit, the tensor product (over Zp) algebra M N is also commutative and associative. By * *using the tensor product of projective resolutions of M and N, we have that Proposition 3.1 H*,*(M N) = H*,*(M) H*,*(N) H*,*(M N) = H*,*(M) H*,*(N) 6 where the tensor products means the tensor product of coalgebras and algebras r* *espectively. Let E(ø) be the exterior algebra generated by ø with |ø| an odd number. A d* *irect calculation shows that Proposition 3.2 H*,*(E(ø)) = P (ø)*, where P (ø)* is the dual of the polynom* *ial algebra P (ø). P (ø)* has a basis øn = {ø___._.-.øz____", n> 1} with the coalgebra map* * defined by n-folds (øn) = n-1i=1øi øn-i. Since E(ø0, . .,.øn, . .).= E(ø0) . . .E(øn) . .,.we have Proposition 3.3 H*,*(E(ø0, . .,.øn, . .).) = P (ø0)* . . .P (øn)*. ... Let T (x) = P (x)=xp be the truncated polynomial algebra generated by x, a k* *nown result is that Proposition 3.4 H*,*(T (x)) = E(y)* P (z)*, where y is represented by x in* * C*(M) and zn in P (z)* is represented by x___xp-1__._.-.xz__xp-1____"in C*(M). n-folds It is well-known that the algebra H*(k(Zp, 0), Zp) = E(v) ( 06i<1T (upi)*), * *where v 2 H1(k(Zp, 1), Zp), upi2 H2pi(k(Zp, 1), Zp), fi(u1) = v (fi denotes the Bockstein* * operation). So by the previous propositions, we have Proposition 3.5 i j H*,*H*(k(Zp, 1)) = P (y)* 06i<1 (E(yi) P (zi)*) where yi is represented by upi in C*(H*(k(Zp, 1))) and zniin P (zi)* is represe* *nted by upi up-1pi . . .upi up-1pi. _____________-z____________" n-folds 4 The case m = 0 We prove that the Main Theorem holds for m = 0. In this case, we take W (0) = K* *(Zp, 0), then E1(W (0)) = k(Zp, 1), and we take f0 = id: k(Zp, 1) ! E1(W (0))= k(Zp, 1).* * It is 7 obvious that (a)0 to (h)0 hold. 5 Comodule H*(W (m), Zp) From now on, we always assume that the Main Theorem holds for m. In this sectio* *n, we prove that H*(W (m), Zp) contains a subalgebra E(ø0, . .,.øn, . .).. Now fm : k(Zp, 1) ! E1(W (m)) induces a map -1f~m: -1~k(Zp, 1) ! W (m). L* *et (f~m)*( -1(upi)) = ø0i. Since æ1. .æ.m -1(f-1m): -1(~k(Zp, 1) ! W (0) is the f* *undamental cohomology class of -1~k(Zp, 0), it follows from [6] that (æ1. .æ.m)*(ø0i) = ø* *i. Since ø0, . .,.øn, . .g.enerate an subalgebra E(ø0, . .,.øn, . .)., we have Proposition 5.1 (a) ø00, . .,.ø0n, . .g.enerate an exterior algebra E(ø00, . .,.ø0n, . .).. (b) H(W (m), Zp)(m+1)= E(ø00, . .,.ø0n, . .).(m+1). 6 The proof of (a)m+1 ,(b)m+1 ,(c)m+1 Let X, Y be two spectra and f: X ! Y be a map. We use C(f) to denote the map co* *ne C(X) [ Y . For ff2 Hm (X, Zp), we also use ff to denote the map from X to K(Zp* *, m) determined by ff. Suppose the Main Theorem holds for m, it follows from Proposition 5.1 that t* *here exists a set of cohomology classes {ffi2Hm+2 (W (m), zp) | 16 i6 s} such that (a) < ffi, E(ø00, . .,.ø0n, . .).(m+2)>= 0, 1 6 i 6 s, where <, > denotes the Kronecker dual product. (b) ff1, . .,.ffs are lineraly independent. (c) dim(H*(W (m), Zp)(m+2)=E(ø00, . .,.ø0n, . .).(m+2)) = s We call ff1, . .,.ffs the Postnikov invariant of W (m + 1) (it is possible t* *hat s is 0), then ff1_ . ._.ffs define a map ff: W (m) ! K(Zp, m + 2) _ . ._.K(Zp, m + 2), w* *e define _______________-z______________" s-folds W (m + 1) = -1C(ff). 8 It can be easily seen that the following is a cofibration sequence W (m + 1) ! W (m) ! K(Zp, m + 2) _ . ._.K(Zp, m + 2) _______________-z______________" s-folds Let æm+1 : W (m + 1) ! W (m) be the natural injection. It follows from the * *above conclusion that H*(W (m), Zp)(m+2) = E(ø00, . .,.ø0n, . .).(m+2) E(ø0, . .,.øn, . .).(m+2) and ßn(W (m+1)) = 0 for n > m+1 and æm+1 is the (m+1) stage Postnikov decomposi* *tion of W (m + 1). So (a)m+1 ,(b)m+1 holds. Now we prove that W (m + 1) is a ring spectrum with unit. Let Mm : W (m) ^ W (m) ! W (m) be the multiplication map of W (m) with unit. Consider the following diagram jm+1^jm+1 W (m+ 1) ^ W (m+ 1) -! W (m) ^ W (m) ? # Mm # jm+1 ff W (m+ 1) -! W (m) -! _s copiesK(Zp, m+ 2) Since (Mm )*(æm+1 ^ æm+1 )*Hm+2 (W (m+ 1) ^ W (m+ 1), Zp) E(ø0, . .).(m+2)* *, we have that (ffi)*(Mm )*(æm+1 ^æm+1 )*Hm+2 (W (m+ 1)^W (m+ 1), Zp) = 0, 06 i6 s, that * *is, we have map equality ffOMm O(æm+1 ^æm+1 ) = 0. Therefore, there exists a map M0m+1: W (* *m+ 1)^ W (m+ 1) ! W (m+ 1) such that æm+1 O M0m+1 = Mm O (æm+1 ^æm+1 ). But we do not know whether M0m+1is a multiplication with unit. Since Mm |S0^W(m) = id|W(m) a* *nd Mm |W(m)^S0 = id|W(m), we have that æm (M0m+1|S0^W(m+1)- id|W(m+1)) = 0 æm (M0m+1|W(m+1)^S0- id|W(m+1)) = 0 Thus, there exist cohomology classes ff, fi 2 Hm+1 (W (m + 1), ßm+1 (W (m + * *1)) such that 9 M0m+1|S0^W(m+1)- id= jff M0m+1|W(m+1)^S0- id= jfi where j: K(ßm+1 (W (m+ 1)), m+ 1) ! W (m+ 1) denotes the natural injection. No* *w we define Mm+1 = M0m+1- j(S*0^ ff) - j(fi ^ S*0) then it is easily seen that Mm+1 is a multiplication with unit and æm : W (m + * *1) ! W (m) is an H-map with respect to Mm+1 and Mm . Thus, (c)m+1 holds. 7 Differences of homotopy First, we construct some spectrum. Since S2 is a co-H-group, let ffi: S2 ! S2 _ S2 be the cogroup map. Let æ: S* *2 ! S2 be the inverse -id of id: S2 ! S2 and ø: S2 _ S2 ! S2 be the map such that the res* *triction of it on every summand S2 is the identity map. It is obvious that ø O (id_ æ) O* * ffi: S2 ! S2 is homotopic to the constant map, so it can be extended to a map L: C(S2) ! S2. Let X be a spectrum. Since X = 2( -2X) = ( -2X ^ S2), X inherit a co-struct* *ure from S2. We also use ~æto denote the map (id)^æ: X= -2X ^S2 ! -2X ^S2= X. The* *n, the map ø(id_ ~æ)ffi: X ! X also can be extended to a map (id^ L): -2X ^ C(S2)* * ! -2X ^ S2 = X. Let f: X ! Y ba a map of spectra. It can easily be seen that the following d* *iagram is commutative C(X) -L! X # # C(Y ) -L! Y As usual, we use S1+ to denote circle with an added base point *. Notice tha* *t X ^S1+ and X ^ S1 = X are two defferent spectra. Proposition 7.1 X ^ S1+ and X _ X are of the same homotopy type. 10 Proof. Notice that X = ( -1X) ^ S1, so we need only prove that space S1 ^ S1* *+ and S2 _ S1 are of the same homotopy type. This is a direct checking. * *Q.E.D. Now let f, g: X ! Y be two maps of spectra. If there exists a map H: X ^ I+ * *such that H|Xx0 = f, H|Xx1 = g, then we say that H is a homotopy from f to g. Let H,* * H0 be two homotopies from f to g, then H and H0 define a T : X ^ S1+ ! Y as follow* *s. T |X^I+1= H T |X^I+2= H0 where we regard S1 as the quotient space I1[ I2= ~ (I1 = I2 = [0, 1]) by identi* *fying {0, 1} of I1 with {0, 1} of I2. Let F : X ! X ^ S1+ be the composite of the natural i* *njection from X to X _ X and the map from X _ X to X ^ S1+, then T F is a map from X to Y . We call T F the difference of H and H0 and denote it by d(H, H0). Hom* *otopy extension theory shows that Proposition 7.2 Let f, g: X ! Y be two maps and H a homotopy from f to g, th* *en for any map ff: X ! Y , there exists a homotopy H0 from f to g such that d(H, H0) * *= ff. We also call H0 the sum of H and ff and denote it by H + ff = H0. If X is a CW-complex and Y is a topological group, f, g: X ! Y are two maps * *and H and H0are two homotopies from f to g, then we define ~d(H, H0)(x, t) = H0(x, t)* *(H(x, t))-1. It is obvious that ~d(H, H0)(x, 0) = f(x)f(x)-1 = y0 = g(x)g(x)-1 = ~d(H, H0)(x* *, 1) where y0 denotes the unit of Y . So, ~d(H, H0) can define a map from X ! Y which w* *e still denote by ~d(H, H0). Let X be a finite spectrum and Y be a connected spectra. It can be easily * *seen that the problem about the homotopies from Er(X) to Y can be reduced to the pro* *blem of homotopies from Er(X) to Er(Y ) for r sufficiently large. Since Er(Y ) is h* *omotopy equivalent to a topological group, we can use d~(H, H0) to define d(H, H0) and * *by this definition, we have the following proposition Proposition 7.3 Let X, Y be two spectra, f, g: X ! Y be two maps. H and H0 a* *re two 11 homotopies from f to g. fi, fi0: X ! Y be two maps, then d(H + fi, H0+ fi0) = d(H, H0) + fi0- fi. Proposition 7.4 Let X, Y be two spectra, f, g: X ! Y be two maps. H and H* *0 are two homotopies from f to g. Then a necessary and sufficient condition for d(H, * *H) = 0 is that one of the following condition holds. (a) H H0 rel X ^ {0, 1}+. (b) The map T : X ^ S1+ defined above may be extended to a map from X ^ (C(S* *1))+ to Y . Let X, Y be two spectra and A be a subpectrum of X. f, g: X ! Y are two maps such that f|A = g|A. If f g relA, that is, there is a homotopy from f to g su* *ch that H(a^ t) = f(a) for all a 2 A and 0 6 t 6 1, we say that H is a stationary homot* *opy from f to g relA or f and g are stationary homotopic relA. We use f~: A^ I+ to deno* *te the homotopy defined by ~f(a^ t) = f(a) for all a 2 A and 0 6 t 6 1. Let H be a hom* *otopy from f to g such that f|A = g|A, then d(H|A^I+, ~f) is defined and is a map fro* *m A to Y . If d(H|A^I+, ~f) 0, we say that H is a quasi stationary homotopy from f t* *o g relA or f and g are quasi stationary homotopic relA. Then, we have the following pro* *position Proposition 7.5 Let f and g be two quasi stationary homotopic map relA, then (a) f and g are stationary homotopic relA. (b) The quasi stationary homotopy H from f to g relA is quasi stationary hom* *otopic rel X^ 0+ [ X^ 1+ to a stationary homotopy from f to g relA. Proof. (a) follows from (b). (b) follows from the homotopy extension propert* *y with respect to pair X^ I+ , the subspectrum X^ 0+ [ X^ 1+ [ A^ I+ and the map H Q.