Filtering the fiber of the pinch map Brayton Gray Dept. of Mathematics, Statistics and Computer Science (M/C 249) University of Illinois at Chicago 851 South Morgan Street Chicago, IL 60607-7045 brayton@uic.edu 1 Introduction In [CMN1 ], the authors analyzed the homotopy type of the mod pr Moore space P 2n+1= S2n [pre2n+1 by constructing a fibration sequence 2S2n+1-@n! Fn -! P 2n+1-! S2n+1. A key result is that both Fn and P 2n+1contain a factor which is the loop * *space on a one point union of Moore spaces that are at least 4n - 2 connected. Removing th* *is factor (which can be inductively considered later) leaves a fibration sequence: _@__ . .!. 2S2n+1- ! F n- ! T 2n+1-! S2n+1 and they show that __ Y k-1 r+1 F n~=S2n-1x S2np {p }. k 1 __ An important remaining question is to understand the components of @: __ 2 2n+1 2n-1 @0 : S -! S __ 2 2n+1 2npk-1 r+1 @k : S -! S {p }. __ @0 is well understood_and plays a key role in homotopy theory. There is much i* *nterest in understanding @k for k > 0; in particular, it is not known whether they are * *all null 1 homotopic [AG], [CMN1],_[G2], [G3], [GT], [N1], [N2]. In case r > 1, Neisendor* *fer ([N1], [N2]) has shown that @k = 0 for k 1, so our emphasis will be on the case r = * *1. A remarkable observation of [CMN2] is that there is an isomorphism of Hopf a* *lgebras (see appendix): __ H*( 2S2n+1; Z=p) ~=H*(F n; Z=p). __ However, the induced homomorphism (@)* = 0. __ The intention of this work is to consider constructions involving Fn which a* *re analogous_ to constructions involving 2S2n+1. We will be able to do this compatibly with * *the map @. In particular, we will look at the filtration of Selick [S] and the classifying* * space of the double suspension [G2]. Our results will follow from the construction of a new "Hopf i* *nvariant" type map: Fn -h! Fnp which induces an epimorphism in p-local homology. Recall that there is a p-loca* *l fibration sequence: Hps 2nps S2n(ps-1)! S2n1-! S1 . By analogy, we compare F(ps-1)to the fiber s of the Hopf invariant hs: s s ! Fn -h! Fnps Theorem 1.1 s ' F(ps-1)x Xs where Xs is a wedge of mod pr Moore spaces. This will appear as 7.1. These same ideas lead to: Theorem 1.2 If k > 1, there is a homotopy commutative diagram: _@ k 2S2n+1 ---k! S2np -1{pr+1} ? x flk?y~= ??_@1 Hpk-1 k-1 2S2n+1 - ---! 2S2np +1 where Hpk is a James-Hopf invariant and flk is a homotopy equivalence. __ k Theorem 1.3 The map @k : 2S2n+1- ! S2np -1{pr+1} is homotopic to a compositi* *on: k-1 r+1 2S2n+1- ! BWn -ffl!S2np {p }. These appear as 8.2 and 9.3 in the sequel. Throughout this paper all spaces will be assumed to be localized at a fixed * *prime p > 2 and all homology will be p-local unless otherwise indicated. 2 2 Filtration of 2S2n+1 H*( 2S2n+1; Z=p) is a free commutative algebra on generators of dimensions 2npi* * - 1, 2npi+1- 2 for i 0. Selick [S] has described a sequence of H-spaces whose hom* *ology filters H*( 2S2n+1; Z=p) by successively adding one generator at each stage. L* *et us write S2n(k)= Jk(S2n) for the kth filtration of the James construction J(S2n) = S2n1.