AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION BRAYTON GRAY AND STEPHEN THERIAULT Abstract.Cohen, Moore, and Neisendorfer's work on the odd primary homotop* *y theory of spheres and Moore spaces, as well as the first author's work on the secon* *dary suspension, predicted the existence of a p-local fibration S2n-1-! T -! S2n+1whose connecting * *map is degree pr. In a long and complex monograph, Anick constructed such a fibration for p* * 5 and r 1. Using new methods we give a much more conceptual construction which is also val* *id for p = 3 and r 1. We go on to establish several properties of the space T. 1. Introduction In [CMN1 , CMN2 , N1] Cohen, Moore, and Neisendorfer proved a landmark result* * concerning the exponent of the homotopy groups of spheres localized at an odd prime p. When p * * 3 and r 1 they constructed a map ssn : 2S2n+1-! S2n-1such that the composition with the doub* *le suspension 2 2S2n+1-ssn!S2n-1-E! 2S2n+1 is homotopic to the pr-power map. The existence of such a map for r = 1 was use* *d to show that pn annihilates the p-torsion in ss*(S2n+1) = 0. In [CMN3 ], the authors raised the question of whether the map ssn occurs in * *a fibration sequence (A) 2S2n+1-ssn!S2n-1-! T -! S2n+1. The first construction of such a fibration was accomplished for p 5 by Anick * *[A ] and was the subject of a 270 page book. There has been much interest in finding a simpler c* *onstruction. It is the purpose of this paper to give an elementary construction of the space T and the* * fibration (A) which is valid for all odd primes. The methods are new and have the advantage of bein* *g straightforward and accessible to nonexperts. It is anticipated that they should be of use for * *other problems as well. A comparison of our methods and Anick's will be given once we state our results. The question of the existence of a fibration as in (A) appeared in another co* *ntext at about the same time. In trying to understand the secondary suspension [C , M], the first * *author [G4 , G5] was led to conjecture the existence of (i-1)-connected spaces Tiwhich fit into seco* *ndary EHP sequences T2n-1-E! T2n-H! BWn 0 H0 T2n-E! T2n+1-! BWn+1 ___________ 2000 Mathematics Subject Classification. Primary 55P45, 55P40, 55P35. 1 2 BRAYTON GRAY AND STEPHEN THERIAULT where BWn is the classifying space of the fiber of the double suspension constr* *ucted in [G3 ]. These EHP fibrations should fit together in such a way that the resulting spectrum {T* *i} is equivalent to the Moore spectrum S0 [pre1. The Ti's would then give a refinement of the secon* *dary suspension into 2p stages. The analysis indicated that T2n is homotopy equivalent to S2n+1* *{pr}, the fiber of the map of degree pr on S2n+1, and that T2n-1would sit in the fibration sequenc* *e (A). Our first objective is to construct a secondary Hopf invariant H : S2n+1{pr}* * -! BWn for p 3. This lets us define T as the homotopy fiber of H. It follows easily that T sati* *sfies the fibration in (A) and the second EHP fibration. We also show that the space we construct is homot* *opy equivalent to Anick's when p 5. The EHP viewpoint also predicted that the Ti's should have a rich structure. * *They should be homotopy associative and homotopy commutative H-spaces enjoying a certain unive* *rsal property. Together, these properties would imply that the mod-pr homotopy classes of the * *Ti's could be rep- resented by multiplicative maps. That is, letting Pi(pr) be the mod-pr Moore sp* *ace of dimension i, there should be a one-to-one correspondence [Pi(pr), Tj] $ {homotopy classes of H-maps from Tito}Tj. The properties were easy to establish when i is even [G4 ]. Subsequent to Anick* *'s work, Anick and the first author [AG ] constructed an H-space structure on T by showing that, f* *or each n, there is a (2n - 2)-connected co-H space G with the property that T is a retract of G and* * G is a retract of T. They also proved a semi-universal property for T. The other properties were* * later established by the second author [T2]. Our second objective is to take advantage of our construction of the space T * *to give a new, simpler construction of the space G, and prove all the properties in [AG ] for p 3. C* *ollectively, our results are as follows. Theorem 1.1. Suppose p 3 and r 1. Then the following hold: (a)there is an H-fibration sequence 2S2n+1-ssn!S2n-1-! T -! S2n+1 where the composition 2 2S2n+1-ssn!S2n-1-E! 2S2n+1 is the pr-power map; (b)there is a fibration sequence G -h!T -! R -! G AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 3 where h has a right homotopy inverse g : T -! G so that G ' T x R with R a wedge of mod-ps Moore spaces for s r; (c)the adjoint of g, "g: T -! G, has a right homotopy inverse f : G -! T and there is a homotopy equivale* *nce T ' G _ W where W is a wedge of mod-ps Moore spaces for s r; (d)there are "EHP fibrations" Wn -P!T2n-1-E! T2n-H! BWn 0 E0 H0 Wn+1 P-!T2n- ! T2n+1-! BWn+1 where T2n= S2n+1{pr}, T2n-1= T, and there is an equivalence of spectra {T* *i} ' S0[pre1. Our methods are simpler and more direct than those of Anick. He constructed T* * as a retract of a loop space D, where D is an infinite dimensional CW-complex whose bottom two* * cells are the mod-pr Moore space P2n+1(pr) and whose other cells come from iteratively attach* *ing certain Moore spaces in a delicately prescribed fashion. A great deal of his effort was direc* *ted towards constructing the attaching maps, and this necessitated the introduction of many new techniqu* *es. The restriction to primes strictly larger than 3 was due to a heavy reliance on differential gr* *aded Lie algebras which require that the primes 2 and 3 be inverted in order for the Lie identiti* *es to be satisfied. By contrast, we construct the space T directly for all p 3 without reference * *to the space D and without reference to differential graded Lie algebras. The main ingredient in t* *his new construction is an extension theorem (presented as Theorem 2.3). This allows for a straightf* *orward extension of the map 2S2n+1-! BWn constructed in [G3 ] to an EHP map H : S2n+1{p} -! BWn. The new methods may be useful in positively resolving a long-standing conject* *ure that the fiber Wn of the double suspension is a double loop space at odd primes. Includ* *ing dimension and torsion parameters, the space T2np-1(p) gives a candidate for a double delo* *oping: potentially Wn ' 2T2np-1(p). Such a homotopy equivalence would have deep implications in h* *omotopy theory, one of which being a much better understanding of the differentials in the EHP * *spectral sequence calculating the homotopy groups of spheres. This paper is the result of combining separate efforts by the two authors. Th* *e second author dis- covered the extension theorem and obtained part (a) of Theorem 1.1 without the * *H-space structure, as well as part (d). The first author later found a different application of th* *e extension theorem to 4 BRAYTON GRAY AND STEPHEN THERIAULT obtain a factorization of the map H, as well as a further application of the ex* *tension theorem to obtain parts (b), (c), and the H-space structure. 