* *E.D. Let H, H0be two quasi stationary homotopy from f to g, then d(H, H0) is defi* *ned and is a map from X to Y . It can be easily seen that the following proposition ho* *lds. Proposition 7.6 Let H and H0 be two quasi stationary homotopies from f to g * *relA, then 12 (a) d(H, H0)| A 0. (b) Let ff: X ! Y be a map such that ff| A 0, then H +ff is also a quasi * *stationary homotopy from f to g relA. For the convenience of later use, we consider d(H, H0) as a map from X to Y* * . Let H be a homotopy from f to f. In what follows, we use d(H) to denote the * *map d(H, ~f): X ! Y , where ~fdenotes the stationary homotopy from f to f relX. I* *t can be easily seen that for maps f, g: X ! Y and homotopies H, H0 from f to g, we * *have d(H, H0) = d(H - H0), where H - H0 denotes the homotopy from f to g defined by ( H(x^ 2t) 06 t6 1_ (H - H0)(x^ t) = 0 1 2 H (x^ 2(1 - t)) _26t6 1 8 Cochain of differences Let X, Y be two connected spectra, {X(m), m> 0} be the usual CW decomposition o* *f X. Let f, g: X ! Y be two maps and H a homotopy from f|X(m)to g|X(m). As usual, we* * define the cochain D(f, g, H) of difference of f and g as follows. Let a be a (m+ 1) c* *ell in X(m+1), using the ordinary orientation we have that a^ 0+ [ @(a)^ I+ [ a^ 1+ is a (m+ 1* *) sphere, so f|a = f|a^0+, g|a = g|a^1+ and H|@a^I+ define an element D(f, g, H)|a 2 ßm+1 (Y* * ). Thus, D(f, g, H)2 Cm+1 (X, ßm+1 (Y )). It is obvious that D(f, g, H) depends on the h* *omotopy H. Now we study the relation between D(f, g, H) and the difference of homotopie* *s. Let x be an (m+ 1) cell in X(m+1), then x may be considered as an element in ßm+1 (X(* *m+1), X(m)). As usual, we use ~@: ßm+1 (X(m+1), X(m)) ! ßm (X(m)) to denote the boundary ope* *ration of relative homotopy group. For x2 Cm+1 (X), we also use ~@(x) to denote the el* *ements in ßm (X(m)). It is obvious that D(f, g, H) is a cocycle. We also use D(f, g, H)* to deno* *te the cohomology class and call it the cohomology of difference of f and g with respe* *ct to H. Obviously, D(f, g, H)* depends on the homotopy H. In the following, it is assum* *ed that pß*(Y ) = 0. We call such a spectrum p spectrum. Let x2 Hm+1 (X, Zp), then x ma* *y be considered as a linear combination of (m+ 1) cells. Let H0 be another homotopy* * from 13 f to g, it may be assumed that ff = d(H, H0). Since on x ^ 0+, x ^ 1+, H and H0 coincide, so D(f, g, H) and D(f, g, H0) differ only on ~@x ^ I+ . It can be eas* *ily seen that D(f, g, H0)(x) = D(f, g, H)(x) + ff*(@~(x)). It should be noticed that ~@(x) i* *s uniquely determined modp in ßm (X(m)). So, ff*(@~(x)) is also uniquely determined. Let `: (X(m)) ! Y be a map. We define _(`) 2 Hm+1 (X, ßm+1 (Y )) by _(`)(* *x) = `*(@~(x)) 2 ßm+1 (Y )) for x 2 Hm+1 (X, Zp). Let G be the subgroup of Hm+1 (X, ßm+1 (Y )) generated by all _(`) with `: * *X(m) ! Y . We define D(f, g)* to be the set {D(f, g, H)*} with H taken over all the homoto* *pies from f to g. It can be easily seen that D(f, g)* is a coset modG. We have Proposition 8.1 Let D(f, g)* be as defined above, then (a) D(f, g)* is uniquely defined by the homotopy classes of f and g. (b) A necessary and sufficient condition for f and g to be m+ 1 homotopy is * *that D(f, g)* = 0 modG. Let A be a subspectrum of X and f, g: X ! Y be two maps such that f|A = g|A.* * Let H be a quasi stationary homotopy from f|X(m)to g|X(m)relA(m), then it can be ea* *sily seen that D(f, g, H)|A(m)= 0. So D(f, g, H) 2 Hm+1 (X, A, ßm+1 (Y )) = Hm+1 (X=A, ßm* *+1 (Y )). Let f: X ! Y be a map of spectra, H be a homotopy from f|X(m) to f|X(m). It * *can be easily seen that D(f, f, H)|a = d(H)*(@~(a)) for any (m+ 1) cell a in X. If * *d(H) 0, then D(f, f, H)|a = 0, so we have the following proposition Propostion 8.2 Let f and H be as above. If d(H) = 0, then D(f, f, H) = 0. Let A be a subspectrum of X, f|A = g|A, and H be a quasi stationary homotopy from f|X(m) to g|X(m) relA(m), then it can be easily seen that D(f, g, H)|A(m) * *= 0, so D(f, g, H) 2 Hm+1 (X, A, ßm+1 (Y )) = Hm+1 (X=A, ßm+1 (Y )). 9 W (m+ 1) is a commutative ring spectrum For any spectrum W , we use T : W ^ W ! W ^ W to denote the map switching the t* *wo factors. Then, the statement that Mm+1 is homotopy commutative is equivalent to* * that Mm+1 OT = Mm+1 . Since S0^W (m+ 1) = W (m+ 1)^S0 = W (m+ 1), Mm+1 T |S0^W(m+1) = 14 Mm+1 |S0^W(m+1) = id and Mm+1 T |W(m+1)^S0 = Mm+1 |W(m+1)^S0 = id. So we have Mm+1 T |W(m+1)^S0_S0^W(m+1)= Mm+1 |W(m+1)^S0_S0^W(m+1). In the following part of this section, we use X to denote W (m+ 1) ^ W (m+ 1* *), A to denote the subspectrum W (m+ 1) ^ S0 _ S0 ^ W (m+ 1). Then, (d)m+1 is included * *by a stronger proposition. Proposition 9.1 Suppose (a) to (c) hold for m+ 1 and (d)m holds, then there * *exists a multiplication Mm+1 : X ! W (m+ 1) such that (a) Mm+1 has a unit. (b) Mm+1 and Mm+1 T are quasi stationary homotopic relA. Using the standard CW decomposition, we have that W (m+ 1)(m+2) is obtained * *by killing (m+ 2) dimensional cohomology classes ff1, . .,.ffs. So W (m+ 1)(m+1)= * *W (m)(m+1). It follows from the induction hypothesis that there exists a multiplication wit* *h unit M~m+1 : X ! W (m+ 1) and a quasi staionary homotopy H from M~m+1|X(m)to M~m+1T * *|X(m) relA(m). So D(M~m+1 , ~Mm+1T, H) is defined. In order to prove Proposition 9.1,* * we need only prove that D(M~m+1 , ~Mm+1T, H) = 0. First, we state some T-properties of homotopy. Let X, Y be two spectra, f, g* *, h: X ! Y be maps with H1 a homotopy from f to g and H2 a homotopy from g to h. We use H1+ H2 to denote homotopy defined by ( H1(x ^ (2t)+) 06 t6 1_ (H1+ H2)(x ^ t+) = + 1 2 x 2 X H2(x ^ (2t - 1) ) _26t6 1 We use -H to denote the homotopy from g to f defined by (-H)(x ^ t+) = H(x ^ (1 - t)+). Suppose H: (W (m+ 1) ^ W (m+ 1))(m)^ I+ ! W (m+ 1) be a homotopy from M~m+1 to M~m+1T , then H O ~Tis also a homotopy from M~m+1 to M~m+1T , where T~* * = T ^ (id): (W (m+ 1) ^ W (m+ 1))(m)^ I+ ! (W (m+ 1) ^ W (m+ 1))(m)^ I+ . We have* * the following proposition. Proposition 9.2 There exists a quasi stationary homotopy H from M~|X(m)to M~* *m+1T |X(m) relA(m) such that d(H, -HT~) 0. 15 Proof. Let H~ be a homotopy from M~m+1|X(m) to M~m+1T |X(m). In general, w* *e do not know whether d(H~, -H~T ) = 0. However, d(H~, -H~T ) is a map from (W (m+ * *1) ^ W (m+ 1))(m)to W (m+ 1), so d(H~, -H~T~) O ( T ) = d(H~T~, ~H~T)c= d(H~T~, -H~). It can be easily seen that d(H~T~, -H~) are obtained from D(H~, -H~T ) by movin* *g the point e0 in S1 to e1. Now the honotopy class of d(H~, -H~T ) is independent of the ch* *oice of e0 and e1. It can be easily seen that H can be extended to a quasi stationary homo* *topy relA from M to MT on X. We define a new quasi stationary homotopy H from M~m+1|X(m) * *to M~m+1T |X(m) relA(m) by H = ~H+ 1_d(H~, -H~T~), then it is obvious that d(H, -H* *T~) 0. 2 Q.E.D. Proposition 9.3 Let H be a quasi stationary homotopy from M~m+1 to M~m+1 T * *and x 2 Hm+1 (W (m+ 1)^W (M+ 1), Zp), D(M~m+1 , ~Mm+1T, H)*.x = -D(M~m+1 , ~Mm+1T, * *H)*. T*(x), then it is equivalent to the statement D(M~m+1 , ~Mm+1T, H)* = -T *(D(M~m+1 , ~Mm+1T, H)*). Proof. It follows by definition of D(M~m+1 , ~Mm+1T, H) and Proposition 9.1.* * Q.E.D. Now we prove Propostion 9.1. For a general M~m+1: (W (m+ 1) ^ W (m+ 1))(m+1* *) ! W (m+ 1) with unit, we do not know whether D(M~m+1 , ~Mm+1T, H) = 0. However, w* *e can define a new multiplication Mm+1 with unit by 1 Mm+1 = M~m+1 - j O __D(M~m+1 , ~Mm+1T, H) 2 i j where j: K ßm+1 (W (m+ 1)), m+ 1 ! W (m+ 1) denotes the natural injection. It * *should be noticed that W (m+ 1) is a p spectrum and ßm+1 (W (m+ 1)) is a Zp vector spa* *ce, that is, 1_22Zp, so 1_2D(M~m+1 , ~Mm+1T, H) is defined. It is obvious that D(Mm+1 , Mm+1 T, H) = D(M~m+1 , ~Mm+1T, H) - D(M~m+1 , ~Mm+1T, H) = 0 16 10 W (m+ 1) is an associative ring spectrum In this section, we always assume Y = W (m+ 1), X = Y ^Y ^Y and A = S0^ Y ^Y [ Y ^S0^ Y [ Y ^Y ^S0. Since S0^ S0^ Y = Y . S0^ Y ^S0 = Y , Y ^S0^ S0 = Y and that Mm+1 is a multi* *plica- tion with unit, so Mm+1 (id^Mm+1 )|S0^S0^Y [S0^Y ^S0[Y ^S0^S0 = Mm+1 (Mm+1 ^id)|S0^S0^Y [S0^Y ^S0[Y ^S0^S0 Similarly, we have Mm+1 (id^Mm+1 )|S0^Y ^Y= Mm+1 |Y ^Y= Mm+1 (Mm+1 ^id)|S0^Y ^Y* *. So we have that Mm+1 (id^Mm+1 )|A = Mm+1 (Mm+1 ^id)|A. Then, (e)m+1 is reduced to* * a more stronger proposition. Proposition 10.1 Suppose (a)m+1 to (d)m+1 hold and (e)m holds, then there e* *xists a quasi stationary homotopy relA from Mm+1 (id^Mm+1 ) to Mm+1 (Mm+1 ^id). Proof. We prove by induction. It is obvious that Proposition 10.1 holds fo* *r m= 0. Since in this case W (1) = W (0 = K(Zp, 0), we may suppose that m> 0. Let ~Mm+1* *: Y ^Y ! Y be a commutative multiplication with unit. By the induction hypothesis, there* * is a quasi stationary homotopy ~Hfrom M~m+1(id^M~m+1 )|X(m)to Mm+1 (Mm+1 ^id)|X(m)relA(m).