* * Then the space capturing the first 2s generators of H*( 2S2n+1; Z=p) is precisely S2n(ps-1). * *Selick [S] defines spaces Gs which contain the first 2s + 1 generators. We define Gs by a diagram * *of fibrations: s-1 ff 2n Gs - --! S2np ---! S(ps-1) ? ? fl ? ? fl y y fl s+1 2n 2n+1 Hnps 2nps+1 2S2n+1 - --! 2S2np ---! S(ps-1)---! S - --! S ? ? ? ? y y BWnps _______ BWnps s-1 2n s 2n where ff : S2np ! S(ps-1)is the attaching map for the 2np cell of S1 , Hnpsi* *s the appro- priate James-Hopf invariant, and BWnpsis the cassifying space for the double su* *spension. Since Gs is the fiber of O Hnps, it is an H space if p > 3 ([G2; Proposition* * 6]). Gs could 0 s+1 also be described as the fiber of a Toda-Hopf invariant S2n(ps+1-1)H-! S2np * *-1using the techniques of ([G2], [MN]). __ Constructing an analogous filtration for F n is not difficult. The key resu* *lt is in the compatibility of the two filtrations. Proof of Theorem CMN In this section we give a brief summary of the proof of the main result of [CMN* *1]. Theorem CMN There is a diagram of fibration sequences: Fn ---! P 2n+1- --! S2n+1 ? ? fl ? ? fl y y fl = S2n-1x Vn ---! T 2n+1- --! S2n+1 ? ? ? ? y i y i0 Pn --=-! Pn ? ? ? _ ? y OE y p 2n+1 Fn ---! P 2n+1- --! S 3 where the maps i and i0are null homotopic, p is the pinch map, Pn is a one poin* *t union of mod pr Moore spaces of dimension 4n and Y k Vn = S2np -1{pr+1}. k 1 Here Sm {d} is the fiber of the degree d map on Sm . The inessentiality of i* * and i0implies that Fn ' S2n-1x Vn x Pn P 2n+1' T 2n+1x Pn. Sketch of the proof. Both the mod pr homotopy and the mod p homology of P 2* *n+1have differential Lie algebra structures, and the Hurewicz map is a Lie algebra homo* *morphism. Furthermore Fn has an extended ideal structure. These structures are obtained * *from the Samelson product and the rth Bockstein fi(r)(fi(i)= 0 fori < r). H*( P 2n+1; Z=p) = U(L) where L is a free Lie algebra on v 2 H2n( P 2n+1; Z=* *p) and u = fi(r)(v) and U(L) is the universal enveloping algebra. L(0) L is the Lie i* *deal generated by xi= adi-1(v)(u) for i > 1 and H*( Fn; Z=p) = U(L(0)). Furthermore the suspensions of the xi, oe(xi) 2 H2ni(Fn; Z=p) ~=Z=p are non zero. u and v lie in the image of the mod pr Hurewicz homomorphism, so * *all of L and hence L(0)does as well L(0) im{ss*( Fn; Z=pr) -! H*( Fn; Z=p)} using the extended ideal structure in homotopy. Both in homotopy and homology, fi(r)(xpk) = 0, so one may construct an exten* *sion: k-1 r+1 ffik P 2np (p ) -! Fn of the mod pr homotopy class k P 2np -1-! Fn representing xpkin homology. The only property of the maps ffik that is needed * *is that their Hurewicz image is xpk. From these maps the authors construct k-1 r+1 e 2npk r+1 _ffik S2np {p } -! P (p ) -! Fn _ where ffikis the adjoint of ffik and e is a particular map described in the nex* *t section. These maps are assembled via loop multiplication to obtain: : S2n-1x Vn -! Fn 4 The Bockstein fi(r)is trivial in H*(S2n-1 x Vn; Z=p) but not in H*( Fn; Z=p). * *However induces an isomorphism in the Bockstein homology of these homology groups (se* *e the appendix for a calculation of the Bockstein homology). Next the authors construct a squence of sub-Lie algebras L(k+1) L(k) L(0)v* *ia short exact sequences of Lie algebras: 0 - ! L(1) - ! L(0) - ! -! 