On Decompositions in Homotopy Theory 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 Abstract We first describe Krull-Schmidt theorems decomposing H spaces and simpl* *y con- nected co-H spaces into atomic factors in the category of pointed nilpoten* *t p-complete spaces of finite type. We use this to construct a 1-1 correspondence betwe* *en homotopy types of atomic H spaces and homotopy types of atomic co-H spaces, and con* *struct a split fibration which connects them and illuminates the decomposition. Var* *ious prop- erties of these constructions are analyzed. The Krull-Schmidt property first arose in the theory of R-modules, and when * *valid, it states that each object decomposes in a unique way into a sum of indecomposable* * objects of the same type. Numerous examples of decomposing the loop space on a co-H space * *can be found in the literature ([Hi], [M ], [CM ], [AG ], [G1 ], [G2 ]). Typically wh* *at happens is that the loop space of an atomic co-H space is a product of various factors, and the lea* *st connected factor is an H space of special interest, while the other factors, in some sens* *e, represent noise. The first general Krull-Schmidt type theorem in homotopy theory was prov* *ed for the p-localization of simply connected finite complexes which are either H spaces o* *r co-H spaces by Wilkerson [W ]. Various stable versions appear in [F ], [Ma ] and [H ]. We will eliminate the finite complex assumption at the expense of retreating* * to the cate- gory C^pof pointed connected nilpotent p-complete spaces with Hi(X; Z=p) finite* *ly generated for each i. Accordingly, we restrict ourselves to this category in the sequel. * *All colimit con- structions will be completed without further notice. In particular, co-H spaces* * will be defined in terms of the coproduct (which is the completion of the one point union), sus* *pensions will be completed and loop spaces will only be considered when the underlying space * *is simply connected. In section 1 we will exploit the strengthened notion of atomicity in this ca* *tegory developed by Adams and Kuhn [AK ], and prove the following Krull-Scmidt theorem. 1 Theorem A Each H space in C^pis homotopy equivalent to the weak direct product* * of atomic H spaces unique up to order. Each simply connected co-H space in C^pis h* *omotopy equivalent to the coproduct of atomic complete co-H spaces unique up to order. In section 2 we give a general correspondence between retracts of n-fold sus* *pensions and retracts of n-fold loop spaces (2.2). In particular, for n = 1 we get Theorem B There is a 1-1 correspondence between homotopy types of atomic H spa* *ces T in C^pand homotopy types of 1-connected atomic co-H spaces G in C^p.1 We call such a pair (G, T ) a corresponding pair. In this corrrespondence, T* * is a retract of G and G is a retract of ST : f g0 0 G -! ST -! G g f ~ 1 g h T -! G -! T hg ~ 1. In fact (2.5) it is possible to choose g and g0 so that they are adjoint. We w* *ill call the maps (f, g, g0, h) structure maps for the corresponding pair (G, T ). A choice* * of structure maps determines an H space structure on T and a co-H space structure on G via * *the compositions: gxg h m : T x T -! G x G -! G -! T f g0_g0 n : G -! ST -! ST _ ST -! G _ G. A surprising amount of the structure of known examples is found in the gener* *al theory. In particular, given structure maps (f, g0, h) there is a fibration: T ---i! R --i-! G with i null homotopic. This leads to 3.2 and 3.12 which we summarize as Theorem C Suppose T is an atomic H space and G is the corresponding atomic co-H space. Then there is a homotopy equivalence G ' T x R whereWR is a retract of the completed join T *T and hence a complete co-H space* *. Furthermore R = Rffwith each Rffatomic and a) if G = SX, Rffis retract of SX(i)for some i 2 b) if G is homotopy co-associative, Sj-1Rffis a retract of G(j)for some j 2 ________________________________ 1Theorems A and B together imply that homotopy types of H spaces in C^pare i* *n 1-1 correspondence with homotopy types of simply connected co-H spaces in C^p. However we will not cons* *ider this correspondence. 