* * So, D(M~m+1 (id^M~m+1 ), Mm+1 (Mm+1 ^id), ~H) is defined. For simplicity, we use xy* * to denote M~m+1 (x^ y) and analogously x(yz) to denote M~m+1(x^ ~Mm+1(y^ z)) and so on. * *Then D(M~m+1 (id^M~m+1 ), Mm+1 (Mm+1 ^id), ~H) can be expressed as D(x(yz), z(xy), ~* *H). Be- fore the proof of Proposition 10.1, we state the relation between D(x(yz), z(xy* *), ~H) and the action of the group Z6 on the factors of X. Let P, T : X ! X be the maps defined by P (x^ y^ z) = y^ z^ x, T (x^ y^ z) =* * z^ y^ x, x, y, z 2 Y . ø: Y ^Y ! Y ^Y be the map defined by ø(x^ y) = y^ x. Let G be the* * quasi stationary homotopy from xy to yx relS0^ Y [ Y ^S0. In what follows, we use x,* * y, z respectively to denote the element of the first, second and third factor Y of * *X. Since i j (xy)^ z ^ idis a map from X^ I+ to Y ^Y ^I+ , so G O ((xy)^ z)^ idis a homotop* *y from (xy)z to z(xy) on X(m). Let i j J1 = H + GO ((xy)^ z)^ id 17 i j J2 = HO(P ^id) + GO ((xy)^ z)^ id i j J3 = HO(P 2^id) + GO ((xy)^ z)^ id , then J1 is a homotopy from x(yz)|X(m)to z(xy)|X(m)and J2 is a homotopy from z* *(xy)|X(m) to y(xz)|X(m)and J3 is a homotopy from y(xz)|X(m)to x(yz)|X(m). Notice that G i* *s defined on Y ^Y ^I+ , so we have the following proposition. Proposition 10.2 D(x(yz), (xy)z, H) = D(x(yz), z(xy), J1). It is obvios that J1+ J2+ J3 is a homotopy from x(yz) to x(yz), we have the * *following proposition. Proposition 10.3 J1+ J2+ J3 is a quasi stationary homotopy relA(m). Proof. Since (S0^ Y ^Y )(m)= (Y ^Y )(m), we have J1 + J2 + J3|(S0^Y ^Y )(m)^I+ = (M~m+1 +G+ M~m+1+ ~Mm+1+ GO(ø^ id)+ M~m+1)|Y ^Y ^I+ = (G+ GO(ø^ id))|Y ^Y ^I+ It should be noticed that the symbol M~m+1 in the above equality is the stat* *ionary homotopy from M~m+1 to M~m+1 relY ^Y . Now by Proposition 9.1, G + GO(ø^ id): S0^ Y ^Y ^I+ ! Y is a stationary homo* *topy from yz to yz rel(S0^ S0^ Y ^I+ [S0^ Y ^S0^ I+ ), by the same argument as above* * we have 0^Y ^I+ 0 0 * * + that (J1+ J2+ J3)|Y ^S is a stationary homotopy from xz to xz rel(S ^ S ^ * *Y ^I [ 0^I+ S0^ Y ^S0^ I+ ) and that (J1+ J2+ J3)|Y ^Y ^S is a stationary homotopy from x* *y to xy rel(Y ^S0^ S0^ I+ [ S0^ Y ^S0^ I+ ). Therefore, the sum of above mentioned homo* *topies satisfies that d(J1+ J2+ J3|A(m)) 0 and J1+ J2+ J3 is a quasi stationary homo* *topy from x(yz) to x(yz) relA(m)on X(m). Since (P ^id) induces a map from X^ I+ to itself and (P ^id) also induces a * *map from X to itself, we also use ~Pto denote the map induced by (P ^id). We have the f* *ollowing proposition. 18 Proposition 10.4 The following diagram is homotopy commutative. d(J1+J2+J3) (X(m)) -! Y # ~P # id d(J1+J2+J3) (X(m)) -! Y Proof. Since d(J1+J2+J3)OP~is obtained from d(J1+J2+J3) by removing the point e of S1+ in X^ S1+, we have d(J1 + J2 + J3)OP~ d(J1 + J2 + J3). We use ~Tto d* *enote the map X ! X to denote the map induced by the map T ^id: X^ I+ ! X^ I+ . It * *is obvious that J1O(ø^ id): X^ I+ ! Y is a homotopy from z(yx) to x(yz), so J1+ J1* *O(ø^ id) is also a homotopy from x(yz) to x(yz). By the same argument as above, we have Proposition 10.5 The following diagram is homotopy commutative. d(J1+J1O(fi^id)) (X(m)) - ! Y # ~T # id d(J1+J1O(fi^id)) (X(m)) - ! Y In what follows, for any homotopy H: X(m)^I+ ! Y from x(yz) to z(xy), we alw* *ays use J1, J2, J3 to denote the homotopy defined above. We have the following prop* *osition Proposition 10.6 There exists a homotopy H from x(yz) to z(xy) on X(m) such * *that (a) d(J1 + J2 + J2) 0. (b) d(J1 + J1 O (T ^ id)) = 0. (c) J1 + J2 + J3 is also a quasi homotopy from x(yz) to x(yz) relA(m). Proof. For a general homotopy H~ from x(yz) to (xy)z on X(m), we do not know whether d(J1+ J2+ J2) 0. Since p> 5 and Y is a ring spectrum with unit, ß( X(* *m), Y ) is also a Zp-vector space. Thus, 1_3d(J1+ J2+ J2) is also a homotopy class from* * X(m) to Y . We define a new homotopy H0 = ~H- 1_3d(J1 + J2 + J2). Let J01, J02, J03be the h* *omotopies defined above for H0, then d(J01+ J02+ J03) i 1 1 1 * * j = d J1 - __d(J1 + J2 + J3) + J2 - __P *d(J1 + J2 + J3) + J3 - __P *2d(J1 + * *J2 + J3) 3 3 3 1i j = d(J1 + J2 + J3) - __d(J1 + J2 + J3) + d(J1 + J2 + J3) + d(J1 + J2 + J3) 3 = 0 19 Then we set H = H~ - 1_2d(H~ + ~H(T ^id)). By the same argument as above, we h* *ave d(J1 + J1O(T ^id)) 0. It is also easy to check that d(J1+ J2+ J3) 0. It is obvious that J1+ J2+ J3 is also a quasi stationary homotopy from x(yz)* * to x(yz) relA(m). It follows from Proposition 7.6 that Proposition 10.7 (a) D(x(yz), z(xy), J1)+ D(z(xy), y(zx), J2)+ D(y(zx), x(yz), J3) = 0 (b) Let a 2 Hm+1 (X, Zp), then D(x(yz), z(xy), J1)|a+ D(x(yz), z(xy), J1)|T*(a)= 0 It is obvious that (a) is equivalent to (1 + P *+ P 2*)D(x(yz), z(xy), J1) =* * 0 and (b) is equivalent to (1 + T *)D(x(yz), z(xy), J1) = 0. It can be easily seen that all the homotopy from x(yz) to (xy)z on X(m)just * *mentioned above are quasi stationary homotopy relA(m). So we have D(x(yz), z(xy), J1)|H*(A,Zp) = D(x(yz), z(xy), H)|H*(A,Zp) = 0 since A ! Z ! (Y=S0)^ (Y=S0)^ (Y=S0) is a cofibration sequence. By the collapse* *d exact sequence mentioned above we have D(x(yz), z(xy), J1) H*((Y=S0)^ (Y=S0)^ (Y=S0* *), ßm+1 (Y )). Since ßm+1 (Y ) is a Zp-vector space, we define ffin: Cn,*(H*(Y, ß* *m+1 (Y )) ! Cn+1,*(H*(Y, ßm+1 (Y )) and D(x(yz), z(xy), J1) 2 C3,m-2(H*(Y ), ßm+1 (Y )), so* * we can de- fine ffi3D(x(yz), z(xy), J1) and have the following proposition. Proposition 10.8 ffi3D(x(yz), z(xy), J1) = 0. Let W = Y ^Y ^Y ^Y and 3,1, 3,2, 3,3, 3,4: W ! X be the map defined by 3,1(a^ b^ c^ d)= (ab^ c^ d) 3,2(a^ b^ c^ d)= (a^ bc^ d) 20 3,3(a^ b^ c^ d)= (a^ b^ cd) 3,4(a^ b^ c^ d)= (da^ b^ c) where a, b, c, d respectively denotes the element in the first, second,third an* *d fourth factors of W . Then ffi3 = ( *3,1- *3,2+ *3,3), Let a(): W ! W (m+ 1) be the homotopy class of the map a(b(cd)). Since x(yz) (xy)z holds on X(m)and W (m+ 1) is homotopy commutative, we have a(b(cd)) a((* *bc)d) a(b0(c0d0)) (b0(c0d0))a holds on W (m+1). Since ßn(W (m+ 1)) = 0 for n > m+ 1* * > 0, we have a(b(cd)) a(b0(c0d0)) (b0(c0d0))a on W for any permutation (b0, c0, d0)* * of (b, c, d). Therefore, a(b0(c0d0)) and (b0(c0d0))a both belong to the homotopy class a[ ]. In the folowing part of this section, we use S to denote ffi3(D(x(yz), z(xy)* *, J1)). Then S = ( *3,1- *3,2+ *3,3)D(x(yz), z(xy), J1) = D((ab)(cd), d[ ], J1O( 3,1^id)) -D(a[ ], d[ ], J1O( 3,2^id)) +D((a[ ], (ab)(cd), J1O( 3,3^id)) For simplicity, we omit the homotopies in D since there is no confusion. So * *S may be expressed as D((ab)(cd), d[ ]) + D(d[ ], a[ ]) + D((a[ ], (ab)(cd)). We use Q to denote the expression D(x(yz), z(xy))+D(z(xy), y(zx))+D(y(zx), x* *(yz)), then Q = 0. So we have *3,i(Q) = 0 for 16 i6 4. So we have D((ab)(cd), d[ ]) + D(d[ ], c[ ]) + D(c[ ], (ab)(cd)) = *3,1(Q) = 0 D(a[ ], d[ ]) + D(d[ ], (bc)(ad)) + D((bc)(ad), a[ ]) = *3,2(Q) = 0 D(a[ ], (ab)(cd)) + D((ab)(cd), b[ ]) + D(b[ ], a[ ]) = *3,3(Q) = 0 D((bc)(ad), c[ ]) + D(c[ ], b[ ]) + D(b[ ], (bc)(ad)) = *3,4(Q) = 0 21 Let L: W ! W be the map defined by L(a^ b^ c^ d) = (b^ c^ d^ a), where a, b* *, c, d respectively denotes the elements of the first, second, third and fourth factor* *s of W . Then, L*(S) = D((bc)(da), c[ ]) + D(c[ ], d[ ]) + D(d[ ], (bc)(da)) L2*(S) = D((ab)(cd), b[ ]) + D(b[ ], c[ ]) + D(c[ ], (ab)(cd)) L3*(S) = D((bc)(da), a[ ]) + D(a[ ], b[ ]) + D(b[ ], (bc)(da)) It is a directing that S + L2*(S) = L*(S) + L3*(S) = D(c[ ]c, d[ ]) + D(d[ ], a[ ]) + D(a[ ], b[ ]) + D(b[ ], c[ ]) i X3 ji * * j ( *3,i) - *3,4D(x(yz), z(xy)) + D(z(xy), y(zx)) + D(y(zx), x(yz)) i=1 = L*(S) + L2*(S) + L3*(S) - S So we have L*(S) + L2*(S) + L3*(S) - S = L1*(S) + L3*(S) + L2*(S) - S = S + L2*(S) + L2*(S) - S = 2L2*(S) = 0 Since p> 5, ßm+1 (W (m+ 1)) is a Zp-vector space, we have L2*(S) = 0. Therefor* *e, S = L2*L2*(S) = 0. Thus, Proposition 10.8 is proved. Q.* *E.D. Now we state the relation between the changing of multiplication Mm+1 and t* *he cohomology class D(x(yz), z(xy)). Let u 2 Hm+1 (Y ^Y, ßm+1 (Y )), u . Hm+1 (S0* *^ Y [ Y ^S0, Zp) = 0, that is, u 2 Hm+1 (Y ^Y=S0^ Y [ Y ^S0, ßm+1 (Y )), then u repre* *sents a map from (Y ^Y ) to the fibre K(ßm+1 (Y ), m+ 1). Let j: K(ßm+1 (Y ), m+ 1) !* * Y be the natural injection, then we define Mm+1 : Y ^Y ! Y by M0m+1= M~m+1+ jOu: Y ^Y ! * *Y . 22 Since u . Hm+1 (S0^ Y [ Y ^S0, Zp) = 0, M0m+1is a multiplication with unit. If * *ø*(u) = u, where ø: Y ^Y ! Y is the map defined by ø(x^ y) = y^ x, since M~m+1 is commut* *a- tive, then it can be easily seen that the new multiplication M0m+1is also commu* *tative. It should be noticed that M0m+1|(Y ^Y )(m)= M~m+1 |(Y ^Y )(m), so the value of * *multiplica- tion M0m+1only differs from M~m+1 on the m+ 1 cell on Y ^Y . To avoid confusio* *n, we use D(x(yz), z(xy), ~Mm+1) to denote the cochain of difference with respect to * *the same homotopy on W (m). We have the following Proposition 10.9 D(x(yz), z(xy), M0m+1)- D(x(yz), z(xy), ~Mm+1) = -ffi2(u* *), where ffi2: C2,*(H*(Y, Zp), ßm+1 (Y )) ! C3,*(H*(Y, Zp), ßm+1 (Y )) is the cobo* *undary opera- tion defined in section 2. So the cohomology class D(x(yz), z(xy), M0m+1) is in* *dependent of the choice of M~m+1. Let T : X ! X be the map defined by T (x^ y^ z) = (z^ y^ x), then we have t* *he following Proposition 10.10 Let u 2 Hm+1 (Y, ßm+1 (Y )), then ffi2ø*(u) = -T *ffi2(u). Proof. Let ff, fi, fl 2 H*(Y=S0, Zp), |ff|+ |fi|+ |fl| = m+ 1, then ø*@2(ff^ fi^ fl) = (-1)|ff||fifl|fifl^ ff - (-1)|fffi||fl|fl^ fffi @2ø*(ff^ fi^ fl) = (-1)|ff||fl|@2(fl^ fi^ ff) = (-1)|ff||fl|(fl^ fiff - flfi^ ff) = (-1)|ff||fl|((-1)|ff||fi|fl^ fffi - (-1)|fi||fl|fifl^ f* *f) = -ø*@2(ff^ fi^ fl) Therefore, ø*@2 = -@2ø*. Dually, we have ffi2ø*(u) = -T *ffi2(u). * * Q.E.D. Now we prove Proposition 10.1. First, we prove that there exists a multiplic* *ation Mm+1 such that D(x(yz), z(xy), Mm+1 ) = 0. Since H*(Y, Zp)(m+2)= E(ø0, . .,.