0 0 - ! L(k+1) - ! L(k) - ! -! 0. Here and are free commutative Lie algebras generated by ok = xpk* *of dimension 2npk - 1 and pk-1` k' 1 X p (0) oek = ___ [xi, xpk-i]fflL . 2p i=1 i This is possible since ok is a generator and oek is decomposible in L(k-1)but n* *ot in L(k)since _1_ pk (k) 2p pk-1 is a p-local unit and xpk-1= ok-1 =2L . It follows that there is a split short exact sequence of differential Hopf a* *lgebras: k-1 r+1 0 -! U(L(k+1)) -! U(L(k)) -! H*(S2np {p }; Z=p) ! 0 and hence U(L(1)) Hp(S2n-1x Vn; Z=p) ~=U(L(0)) = H*( Fn; Z=p) T where L(1)= L(k). k 0 Consequently Ht(U(L(1)); fi(r)) = 0 for t > 0 and hence L(1) has a basis con* *sisting of classes {xff, fi(r)xff}. Since xff2 L(1) L(0)one can choose maps OEff: P nff-! Fn whose homology image is xffand fi(r)xffusing the extended ideal structure. Asse* *mbling these one produces a one point union of mod pr Moore spaces, Pn and a map: __ OE: Pn -! Fn such that the homology image of __ ( OE)* : H*( Pn; Z=p) -! H*( Fn; Z=p) is exactly U(L(1)). From this the authors produce a homotopy equivalence S2n-1x Vn x Pn -! Fn __ __ by_multiplying the maps and OEvia the H space structure in Fn. Let Fn be the* * fiber of OE. Then __ S2n-1x Vn -! Fn -@! Fn is a homotopy equivalence and i is inessential. Thus completes the outline of t* *he proof. 5 3 Combinatorial description of Fn In [G1 ] a combinatorial description of the fiber of the pinch map: F -! X [ CA -ss!SA was described in the spirit of the James construction X1 for SX. The model, de* *signated (X, A)1 consists of all words in X1 with the property that all letters after * *the first letter are required to lie in A, where A X. Alternatively, this can be described by * *a push out diagram: X x A1 ---! (X, A)1 x x (3.0) ?? ?? A x A1 ---! A1 Proposition 3.1 ([G1]). There is a map (X, A)1 ! F which is a homotopy equiv* *alence when the inclusion A X is a cofibration. There is an action of the monoid A1 on (X, A)1 and (X, A)1 can be thought of* * as a universal space in the following sense. If Y is any space on which A1 acts and * *g : X ! Y is any map such that g(a) = a . * for some point * 2 Y , there is a unique A1 equi* *variant map g1 : (X, A)1 ! Y. (See [G; 3.2]). The map (X, A)1 ! F is constructed from the action A1 xF ! SAx* *F ! F . The orbit space of (X, A)1 under the action of A1 is X=A and we may use the * *universal property to establish the following diagram (X, A)1 --e-! X1 ? ? (3.2) ae?y ?y X=A ---! (X=A)1 Note that the inclusion e : (X=A)1 ! X1 is the unique A1 equivariant extensio* *ns of the inclusion of X in X1 . Example 3.3 Let A = S2n-2 P(2n-1pt)= X. Then (X, A)1 is the homotopy fiber * *of the map of degree pt pt 2n-1 (X, A)1 = S2n-1{pt} -! S2n-1- ! S and the map e : S2n-1{pt} -! (P 2n-1(pt))1 ~= P 2n(pt) is uniquely determined * *as a S2n-1 equivariant map extending the inclusion of P 2n-1(pt). 