2.The extension theorem We begin by restating a theorem of the first author [G3 ] which identifies ce* *rtain homotopy pull- backs as homotopy pushouts. A homotopy fibration X -! Q -! A has a trivializati* *on if there is a homotopy equivalence Q ' A x X in which the map Q -! A becomes the projection A* * x X -ss1!A. Theorem 2.1. Suppose X -! F0 -! E0 is a homotopy fibration and there is a map * *A -! E0. Let Q be the homotopy pullback Q ____//_F0 | | | | fflffl|fflffl| A ____//_E0 and let E be the homotopy cofiber of A -! E0. Then the homotopy fibration X -! * *Q -! A has a trivialization if and only if there is a homotopy pullback F0 ____//_F | | | | fflffl|fflffl| E0 ____//_E for some space F. Further, if the trivialization exists, there is a homotopy pu* *shout Q ' A x X ____//_F0 ss2|| || fflffl| fflffl| X ________//_F where ss2 the projection onto the second factor. There is a special case of Theorem 2.1 in the context of principal fibrations* * which is the key tool used to construct T and prove Theorem 1.1. In general, suppose there is a * *homotopy fibration sequence B -@!F -! E -! B. Then there is a canonical homotopy action ` : F x B -! F satisfying homotopy * *commutative diagrams ~ ` B x B ____//_ B F x B ____//_F |@x1| @|| |ss1| || |fflffl` fflffl| fflffl| fflffl| F x B ______//F F _______//E AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 5 where ~ is the loop multiplication and ss1 is the projection. Note that both sq* *uares are homotopy pullbacks. Now suppose there is a homotopy cofibration A -b!E0- ! E. Define spa* *ces Q and F0 by the iterated homotopy pullback diagram Q _____//F0____//F | | | | | | fflffl|fflffl|bfflffl| A _____//E0____//E | | | | | | fflffl|fflffl| fflffl| B ______B ______B. In particular, the map A -! B is null homotopic as it factors through the middl* *e row which consists of two consecutive maps in a homotopy cofibration. So Q ' Ax B. This lets us ap* *ply Theorem 2.1 to see that there is a homotopy pushout A x B ____//_F0 ss2|| || fflffl| fflffl| B ______//_F. What we wish to do is choose a particular trivialization of Q which lets us ide* *ntify the map A x B -! F0. The fact that there is some decomposition Q ' A x B implies that we can choo* *se a lift a : A -! F0 of b. There may be many choices of a lift, but for the moment any choice suffic* *es. The definition of F0 as a homotopy pullback results in a homotopy fibration sequence B -! F0 * *-! E0- ! B. _ This determines a homotopy action ` : F0x B -! F0. Let `be the composite _ ax1 0 ` 0 `: A x B -! F x B -! F . Proposition 2.2. Let F -! E -! B be a homotopy fibration and suppose there is* * a homotopy _ cofibration A -b!E0-! E. Define the space F0 and the map `as above. Then there* * is a homotopy pushout _` A x B ____//_F0 ss2|| || fflffl| fflffl| B ______//_F. Proof.Consider the diagram ax1 ` A x B ____//_F0x B____//_F0 |ss1| |ss1| || fflffl|a fflffl| fflffl| A _________//_F0_____//E0. 6 BRAYTON GRAY AND STEPHEN THERIAULT The right square is a homotopy pullback as it is one of the canonical propertie* *s of the homotopy action `. The left square is a homotopy pullback by the naturality of the proje* *ction. So the outer * * _ rectangle is also a homotopy pullback. Observe that the top row of the rectangl* *e is the definition of ` while the bottom row is the given map b by the definition of a as a lift. Thus * *if Q is the homotopy pullback f Q ____//_F0 g|| || fflffl|fflffl| A ____//_E0 then there is a homotopy equivalence e : A x B -! Q such that g O e ~ ss1 - s* *o the homotopy _ fibration B -! Q -! A has been trivialized - and g O e ~ `. Therefore Theorem * *2.1 implies the existence of the asserted homotopy pushout. We now state Theorem 2.3, which uses Proposition 2.2 to construct an extensio* *n under certain conditions. The conditions involve exponent information, so we first make two d* *efinitions. If A is a co-H space, let p_r: A -! A be the map of degree pr. If Z is an H-space, let pr* * : Z -! Z be the pr-power map. Theorem 2.3. Let B _____//F0___//_E0___//_B || | | || || | | || || fflffl| fflffl||| B _____//F_____//E____//_B be a homotopy fibration diagram and suppose there is a homotopy cofibration A -* *b!E0-! E where A is a suspension. Observe that the map b lifts to F0; suppose there is a choic* *e of lift a : A -! F0 with the property that a ~ t O p_rfor some map t. Suppose there is a map f0: F* *0- ! Z where Z is a homotopy associative H-space whose pr-power map is null homtopic. Then there * *is an extension f0 F0 ____//_Z | || | || fflffl|||f F _____//Z for some map f. Before beginning the proof, we state a Theorem of James [J] and prove two pre* *liminary Lemmas. If X is a space, let E : X -! X be the suspension. Theorem 2.4. Let X be a path-connected space and Z be a homotopy associative H-* *space. Let __ __ f : X -! Z be a map. Then there is a unique H-map f: X -! Z such that fO E ~ * *f. AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 7 __ We say that f is the multiplicative extension of f. To prepare for Lemmas 2.5 and 2.6 we establish some notation. Let X and Y be * *spaces. Let i1 : X -! X x Y and i2 : X -! X x Y be the inclusions, and let ss1 : X x Y -* *! X and ss2 : X x Y -! Y be the projections. It is well known that there is a natural * *homotopy equivalence e : X _ Y _ ( X ^ Y ) -! (X x Y ) such that the restrictions of e to X and Y are i1 and i2, and ss1Oe and s* *s2Oe are homotopic to the pinch maps onto X and Y . There may be many choices of such a homotopy* * equivalence; any fixed choice will do. Let j be the restriction j : X _ ( X ^ Y ) ,! X _ Y _ ( X ^ Y ) -e! (X x Y ). Lemma 2.5. Let Z be a homotopy associative H-space. Suppose there is a map f : * *X x Y -! Z __ whose multiplicative extension f: (X x Y ) -! Z has the property that the com* *posite _f X _ (X ^ Y ) -E! (X _ (X ^ Y )) -j! (X x Y ) -! Z is null homotopic. Then there is a homotopy commutative diagram _f (X x Y )_____//Z || ss2 |||| fflffl|_fY || Y _______//Z __ i2 f where fY is the multiplicative extension of fY : Y -! X x Y -! Y . Proof.Consider the diagram _ e f (X _ Y _ (X ^ Y ))__//_ (X x Y_)__//_Z || q ||ss2 |||| fflffl| fflffl|_fY || Y _______________ Y _______//Z where q is the pinch map. By definition of e, we have ss2 O e ~ q, so the lef* *t square homotopy commutes. The assertion of the Lemma is that the right square homotopy commutes* * as well. As e is a homotopy equivalence, it is equivalent to show that the outer rectangle homot* *opy commutes. Since all maps are multiplicative and Z is homotopy associative, Theorem 2.4 implies * *that it is equivalent __ __ to show that fO e O E ~ fYO q O E, where E : X _ Y _ (X ^ Y ) -! (X _ Y _ * *(X ^ Y )) is the __ suspension. By hypothesis and the naturality of E, the restriction of fO eOE to* * X _(X ^Y ) is null __ q fY homotopic, so fO eOE factors as the composite X _Y _(X ^Y ) -! Y -! Z. On the * *other hand, __ __ as fY is the multiplicative extension of fY, we have fYO E ~ fY. The naturality* * of the suspension __ __ __ * * __ therefore implies that fYO q O E ~ fYO E O q ~ fY O q. Hence fO e O E ~ fY O* * q ~ fYO q O E, as required. 8 BRAYTON GRAY AND STEPHEN THERIAULT @ Lemma 2.6. Let B -! F -! E -! B be a homotopy fibration sequence and let ` : F* * x B -! F be the associated homotopy action. Suppose A is a suspension and there is a map* * a : A -! F such _ that a ~ t O p_rfor some map t. Let `be the composite _ ax1 ` ` : A x B -! F x B -! F. Suppose there is a map f : F -! Z where Z is a homotopy associative H-space wh* *ose pr-power map is null homotopic. Then there is a homotopy commutative diagram _` A x B _____//F |ss2| |f| fflffl|fO@fflffl| B ______//_Z. Proof.First suspend and look at (a x 1). Consider the diagram p_r_( p_r^1) A _ ( A ^ B)___________// A _ ( A ^ B) || | || |t_(t^1) || a_( a^1) fflffl| A _ ( A ^ B)___________// F _ ( F ^ B) |j| |j| fflffl| (ax1) fflffl| (A x B) _______________// (F x B). The top square homotopy commutes by the hypothesis that a ~ t O p_r. The bott* *om square homotopy commutes by the naturality of the map j. Looping this diagram and using the naturality of the suspension, we obtain a * *homotopy commu- tative diagram p_r_(p_r^1) A _ (A ^ B)______________//_A _ (A ^ B) |E| |E| fflffl| (pr_(pr^1)) fflffl| (1) (A _ (A ^ B))__________//_ (A _ (A ^ B)) ||j ||(jO(t_(t^1))) fflffl| (ax1) fflffl| (A x B) ______________//_ (F x B). Let OE = (j O (t _ (t ^ 1))) O E O (p_r_ (p_r^ 1)) be the upper direction arou* *nd (1), and let ' = (a x 1) O j O E be the lower direction around (1). So OE ~ '. Now compose to* * Z as follows. By hypothesis, Z is homotopy associative, so by Theorem 2.4 the identity map on* * Z extends to an H-map r : Z -! Z such that r O E ~ 1. Define fl by the composite fl : (F x B) --`k-1--! F ---f-! Z --r--!Z. AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 9 Observe that fl O OE is an element of N = [A _ (A ^ B), Z] which is divisible by pr. Here, the group structure on N is determined by A _ * *(A ^ B) being a suspension. As Z is homotopy associative, this group structure on N is equiv* *alent to the one determined by the H-structure on Z. By hypothesis, the pr-power map on Z is nul* *l homotopic, and so N has exponent pr. Thus fl O OE is null homotopic, and so fl O ' is null hom* *otopic. Now we set up to apply Lemma 2.5. Consider the diagram ax1 ` f A x B ________//_F x B______//_F______//ZDD DDDDDD |E| |E| |E| |E| DDDDDDD fflffl| (ax1) fflffl| ` fflffl| ffflffl|DDDr (A x B) ____//_ (F x B)___//_ F____//_ Z____//_Z. The three squares homotopy commute by the naturality of E. The right triangle h* *omotopy commutes by the definition of r. So the entire diagram homotopy commutes. Let g = f O ` * *O (a x 1) be the composite along the top row, and let _g= r O f O ` O (a x 1) be the compo* *site along the bottom row. Observe that _gis an H-map as it is the composite of H-maps, an* *d _gO E ~ g. Thus _gis the multiplicative extension of g. Further, by their definitions, _g=* * fl O (s x 1) and ' = (s x 1) O j O E, so the null homotopy for fl O ' implies that the compos* *ite _g A _ (A ^ B) -jOE-! (A x B) --! Z is null homotopic. Therefore, by Lemma 2.5, there is a homotopy commutative squ* *are _g (A x B) ____//_Z || ss2 |||| fflffl|_h || B _______//Z __ where his the multiplicative extension of h : B -i2!A x B -g!Z. Finally, the previous square and the naturality of the suspension give a homo* *topy commutative diagram E _g A x B ____//_ (A x B)___//_Z ss2|| || ss2 |||| fflffl|E fflffl|_h || B _________//_ B_______//Z. __ * * _ Since _gand h are the multiplicative extensions of g and h respectively, we hav* *e gO E ~ g and __ hO E ~ h, and so the homotopy commutativity of the diagram implies that g ~ h O* * ss2. Now we _ * * _ untangle definitions. By their definitions, g = f O ` O (a x 1) and `= ` O (a x* * 1). So g ~ f O `. By their definitions, h = g O i2 and g = f O ` O (a x 1). The definition of ` as a* * homotopy action implies 10 BRAYTON GRAY AND STEPHEN THERIAULT _ that ` O (a x 1) O i2 ~ @. Thus h ~ f O @. Hence f O `~ f O @ O ss2, precisely * *as asserted by the Lemma. Proof of Theorem 2.3: The given diagram of principal fibrations and homotopy co* *fibration let us apply Proposition 2.2 to obtain a homotopy pushout _` A x B ____//_F0 ss2|| || fflffl| fflffl| B ______//_F. Since A is a suspension, the lift A -a!F0 has the property that a ~ t O p_r, a* *nd Z is a homotopy associative H-space whose pr-power map is null homotopic, we can apply Lemma 2.* *6 to the homotopy 0 fibration sequence B -! F0 -! E0 -! B and the given map F0 -f!Z in order to o* *btain a homotopy commutative diagram _` A x B ____//_F0 |ss2| |f0| fflffl| fflffl| B ______//_Z. Therefore there is a pushout map f : F -! Z with the property that the composit* *e F0- ! F -f!Z is homotopic to f0, as required. 