øn, . .)* *.(m+2), we 23 i j(m+2) i j(m+2) have H*,*H*(Y, Zp) = H*,*E(ø0, . .,.øn, . .). . Notice that i j __ __ H*,*E(ø0, . .,.øn, . .). = P(ø0) . . .P(øn) . . . We divide the proof into the following cases. If there is no integers i, j, k> 0 such that m+ 1 = (2pi-1)+ (2pj- 1)+ (2pk-* * 1), then H3,m-2(E(ø0, . .,.øn, . .)., ßm+1 (Y )) = 0. So D(x(yz), z(xy), ~Mm+1)* 0 in * *C*(H*(Y, Zp), ßm+1 (Y )) = C*(E(ø0, . .,.øn, . .)., ßm+1 (Y )) and there is u 2 Hm+1 (Y=S0^ Y* *=S0, ßm+1 (Y )) such that D(x(yz), z(xy), ~Mm+1) = ffi2(u), then ø*(1_2(u + ø*(u)) = 1_2(u + ø** *(u)). We define a new multiplication Mm+1 : Y ^Y ! Y by Mm+1 = M~m+1+ jO 1_2(u+ ø*(u)), then it* * follows easily that Mm+1 is also commutative. We have D(x(yz), z(xy), Mm+1 ) - D(x(yz), z(xy), ~Mm+1) 1 1 * = -ffi2(__u+ __ø (u)) 2 2 1 i * j = -__ ffi(u)+ ffi(ø (u)) 2 i 1 j i1 j = - __D(x(yz), z(xy), ~Mm+1) - __T *D(x(yz), z(xy), ~Mm+1) 2 2 i 1 j i1 j = - __D(x(yz), z(xy), ~Mm+1) - __D(x(yz), z(xy), ~Mm+1) 2 2 = -D(x(yz), z(xy), Mm+1 ) Therefore, we have D(x(yz), z(xy), Mm+1 ) = 0 If there exist integers i, j, k> 0 such that m+ 1 = (2pi-1)+ (2pj- 1)+ (2pk-* * 1). For __ any homogeneous element u = øi1. .ø.ik2 E (ø0, . .,.øn, . .)., we define the we* *ight of u to __ be w(u) = k. Let u1 . . .ul be any homogeneous element of l copiesE(ø0, . .,* *.øn, . .)., we define w(u1 . . .ul) = li=1w(ui). We use U(l, k) to denote the subvector * *space of __ l copiesE(ø0, . .,.øn, . .).spanned by all homogeneous elements with weight k.* * Let U(k) = 06l<1U(l, k). It is obvious that U(l, k) = 0 for l> k, that is, w(u1 . . .ul)* *> l. It can be easily seen that U(k) is a subcomplex of C*(H(Y ), Zp). Now that H*,*(E(ø0, . .* *,.øn, . .).= __ __ P (ø0) . . .P(øn) . .,.we have the following proposition. Proposition 10.11 (a) Hl,*(H*(E(ø0, . .,.øn, . .).)) = Hl,**,*(U(l)). 24 (b) Hl,*n,*(U(l)) = 0 for n< l. We also use U(l, k)* to denote the dual of U(l, k), U(k)* to denote the dual* * of U(k) and (øi1. .ø.il)* to denote the dual of øi1. .ø.il). Then D(x(yz), z(xy), ~Mm+1) may be written in the form u+ v with w(u) = 3 and w(v) > 3. It follows from Proposition 10.11 that ffi3(u) = ffi3(v) = 0. Since* * v is a co- __ __ cycle, there exists ~v2 E (ø0. .ø.n) E(ø0. .ø.n), w(~v) > 3 such that ffi2(~v* *) = v. Then, since T *D(x(yz), z(xy), ~Mm+1) = -D(x(yz), z(xy), ~Mm+1), we have T *(u) = -u,* *T *(v) = -v. So 1_2ffi(~v+øv~) = 1_2v+ 1_2T *(v) = 1_2v+ 1_2v = v. Now we define a * *new multiplica- tion M~0m+1by M~0m+1+j(1_2(~v+øv~)). It can be easily seen that D(x(yz), z(xy)* *, ~M0m+1) - D(x(yz), z(xy), ~Mm+1) = v, so D(x(yz), z(xy), ~M0m+1) = u. Thus, u can be expr* *essed in the following form ~(i0, j0, k0)ø*i0ø*j0ø*k0, where ~(i0, j0, k0) 2 Zp and the* * set {i0, j0, k0, } = {i, j, k} We also divide the proof into three cases (a) i= j= k (b) i= j, j6=k. (c) i, j, k are mutually different. Case (a). Since i= j= k, U(3, 3)(m+1) contains only one m+ 1 cell øi^øi^øi,* * we have P*(øi^øi^øi) = (øi^øi^øi). Since D(x(yz), z(xy), ~M0m+1)|(a+p*+p2*)(fii^fii^fi* *i)= 0, we have 3D(x(yz), z(xy), ~M0m+1)|fii^fii^fii= 0. Therefore, D(x(yz), z(xy), ~M0m+1)|fii* *^fii^fii= 0. Case (b). Since i= j, j6=k, U(3, 3)(m+1)contains three m+ 1 cells øk^ øi^øi,* * øi^øk^ øiand øi^øi^øk, we have P*(øk^ øi^øi) = (øi^øi^øk), P*2(øk^ øi^øi) = (øi^øk^ øi), T*(* *øk^ øi^øi) = -(øi^øi^øk). We define a new multiplication map Mm+1 by ji * * * *j Mm+1 = M~0m+1+__~i,i,køi^ (øi^øk) + ~k,i,i(øi^øk) ^øi 2 where ~i,i,k= ~k,i,iand ~i,i,k+~i,k,i+~k,i,i= 0. It can be easily seen that Mm+* *1 is commu- tative and D(x(yz), z(xy), ~M0m+1)|fik^fii^fii = D(x(yz), z(xy), ~M0m+1)|fii^fik^fii = D(x(yz), z(xy), ~M0m+1)|fii^fii^fik = 0 25 and so D(x(yz), z(xy), ~M0m+1) = 0. Case (c). Since i, j, k are mutually different, there are six cells øi^øj^øk, øj^øk^ øi,øk^ øi^øj, øk^ øj^øi,øi^øk^ øj,øj^øi^øk. It follows from Proposition 10.5 that D(x(yz), z(xy), ~M0m+1)(øi^øj^øk+ øj^øk^ øi+øk^ øi^øj) = ~i,j,k+~j,k,i+~k,i,j= 0 Since T *D(x(yz), z(xy), M0m+1) = -T *D(x(yz), z(xy), M0m+1) and T*(øi^øj^* *øk) = -(øk^ øj^øi), T*(øj^øk^ øi) = -(øi^øk^ øj), T*(øk^ øi^øj) = -(øj^øi^øk), we als* *o have ~i,j,k= -~k,j,iand ~j,k,i= -~i,k,j,. We now define a new multiplication Mm+1 by j i * * * * * * * * * ** j Mm+1 = M0m+1+__ ~i,j,k((øi(øjøk) + (øjøk) øi)) + ~k,i,j((økøi) øj+ øj(økø* *i)) 2 It is a direct checking that Mm+1 is also commutative and D(x(yz), z(xy), Mm* *+1 ) = 0. Therefore, Mm+1 is associative. Since D(x(yz), z(xy), H) = D(x(yz), z(xy), Mm+1 ) = 0, the stationary homoto* *py H from x(yz) to (xy)z relA(m) on X(m) can be extended to a stationary homotopy fr* *om x(yz) to (xy)z relA on X^ I+ . So Proposition 10.1 is proved. 11 The proof of (g)m+1 . In what follows, we will prove that æm : k(Zp, 1) ! E1(W (m)) can be lifted to * *a map æm+1 : k(Zp, 1) ! E1(W (m)). Let ( -1E1)*(ff1), . .,.( -1E1)*(ffs) 2 Hm+1 (E1(W (m)), Zp) be the image of* * ff1, . .,. ffs in Hm+3 (E1(W (m), Zp). Since E1(W (m+ 1), Zp) is the fibre space obtained * *by killing the cohomology classes ( -1E1)*(ff1), . .,.( -1E1)*(ffs), it is obvious that ( * *-1E1)*(ff1), . .,.( -1E1)*(ffs) are additive cohomology classes in H*(E1(W (m), Zp). By the * *induction hypothesis (h)m , fm : k(Zp, 1) ! E1(W (m)) is an H-map, so f*m( -1E1)*(ff1), .* * .,. f*m( -1E1)*(ffs) are additive cohomology classes. We have H*(k(Zp, 1), Zp) = E* *(ff) P (fi(ff)), so the only additive cohomology classes in H*(k(Zp, 1), Zp) are mul* *tiples of 26 n n ff, fi(ff) and (fi(ff))p . Therefore, if m + 3 = 2p , n = 1, 2, . .,.by the as* *sumption of ff1, . .,.ffs, ffiø0n = 0, 16 i6 s. Since ø0nis the image of upn in H*(W (* *m), Zp), it follows that ( -1E1)*(ffi)(fm )*(H2pn(k(Xp, 1), Zp)) = 0, so f*m( -1E1)*(ff1) =* * 0 ,. .,. f*m( -1E1)*(ffs) = 0 and fm can be lifted to a map fm+1 : k(Zp, 1) ! E1(W (m+ 1* *)). If m+ 3 6= 2pn, n = 1, 2, . .,.it is obvious that f*m( -1E1(ffi)) = 0, 16 i6* * s. So fm can also be lifted to a map fm+1 : k(Zp, 1) ! E1(W (m+ 1)). 12 Semi-product of CW complexes. In order to prove (h)m+1 , we introduce the notion of semi-product of CW comple* *xes. Let X be a CW complex, as usual we use X to denote the suspension X ^ S1. S* *ince S1 = I={0, 1}, any point of X can be expressed as x ^ t, 06 t6 1. Let x02X be * *the base point of X. It should be noticed that X^ 0 [ x0^I [ X^ 1 collapse to the base p* *oint of X. Let Y be another CW complex, as usual we use X#Y to denote Join(X, Y ), the * *space obtained by all the segment joining the point of X to the point of Y . All the* * point of X#Y can be expressed as x ^ t ^ y with x2 X, y2 Y , 06 t6 1. We define two* * maps `1, `2: X#Y ! (X x Y ) as follows `1(x ^ y ^ t) = (x, y) ^ t, `2(x ^ y ^ t) = * *(x, y) ^ (1- t). In general, `1 and `2 are not homotopic. It can be easily seen that X#Y = (X ^ Y ). We have the following proposition Proposition 11.1 (i) _ (j) _ `1, (i) _ (j) _ `2: X _ Y _ X#Y ! (X x Y* * ) are homotopy quivalences, where i and j are respectively the natural injections fro* *m X and Y to X x Y . ! We define semi-product X x Y and X x Y of X and Y as follows. X x Y = {(x ^ r, y ^ s) | x2 X, y2 Y, 06 r6 s6 1} Xx Y ! X x Y = {(x ^ r, y ^ s) | x2 X, y2 Y, 06 s6 r6 1} Xx Y ! We call X x Y the minus product of X and Y and X x Y the plus product of X a* *nd Y . It can be easily seen that ! (1) X x Y = (X x Y ) [ (X x Y ). 