6 One of the main features of the construction (X, A)1 is that we can define * *functorial Hopf invariants using the same formulas as in James [J]. In particular we have * *a pointwise commutative diagram: Hk (k) A1 ---! (A )1 ? ? ? ? y y Hk (k-1) (X, A)1 ---! (X ^ A )1 ? ? ? ? y y Hk (k) X1 ---! (X )1 It would be desireable to construct functorial compressions of these maps: h : (X, A)1 - ! (X ^ A(k-1), A(k))1 (X ^ A(k-1))1 but I have been unable to do this. In the next section we will construct a map* * h of this form, but we have no knowledge of how it relates to Hk. 4 Construction of h : Fn - ! Fnp In this section we will define a kind of Hopf invariant which is key for the re* *sults of this paper. Theorem 4.1 Suppose (X, A) is a suspension pair and r 1. Then there is a ma* *p: h : (X, A)1 - ! (X ^ A(r-1), A(r))1 such that the composition: p (r-1) (r) X x A1 - ! (X, A)1 - h!(X ^ A(r-1), A(r))1 - ! X ^ A =A is homotopic to the composition: 1xHr-1 (r-1) (r-1) (r-1) (r-1) (r) X x A1 - ! X x A1 -! X ^ A1 -! X ^ A -! X ^ A =A 1^ffl where Hr-1: A1 ! A(r-1)1is any map and ffl uses the suspension structure of X t* *o collapse of X ^ A(r-1)1to X ^ A(r-1). Corollary 4.2 Let Fn be the fiber of the pinch map p : P 2n-1(pr) -! S2n+1 for each n and Hr-1 be any choice of James-Hopf invariants. Then there is a map: h : Fn -! Frn 7 such that the composition: S2n x S2n+1- ! Fn -h! Frn- ! P 2rn+1 is homotopic to: 1xHr-1 2n 2(r-1)n+1 2rn 2rn+1 S2n x S2n+1 -! S x S -! S - ! P ; in particular, h* : H2rn(Fn) -! H2rn(Frn) is an isomorphism. Note: It is an easy calculation to see that ae Z(p) n | i Hi(Fn) = 0 n - i and the map Fn -ss!P 2n+1(pr) is reduction mod pr in homology. In particular w* *e have defined a map: Fn -! P 2nr+1(pr) for each r 1 which is onto in p-local homology. Proof of 4.1 We begin by constructing a map ~ : (X, A)1 - ! (X, A)1 =A1 - ! X x A1 =A o A1 X o A1 =A o A1 which follows from the push out diagram (3.0). This map is functorial and we h* *ave a commutative diagram: ~ (X, A)1 ---! X o A1 =A o A1 x x ? ? ? ? X x A1 ---! X o A1 using the functorial property, we have the commutative diagram: (X, A)1 - --! X o A1 =A o A1 ? ? ? ? y y (CX, A)1 - --! CX o A1 =A o A1 ' S(A o A1 ) however (CX, A)1 is the fiber of CX=A -! SA and hence is contractible. Conseque* *ntly ~ lifts to (X o A1 , A o A1 )1 which is the homotopy fiber of X o A1 =A o A1 - ! SA o A1 ; we now choose a map Hr-1 : A1 -! A(r-1)1and use the suspension structure of th* *e pair (X, A) to obtain a map (X o A1 , A o A1 )1 -! (X ^ A(r-1)1, A ^ A(r-1)1)1 -! (X ^ A(r-1), A(r))1 ; composing with ~0defines h. 8 Proposition 4.3 The map h : Fn -! Fnpsinduces an epimorphism in p-local homolo* *gy. Proof: We use the map S2n+1- @!Fn which has degree p in H2nkfor each k > 0 to * *calculate the cup product structure in H*(Fn). We choose generators ei2 H2ni(Fn) such tha* *t @*(ei) is p times the generator of H2ni( S2n+1) dual to the ith power of a homology ge* *nerator in H2n( S2n+1). Then it is easy to see that ` ' i + j eiej = p ei+j. i s Let us designate di2 H2nip(Fnps) for the corresponding generator; then h*(d1) = u1eps for some p local unit u1 by 4.2. We show that h*(di) = uieips where uiis a p-local unit for each i 1 by induction. Using the product struct* *ure we have pidi= d1di-1, so pih*(di) = h*(d1di-1) = h*(d1)h*(di-1) = (u1eps)(ui-1e(i-1)ps) ` ' ips = pu1ui-1 eips. ps ips It suffices to show that 1_ipsis a p-local unit. Now let vp(m) be the exponent of p in m and [x] be the greatest integer less* * than or equal to x. Then ~ ~ X n vp(n!) = __i . i 1 p Consequently vp((psi)!) = ps-1i + ps-2i + . .