2 W c)if G = SX and G ^ G ' ff2ASnffG, R ' G ^ W where W is a wedge of spheres* * and t(PT - 1) PW = PT - _________+ 1 PG - 1 where PX is the Poincar'e series for X. This result should be compared with the results of Selick and Wu [SW ]. The* *y have a similar decomposition in case G as the double suspension of a p tosion space: S2X = A x B where X is contained in A. A however is much larger than T . In fact, T is a re* *tract of A. In section 3 we construct the fibration which controls the splitting. Sectio* *n 4 deals with the possibility of dualizing the material in section 3 and gives a counter-exam* *ple to a con- jecture of Ganea. Section 5 discusses naturality and section 6 constructs a ref* *inement in the case that T has a homotopy associative multiplication. In section 7 we discuss * *criteria which are needed for the spece T to have a homotopy associative H space structure. Fi* *nally, in an appendix, we collect some results of a general nature regarding fibrations over* * suspensions. x1. Homology and cohomology groups will be with Z=p coefficients unless oth* *erwise noted. We will work in the category C^pof pointed connected nilpotent p-complet* *e spaces of the homotopy type of a CW complex X with Hi(X) finitely generated for each i. W* *e need to arrange our definitions so that p-completion preserves the usual operations * *of homotopy theory. All spaces we consider will either be simply connected or admit H space* * structures, so they will be Z=p good in the sense of [BK ]. All fibrations we consider will* * have a simply connected base, so p-completion preserves fibrations. The p-completion of a co-* *H space is often not a co-H space. However, the category of p-complete spaces does have a * *co-product - the p-completion of the one point union: (X _ Y )^p and we will write X _ Y for this co-product. With this in mind, we define a co-* *H space in C^pto be a space G together with a map : G -! G _ G such that the composition with the natural map G _ G -! G x G is homotopic to t* *he diagonal. Thus the p-completion of a co-H space is a co-H space in this sense.* * If X is simply connected, we will consider X and ( X)^p= (X^p). The functor has a * *left adjoint given by the completion of the suspension, and we write SX for the spac* *e (SX)^p. Then SX is a complete co-H space. Similarly we write X ^Y and X *Y for the p co* *mpletions of the smash and join of two spaces. Then all of these operations commute with * *p-completion. Adams and Kuhn discuss self maps of such spaces [AK ]. They show a close rel* *ationship between irreducibility of such spaces (having no nontrivial retracts) and atomi* *city. We summarize the results of [AK ] that we will need. 3 Definition 1.1 A based map f : X -! X is called topologically nilpotent if the * *sequence {fn} conveys to the constant map in the profinite topology on [X, X]. In particular, if ßk(X) is finite for each k, this is equivalent to the stat* *ement that for all k there is an n such that fn is null homotopic when restricted to the k-skeleto* *n of X. On f* k the other hand, if for some k > 0 f* induces an isomorphism Hk(X) -! H (X) 6= * *0 or f* induces an isomorphism f* : ßk(X) -! ß*(X), then f is not topologically nilpote* *nt. Definition 1.2 A space is atomic iff every self map is either an equivalence or* * is topologi- cally nilpotent. This seems stronger than the usual definition [CM ]. That it isn't (in the * *case of p-complete spaces of finite type) follows from: Theorem 1.3 (Adams and Kuhn) If X is not atomic, there is a nontrivial ide* *mpotent e : X -! X; i.e., e2 = e, e 6= *, e not an equivalence. As an immediate corollary we have a generalization of a result of Wilkerson * *[W ] in the p-complete case. Corollary 1.4 Let X be an H space. Then there are atomic spaces Xi such that X* * is homotopy equivalent to the weak direct product of the Xi: X ' Xi Furthermore, the Xi are unique up to order and do not depend on the H space str* *ucture of X. Corollary 1.5 Let X be a simply connected co-H space. Then there are atomic spa* *ces Xi such that X is homotopy equivalent to the co-product of the Xi in C^p: i` j^ X ' Xi . p Furthermore the Xi are unique up to order and do not depend on the co-H structu* *re on X. Proof. We consider