27 ! (2) (X x Y ) = (X x Y ) \ (X x Y ). We have the following proposition Proposition 11.2 (1) X x Y = C(`2). ! (2) X x Y = C(`1). Proof. The proof is straightfowward. We define two homotopy equivalences _1 * *and ! _2 respectively from C(`1) and C(`2) to X x Y and X x Y by ! _1((x ^ t ^ y), l)= (x ^ ((1- l)t + l), g ^ (1- l)t) 2 X x Y ! _2((x ^ t ^ y), l)= (x ^ ((1- t)(1 - l), g ^ ((1- t)(1- l) + l)) 2 X x Y x 2 X, y 2 Y 06 t6 1, 06 l6 1. Q.E.D. Let Z be another CW complex, since (X x Y )#Z = ((X x Y ) ^ Z) = ( X _ Y _ (X ^ Y ) ^ Z = (X ^ Z) _ (Y ^ Z) _ (X ^ Y ) ^ Z, we have ! (X x Y ) x Z ((X x Y ) x Z) [ C((X x Y )#Z) = ( X _ Y _ (X ^ Y ) _ Z _ X ^ Z _ Y ^ Z _ (X ^ Y ) ^ Z) [C ( (X ^ Z) _ (Y ^ Z) _ (X ^ Y ) ^ Z) = X _ Y _ Z _ ( (X ^ Z) [ C( X ^ Z)) _( (X ^ Z) [ C( X ^ Z)) _ C ( (X ^ Y ) ^ Z ^ C((X ^ Y ) ^ Z))(I) Similarly, we also have ! X x (Y x Z) = X _ Y _ Z _ ( (X ^ Y ) [ C( X ^ Y )) _( (X ^ Z) [ C( X ^ Z)) _ C ( (X ^ (Y ^ Z)) ^ C(X ^ (Y ^ Z))) (II) Since there is a natural topological isomorphism between (X x (Y x Z)) and ((X x Y ) x Z) such that ( (X ^ (Y ^ Z)) = ((X ^ Y ) ^ Z), we may identify 28 ! the subspace (X x (Y x Z)) of X x (Y x Z) with the subspace ((X x Y ) x Z) of ! (X x Y ) x Z). With this identification ~, we define ! ! W (X, Y, Z) = X x (Y x Z) [ (X x Y ) x Z)= ~ . It follows from (I) and (II) and the following equalities (X ^ Z) i j = X ^ Z [ C X ^ Z [ C X ^ Z) (X ^ Y ^ Z) i j = (X ^ Y ) ^ Z [ C (X ^ Y ) ^ Z [ C( (X ^ (Y ^ Z)) we have the following proposition Proposition 11.3 Let X, Y, Z be three CW complexes, then W (X, Y, Z) and ( X* *) _ i j ( Y ) _ ( Z) _ (X ^ Z) _ (X ^ Y ^ Z) are of the same homotopy type. In what follows, we use _: (X ^ Y ^ Z) ! W (X, Y, Z) to denote the natural injection. We call W (X, Y, Z) the semi-product of X, Y and Z. 13 The Milnor construction of topological group. Let G be a topological group. J. Milnor introduce the notion Br(G), r> 0, to st* *udy the relation between G and its classifying space BG. In what follows, we need only* * B1(G) and B2(G). As we know, B1(G) = G. We have a map M~: G#G ! G defined as follows. M~(g1 ^ t ^ g2) = (g1g2 ^ t) 2 G (g1g2 is the product of G), then B2(G) is the* * mapping cone C(M~) = G [ C(G#G) of M~. ! ! Since G x G = C(`1) = (G x G) [ C(G#G), we may define "+: G x G ! B2(G) by "+| (GxG) = M: (G x G) ! G "+|C(G#G) = id: C(G#G) ! C(G#G) B2(G) 29 where M is the product of G and define ! ! j1: G x (G x G) ! G x G ! ! j2: (G x G) x G ! G x G by that for any g1, g2, g3 2 G and 06 s6 r6 1, j1(g1 ^ r, (g2, g3) ^ s) = (g1 ^ r, g2g3 ^ s) j2((g1, g2) ^ r, g3 ^ s) = (g1g2 ^ r, g3 ^ s) Since G is associative, we have that "+j1 (G, (G, G)) = "+j1 (G, G, G) = "+j2 ((G, G), G) and so "+j1 and "+j2 define a map j: W (G, G, G) ! B2(G) by j|Gx!x(GxG)= "+j1 j|(GxG)!xG= "+j2 Let T be a topological group. t0 2 is the unit of T . G is the loop space * * (T ) of T with base point t0. The multiplication map M of G inherits from that of T , t* *hat is, M(~1, ~2)(t) = ~1(t)~2(t) for any ~1, ~2 2 (T ). It is well known that M and t* *he loop multiplication of G are homotopic. We define oe1: B1(G) = (G) ! T , oe2: B2(G)* * ! T as follows. For any ~ 2 (T ), g1, g2 2 G, and 06 t, l6 1, oe1(~ ^ t) = ~(t) oe2| G = oe1 oe2((g1 ^ t ^ g2), l) = g1((1- l)t + l)g2((1- l)t) It can be easily seen that oe2 is an extension of oe1. 30 Now, let _: (G^G^G) ! W (G, G, G) be as defined in the last part of the pr* *evious section, then Proposition 11.4 (oe2j_)* = 0: H*( (G ^ G ^ G), Zp) ! H*(T, Zp). In fact, Proposition 11.4 is the law of associativity for the relation betwe* *en the ho- mology group of loop space and its classifying space. Since we only concern about the homology group of product of spaces, we use * *here Serr's cubic singular homology theory. Before the proof of Proposition 11.4, we* * introduce the semi-product of singular cubes. Let X, Y be two topological spaces, l, m positive integers, ff: Il ! X, fi: * *Im ! Y are ! respectively l, m cubes in X and Y . We use ff x fi to denote the (l+ m+ 2) sin* *gular cube defined as follows. Let A = (0, 0), B = (1, 0), C = (1, 1) and oe: I2 ! 4ABC be* * the map that leaves the segment AB and BC invariant (we regard 4ABC as a subspace of I2) and sends the segment from (1, 1) to (0, 1) linearly to CA and collapse the seg* *ment from (0, 1) to (0, 0) to point A. It is required that oe is a topological map from t* *he interior of I2 to the interior of 4ABC. Suppose oe(r, s) = (r0, s0) 2 4ABC, then 1> r0>s0>0* *, and we define ! (ff x fi)(x, y, r, s) ! = (ff(x) ^ r0, fi(y) ^ s0) 2 X x Y ! ! So ff x fi is a (l+ m+ 2) cube of X x Y . We call it semi-product of ff and fi. For chains a = ~iffi and b = ~jfij, where ffi and fij are respectively sin* *gular cubes ! ! in X and Y , we define a x b = ~i~jffix fij. Let fl: In ! Z be a cube in Z, we define the semi-mixed product W (ff, fi, * *fl) by ! ! W (ff, fi, fl) = ff x (fi x fl)- (ff x fi) x fl We must point out that W (ff, fi, fl) is not a cub but a (l+ m+ 2) chain in * *W (X, Y, Z). We can similarly define the semi-mixed product of chains in X, Y and Z. Let a, b, c be respectively the l, m, n-cycles modp in X, Y, Z. It can be ea* *sily seen that W (a, b, c) is also a cycle modp in W (X, Y, Z) and any homology class in _*H*(* * X ^ Y ^ Z, Zp) can be expressed in the form ~W (a, b, c)*, where as usual, we use * *W (a, b, c)* 31 to denote the homology class containing W (a, b, c). Now we prove Proposition 11.4. Let ff, fi, fl be respectively l, m, n singul* *ar cubes in G. We define a singular (l+ m+ n+ 3) singular cube D(ff, fi, fl) by D(ff, fi, fl)(* *x, y, z, r, s, t) = ff(x)(r0)fi(y)(r0t+ s0(1- t))fl(z)(s0) 2 T , where r, s, t2 I, x2 Il, y2 Im ,* * z2 In, oe(r, s) = (r0, s0). Specifically, we have that D(ff, fi, fl)(x, y, z, r, s, 0) = ff(x)(r0* *)fi(y)(s0)fl(z)(s0), and that D(ff, fi, fl)(x, y, z, r, s, 1) = ff(x)(r0)fi(y)(r0)fl(z)(s0). Thus, D(ff, fi, * *fl) is a homotopy from ! ! oe2j1(ff x (fi x fl)) to oe2j2((ff x fi) x fl). Let a, b, c be as above, it follows from the construction of D(a, b, c) that* * @(D(a, b, c)) = (oe2)*j*W (a, b, c)+ some degerated chains. Since in singular cubic chain compl* *ex of topo- logical space, degenerated chains = 0, we have (oe2)*j*W (a, b, c) = 0. Since a* *ny homology class in _*H*( G ^ G ^ G) can be expressed in the form ~W (a, b, c)*, we have* * that (oe2j_*)* = 0 Q.E.* *D. Let ": G ! g0 be the augmentation map. As usual, we use ~H(G, Zp) to denote* * the subgroup "-1*(0). We also denote group homomorphisms induced by M respectively * *by M*: ~H*(G, Zp) ~H*(G, Zp) ! ~H*(G, Zp) and M*: ~H*(G, Zp) ! ~H*(G, Zp) ~H*(* *G, Zp). ~M fi ff Consider the following sequence G#G -! G -! B2(G) -! (G#G) -! where ø denotes the natural injection and oe denotes the natural map from B2(G) = G[C(* *G#G) to (G#G) = G ^ G. Since it is a cofibration, we have Proposition 11.5 (a) H*(B2(G), Zp) i j = ~H*( G, Zp)=M* (H~*(G, Zp) ~H(G, Zp)) + M-1*(0) H*(B2(G), Zp) i j i j = H~*(G, Zp) ~H*(G, Zp) =M* H*( G, Zp) + M*-1(0) (b) oej_ = @3: G ^ G ^ G ! (G ^ G) = (G#G), where @3 is the boundary map in the chain complex C*(H*(G, Zp)). By this proposition and the definition of H*,*(H*(G), Zp), we have the follo* *wing propo- 32 sition Propoition 11.6 (a) ø*H*( G, Zp) = H1,*(H*(G, Zp)). (b) 0 ! j*_*~H( (G ^ G ^ G), Zp)) ø*(H*( G, Zp)) ! ~H(B2(G), Zp) ! H2,*(H*(G, Zp)) ! 0 is an exact sequence. Let Y be a space, f, g: Y ! G be maps. Since G is a topological group, we ca* *n define fg: Y ! G by fg(y) = f(y)g(y). We denote f, g: (fg): Y ! G as usual. Since G is a cogroup, we can define f _ g: Y ! G. We have the following proposit* *ion Proposition 11.7 ø( f _ g) ø( (fg)). Proof. Let d: Y ! Y xY be the diagonal map, then fg is defined to be M O(f x* *g)Od. Since (Y x Y ) = Y _ Y _ Y #Y , we have ø (fg) = ø( f _ g + (f#g)). Since B2(G) = G [ C(Y #Y ), so (f#g) = 0. Thus ø( f _ g) = ø( (fg)). Q.E.D. Let X, Y be two homotopy commutative H-space. For any x1, x2 2 X and y1, y2 * *2 Y , we use x1x2 and y1y2 to denote the their products of the H-spaces. Let f: X ! Y* * be an H-map, then there exists a homotopy H from f(x1)f(x2) to f(x1x2). We define a m* *ap B2(f, H): B2(X) ! B2(Y ) as follows. ( f(x1) ^ (2t- 1) ^ f(x2)1_6t6 1 B2(f, H)(x1 ^ t ^ x2)= 2 1 H(x1, x2, 1- 2t) 06 t6 _2 B2(f, H)| X = (f) It can be easily seen that B2(f, H) is uniquely determined by f and H. In w* *hat follows, we simply use B2(f) to denote B2(f, H). 14 The proof of (h)m+1 for m+ 2 6= 2pi+ 2pj- 2, 2pi, 2pi- 1. Since W (m) is an infinite loop space, E1(W (m)) = (E2(W (m)), E2(W (m+ 1)) * * = (E3(W (m+ 1)), E3(W (m+ 1)) = (E4(W (m+ 1)). We may assume that E4(W (m+ 1)) is a simplicial complex and E3(W (m+ 1)) is the Milnor's simplicial loop space * *defined in 33 [5] and is thus a topological group. Therefore, E2(W (m+ 1)) = (E3(W (m+ 1)) * *inher- its a multiplication map from that of E3(W (m+ 1)) (see section 11) and E1(W (m* *)) = (E2(W (m)) also inherits a multiplication map from that of E2(W (m)), that is,* * they are both topological groups. So all the conclusions in section 11 hold for G = E1(W* * (m)) and T = E2(W (m)). For two H-spaces X, Y , we use X x Y to denote the H-space whose product is* * in- duced by those of each factors. Suppose (a)m+1 to (g)m+1 hold, we will prove (* *h)m+1 , that is, fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map. We reduce the problem to * *the property of H*(k(Zp, 1), Zp). Let gm : C(m) ! W (m) be the universal covering o* *f W (m), then E1(gm ): E1(C(m)) ! E1(W (m)) is also the universal covering of E1(W (m)).