+.i + vp(i!) so ` ` '' psi s s s vp = vp((p i)!) - vp((p (i - 1))!) - vp(p !) ps = vp(i!) - vp((i - 1)!) = vp(i). and we are done. By 4.2, the composition 2+1 S2n+1- @!Fn -h! Fnps-! P 2np 9 is null homotopic, so there is a lifting eHps Heps s S2n+1 ---! S2np +1 ? ? @?y ?y@ Fn --h-! Fnps. s+1 Proposition 4.4 (Heps)* : H*( S2n+1) ! H*( S2np ) is an epimorphism. Proof: This follows since both maps labeld @ have degree p in homology and h* ** is an epimorphism. In particular, the fiber of Hepsis S2n(ps-1)as if Hepswere the James-Hopf in* *variant Hps. Both Hepsand Hps can be placed in the fibration sequence induced by the inclu* *sion S2n(ps-1) S2n+1, so there is an equivalence fls : 2S2n+1! 2S2n+1 such that Heps= HpsO fls hence we have Corollary 4.5 There is a homotopy commutative diagram 2S2n+1 - @n--! Fn ? ? Heps?y ?y hs s+1 @np 2S2np - --! Fnps where Heps~ HpsO fls for some homotopy equivalence fls : 2S2n+1- ! 2S2n+1. 5 A Filtered Decomposition Let s be the homotopy fiber of hs : Fn ! Fnps. In this section we will compar* *e the decompositions of Fn and Fnpsand prove Q k s Theorem 5.1 s ~=S2n-1x S2np -1{pr+1} x S2np -1x Rs where Rs is a wed* *ge 1 kFrom this we construct _ k+1-1 r+1 e 2npk+1r+1 ffik+1 S2np {p } ---! P (p ) - --! Fn ? ? ? ? ? ? y= y= y h _ k+1-1 r+1 e 2npk+1r+1 ffik(np) S2np {p } ---! P (p ) ----! Fnp where k > 0, and _ S2np-1{pr+1}---! P 2np(pr+1)--ffi1-! Fn ? ? ? ? ? ? y y y h S2np-1 ---! S2np ---! Fnp using diagram 3.2. Multiplying these together in order gives the result. __ Lemma 5.3 The map OEn: Pn ! Fn can be chosen so that Wn = Pnp_ Qn and there i* *s a homotopy commutative diagram: _OE Pnp_ Qn ---n! Fn ? ? p?y ?yh _OE np Pnp ---! Fnp where p is the projection. Proof: Since h : H*(Fn; Z=p) ! H*(Fnp; Z=p) is onto the same holds for U(L(0)(n)) ~=H*( Fn; Z=p) ! H*( Fnp; Z=p) ~=U(L(0)(np)) in fact the generators xip2 L(0)(n) satisfy h*(xip(n)) = uixi(np) 11 where uiis a p-local unit (see 4.3). Now given a basis {xff, fi(r)xff} for L(1)(np), each xffis a Lie bracket in * *the xi(np) and this element consequently lifts to the same bracket in xip(n). Thus these lifti* *ngs are linearly independent and can be choosen as part of a basis. They are all in the image of* * the mod pr Hurewicz homomorphism. Thus we have choosen generators for L(1)(n) which sor* *t into those which are lifting of the generators_for L(1)(np) and the others. Realizin* *g these via the Hurewicz homomorphism gives the maps OEn. By a change in basis for Qn, we can a* *ssume that the map p : Pn ! Pnp is trivial on Qn. We now use 5.3 to construct the following ladder of fibrations: _@ __ _OE Fn --n-! F n ---! Pn - -n-! Fn ? ? ? ? (5.4) ?y ?yfls ?y ?yhs _@ _ nps __ OEnps Fnps ---! Fnps ---! Pnps- --! Fnps and we use 5.2 to construct compatible equivalences: __ S2n-1x Vn - --! Fn -@n--! Fn ? ? ? (5.5) ?yOEs ?y hs ?yfls s-1 __ S2np x Vnps- --! Fnps ---! Fnps. Taking fibers vertically in (5.4), we have a fibration sequence: s ! Ks ! Rs ! s where Rs = (Qn _ . ._.Qpns-1) o Pnpsis a wedge of mod pr Moore spaces and Y k s Ks = S2n-1x S2np -1{pr+1} x S2np -1. 