3.The construction of the space T The purpose of this section is to construct the spaces T and produce several * *fibration sequences. We begin our discussion with the Moore space Pk(pr) = Sk-1[prek which we will abbreviate as Pk. Let us fix some notation by defining a diagram * *of fibration sequences induced by the lower right hand corner @ ss 2S2n+1________//E_________//_F_____//_ S2n+1 | oe| | | | | | | fflffl| fflffl| fflffl| fflffl| (B) *_________//P2n+1_______P2n+1_______//* | | | | | | | | fflffl| fflffl| fflffl|pr fflffl| S2n+1 _____//S2n+1{pr}___//S2n+1____//S2n+1. The spaces E and F were first introduced in [CMN2 , CMN1 ]. It is easy to see t* *hat 8 > :0 i 6= 2kn AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 11 2n+1-! F is divisible by pr in each nonzero* * degree in integral and the fibration connecting map S cohomology. In their work [CMN2 ], the authors introduced certain maps xi: P2ni* *-1-! F whose adjoints "xi: P2ni-! F induce epimorphisms in integral cohomology. For each i * *> 1, xiis a relative Samelson product, so the composition P2ni-"xi!F -! P2n+1 is an iterated Whitehead product. Since S2n+1{pr} is an H space, these classes * *lift to E, giving diagrams yi P2ni ______//_E |"xi| |oe| fflffl| fflffl| F _____//_P2n+1 for some maps yi. In particular, ssyi- "xi: P2ni-! F composes trivially to P2n* *+1 and so factors through S2n+1. Thus the induced homomorphism in 2ni dimensional cohomology is * *divisible by pr and hence trivial. Consequently, we obtain the following. * y*i Lemma 3.1. The composite H2ni(F) ss-!H2ni(E) -! H2ni(P2ni) is an epimorphism* * for each i > 1. We require one more lemma to apply the results of Section 2. Lemma 3.2. Suppose X is 2-connected and M is either a sphere or a Moore space. * *Let f : M -! X be given. Define A by the cofibration sequence s i j M -p!M -! A -! M. Suppose there is a commutative diagram j A _______//_ 2M x|| |||| fflffl|ae || X [f C M ____//_ 2M for some map x, where ae is the quotient map. Then there is a commutative diagr* *am i M ________// A |x0| |x| fflffl| fflffl| X _____//_X [f C M for some map x0, and f is homotopic to ps. x0. 12 BRAYTON GRAY AND STEPHEN THERIAULT Proof.Consider the standard map from a cofibration sequence to a fibration sequ* *ence defined by the right hand square ps i j M ________// M __________// A________// 2M |x00| x0|| |x| |||| fflffl| fflffl| fflffl|ae || J( M) ____//_J(X, M)___//_X [f C M____//_ 2M. Here J( M) is the James construction and J(X, SM) is the fiber of ae [G2 ]. The* * map x00is the adjoint to the identity for an appropriate choice of x0. Suppose M has dimens* *ion k. Since X is 2-connected, the k + 1 skeleton of J(X, M) is X and x0 factors through X up* * to homotopy. Since x00factors through M as well we have a homotopy commutative square ps M ____//_ M || | 0 || |x || f fflffl| M _____//_X which proves the Lemma. We apply these results as follows. Let F(i)be the 2ni skeleton of F, so F(i)= F(i-1)[flie2ni where fliis the attaching map. Now combining 3.1 and 3.2 with M = S2ni-2, X = F* *(i-1), f = fli, s = r, and x = ssyiwe obtain the following. Corollary 3.3. For each i > 1 we have a homotopy commutative diagram pr S2ni-1_____//S2ni-1 || | || |ffii || fli fflffl| S2ni-1_____//F(i-1) where ffiisatisfies a homotopy commtutative diagram S2ni-1____//_P2ni___//E ffii|| |ss| fflffl| fflffl| F(i-1)______________//_F. We now set up to apply Theorem 2.3. Define the space E(i)as the homotopy pull* *back E(i)____//_E | || | ss| fflffl| fflffl| F(i)____//_F. AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 13 Observe that there is a homotopy pullback diagram 2S2n+1 ____//_E(i-1)__//_F(i-1)__// S2n+1 || | | || || | | || || fflffl| fflffl| || 2S2n+1 _____//_E(i)____//_F(i)___// S2n+1. By the definition of F(i)there is a homotopy cofibration S2ni-1-fli!F(i-1)-! F(* *i). Thus fli lifts to E(i-1). A lift can be chosen which is divisible by pr. Specifically, by Coro* *llary 3.3, fli~ ffiiO pr. Moreover, S2ni-1-ffii!F(i-1)composed to F factors through E -ss!F. Thus there i* *s a pullback map ~yi: S2ni-1-! E(i-1)such that the composite S2ni-1-~yi!E(i-1)-! F(i-1)is homot* *opic to ffii. Hence a = ~yiO pr is a lift of fli. Theorem 2.3 now immediately implies the fol* *lowing. Theorem 3.4. If Z is a homotopy associative H space whose pr-power map is null * *homotopic, then __ for i > 1 any map E(i-1)OE-!Z extends to a map OE: E(i)-! Z. In [G3 ], a classifying space BWn of the fiber of the double suspension was c* *onstructed, along with a fibration sequence 2 S2n-1-E! 2S2n+1-! BWn. Corollary 3.5. There is a map E : E -! BWn such that the composition E 2S2n+1-@! E -! BWn is homotopic to . Proof.Since F(1)= S2n, we have the fibration 2S2n+1-! E(1)-! S2n -! S2n+1. This fibration was analyzed in [G3 ] and it was shown that E(1)' S4n-1x BWn i* *n such a way that the composition 2S2n+1-@! S4n-1x BWn ss2-!BWn is homotopic to . It was also shown that for p 5 BWn is a homotopy associati* *ve H space. The H space structure on BWn was shown to be homotopy associative for p = 3 and tha* *t the pth-power map on BWn is null homotopic in [T5]. Thus for i > 1 we can apply Theorem 3.4 t* *o construct maps i: E(i)-! BWn by induction such that i@i~ . Since E = [E(i), we define E : * *E -! BWn by E | Ei= i. 14 BRAYTON GRAY AND STEPHEN THERIAULT Theorem 3.6. There is a diagram of fibrations S2n-1_____// 2S2n+1___//_BWn |i| |@| |||| fflffl| fflffl| E || R0 ________//_E______//BWn | | | | fflffl| fflffl| F __________F with i null homotopic and so F ' S2n-1x R0. Proof.The space R0 is defined as the fiber of E . Since the fibration 2S2n+1-@! E -! F is induced by a map to S2n+1 which induces an isomorphism in H2n( ), the map * *F -! S2n-1 induces an isomorphism in H2n-1( ) and hence has a right homotopy inverse. It is worth noting at this point that the space R0 is split in [CMN1 ]; ther* *e is a homotopy decomposition Y i R0 ' S2np -1{pr+1} x P(n, r) i 1 where P(n, r) is a complicated wedge of mod-pr Moore spaces. The fact that the * *product on the right is a loop space and is mapped to F by a loop map is not obvious from the* *ir analysis. The structure of R0 is rather simple. Proposition 3.7. We have 8 > :0 otherwise. i+j Furthermore, there is a choice of generators ei2 H2mi(X) such that eiej= pr i * *ei+j. Proof.Apply the Serre spectral sequence to the fibration S2n-1-! R0 -! F in Th* *eorem 3.6. We now construct the space T in Theorem 1.1 and prove the existence of the fi* *brations in parts (a) and (d), leaving the H-structure to the next section. By Diagram (B) there is a* * fibration sequence S2n+1{pr} -o!E -oe!P2n+1- ! S2n+1{pr}. Define H by the composition E H : S2n+1{pr} -o!E -! BWn. Note that H can be regarded as a secondary Hopf invariant. Define T as the homo* *topy fiber of H. Then Theorem 3.6 implies the following. AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 15 Theorem 3.8. There is a diagram of fibrations H T ______//_ S2n+1{pr}___//BWn | | || | |o || fflffl| fflffl| E || R0 _________//_E________//BWn | | | |oe fflffl| fflffl| P2n+1 ________P2n+1. The connecting maps for the vertical fibrations in Theorem 3.8 immediately gi* *ve the following. Corollary 3.9. There is a homotopy commutative diagram P2n+1 _______ P2n+1 | | | | fflffl| fflffl| T _______// S2n+1{pr} where the right map is the loop of the inclusion of the bottom Moore space. Continuing the diagram in (B), we have ae 2S2n+1 ____//_ S2n+1{pr} || | || |o || @ fflffl| 2S2n+1 ________//_E. Observe that Hae ~ E oae ~ E @ ~ . Theorem 3.8 therefore implies the followi* *ng. Theorem 3.10. There is a diagram of fibrations 2S2n+1 _______ 2S2n+1 |ssn| |pr| fflffl|E2 fflffl| S2n-1_______// 2S2n+1_____//BWn | |ae || | | || fflffl| fflffl|H || T _______//_ S2n+1{pr}___//BWn | | | | fflffl| fflffl| S2n+1 ________ S2n+1. In particular, the top square in Theorem 3.10 is Cohen, Moore, and Neisendorf* *er's factorization of the pr-power map on 2S2n+1. Since ssn has degree pr, we have the following * *corollary. 16 BRAYTON GRAY AND STEPHEN THERIAULT Corollary 3.11. There is a homotopy commutative diagram pr S2n-1_______//S2n-188 | ssnrrrrr| | rrr | fflffl|prrr fflffl| 2S2n+1_____// 2S2n+1 for each r 1. 4.The construction of G and the H-space structure on T In this section we construct an H-space structure on T. In fact we do more t* *han that. We construct a corresponding co-H space G in the sense of [G7 ]; i.e., we construc* *t a (2n - 2)-connected space G and maps f : G -! T g : T -! G h : G -! T such that the compositions G -f! T -"g!G T -g! G -h!T are homotopic to the identity, where "gis the adjoint of g. We go on to derive * *several interesting results from this structure. We will write Tm for the m-skeleton of T. We will also reintroduce the torsi* *on parameter for Moore spaces as we will need to consider mod-ps Moore spaces Pm (ps) for s 6= r* *. The space G will be filtered by subcomplexes Gk which will be constructed inductively starting w* *ith G-1 = *. We will construct a map k r+k ffk : P2np (p ) -! Gk-1 and define Gk as the mapping cone of ffk. The induction proceeds through 14 steps for each k, and we collect some infor* *mation outside of the induction first. Proposition 4.1. As an algebra, H*(T; Z=p) is generated by classes u of dimensi* *on 2n - 1 and vi of dimension 2npifor each i 0 subject to the relations vpi= 0 and u2 = 0. For* * each i define ui= uvp-10vp-11. .v.p-1i-1 AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 17 (r+i)u = v . As a vector space "H*(T; Z=p) is generated by classes v(m) * *of dimension 2mn Then fi i i and u(m) of dimension 2mn - 1 for each m 1 where v(m)= vess.v.e.tt= fi(r+s)u(s) u(m) = usvessves+1s+1.v.e.tt P t and m = i=seipi, 0 ei< p, es 6= 0. Proof.We apply the Serre spectral sequence for the cohomology of the fibration S2n-1-! T -! S2n+1 Using Z=p coefficients we see that 2n-1 * 2n+1 H*(T; Z=p) ~=H* S ; Z=p H S ; Z=p as algebras. Using integer coefficients we see that (m) is the reduction of a * *class of order pr+sso v(m) = fi(r+s)u(s) 6= 0. We define vs = fi(r+s)us. Note that dually the homology of T has a very simple description. There is a* * Hopf algebra isomorphism H*(T) ~= (~u) Z=pZ[~v] where ~uand ~vare dual to u and v respectively, and the dual Bocksteins are det* *ermined by fi(r+i)~vpi= ~u~vpi-1for i 0. Anick [A ] introduced the notation Wbafor the class of all spaces that are lo* *cally finite wedges of mod-ps Moore spaces for a s b. Note that any simply connected Moore space i* *s a suspension, so any simply connected space in Wbais a suspension. Recall that the smash of t* *wo Moore spaces is homotopy equivalent to a wedge of Moore spaces: if s t then there is a homoto* *py equivalence Pm (ps) ^ Pn(pt) ' Pm+n (ps) ^ Pm+n-1 (ps). In particular, Wbais closed under smash products. Recall also that any retract * *of a wedge of Moore spaces is homotopy equivalent to a wedge of Moore spaces, so Wbais closed under* * retracts. Lemma 4.2. Suppose W 2 Wbais simply connected and f : Pk(pt) -! W is divisible * *by pb. (a) Write W = W1 _ W2 with W1 2 Wb-1aand W2 2 Wbb. Then f factors through W2* * up to homotopy. (b) Suppose in addition that W2 is (d - 1) connected and k < pd. Then f ~ *. Proof.Since W is a wedge, there is a homotopy equivalence W = W2x (W1o W2) * *(see, for * * __ example, [G1 ]). Since W1, W2 2 Wba, both spaces are suspensions, and we can wr* *ite W1 = W 1 and 18 BRAYTON GRAY AND STEPHEN THERIAULT __ W2 = W 2. Since W1 is a suspension, we haveWW1o W2 ' W1_ (W1^ W2). For the r* *ight wedge summand, the James splitting of X as X(i)gives __ __ __ i` __(i)j W1^ W2 ' W 1^ W 2' W 1^ W 2 . Combining, we have i i` j j W1o W2 ' W1_ W1^ W(i)2 . In particular, since Wbais closed under smash products, we have W1 o W2 2 Wba.* * Applying the Q Hilton-Milnor theorem therefore implies that (W1o W2) ' i Pni(psi) with a * *s b - 1. By [N3], the pr+1-power map on 2Pm (pr) is null homotopic for any r 1 and * *m 3. Thus Pm (pr) admits no nontrivial maps which are divisible by pr+1. In our case, th* *is implies that Q ni si b i P (p ) admits no nontrivial maps which are divisible by p . Thus the adjo* *int of f, which is divisible by pb, is trivial on (W1 o W2) and so factors through the inclus* *ion W2 -! W. Hence, adjointing, f factors through the inclusion W2 -! W, proving part (a). For part (b), since W2 2 Wbband W2 is (d - 1)-connected, the Hilton-Milnor th* *eorem implies Q that W2 = Pni(pb) where ni > d. By [CMN1 , N3], P2m+1(pr) admits no nontri* *vial maps which are divisible by pr from a CW-complex of dimension t < 2mp, and P2m (pb))* * admits no nontrivial maps which are divisible by pr from a CW-complex of dimension t < 2(* *2m - 1)p. In our case, the CW-complex is Pk(pt), the domain of f, and the target Moore space* *s are the Pni(pb) in the decomposition of W2. Since ni> d for each i, the hypothesis k < pd guar* *antees that the component of f on Pni(pb), being divisible by pb, is null homotopic. Hence f is* * null homotopic. Theorem 4.3. For each k 0 there are spaces Gk and Wk 2 Wr+k-1rsatisfying the * *following conditions: (a) T2npk-2' Gk-1_ Wk; (b)there are maps gk : T2npk-2-! Gk-1 and hk-1 : Gk-1 -! T such that hk-* *1gk is homotopic to the inclusion of T2npk-2into T; (c)there is a homotopy commutative diagram of cofibration sequences which de* *fines Gk mk k k P2npk(pr+k)____//_ T2np -2__//_ T2np || | | 0 || |"gk |gk || ffk fflffl| fflffl| P2npk(pr+k)______//_Gk-1______//Gk where "gkis the adjoint of gk; (d)there is a map e : P2npk(pr+k-1) _ P2npk+1(pr+k-1) -! T2npkwhich induces* * an epimor- phism in mod-p cohomology; (e)the map mk : P2npk(pr+k) -! T2npk-2is divisible by pr+k-1; (f)there is a map 'k : Gk -! S2n+1{pr} extending 'k-1; AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 19 r+k; (g) Gk 2 Wr (h)there is a homotopy commutative diagram of fibration sequences hk Gk ________//_T_________//Rk_____//_Gk | | || | | || fflffl| fflffl| || S2n+1{pr}______//Ek_____//_Gk___//_S2n+1{pr} |H| ||k fflffl| fflffl| BWn ________BWn; (i) 2 Gk-12 Wr+k-1r; (j)the equivalence in (a) extends to an equivalence T2npk' Gk _ Wk; (k) 2T2npk2 Wr+kr; (l)Gk ^ T2npk2 Wr+kr; (m) T2npk^ T2npk2 Wr+kr; (n)there is a map ~k : T2npkx T -! T which is the inclusion on the first axi* *s and the identity on the second. Furthermore there is a homotopy commutative square ~k T2npkx T __________//T | | | | fflffl| fflffl| S2n+1x S2n+1 _____// S2n+1. Proof.With G-1 = * and G0 = P2n+1 these statements are all immediate for k = 0 * *with '0 : P2n+1 -! S2n+1{pr} the inclusion, E0 = E from Theorem 3.6, 0 = E , ~0 : * *P2n x T -! T obtained from the action of P2n+1on T defined by the fibration in Theorem 3.6.* * We now supposed that (a)-(n) are all valid with k - 1 in the place of k and we proceed to prove* * them for k. Proof of (a).We will construct a map k-2 fm : P2mn+1(pr+s) -! T2np which induces a monomorphism in mod-p homology for each m satisfying pk-1 < m <* * pk, where s = p(m). We then assemble these into a map 0 k 1 k-1 p`-1 2mn+1 r+s 2npk-2 T2np _ @ P (p )A-! T m=pk-1+1 which induces an isomorphism in mod-p homology. By applying (j) in the case k -* * 1 we are done. 20 BRAYTON GRAY AND STEPHEN THERIAULT To construct the maps fm we appeal to (n) in the case k -1 and iterate this t* *o produce a diagram with p factors T2npk-1x . .x.T2npk-1______//_T2npk | | | | fflffl| fflffl| J(S2n)pk-1x . .x.J(S2n)pk-1__//_J(S2n)pk where J(S2n)j is the 2nj skeleton of S2n+1. Since pk-1< m < pk,we can write m * *= asps+ . .+. ak-1pk-1 with as > 0 and ak-1> 0. Write l = asps+ . .+.ak-2pk-2 so that m = l +* * ak-1pk-1 and further restrict the above diagram to one with ak-1+ 1 factors _~ T2nlx T2npk-1x . .x.T2npk-1_______//_T2nm | | | | fflffl| fflffl| J(S2n)lx J(S2n)pk-1x . .x.J(S2n)pk-1_//_J(S2n)m . By applying the maps in this diagram to a generator of H2mn(J(S2n)m ; Z=p) we s* *ee that (__~)*(v(m))= v(l) vk-1 . . .vk-1. Now v(m) = fi(r+s)u(m) and v(l) = fi(r+s)u(l), so v(l) vk-1 . . .vk-1= fi(r+s)(u(l) vk-1 . . .vk-1). Applying (k) and (l) in case k - 1 we see that i k-1 k-1j T2nlx T2np x . .x.T2np 2 Wr+k-1r. Now given any space W 2 Wr+k-1rand any class , 2 Hi(W; Z=p) with fi(j), 6= 0, t* *here is a map f, : Pi+1(pj) -! W with f*,an epimorphism. Thus for each m satisfying pk-1 < m < pk we may choose * *such a map corresponding to , = u(l) vk-1 . . .vk. The composition i k-1 k-1j _~ Pm (pr+s) f,-! T2nlx T2np x . .x.T2np -! T2mn therefore gives the desired map fm . Proof of (b).From part (a) we obtain a map T2npk-2-! Gk-1 which induces an is* *omorphism in ss2n-1( ). The composition k-2 hk-1 T2np -! Gk-1 -! T factors through T2npk-2and provides a self map of T2npk-2which induces an isomo* *rphism on ss2n-1( ). Calculations with cup products and Bocksteins show that this map is a homotopy * *equivalence, so composing with the inverse provides a possibly different map k-2 gk : T2np -! Gk-1 AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 21 such that hk-1gk is homotopic to the inclusion. Proof of (c).Using the map gk from (b) we construct a commutative diagram where* * the bottom row is the fibration sequence from (h) in case k - 1 and the middle row is a co* *fibration sequence T2npk=T2npk-2_____P2npk(pr+k) | | | | |fflffl fflffl| k-2_____//_ k T2npk-2_____//T_____//_T=T2np T2np -2 || |gk| |||| || |"gk| fflffl|hk-||1 |fflffl fflffl| Gk-1 _____//T________//_Rk-1_________//Gk-1. Define ffk : P2npk(pr+k) -! Gk-1as the vertical composition on the right. Defin* *e g0kby the diagram of cofibration sequences mk k k P2npk(pr+k)____//_ T2np -2__//_ T2np || | | 0 || |"gk |gk || ffk fflffl| fflffl| P2npk(pr+k)______//_Gk-1______//Gk. Proof of (d).As in part (a), we consider the diagram: "~ k T2npk-1x . .x.T2npk-1______//_T2np | | | | fflffl| fflffl| J(S2n)pk-1x . .x.J(S2n)pk-1__//_J(S2n)pk with p factors on the left. This is defined by iterated application of part (n)* * in case k - 1. Clearly "~*(vk) = vk-1 . . .vk-1= fi(r+k-1)(uvp-11. .v.p-1k-2 vk-1 . . .vk-* *1). As before there is a map k+1 r+k-1 i 2npk-1 2npk-1j q : P2np (p ) -! T x . .x.T such that ( ("~)q)* is an epimorphism in Z=p cohomology obtained by applying (k* *) and (m) in case k - 1. Similarly, vk = fi(r+k)uk and X ("~)*uk = vk-1 . . .vk-1 uk-1 vk-1. . .vk-1. p terms In particular, the map k-1-1 2npk-1 2npk-1 "~02npk T2np x T x . .x.T -! T 22 BRAYTON GRAY AND STEPHEN THERIAULT has the property that ("~0)*(uk)= uk-1 vk-1 . . .vk-1 = fi(r+k-1)(uk-1 uk-1 vk . . .vk). It follows, as before, that there is a map k r+k-1 i 2npk-1-1 2npk-1 2npk-1j r : P2np (p ) -! T x T x . .x.T such that ( ("~0)r)* is an epimorphism in Z=p cohomology. We construct e as the* * wedge sum k r+k-1 2npk+1 r+k-1 2npk e = ( "~0)r _ ( "~)q : P2np (p ) _ P (p ) -! T . Proof of (e).We apply Lemma 3.2 with x = e, s = r + k - 1, X = T2npk-2, M = P2* *npk-1(pr+k), and f = mk : M -! X. In this case A = P2npk-1(pr+k-1) _ P2npk(pr+k-1), which i* *s the cofiber of pr+k-1on M = P2npk-1 pr+k. It follows that mk is divisible by pr+k-1. Proof of (f).To show that there is an extension of 'k-1 to 'k, ffk k P2npk(pr+k)______//Gk-1______//Gk = Gk-1[ffkCP2np (pr+k) kkk 'k-1|| k'kkkkkkk fflffl|uukkkk S2n+1{pr} it suffices to show that ffk is divisible by pr. This holds by (e) since r + k * *- 1 r. k-1* *W i Proof of (g).By part (g) for k - 1, there is a homotopy equivalence SGk-1 ' * *P2np +2pr+i . k i=0 Also, by definition, SGk = SGk-1[SffkCP2np pr+k. By part (e) ffk = "ffO pr+k-1* *' so Sffk = "ffO pr+k-1' ~ pr+k-1'O "ff. However k-1` i k-1` i pr+k-1': P2np +2pr+i -! P2np +2pr+i i=0 i=0 is null homotopic since the order of the identity map on a mod-pr Moore space i* *s pr. AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 23 2n+1{pr} in part (f), we have a* * pullback diagram Proof of (h).As we have constructed 'k : Gk -! S of principal fibrations S2n+1{pr}______ S2n+1{pr} | | | | fflffl| fflffl| Ek-1 __________//_Ek | | | | fflffl| fflffl| Gk-1 __________//_Gk |'k-1| |'k| fflffl| fflffl| S2n+1{pr}_______S2n+1{pr}. We wish to apply Theorem 2.3 to extend k-1: Ek-1- ! BWn to Ek. It suffices to * *show that there is a lifting ffi of ffk, Ek-199 ss ffisss | ssss | k s ffk fflffl| P2np (pr+k)___//_Gk-1 which is divisible by p. Since ffk ~ pr+k-1"ff, it follows that pr"fflifts to a* * map ffi0: P2npk(pr+k) -! Ek-1 with pk-1ffi0 = ffi a lifting of ffk. Thus as long as k > 1 we can constr* *uct ffi with the requisite property. When k = 1, we appeal to [CMN1 ] where it is shown that ff* *1 = pffi1 with ffi1 : P2np(pr+1) -! P2n+1(pr) lifting to E0. Proof of (i).By part (j) in case k -1, 2 Gk-1is a retract of 2 T2npk-1. The * *latter space splits since the loop space can be approximated by the James construction [J], giving 0 1 k-1 2 ` 2npk-1(i) 2 T2np ' @ (T ) A i 1 which is in Wr+k-1rby (k) and (l) in case k - 1. Since Wr+k-1ris closed under * *retracts we are done. Proof of (j).By part (a), we have T2npk-2' Gk-1_ Wk and by (e), we have k i 2npk-2j 2npk r+k T2np ' T [mk CP (p ) with mk divisible by pr+k-1. It suffices to show that the map k r+k 2npk-2 mk : P2np (p ) -! T ' Gk-1_ Wk factors though Gk-1. To this end, observe that there is a homotopy decomposition (Gk-1_ Wk)' Gk-1x (Wk o Gk-1). 24 BRAYTON GRAY AND STEPHEN THERIAULT 2npk(pr+k) -! W o G which is divisible by pr+k* *-1 is null We will show that any map P k k-1 homotopic. Since Wk is (4n - 1)-connected, the Moore spaces in Wk are double s* *uspensions, so Wk o Gk-12 Wr+k-1r. In fact, Wk o Gk-1' W1_ W2 with W1 2 Wr+k-2rand W2 a retr* *act of p-1` k-1 P2np +1(pr+k-1) o Gk-1 r=2 which is 4npk-1- 1 connected. The result follows from Lemma 4.2. Proof of (k).This follows immediately from (g) and (j). Proof of (l).This follows from 3 steps based on an analysis which first appeare* *d in [T1]. Step 1: Gk ^ T2npk-12 Wr+k-1r. Consider the cofibration sequence k r+k 2npk-1 ffk^1 2npk-1 2npk-1 P2np (p ) ^ T ----!Gk-1^ T ----!Gk ^ T . We have P2npk(pr+k)^T2npk-12 Wr+k-1rand ffk^1 is divisible by pr+k-1. Consequen* *tly, ffk^1 ~ * and so there is a homotopy decomposition k-1 2npk-1 2npk+1 r+k 2npk-1 Gk ^ T2np ' (Gk-1^ T ) _ (P (p ) ^ T ) which is in Wr+k-1rby (k) in case k - 1. Step 2: Gk-1^ T2npk2 Wr+k-1r. By (j) in case k - 1, Gk-1^ T2npkis a retract of T2npk-1^ T2npk. But k-1 2npk 2npk-1 2npk 2npk-1 T2np ^ T ' T ^ T ' T ^ (Gk _ Wk) by (j). By Step 1 and (k) in case k - 1, the latter space is in Wr+k-1r. Since * *Wr+k-1ris closed under retracts, we therefore have Gk-1^ T2npk2 Wr+k-1r. Step 3: Gk ^ T2npk2 Wr+kr. Consider here the cofibration sequence k r+k 2npk ffk^1 2npk 2npk P2np (p ) ^ T ----! Gk-1^ T ----! Gk ^ T . The first space is in Wr+krby (k) and the second is in Wr+k-1rby Step 2. In fac* *t, Gk-1^ T2npk' (P2npk-1+1(pr+k-1) ^ T2npk) _ W0 with W0 2 Wr+k-2r. Here, the projection onto t* *he first factor is aek-1^ 1, where aek-1 is obtained by collapsing Gk-2 to a point. Applying Le* *mma 4.2(b), we see that if ffk ^ 1 is nontrivial, so is the composition k r+k 2npk ffk^1 2npkaek-1^12npk-1 r+k-1 2npk P2np (p ) ^ T ----! Gk-1^ T ----! P (p ) ^ T . We will show that this composition is null homotopic. Let ffi = aek-1ffk, which* * is divisible by pr+k-1 because ffk is. According to [N1], the pr+k-1-power map on S2npk-1+1{pr+k-1} is* * null homotopic. AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 25 Therefore the composition k r+k ffi2npk-1+1 r+k-1 2npk-1+1 r+k-1 P2np (p ) -! P (p ) -! S {p } is null homotopic. It follows that the composition k r+k ffi2npk-1+1 r+k-1 ae 2npk-1+1 P2np (p ) -! P (p ) -! S is null homotopic. Since the map k-1+1 r+k 2npk ae^1 2npk-1+1 2npk P2np (p ) ^ T ----! S ^ T has a left homotopy inverse, the map k r+k 2npk ffi^1 2npk-1+1 r+k 2npk P2np (p ) ^ T ----! P (p ) ^ T is null homotopic. Since ffk ^ 1 is the composition k r+k 2npk ff^1 2npk ffi^1 2npk-1+1 r+k 2npk P2np (p ) ^ T ----! Gk-1^ T ----! P (p ) ^ T , it is null homotopic as well. Consequently, there is a homotopy decomposition k 2npk 2npk+1 r+k 2npk Gk ^ T2np ' (Gk-1^ T ) _ (P (p ) ^ T ). Both terms on the right are in Wr+krby (k) and Step 2. Proof of (m).By (j), T2npk^ T2npk' (Gk_ Wk) ^ T2npk. By (l), Gk^ T2npk2 Wr+kr,* * and as Wk is a wedge of Moore spaces which are at least (4n-1)-connected, it is a double * *suspension, so by (k) we have Wk ^ T2npk2 Wr+kr. Thus T2npk^ T2npk2 Wr+kr. Proof of (n).Since the composite Rk -! Ek -! Gk -! S2n+1{pr} -! S2n+1 is null h* *omotopic by (h), there is a commutative diagram of principal fibrations: Gk ____//_ S2n+1 hk|| |||| fflffl| || T _____//_ S2n+1 | | | | fflffl| fflffl| Rk _____//PS2n+1 | | | | fflffl| fflffl| Gk ______//S2n+1 where PS2n+1is the path space on S2n+1. Consequently the actions are compatible Gk x T ____//_ S2n+1x S2n+1 | | | | |fflffl fflffl| T ___________// S2n+1. 26 BRAYTON GRAY AND STEPHEN THERIAULT 2npk-! G such that the composition Using (j) we construct a map gk : T k k gk hk T2np -! Gk -! T is homotopic to the inclusion as in (b). This gives a homotopy commutative diag* *ram gkx1 T2npkx T ____//_ Gk x T |~k| |a| fflffl| fflffl| T ____________T. Combining the preceeding two diagrams gives the result and completes the induct* *ion. S S S We now consider the limiting case. Write G = Gk, R = Rk and E1 = Ek. Theorem 4.4. There is a diagram of fibration sequences h i G _________//T_________//_R_____//G |E| || |||| fflffl| fflffl| || ' S2n+1{pr} _____//E1_____//G____//_S2n+1{pr} |H| || fflffl| fflffl| BWn ________BWn and there are maps "g: T -! G and f : G -! T such that the composites G -f! T -g!G T -"g! G -h!T are homotopic to the identity maps. Proof.The diagram is the direct limit of the diagrams in Theorem 4.3 (h) with h* * = lim!hk, g = lim!gk and f = lim!fk, where fk : Gk -! T2npkis a right inverse for gk given by Theor* *em 4.3 (j). Theorem 4.5. The following space belong to W1r: 2 G, G, G ^ T, T ^ T, and W * *where T ' G _ W. Proof.This follows immediately from the results in Theorem 4.3 by taking limits. The retraction of T off G in Theorem 4.4 induces an H-structure on T by the * *composite m : T x T "gx"g-! G x G -! G -h!T. The following proposition establishes the H-fibration property in Theorem 1.1 (* *a) as a consequence of a slightly stronger result. Proposition 4.6. The map T -E! S2n+1{pr} is an H map with respect to the H-spac* *e structure m on T. Consequently, there is an H-fibration sequence S2n-1-! T -! S2n+1. AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 27 Proof.Filling in the fibration diagram in Theorem 4.4 