* * It can be easily seen that fm x E1(gm ): k(Zp, 1) x E1(C(m)) ! E1(W (m)) and fm+1 x E1(gm+1 ): k(Zp, 1)xE1(C(m+ 1)) ! E1(W (m+ 1)) are both homotopy equivalences. * *Since fm is an H-map, so is fm x E1(gm ). Thus, the statement that fm+1 is an H-map i* *s equiv- alent to the statement that fm+1 x E1(gm+1 ) is an H-map. Since B2(G) = C(M~) w* *here M~: G#G ! G is induced by M: G x G ! G, the homotopy type of B2(G) is uniquely determined by that of M. We may assume that E1(W (m)) = k(Zp, 1) x E1(C(m)), th* *en fm : k(Zp, 1) ! E1(W (m)) induces a map B2(fm ): B2(k(Zp, 1)) ! B2(E1(W (m))). * *It can be easily seen that the following diagram is commutative. fm^fm^fm k(Zp, 1) ^ k(Zp, 1) ^ k(Zp, 1) -! E1(W (m)) ^ E1(W (m)) ^ E1(W (m)) j_ # # j_ B2(fm) B2(k(Zp, 1)) -! B2(E1(W (m))) We have the following proposition Proposition 12.1 The following statements are equivalent. (a) fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map. (b) oe2B2(fm ): B2(k(Zp, 1)) ! E2(W (m)) can be lifted to a map from B2(k(Zp* *, 1)) to E2(W (m+ 1)). (c) B2(fm )*oe*2E2( 2ffi) = 0, 16 i6 s, where ff1, . .,.ffs 2 Hm+2 (W (m), Z* *p)) are the Postnikov invariants and E2( 2ffi) are the image of ffi in Hm+4 (E2(W (m)), Zp). Proof. First, we prove that (a) implies (b). Suppose that (a) holds, then B2* *(fm+1 ): 34 B2(k(Zp, 1)) ! B2(E(W (m+ 1))) exists. It is obvious that E2(æm+1 )*B2(fm+1 ) =* * oe2B2(fm ), where æm+1 : W (m+ 1) ! W (m) is the map in (c) and E2(æm+1 ): E2(W (m+ 1)) ! E* *2(W (m)) is the map induced by æm+1 . So (b) holds. Now we prove that (b) implies (a). Let M0: k(Zp, 1) x k(Zp, 1) ! k(Zp, 1) b* *e the multiplication of k(Zp, 1) and M: E1(W (m+ 1)) x E1(W (m+ 1)) ! E1(W (m+ 1)) be* * the multiplication map inheriting from E2(W (m+ 1)), we will prove that the followi* *ng diagram is homotopy commutative. fm+1xfm+1 k(Zp, 1) x k(Zp, 1) - ! E1(W (m + 1)) x E1(W (m + 1)) # M0 # M fm+1 k(Zp, 1) - ! E1(W (m+ 1)) We have oe2Oø O M O (fm+1 xfm+1 ) = oe2O( fm+1 _ fm+1 ) = oe1O( fm+1 _ fm+1 * *) and i j M0 = (id_id_M~0): k(Zp, 1)xk(Zp, 1) = k(Zp, 1)_ k(Zp, 1)_k(Zp, 1)#k(Zp, 1)* * ! k(Zp, 1), so ( fm+1 )( M0) = fm+1 _ fm+1 _ ( fm+1 )( M~0). Since oe2B2(fm ) * *can be lifted to a map from B2(k(Zp, 1)) to E2(W (m+ 1)) and so øM~0 = 0 in B2(k(Zp, 1* *)). Thus, in E2(W (m+ 1)), ø( fm+1 )(M~0) = 0. So we have oe1( fm+1 )( M0) = oe1( fm+1 _ fm+1 ) = oe2øM (fm+1 x fm+1 ) Since E1(W (m+ 1)) = E2(W (m+ 1)), we have fm+1 OM = MO (fm+1 x fm+1 ). So fm+1 is an H-map, that is, (b) implies (a). Since (b) and (c) are equivalent, (a), (b) and (c) are all equivalent. * * Q.E.D. So to prove (h)m+1 , we need only prove B2(fm )*oe*2( 2E)*(ffi) = 0. According to Proposition 11.5, we discuss the problem in four cases; (a) m+ 1 6= 2pi+2pj- 2, 2pi, 2pi-1 for any i, j> 0. (b) m+ 1 = 2pi+2pj- 2 for some i, j> 0. (c) m+ 1 = 2pi-1 for some i> 0. (d) m+ 1 = 2pi for some i> 0. Now we prove the first case (a). In this case, m+ 1 6= 2pi+2pj- 2, 2pi, 2pi-* *1 for any i, j> 0, then it follows from Proposition 11.6 that H2*,m+2(H*(k(Zp, 1))) = 0. * *So we have 35 Hm+2 (B2(k(Zp, 1), Zp) = j*_*H*( k(Zp, 1) ^ k(Zp, 1) ^ k(Zp, 1), Zp). Therefor* *e, oe2*B2(fm )*(Hm+2 (B2(k(Zp, 1), Zp) = oe2*j*_* (fm ^ fm ^ fm )*H*( k(Zp, 1) ^ k(Zp, 1) ^ k(Zp, 1), Zp) = 0 i j So we have oe2* Hm+2 (B2(k(Zp, 1), Zp)) = 0. That is, B*2((fm )*oe*2( 2E2)*(f* *fi) = 0. It follows from Proposition 12.1 that fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map. Now we prove the case (c). In this case, Hm+4 (B2(k(Zp, 1)), Zp) = j*_*Hm+4 * *(k(Zp, 1)^ k(Zp, 1)^k(Zp, 1), Zp)+ø*(Hm+1 ( k(Zp, 1), Zp)). Since fm+1 exists, ø*(B2(fm ))* **( 2E2)*(ffi) = 0. By the same argument as above and the reason that oe2*j*_*( fm ^ fm ^ fm * *)* = 0, we have that B*2(fm )oe*2( 2E2)*(ffi) = 0 and so fm+1 : k(Zp, 1) ! E1(W (m+ 1))* * is also an H-map. 15 The proof of (h)m+1 for m+ 1 = 2pi + 2pj - 2. Since for i 6= j, upi upj- upj upi is a generater of H2,*(H*(k(Zp, 1), Zp)), * *to prove (h)m+1 we need only prove that upi upj- upj upiis the product of some homolog* *ical classes in H*(E2(W (m)), Zp). First, we deduce some properties of Pontrjagin pr* *oduct in H*(E2(W (m)), Zp). First, we define a map from E1(W (m)) x E1(W (m)) to E2(W (m)). Let X be an arcwise connected Hausdorff space. A, B X are two closed arcw* *ise connected subspaces of X such that X = A[ B. Let C = A\ B and T (A, B) = Ax {0}* * [ Bx {1} [ Cx I Xx I. We define æ: T (A, B) ! X by æ(a, 0) = aa 2 A æ(b, 1) = bb 2 B æ(c, t) = cc 2 C 0 6 t 6 1 It can be easily seen that æ is a weak homotopy equivalence and a homotopy e* *quiva- lence if X is a CW complex and A, B are sub CW complexes of X. Let Y be another Hausdorff space and f: A ! Y and f: B ! Y be two maps, then f|C and g|C are two maps from C to Y . If f|C g|C and H is the homotopy from * *f|C to 36 g|C, then we define a map s(f, g, H): T (A, B) ! Y as follows. s(f, g, H)(a, 0) = f(a)a 2 A s(f, g, H)(b, 1) = g(b)b 2 B s(f, g, H)(c, t) = H(c,ct)2 C 0 6 t 6 1 ! ! Let X be a topological group. Then Xx X = Xx X [ Xx X, Xx X \ Xx X = ! (Xx X) = (Xx X) ^ S1. We define `1: Xx X ! B2(X) and `2: Xx X ! B2(X) by that for any x, y 2 X, `1(x ^ r, y ^ s) = (x ^ r, y ^ s) 2 B2(X)r > s `1(x ^ r, y ^ s) = (xy ^ s) 2 X 2 B2(X)r = s `2(x ^ r, y ^ s) = (y ^ s, x ^ r) 2 B2(X)r < s `2(x ^ r, y ^ s) = (yx ^ s) 2 X 2 B2(X)r = s Suppose that X is a homotopy commutative topological group and H is a homoto* *py from (Xx X)x I to X between the maps (xy)^idto (yx)^id, then s(`1, `2, H) is * *defined ! and is a map from T (Xx X, Xx X) to B2(X). If X is of the same homotopy type wi* *th a ! CW complex, then T (Xx X, Xx X) and B2(X) are both of the same homotopy type wi* *th ! CW complexes. Thus, T (Xx X, Xx X) and ( Xx Y ) are of the same homotopy type. Therefore, s(`1, `2, H) determine a homotopy class of maps from Xx Y to B2(X). Let T be a homotopy commutative topological group. X = (T ) be the loop spa* *ce of T with the end point the unit e of T . The multiplication of X is induced from * *that of T as we defined previously. Let ~H: T xT xI ! T be a homotopy from t1t2 to t2t1, t1, t2 2 T and H: Xx Xx* * I ! X is the homotopy induced by ~H, that is, H(~1, ~2, t)(s) = ~H(~1(s), ~2(s), t) f* *or 06 s, t6 1. ! Let æ: T ((Xx X, Xx X) ! Xx Y be the map defined above and oe: X = (T )* * ! T be the map defined by oe(~ ^ s) = ~(s). We have the following proposition Proposition 14.1 The following homotopy relation holds. oe2 O s(`1, `2, H) M O (oex oe) O æ ! j T (Xx X, Xx X) -! Xx X # oex oe # s(`1, `2, H) T xT # M B2(X) -ff2! T 37 where oe2 is as defined in section 12 and M is the product map of T and æ is th* *e homotopy equivalence defined in the beginning part of this section. ! ! Proof. Since T (Xx X, Xx X) = Xx Xx {0}\Xx Xx {1}\ (Xx Y )x I, we construct ! the required homotopy piecewisely on the three component Xx Xx {0}, Xx Xx {1} a* *nd (Xx Y )x I. It is obvious that for x, y 2 X and 06 s6 r6 1, i j oe2s(`1, `2, H ^ id) (x ^ r, y ^ s)x {0} = oe2`1(x ^ r, y ^ s) = x(r)y(s) i j = M(oex oe)æ (x ^ r, y ^ s)x {0} ! So we define ~H: Xx Xx {0}x I ! T by ~H(x ^ r, y ^ s, 0, t) = x(r)y(s), 06 r6 s* *6 1, 06 t6 1. i j Now we have oe2s(`1, `2, H) (x ^ r, y ^ s)x {1} = y(s)x(r) and M(oex oe)æ(x ^ * *r, y ^ s) = x(r)y(s), so we can define H~: Xx Xx {1}x I ! T for x, y 2 X, 06 r6 s6 1, 06 t6* * 1 by H~(x ^ r, y ^ s, 1, t) = ~H(x(r), y(s), 1- t). Now on (Xx Y ) = (X!xX) \ (Xx X* *) we have H~(x ^ r, y ^ s, 0, t) = x(r)y(s) H~(x ^ r, y ^ s, 1, t) = H~(x(r), y(s), 1- t) i j oe2s(`1, `2, H) (x ^ r, y ^ t), u = H~(x(r), y(s), u) = M(oex oe)æ(x ^ r, y ^ s, u) = x(r)y(s) These maps can define a map from (Xx Y )x @(Ix I) to T . It can be easily s* *een that the map can be extended to a map ~Hfrom (Xx Y )x (Ix I) to T . So oe2Os(`1, `2* *, H^id) M O (oex oe) O æ. Q.* *E.D. Since æ is a homotopy equivalence, Proposition 14.1 says that the product M(* *oex oe) can be determined by oe2s(`1, `2, H). 38 Let X, W be two homotopy commutative H-space, f: X ! W be an H-map. We now study that under what conditions the following diagram ! ! T (Xx X, Xx X) - ! T (W xW, W xW ) Xx X W x W # # B2(X) - ! B2(W ) is homotopy commutative. ! ! ! Since T (Xx X, Xx X) = (Xx X)x 0[ (Xx X)x I [(Xx X)x 1, any point of T (Xx X, Xx X) can be written in the form (x1^s^ x2^t)x l, where x1, x22X and if l= 0, t* *hen 06 t6 s6 1 and if l= 1, then 06 s6 t6 1 and if 0< l< 1, then 06 s= t6 1. Thus,* * we can define ! ! T (f): T (Xx X, Xx X) ! T (W xW, W xW ) by T ((x1^s^ x2^t)x l) = (f(x1)^ s^ f(x2)^ t)x l). Since X, W are homotopy commutative, there exist a homotopy H from x1x2 to x* *2x1 and a homotopy ~~Hfrom u1u2 to u2u1. Since f is an H-map, there exists a homoto* *py ~H from f(x1)f(x2) to f(x1x2). So both fH + ~Hand H~+ ~~Hare homotopies from f(x1x* *2) to f(x2)f(x1). Thus d(fH + ~H, ~H+ ~~H) is defined. If d(fH + ~H, ~H+ ~~H) 0,* * we say that f is a strong H-map with respect to H, ~H, ~~H. According to the construc* *tion of ! T (Xx X, Xx X) and T (f), we can easily obtain the following result. Proposition 14.2 If f is a strong H-map with respect to H, ~H, ~~H, then the* * following diagram is homotopy commutative; ! T(f) ! T (Xx X, Xx X) -! T (W xW, W xW ) # # B2(f) B2(X) -! B2(W ) Now we study under what conditions f can be a strong H-map. Let ø: Xx X ! Xx X be the map defined by ø(x1, x2) = (x2, x1). We also use ø* * to denote the similar map from W xW to W . If d(H, -H(ø^ id)) 0, we say that H i* *s a ø-homotopy from x1x2 to x2x1. An H-space X is a p-H-space if pß[Q, X] = 0 for a* *ny CW complex Q. It is obvious that Er(W (m)) is a p-H-space for r > 1. We have the f* *ollowing proposition 39 Proposition 14.3 Let X be a homotopy commutative p-H-space, then there exist* *s a ø-homotopy from x1x2 to x2x1. Proof. Let H~ be n homotopy from x1x2 to x2x1, we define a new homotopy H by H = ~H- d(H~, (-H~)(ø^ id)). It must be pointed out that (-H~)(ø^ id) is also a* * homotopy from x1x2 to x2x1. So d(H~, -H~(ø^ id)) is defined and is a map from (Xx X) to* * X. Since X is a p-H-space, 1_2d(H~, (-H~)(ø^ id)) is defined . Therefore, we define a ne* *w homotopy H = ~H- 1_2d(H~, (-H~)(ø^ id)). It is obvious that H is a ø-homotopy. * * Q.E.D. Now we have Propotition 14.4 Let W be a p-H-space and H be a ø-homotopy from x1x2 to x2x* *1 and H~~be a ø-homotopy from u1u2 to u2u1 and f: X ! W be an H-map, then there exist* *s an homotopy H~ from f(x1)f(x2) to f(x1x2) such that f is a strong H map with respe* *ct to H, ~H, ~~H. Proof. Let ~Hbe a homotopy from f(x1)f(x2) to f(x1x2). In general, d(fH + ~H* *, ~H+ H~~) 6 0. We define a homotopy f(x1)f(x2) to f(x1x2) by ~H= ~H- 1_d(fH + ~H, ~H+ ~~H). 