1 k -! 0 ok+1 0 -! L(k+1) -! L(k) -! -! 0 where ssk+1 is any map of Lie algebras such that ssk+1(oek)= oek ssk+1(ok)= ok where oek, ok 2 L(k) L(0)are given by the formulas: ok = xpk pk-1` k' 1 X p oek= ___ [xi, xpk-i]. 2p i=1 i Nothing is said about the value of ssk+1 on the other generators. We need to b* *e more specific at this point. Let us define the weight of an element in a free Lie a* *lgebra to be the minimal number of brackets in any term; in particular, L(0)is free on gener* *ators xi, we define !(xi) = 1, ![x, y] = !(k) + !(y) and !( ai) = min!(ai). For an element z* * 2 L(k)we define the weight of z to be the weight considered as an element of L(0). Thus* * !(ok) = 1 and !(oek) = 2. We further specify the Lie algebra homomorphism ssk+1 by demand* *ing that 13 ssk+1(z) = 0 if !(z) > 2. Now define L(k)s= Ls\ L(k)for k s. Since oek, ok 2 * *L(k)s, we have short exact sequences: 0 ---! L(1)s---! L(0)s---! ---! 0 ? ? ? fl ? ? ? fl y y y fl 0 ---! L(1) ---! L(0) ---! ---! 0 0 ---! L(k+1)s---! L(k)s---! ---!0 ? ? ? fl ? ? ? fl y y y fl 0 ---! L(k+1)---! L(k) ---! ---!0 for k < s. Define L(s+1)sto be the kernel of L(s)s! ! 0. In fact we have L(s+1)s* * L(1). To see this we need to show that ssr+1(L(s+1)s) = 0 for r s. The generators * *of L(s+1)sof filtration 1 are of the form xi for i < ps and those of filtration 2 are of the* * form [xi, xj] for i, j < ps. None of these have dimension 2nps - 1, so L(s+1)slies in the ke* *rnel of the composition L(s+1)! ! . Consequently L(s+1)s L(s+1). Similarly f* *or r > s the generators of L(s+1)sof weight 1 and 2 have dimensions 4n(ps- 1) and conseque* *ntly their images are 0 in for r > s. It follows that we may first choose a basis for L(s+1)sand then choose a bas* *is for L(1) containing these elements. This completes the proof. Corollary 6.2 There is a commutative diagram of fibration sequences: Fn ---! Vn - -*-! Pn ---! Fn x x x x ? ? ? ? ? ? ? ? F(ps+1-1)---! Ks+1 - -*-! Qs+1 ---! F(ps+1-1) x x x x ? ? ? ? ? ? ? ? F(ps-1) ---! Ks - -*-! Qs ---! F(ps-1) where all the vertide maps are mod p homology monomorphisms and each Qs is a we* *dge of mod pr Moore spaces. 7 On the sequence F(ps-1) ! Fn ! Fnps Recall that there is a p-local fibration sequence Hps 2nps S2n(ps-1)! S2n1-! S1 . 14 By analogy, we compare F(ps-1)to the fiber s of the Hopf invariant hs: s s ! Fn -h! Fnps Theorem 7.1 s ' F(ps-1)x Xs where Xs is a wedge of mod pr Moore spaces. Proof: For dimensional reasons, the inclusion F(ps-1) Fn lifts to s. Consi* *der the pull back diagram: Ys ---! F(ps-1)- --! Ks ---! Ys - --! F(ps-1) ? ? ? ? ? ? ? ? ? ? y y y= y y Rs ---! s - --! Ks --*-! Rs - --! s. The map Ks ! s ! Fn constructed in section 6 is obtained from maps: _ k ffik: P 2np-! Fn k < s s-1 2nps _ffis S2np - ! P -! Fn and all of these maps factor through F(ps-1)for dimensional reasons. Consequen* *tly the retraction Ks ! s factors through F(ps-1)as well and hence the map Ks ! Ys i* *s null homotopic. Consequently Qs ' Ys and the map Qs ! F(ps-1)lifts to an equivale* *nce Qs ' Ys. Now the inclusion Qs ! Pn factors through Rs, so Rs = Qs _ Es and we h* *ave a diagram of fibrations: Ks ---! Qs ---! F(ps-1) ? ? ? =?y ?y ?y Ks ---! Qs_ Es ---! s. Now the fiber of Qs ! Qs_ Es is (Eso Qs) which is a retract of (Qs_ Es) = R* *s and hence a retract of s. This completes the proof with Xs = Eso Qs. __ 8 Factorization of @ In this section we will consider the components of __ 2 2n+1 2n-1 2n-1 Y 2npk-1 r+1 @n : S - ! S x Vn = S x S {p } k 1 k-1 r+1 * *__ we will write @n(k) : 2S2n+1- ! S2np {p } for the kth component, k > 0 and * *@n(0) for the projection 2S2n+1- ! S2n-1. We begin by combing the diagram from 4.5: __ 2S2n+1 --@n-! Fn - --! Fn ? ? ? Heps?y ?y hs ?yh0 s+1 __ 2S2np ---! Fnps - --! Fnp 15 with the equivalences from 5.2 __ S2n-1x Vn --n-! Fn - --! Fn ? ? ? ? ? ? y_ y h y h0 np __ S2np-1x Vnp ---! Fnp - --! F np where h0is defined in 5.3, to get a homotopy commutative diagram: _@ 2S2n+1 --n-! S2n-1x Vn ? ? Heps?y ?y_ _ s+1 @nps 2nps-1 2S2np ---! S x Vnps >From this we see that __ __ @n(k + 1) = @npk(1) O HpkO flk, k 0 while _@(1) 2S2n+1 --n-! S2np-1{pr+1} ? ? (8.1) Hep?y ?yP _@ np(0) 2np-1 2S2np+1 ---! S . Putting these together gives: __ __ 1X __ (8.2) @n = @n(0) + @npk(1) O HpkO flk k=0 where flk : 2S2n+1! 2S2n+1 is an equivalence. Remark: If we had better control over the relationship between the Hopf invaria* *nts hs and the James-Hopf invariant, we could remove the factor flk. __ Proposition 8.3 Suppose @n(1) : 2S2n+1 ! S2np-1{pr+1} is null homotopic for e* *ach n. Then __ a) @n factors through S2n-1 b) The loops on the pth James-Hopf invariant: Hp : 2S2n+1- ! 2S2np+1 is homotopic to a composition: 2p 2 2np+1 2S2n+1- ! 2P 2np+1-! S __ c) The fiber B of @np(0) : 2S2np+1! S2np-1is a retract of 2P 2np+1 d) B ' BWn. 16 Proof: a) follows directly from 7.2. To verify b), note that by hypothesis Hp : 2S2n+1- ! 2S2np+1 __ lifts wo the fiber B of @np(0). The following diagram provides a map from B to * * 2P 2np+1: B ---! 2P 2np+1 ? ? ? ? y y 2S2np+1 --1-! 2S2np+1 ? ? ?_ ? y@np(0) y S2np-1 ---! Fnp c) follows since Fn ' S2np-1x Vnpx Pnp, so 2P 2np+1' B x Vnpx 2Pnp. To verify d), note that S2BWn is a retract of S2 2S2n+1by [G2], so there is a m* *ap S2BWn ! P 2np+1which is non zero in homology, It's adjoint provides an equivalence: BWn ! 2P 2np+1! B. 9 Relationship with BWn __ In this section we describe a factorization of @using the classifying space BWn* * for the double suspension. Recall from [G2] Proposition 9.1 Suppose ff : S2n ! Y and fi : S2n-1Y ! S4n-1 are maps such t* *hat fi OS2n-1ff : S4n-1! S2n-1Y ! S4n-1 is homotopic to the identity. Then there is* * a fibration sequence: _ S2n-1- ff! Y - ! B. This is natural in the following sense: Suppose we are also given ff0: S2n -! Z fi0: S2n-1Z -! S4n-1 f : Y - ! Z such that fi ~ fi0O S2n-1f and ff0~ fff. Then we have a homotopy commutative diagram; _ff S2n-1 ---! Y ---! B ? ? ? 1?y ?y f ?y_ _ff0 S2n-1 ---! Z ---! B0 17 Corollary 9.2 There is a homotopy commutative diagram: S2n-1 ---! 2S2n+1 ---! BWn ? ? ? ? ? ? y y y S2n-1 ---! Fn ---! B Since S2n-1 is a retract of Fn, B ' Vn x Wn and we have Corollary 9.3 There is a homotopy commutative diagram: 2S2n+1 ---! BWn ? ? ? ? y@ y __ Fn ---! Fn. __ k In particular, @n(k) : 2S2n+1! S2np -1{pr+1} factors through BWn, proving Theo* *rem 1.2. 