on the right, we obtain a* * homotopy commu- tative square h G ________//_T || | || |E || ' fflffl| G ____//_ S2n+1{pr}. Now consider the following diagram "gx"g h T x T ___________//_RR G x__G____________// G________//Tt RRRR | | E tttt RRRR | 'x ' | ' tt ExE RRR))R fflffl| fflffl|yyttt S2n+1{pr} x S2n+1{pr}___//_ S2n+1{pr}. The middle square commutes as ' is an H-map and we have just seen that the rig* *ht triangle commutes. The left triangle commutes since ' ~ Eh, so '"g~ E. As the top row is* * the definition of the multiplication m on T, the commutativity of the diagram implies that E i* *s an H-map. Consequently, the composition T -E! S2n+1{pr} -! S2n+1is an H-map as it is a* * composite of H-maps, and so the homotopy fibration S2n-1-! T -! S2n+1is of H-spaces and* * H-maps. The next proposition and the following corollary give structural properties o* *f the spaces T, G, and R. Proposition 4.7. The spaces T and G are atomic. Proof.It is easy to see that T is atomic using the product structure and the Bo* *ckstein relations. The case of G is more difficult. We first show that if G is not atomic then the map* * P2npk(pr+k) ffk-!Gk-1 is null homotopic for some k. Suppose fl : G -! G is a map with the property th* *at fl|Gk-1: Gk-1- ! Gk-1 is a homotopy equivalence and ffk has order p. Consider the diagram ffk P2npk(pr+k)____//_Gk-1___//_Gk | |d| |fl| |fl| fflffl|ffk fflffl| fflffl| P2npk(pr+k)____//_Gk-1___//_Gk. Since fl|Gk-1: Gk-1- ! Gk-1 is an equivalence, flffk has the same order as ffk.* * Consequently d 6 0 (mod p) and hence d is an equivalence. It follows that fl|Gk : Gk -! Gk is an e* *quivalence. Suppose now that ffk ~ *. Then we can construct a map k+1 r+k s : P2np (p ) -! Gk which induces an isomorphism in H2npk+1( ). We now show that the composite k h*k 2npk ( s)*2npk 2npk+1 r+k H2np (T) -! H ( Gk) -! H ( P (p )) 28 BRAYTON GRAY AND STEPHEN THERIAULT * is a direct summand s* *ince g : is an isomorphism. First we note that the image of (hk) * * k+1 T2npk+1-2-! Gk induces a left inverse in cohomology. However, there is an isom* *orphism k 2npk 2npk 2npk+1 r+k H2np ( Gk) ~=H ( Gk-1) H ( P (p )) since the inclusion Gk-1_ P2npk+1(pr+k) -! Gk-1x P2npk+1(pr+k) is an equivalenc* *e in this range. However, H2npk( Gk-1) contains no elements of order pr+k since 2 Gk-1 2 Wr+k-1* *rby Theo- rem 4.3 (i). We conclude that the composition is an isomorphism. Now apply cell* *ular approximation to obtain a homotopy commutative diagram s0 k P2npk(pr+k)______________//_T2np | E|| | fflffl| s fflffl|| P2npk+1(pr+k)____//_ Gk_____//T where E is the suspension and s0is a skeletal factorization. It follows that s0* *induces an isomorphism in H2npk( ) = Z=pr+k. From this we see that k r+k (s0)* H2npk-1(P2np (p ) : Z=p) -! H2npk-1(T; Z=p) induces an isomorphism as well because of the Bockstein structure. However H2n* *pk-1(T; Z=p) is generated by uk which is decomposable if k > 0. This is a contradiction which i* *mplies that ffk is essential and G is atomic. The fact that G and T are atomic and the maps f, g, * *h exist implies that (G, T) is a corresponding pair in the sense of [G7 ]. Corollary 4.8. R 2 W1r. Proof.According to [G7 , Theorem 3.2], R is a retract of T ^ T 2 W1r. The next proposition implies that the space T constructed in this paper is ho* *motopy equivalent to the space Anick constructed in [A ] when p 5 (the primes for which Anick's* * construction holds). Proposition 4.9. Suppose X is an H space and there is a fibration sequence: 2S2n+1-'! S2n-1-!i X such that the composite 2 2S2n+1-'! S2n-1-E! 2S2n+1 is homotopic to the pr power map. Then X ' T. AN ELEMENTARY CONSTRUCTION OF ANICK'S FIBRATION * * 29 Proof.Consider the diagram of fibrations Wn ______//_ X______//_ 2S2n+1{pr} | | | | | | fflffl| fflffl| fflffl| PWn ____//_ 2S2n+1_______ 2S2n+1 | | | r | |' |p fflffl| fflffl|E2 fflffl| Wn ______//_S2n-1______//_ 2S2n+1 | | fflffl| X. Since p . ss*(Wn) = 0 and pr . ss* S2n+1{pr} = 0 we conclude that pr+1. ss*(X) * *= 0. Since ss2np-1(Wn) = 0, we also see that pr . ss2np-1(X) = 0. According to [AG , Coro* *llary 4.2] this is sufficient to construct a map ' : G -! X which induces an isomorphism in ss2n. The construction given in [AG ] depends o* *nly on the co-H space structure on G and the fact that ffk is divisible by pr+k-1, so the proof* * works in this context as well. From this we construct the composition T -g! G -'! X -! X. It is an easy calculation with the Serre spectral sequence that H*(X; Z=p) ~=H** *(T; Z=p), so this map is a homotopy equivalence. References [A]David Anick, Differential algebras in topology, Research Notes in Mathematic* *s, 3, A K Peters, Ltd., Wellesley, MA, 1993, xxvi+274 pp. [AG]David Anick and Brayton Gray, Small H spaces related to Moore spaces, Topol* *ogy 34 (1995), no. 4, 859-881. [C]Frederick R. Cohen, The unstable decomposition of 2 2X and its applications* *, Math. Z. 182 (1983), no. 4, 553-568. [CMN1]F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy gro* *ups, Ann. of Math. (2) 109 (1979), no. 1, 121-168. [CMN2]_____, The double suspension and exponents of the homotopy groups of sphe* *res, Ann. of Math. 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[N5]____, A survey of Anick-Gray-Theriault constructions and applications to th* *e exponent theory of spheres and Moore spaces, Une d'egustation topologique [Topological morsels]: homotopy t* *heory in the Swiss Alps (Arolla, 1999), Contemp. Math., vol. 265, Amer. Math. Soc., Providence, RI, 2000, pp.* * 159-174. [S]Manfred Stelzer, On certain co-H spaces related to Moore spaces, Trans. Amer* *. Math. Soc. 354 (2002), no. 8, 3085-3093. [T1]S. D. Theriault, PhD Thesis, University of Toronto, 1997. [T2]____, Properties of Anick's spaces, Trans. Amer. Math. Soc. 353 (2001), no.* * 3, 1009-1037. [T3]____, Anick's spaces and the double loops of odd primary Moore spaces, Tran* *s. Amer. Math. Soc. 353 (2001), no. 4, 1551-1566. [T4]____, Proofs of two conjectures of Gray involving the double suspension, Pr* *oc. Amer. Math. Soc. 131 (2003), no. 9, 2953-2962. [T5]____, The 3-primary classifying space of the fiber of the double suspension* *, accepted by Proc. Amer. Math. Soc. Department of Mathematics, Statistics and Computer Science, University of Ill* *inois at Chicago, 851 S. Morgan Street, Chicago, IL, 60607-7045, USA E-mail address: brayton@uic.edu Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3U* *E, United Kingdom E-mail address: s.theriault@maths.abdn.ac.uk