2 It can be easily seen by simple calculation that d(fH + ~H, ~H+ ~~H) 0. So f * *is a strong H-map with respect to H, ~H, ~~H. Q* *.E.D. Apply the above theory to fm : X= k(Zp, 1) ! W =E1(W (m))= E2(W (m)), then * *any homotopy from x1x2 to x2x1 is a ø-homotopy. Let ~~Hbe a homotopy from u1u2 to u* *2u1 (u1, u2 2 W ) inheriting from a ø-homotopy from t1t2 to t2t1 with t1, t2 2 E2(W* * (m)). Then, ~~His also an ø-homotopy. So we have Proposition 14.5 Let X = k(Zp, 1), W = E1(W (m)), then there exists a homoto* *py H~ from f(x1)f(x2) to f(x1x2) such that the following diagram is homotopy commutat* *ive; ! T(fm) ! T (Xx X, Xx X) -! T (W xW, W xW ) # s(`1, `2, H) # s(`1, `2, ~~H) B2(fm,H) B2(X) -! B2(W ) 40 By Propotition 14.1 to Proposition 14.5, we have the following proposition Proposition 14.6 i j oe2*O B2(fm )* O s*(`1, `2, H)(æ-1)* ( upi)x ( upj) i j i j = oe1*( fm )*(upi) . oe1*( fm )*(upj) ! where æ: T (k(Zp, 1)x k(Zp, 1), k(Zp, 1)x k(Zp, 1)) ! ( k(Zp, 1))x ( k(Zp, 1)) * *denotes the map defined at the beginning of this section and . denotes the Pontrjagin produ* *ct of ho- mology group in E2(W (m)). The following is the proof of (h)m+1 for m+ 1 = 2pi+2pj- 2. Since ( -2E2)*(ffi) = oe*1 -3E*3(ffi) = 0, 16 i6 s, where oe1: E2(W (m)) ! * *E3(W (m+ 1)) i j i j denotes the adjoint map, we have oe1*( fm )*(upi) . oe1*( fm )*(upj) E*2( 2f* *fi) = 0. By the same argument as above, we have (B2(fm )*oe*2E*2( 2ffi)) = 0. Therefore, fm* *+1 is an H-map for m+ 1 = 2pi+2pj- 2. 16 The N (Zp, 0) action on spectra. Before we prove (h)m+1 , we state some properties of N(Zp, 0) action on spectra* *. In what follows, we always use N(Zp, 0) to denote the Moore spectrum and n(Zp, 1) to de* *note the Moore space, then N(Zp, 0) = -1~n(Zp, 1) For X = W (m), fm : k(Zp, 1) ! E1(W (m)), and f~mdenote the map from ~k(Zp, * *1) to E~1(W (m)), then E1f~mis a map from ~k(Zp, 1) to W (m) and E2~oe2is a map f* *rom B~2(~k(Zp, 1)) to 2W (m). It may be assumed that W (m) =-lim! -rE~r(W (m)) and -rE~r(W (m)) m = 0, 1,* * . . . are all CW subspectra of W (m). It can be easily seen that the following diagram is commutative. E1(f~m) 2 ~k(Zp, 1) -! W (m) # ~ø # id B~2(k(Zp, 1)) -~ff2! 2W (m) where ø denotes the natural injection. 41 Since W (m) is a ring spectrum with unit, ß0(W (m)) = Zp, ß1(W (m)) = 0, N(Z* *p, 0) = V (0), N(Zp, 0) may be considered naturally as a subspectrum of W (m). Let Mm :* * W (m)^ W (m) ! W (m) be the multiplication of W (m), then P (m) = Mm |N(Zp,0)^W(m)is a map from N(Zp, 0) ^ W (m) to W (m). In general, any map S: N(Zp, 0) ^ X ! X that satisfies S|S0^X = S|X = id|X is called an action of N(Zp, 0) on X. The map P (* *m) is an action of N(Zp, 0) on W (m) which we call the natural action of N(Zp, 0) induce* *d by the multiplication Mm . For the space k(Zp, 1). Since k(Zp, 1) is a topological group, let d > 1, w* *e use ed: k(Zp, 1) ! k(Zp, 1) to denote the map defined by ed(x) = xd, x 2 k(Zp, 1). * *It follows from Proposition 11.6 that in the space B2(k(Zp, 1)), we have ø(d . id k(Zp,1))* * ø( ed). Now in k(Zp, 1), ep 0, so we have ø(p . id k(Zp,1)) 0 in B2(k(Zp, 1)). * *So the map ø: k(Zp, 1) ! B2(k(Zp, 1)) can be extended to a map S: n(Zp, 1) ^ k(Zp, 1)* * ! B2(k(Zp, 1)) (n(Zp, 1) is the Moore space), then we have the following sequence* * of maps ~n(Zp, 1) ^ ~k(Zp, 1) -! N(Zp, 0) ^ ~E1(W (m)) -! N(Zp, 0) ^ W (m)= 2N(Zp, 0) ^ W (m) -! ~S E2 2 2W (m) ^ ~n(Zp, 1) ^ ~k(Zp, 1) -! ~B2(k(Zp, 1)) -! W (m) So we have the following diagram F^E1Of~m 2 ~n(Zp, 1) ^ ~k(Zp, 1)-! N(Zp, 0) ^ W (m) # ~S # 2P (m) ~B2(k(Zp, 1)) E2~ff2~B2(fm)-! 2W (m) where F : ~n(Zp, 1) ! N(Zp, 0) is the natural injection. Then, we have the fo* *llowing proposition Proposition 15.1 Suppose (a)m to (h)m holds for m, then there exists a commu* *tative associative multiplication M~m: W (m) ^ W (m) ! W (m) such that the above diagr* *am is homotopy commutative for the natural N(Zp, 0) action P (m) induced by M~m. Proof. We prove it by induction. If m = 0, then in this case, W (0) = k(Zp, * *0) and the conclusion follows easily from the fact that 2P (0) O (F ^E1f0) and EB~2(f* *0)S~induce 42 the same homomorphisms on the first non-zero homology group modp. Suppose that * *the Proposition holds for m, we will prove it for m+ 1. Before we prove it, we introduce the notations which will be used later. Let* * X be a spectrum, A be an Abellian group, ff: X ! K(A, m+ 2) be a map. Let P (X, ff) de* *note the map cone C(X) [ K(Z, m+ 2). Then, we have the following cofibration sequence j ff -1P (X, ff) -! X -! K(A, m+ 2) where æ is the natural identification map by collapsing K(A, m+ 2) to the base* * point. Let B be a CW subspectrum of X, oe: B ! X be the natural injection, ~: X ! X* *=B be the natural identification map. Suppose that there exists a map ~ff: X=B ! K* *(A, m+ 2) such that ff = ~ff~, we define a map J(~ff): B ! P (X, ff) as follows. J(~ff)(* *b^ t) = (b^ t) 2 C(X) P (X, ff) for 06 t6 1. It should be noticed that J(~ff)(b^ 0) = J(~ff)(* *b^ 1) = base point. So J(~ff) define a map from B to P (X, ff) Let ff0, ff00: X=B ! K(A, m+ 2) be two maps such that ff0~ ff00~, then P (* *X, ff0~) and P (X, ff00~) are of the same homotopy type. There exists a homotopy equiva* *lence l: P (X, ff0~) ! P (X, ff00~) such that the following diagrams are commutative 1) P (X, ff0~)-!l P (X, ff00~) # æ æ X 2) K(A, m+ 2) q # q P (X, ff0~)-!l P (X, ff00~) where q: K(A, m+ 2) ! P (X, ff0~), P (X, ff00~) denotes the natural injection* *s of the fibre. From the conclusion that ff0~ and ff00~ denote the same cohomology class if * *and only if ff0-ff002 im Hm+2 ( B, A) we have the following proposition. Proposition 15.2 J(ff0)-J(ff00) q< ff0-ff00>, where denotes the * *cohomology class in Hm+2 ( B, A) which is map to ff0- ff00by the map X=B ! B. 43 Let ff 2 Hm+2 (X, A), then the homotopy type of P (X, ff) is uniquely determ* *ined by the cohomology class ff. Suppose that oe*(ff) = 0 and J: X ! P (X, ff) be a ma* *p such that (æ) O J (oe). It follows from Proposition 15.2 that Proposition 15.3 There exists a ff02 Hm+2 (X=B, A) such that ff ff0~ and J* *(ff0) J: B ! P (X, ff0~) = P (X, ff). To prove Proposition 15.1, we must define a new map homotopic to P (m). Let P (m): N(Zp, 0) ^ W (m) ! W (m) be the N(Zp, 0) action induced by Mm . Let ______ i j W (m) = W (m) [ N(Zp, 0)^ W (m) ^ I+ i j i * * j where N(Zp, 0)^ W (m) ^0+ is identified with the subspectrum P (m) N(Zp, 0)^ W* * (m) , that is, W~(m) is the mapping cylinder of P (m). In what follows, we set A = ßm+1 (W (m+ 1)) = Zp . . .Zp, ff = ff1+ . .+.f* *fs 2 ______-z_____" s-copies____ Hm+2 (W (m), A). Since W (m) is a deformation retract of W (m) , we may also s* *et ff 2 ______ ______ Hm+2 (W (m) , A) and ff|W(m) = ff. Let P~(m): N(Zp, 0) ^ W (m) ! W (m) be the* * map ______ defined by P~(n ^ u) = n^ u^ 1 2 W (m) for n 2 N(Zp, 0), u 2 W (m). Let _: W (m* *) ! ______ _____ W (m) be the natural injection. It is obvious that _P (m) P (m). Now, W (m* *+ 1) = _________ ______ -1(P (W (m), ff)), we may define W (m+ 1) = -1(P (W (m) , ff)). It is obvio* *us that _________ W (m+ 1) is a deformation retract of W (m+ 1). It follows from the definition o* *f W (m+ 1) that P (m+ 1): N(Zp, 0)^ W (m+ 1) = N(Zp, 0)^ P (W (m), ff) ! W (m+ 1) = P (* *W (m), ff) may be constructed as follows. For any n 2 N(Zp, 0), w 2 W (m), P (m+ 1)((n^ w)^ t) = P (m)(n^ w)^ (2t- 1) 2 C(W (m)) P (W (m), ff)1_26t* *6 1 P (m+ 1)((n^ w)^ t) = H(n^ w)^ 2t 2 K(A, m+ 2) 06 t6 1_2 where H: N(Zp, 0) ^ W (m) ^ I+ is a homotopy from the composed map MO (id|N(Zp,* *0)^ff): N(Zp, 0) ^ W (m) ! N(Zp, 0) ^ K(A, m+ 2) ! K(A, m+ 2) to the composed map ffP (* *m): N(Zp, 0)^W (m) ! K(A, m+ 2), where M denotes the natural N(Zp, 0) action of K(A* *, m+ 2) and P (m+ 1)|N(Zp,0)^K(A,m+2)= M. _________ Now we define a map P~(m+ 1): N(Zp, 0) ^ W (m+ 1) ! W (m+ 1) related to P (m+ 1). 