18 A Appendix The following observation appears in the work of Cohen, Moore and Neisenderfer.* * We repeat it here because it sheds light on their results and is easily extended t* *o the case of the filtrations in section 6. We begin by looking at the homology Serre spectral sequence for the fibering Fn -! P 2n+1-! S2n+1. ss This is a multiplicative spectral sequence with E2p,q~=Hp( S2n+1; Z=p) Hq( Fn* *; Z=p). Since all elements of E2p,0are permanent cycles, E2 = E1 . Restricting this fib* *ration to S2n(k) leads to fibration Fn -! Ek -! S2n(k) and H*(Ek; Z=p) H*(Ek+1; Z=p) . . .H*( P 2n+1; Z=p). Thus H*( P 2n+1; Z=p) is a filtered differential group using fi(r). It has a* * spectral sequence: E2p,q= Hp( S2n+1; fi(r)) Hq(H*( Fn; Z=p), fi(r)) converging to Hp+q( P 2n+1; Z=p) = 0 if p + q > 0. This gives a multiplicative* * spectral sequence: E2p,q= Zp[u2n] Hq(H*( Fn; Z=p); fi(r)) ) 0. This spectral sequence has the same form as the Serre Spectral sequence for the* * homology of the fibration: 2S2n+1- ! P S2n+1- ! S2n+1. The differentials are forced by the multiplicative structure and consequently t* *hese spectral sequences are isomorphic. From this we conclude Proposition A.1 ([CMN1]) H*(H*( Fn; Z=p); fi(r)) ~= H*( 2S2n+1; Z=p) as alegbr* *as and co-algebras. Corollary A.2 H*(H*( F(ps-1); Z=p); fi(r)) ~=H*( S2n(ps-1); Z=p). The later follows by restricting the spectral sequence to the first ps- 1 colum* *ns. References [G] Anick, David; Gray, Brayton: Small H spaces related to Moore spaces. To* *pology 34 (1995), no. 4, 859-881. [CMN1] Cohen, F. R.; Moore, J. C.; Neisendorfer, J. A.: Exponents in homotopy * *theory. Algebraic topology and algebraic K-theory (Princeton, N.J.,1983), 3-34,* * Ann. of Math. Stud., 113, Princeton Univ. Press, Princeton, NJ, 198. 19 [CMN2] Cohen, F. R.; Moore, J. C.; Neisendorfer, J. A.: Torsion in homotopy gr* *oups. Ann. of Math. (2) 109 (1979), no. 1, 121-168. [G1] Gray, Brayton: On the homotopy groups of mapping cones. Proc. London Ma* *th. Soc. (3) 26 (1973), 497-520. [G2] Gray, Brayton: On the iterated suspension. Topology 27 (1988), no. 3,30* *1-310. [G3] Gray, Brayton EHP spectra and periodicity. I. Geometric constructions. * *Trans. Amer. Math. Soc. 340 (1993), no. 2, 595-616. [G4] Gray, Brayton,; Theriault, Stephen: On the double suspension and the M* *od-p Moore space. Contemporary Mathematics Voll 399, 2006, 101-121. [J] James, I. M.: Reduced product spaces. Ann. of Math. (2) 62 (1955), 170-* *197. [MN] Moore, John C.; Neisendorfer, Joseph A.: Equivalence of Toda-Hopf inva* *riants. Israel J. Math. 66 (1989), no. 1-3, 300-318. [N1] Neisendorfer, Joseph: Product decompositions of the double loops on odd* * primary Moore spaces. Topology 38 (1999), no. 6, 1293-1311. [N2] Neisendorfer, Joseph: James-Hopf invariants, Anick's spaces, and the do* *uble loops on odd primary Moore spaces. Canad. Math. Bull. 43 (2000), no. 2, 226-2* *35. [S] Selick, Paul: A spectral sequence concerning the double suspension. Inv* *ent. Math. 64 (1981), no. 1, 15-24. [T] Theriault, Stephen D.: Proofs of two conjectures of Gray involving the* * double suspension. Proc. Amer. Math. Soc. 131 (2003), no. 9, 2953-2962. 20