44 Let G: N(Zp, 0) ^ W (m) ^ I+ be the homotopy from _P (m) to P~(m) defined by ______ G(n^w ^s) = (n^w ^s) 2 N(Zp, 0)^W (m)^I+ W (m), 06 s6 1. We define P~(m+ 1) as follows. For any n 2 N(Zp, 0), w 2 W (m), ______ ______ P~(m+ 1)((n^ w)^ t) = ~P(m)(n^ w)^ (2t- 1) 2 C(W (m) ) P (W (m) ,1ff)_26t* *6 1 P~(m+ 1)((n^ w)^ t) = ff(G(n^ w^ 4t- 1)) 2 K(A, m+ 2) 1_46t6 1_2 P~(m+ 1)((n^ w)^ t) = H(n^ w^ 4t) 2 K(A, m+ 2) 06 t6 1_4 We also define P~(m+ 1)|N(Zp,0)^K(A,m+2)= M. It can be easily seen that the following proposition holds ______ Proposition 15.4 ~_ P (m+ 1) P~(m+ 1), where _~: P (W (m), ff) ! P (W (m)* * , ff) = __________ W (m + 1) denotes the natural homotopy equivalence. Now we prove proposition 15.1. We prove it by induction. If m = 0, then in t* *his case, W (0) = K(Zp, 0) and the conclusion follows easily from the fact that 2P (0) O* * (F ^E1f0) and Eoe2S induces the same homomorphisms on the first non-zero homology group m* *odp. Suppose that the proposition holds for m 6= 2pi, i = 0, 1, . .,.we shall prove * *that it holds for m+ 1. Since fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map, oe2: B2(k(Zp* *, 1)) ! E2(W (m+ 1)) is defined. For simplicity, we use J to denote the map ~E2Ooe2: ~B* *2(k(Zp, 1)) ! 2W (m+ 1). Since H*(N(Zp, 0)^B~2(k(Zp, 1)), A) = H*(B~2(k(Zp, 1)), A)+ H*(B~2(* *k(Zp, 1)), A) ~ø0and ff . 2H*(N(Zp, 0) ^ ~B2(k(Zp, 1), A) = 0, B~2(k(Zp, 1)) = S0 ^ ~B2(* *k(Zp, 1)) * * ______ N(Zp, 0)^B~2(k(Zp, 1)) and B2(k(Zp, 1))^ I+ [N(Zp, 0)^ 1+ is a CW subspectrum o* *f W (m), ______ i it follows from Proposition 15.2 that there exists a map ff0: 2W (m) = B2(k(Zp* *, 1))^ I+ [ j N(Zp, 0)^ 1+ ! K(A, m+ 2) such that J J(ff0~) where ~ denotes the natural map ______ ______ i j from 2W (m) to 2W (m) = B2(k(Zp, 1))^ I+ [ N(Zp, 0)^ 1+ . By induction hypothesis and homotopy extension properties with respect to th* *e pair N(Zp, 0) ^ ( W (m), ~k(Zp, 1)), it may be assumed that 2P (m)|N(Zp,0)^~k(Zp,1)* *= E2~oe2~S. Since ~ø: ~k(Zp, 1) ! B2(k(Zp, 1)) is the natural injection, we have E1f~m* *+1= -1~øJ = -1~øJ(ff0~). Therefore, E1f~m+1~k(Zp, 1) is on the spectrum J(ff0)(B~2(k(Zp, 1* *))). It follows from the construction of P (m+ 1) and ~P(m+ 1) just mentioned. 45 For t = 0, 1_2, n 2 N(Zp, 0), b 2 B~2(k(Zp, 0)), we have P~(m+ 1)(n ^ b ^ t)* * = the base point of K(A, m+ 1), so P~(m+ 1) induces a map oe from N~(Zp, 0) ^ ~B2(k(* *Zp, 0)) to K(A, m+ 2). It follows from the construction of P (m+ 1) and P~(m+ 1), we h* *ave ~P(m+ 1)|N(Z = E ~oe~S+ ø*(oe). Since m 6= 2pi, ø* = 0: Hm+2 (N(* *Z , 0) ^ p,0)^~k(Zp,1) 2 2 * * p B2(k(Zp, 0)), A) ! Hm+2 (N(Zp, 0) ^ ~k(Zp, 1), A), so we have P~(m+ 1)|N(Zp,0)^* *~k(Zp,1)= E2~oe2~S. So Proposition 15.1 holds. If m+ 1 = 2pi, by the induction hypothesis we have E2~oe2~SPm (F ^ ~E1~fm) o* *n 2W (m), so E2~oe2~Sand 2Pm+1 (F ^E~1~fm+1) are m+ 2 homotopic. Thus, we can define the* * cocycle of difference D(E2~oe2~S, 2Pm+1 (F ^ ~E1~fm+1)) 2 Hm+3 (~n(Zp, 1) ^ ~k(Zp, 1), ßm* *+1 (W (m+ 1))). Since Pm+1 |S)^W(m+1) = id, S|S1^k(Zp,1)= S| k(Zp,1)= ø: k(Zp, 1) ! B2(k(Zp, 1* *)), we have D(E2~oe2~S, 2Pm+1 (F ^ ~E1~fm+1))|~S1^~k(Zp,1)= 0. Let D(E2~oe2~S, 2Pm+1 (F ^ ~E1~fm+1))|fi00^~upn= ` 2 2ßm+1 (W (m+ 1)) (I)* *. We define a new multiplication M~(m+ 1) from W (m+ 1) ^ W (m+ 1) to W (m+ 1) by M~(m+ 1) = i j M(m+ 1)+j -2(`)(ø*i^ø*0+ø*0^ø*i) , where M(m+ 1): W (m+ 1)^W (m+ 1) ! W (m+ 1) denotes the multiplication of W (m+ 1) and j: K(A, m+ 1) ! W (m+ 1) denotes the* * natural injection. It can be easily proved that M~(m+ 1) is also homotopy commutative * *and homotopy associative. Let ~Pm+1be the N(Zp, 0) action defined by M~(m+ 1). It c* *an be proved from (I) that E2~oe2~S 2P~m+1(E1^E~1~fm), that is, Prposition 15.1 hol* *ds for m+ 1 with respect to the multiplication M~(m+ 1). 17 The proof of (h)m+1 for m+ 1 = 2pi. It can be easily seen that E2*~oe2*~S*(~ø00^ ~upi) 2P~m+1(F ^ ~E1~fm)*(~ø0^ ~upi) = 2Pm+1 (~ø0^ ~øi) = 2(~ø0, ~øi) where ~ø02 H2(n(Zp, 1), Zp) denotes the generater of H2(n(Zp, 1), Zp) which sen* *ds ~ø0of i * * 2+2pi K(Zp, 0). So, S*(ø00^ upi) 6= 0. Since H2+2p(B2(k(Zp, 1), Zp) is generated by * *j*_*H 46 (E1(W (m)) ^ E1(W (m)) ^ E1(W (m)), Zp) and s*(ø00^ upi) and oe2*j*_* = 0, by t* *he construction of ff = (ff1, . .,.ffs) we have ffk(~ø0, ~øi) = 0, k = 1, . .,.s. * * Thus, we have ff(~ø0, ~øi) = 0. Therefore, E*2( 2ff)s(ø00^ upi) = 0. So we have oe*2E*2( 2f* *f) =). So, fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map. 18 Appendix In what follows, we use D to denote the statement ff1fip1= 0. There are two di* *fferent proofs of D. The first was given in [14]. The second was given in [15]. We will* * indicate that both are incorrect. We follow the notation in [14]. In [14] (P.841), a (2m+ 1) sphere bundle Bm* * (p) over S2m+2p+1 was defined and it satisfies that H*( (Bm (p)), Zp) = p(ai, bi).* * Notice that Bm (p) and S2m+1 x S2m+2p+1 are of different homotopy type and so (Bm (p)* *) and (S2m+1 x S2m+2p+1) are of different homotopy type, too. We only know that the* *re is a spectral sequence Er converging to H*( (Bm (p)), Zp) with E2 = H*( (S2m+1, Zp) H*( (S2m++2p+1, Zp)) = p(ai) p(bi) = p(aibi). But we do not know whether * *the equality H*( (Bm (p)), Zp) = p(ai, bi) holds. So the proof is incorrect. Now we will show that the second proof of D in [15] (P.198-203) is also inco* *rrect. In what follows, we use the same notations as that in [15]. The proof of D depends* * on the following statement. For p> 3 and n> 2, we have P*1Hpn+2(p-1)(epp-2(M(Zp, n), Z* *p)) 6= 0. However, using some elementary method, we can show that this conclusion is wron* *g. We have the following propositions Proposition 19.1 For p= 3 and l> 2, we have P*1H3l+4(epp-2(M(Zp, l), Zp)) = * *0. Recall the definition of ep1(X). It is the identification space X3^ I+ = ~,* * where the equivalence relation ~ is defined by x1^x2^x3^0 ~ x2^x3^x1^1 and e^ e^ e^ t ~ e* *0 (e and e0 are respectively the base points of X and ep1(X)). Let P : X3 ! X3 be de* *fined by P (x1^x2^x3) = x2^x3^x1. It is obvious that if X is a cogroup, then ep1(X) i* *s of the same homotopy type as that of the mapping cone C(P - id) where id is the identi* *ty map of X3. So when X is taken to be M(Zp, n) for n> 2, ep1(M(Zp, n)) and C(P - id) * *are of the same homotopy type. 47 Before proving Proposition 19.1, we state some properties of Moore space. Fo* *r sim- plicity, we abbreviate M(Zp, n) to M(n) in what follows. M(n) = an [ bn, where * *an is the n-sphere and bn is the cone C(p(id)). Let l, m> 2 and !, ` are homotopy classes* * of map from M(l+ m) and M(l+ m+ 1) to M(l)^ M(m) determined by the following relations !*(al+m) = al^ am !*(bl+m) = bl^ am `*(al+m+1) = al^ bm + (-1)l+1bl^ am `*(bl+m) = bl^ bm It is easy to check that the map ! _ `: M(l+ m) _ M(l+ m+ 1) ! M(l)^ M(m) is a homotopy equivalence. Let X, Y be two spaces and ff, fi be respectively hom* *otopy J classes of maps from M(l) and M(m) to X and Y . We use ff fi and ff fi to d* *e- note respectively the homotopy classes (ff^ fi)*(!) and (ff^ fi)*(`). It is ea* *sy to check that for any maps f: X ! X~ and g: y ! Y~, we have (f^ g)*(ff fi) = f*(ff) g** *(fi), J J (f^ g)*(ff fi) = f*(ff) g*(fi). Let l, m, n > 2, ff, fi, fl be respectively the identity maps of M(l), M(m),* * M(n). We * * J use q1, q2, q3, q4 to denote respectively the homotopy class of maps (ff fi) * *fl, (ff fi) fl, J J J (ff fi) fl, (ff fi) fl. Then, it is easy to check that q1 _ q2 _ q3 _ q4 fr* *om M(l+ m+ n) _ M(l+ m+ n+ 1) _ M(l+ m+ n+ 1) _ M(l+ m+ n+ 2) to M(l) ^ M(m) ^ M(n) is a homoto* *py equivalence. In what follows, we always assume that l = m = n and so ff = fi = * *fl. Let J * * J P from M(l)3 to itself be defined by P (x ^ y ^ z) = y ^ z ^ x, then P*((ff fi* *) fl) = J J J J J J J J * * J J (fi fl) ff = (ff ff) ff = (ff fi) fl = (id)*((ff fi) fl). So we have ((* *ff fi) fl)P = J J 3 ((ff fi) fl)id, where id is the identity map of M(l) . Now we prove Proposition 19.1. Let T1 = M(3l) _ M(3l + 1) _ M(3l + 2) and T2* * = M(3l + 2) and W = M(l) ^ M(l) ^ M(l) and q = q1 _ q2 _ q3 _ q4. Since q: T ! W is a homotopy equivalence, we have that ep1(M(l)) = C(P - id) and C((P - id)q) * *are of the same homotopy type and therefore the two spaces may be considered the sa* *me space in homotopy theory. Let ~q= q1 _ q2 _ q3, then q = ~q_ q4 and (P - id)q = (P - id)~q_ (P - id)q4. It follows from the previous argument that (P - id)q4 * * 0. So P 1 P C((P -id)q4) = M(3l+2)_W and we have that ep (M(l)) = C((P -id)~q)_ M(3l+2). Therefore, H3l+4(ep1(M(l)), Z3) = H3l+4(C((P - id)~q), Z3) H3l+4((M(3l + 2)),* * Z3) = P H3l+4((M(3l + 2)), Z3) = Z3, so we have P*1H3l+4(ep1(M(l)), Z3) = P*1H3l+4( M(* *3l + 48 2), Z3) = P*1H3l+4(M(3l + 2), Z3) = 0. This proposition shows that the statement P*1H3l+2(p-1)(epp-2(M(l)), Zp) 6= * *0 is false. So, the second proof of D is also incorrect. Since V (0) = {M(1), M(2), . .,.M(n), . .}., the above proposition implies t* *he follow- ing proposition Proposition 19.2. P*1H4(ep1(V (0)), Z3) = 0. References [1]Adams,J.F.,On the non-existence of Hopf invariant one, Ann. of Math.72.(196* *0),20- 103. [2]Husemoller,D.,Fibre Bundles, Second Edition, Graduate Text in Mathematics, * *Vol. 20.(1985),Springer Verlag. [3]Miller, H.R., Ravenal, D.C. and Willson, Period phenomena in the Adams Novi* *kov spectral sequence, 105 (1977) 469-516. 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[13]Zhou,X.G.,Higher cohomology operations that detect homotopy classes, Lectu* *re Notes in Mathematics1340,416-436. [14]Toda, H. An important relation in homotopy group of spheres,Proc. Japan Aca* *d. 43 (1967) 839-842. [15]Toda, H. Extended p-th power of complexes and applications to homotopy theo* *ry. Proc. Japan Acad. 44 (1968) 198-203. Springer-Verlag, (1989). 50