Corollary 1.4 first. If X is not atomic, choose a nontriv* *ial idempotent self map by Theorem 1.3; write Tel(e) for the telescope: X -e! X -e! X -e! . ... And similarly for Tel(1 - e) where the H space structure and inverse map are us* *ed to construct 1 - e : X -! X. Then the natural map: X -! Tel(e) x Tel(1 - e) induces isomorphisms in all homotopy groups. Thus both telescopes are retracts * *of X and so are themselves complete. Thus every non atomic H space can be split as a pr* *oduct. Since X is of finite type, iteration of this process converges, and the limit i* *s complete. The uniqueness assertion follows from the following argument essentially due to Wil* *kerson [W; 1.6]. 4 Lemma 1.6 Suppose X is atomic and X is a retract of X1 x X2 where X1 and X2 a* *re H spaces. Then either X is a retract of X1 or a retract of X2. Proof. Let ei : X1 x X2 -! X1 x X2 be projection through Xi, i = 1, 2. Then (e1)* + (e2)* = 1 in homotopy. Let fi= X -! X be the composition ei X -! X1 x X2 -! X1 x X2 -! X. SoP(f1)* + (f2)* = 1. This implies that (f1)*(f2)* = (f2)*(f1)*. So 1 = [(f1)* * *+ (f2)*]N = N i N-i i (f1)*(f2)* . If both f1 and f2 were topologically nilpotent, this sum wo* *uld be trivial for large N. Thus one of them is not topologically nilpotent and hence an equiv* *alence. The argument for Corollary 1.5 is similar. We replace homotopy by homology, * *products by coproducts and use the co-H structure to add and subtract. x2. We will often require that the space T admits an H space structure and t* *he space G admits a co-H space structure, but will have no need to specify any particula* *r structure. This is equivalent to assuming that the inclusion ' : T -! ST has a left homot* *opy inverse and that the evaluation map ffl : S G -! G has a right homotopy inverse. For t* *he later result we use the fact that G is simply connected to conclude that the completi* *on of the pullback square [Ga ]: S G ---! G _ G ? ? ? ? y y G ---! G x G is again a pullback square. More generally we have (see [KSW ]) Definition 2.1 A space T is an Hn space if the inclusion ' : T -! nSnT has a * *left homotopy inverse and a space G is a co-Hn space if the evaluation ffl : Sn nG -* *! G has a right homotopy inverse. Theorem 2.2 There is a 1-1 correspondence between homotopy types of connected* * atomic Hn spaces and homotopy types of n-connected atomic co-Hn spaces for n 1. Proof. Both an n-fold suspension is a co-Hn space and retract of a co-Hn sp* *ace is a co-Hn space. Thus, for any space X we can apply Corollary 1.5 to SnX: ` SnX ' Gi and each Gi is a co-Hn space. Choose one of these Gi with the least connectivit* *y. We shall see that in case X is a connected atomic Hn space, there is only one such choic* *e. Similarly, if X is n-connected choose an atomic factor T of nX of least connectivity. We* * shall see that in case that X is an atomic co-Hn space, again, there is only one choice. 5 Now let us begin with an n-connected atomic co-Hn space G. Choose T atomic o* *f least connectivity as above. We have g0 n h0 T -! G -! T where h0g0' 1. Next choose G0atomic and of least connectivity: f00 n g00 0 G0- ! S T -! G with g00f00' 1. We now construct a homotopy equivalence between G and G0. Thi* *s will complete the proof. The maps are: nh0 g00 ff :G -! Sn nG S-! SnT -! G0 f00 n Sng0 n n ffl fi :G0- ! S T -! S G -! G where ffl is the evaluation map and is any right inverse to ffl. We will show* * that fffi is an equivalence. Suppose that G is (k + n - 1) connected (k 1) and ßk+n(G) 6= 0. Then * a* *nd ffl* are inverse isomorphisms between ßk+n(G) and ßk+n(Sn nG). Since (Snh0)(Sn* *g0) ' 1 and g00f00' 1, it follows that (fffi)* : ßk+n(G0) -! ßk+n(G0) is an isomorphism* *. nG is k - 1 connected and ßk( nG) 6= 0. By choice of T having minimal connectivity, T* * is k - 1 connected and ßk(T ) 6= 0. Hence SnT is k + n - 1 connected and ßk+n(SnT ) 6= * *0. By choice of G0having minimal connectivity, ßk+n(G0) 6= 0. Therefore fffi is not t* *opologically nilpotent. Since G0 is atomic, fffi is an equivalence. Let e = fi O (fffi)-1 * *O ff : G -! G. Then e is an idempotent. Since G is atomic and e induces an isomorphism in ßk+n* *, e is an equivalence. Consequently fi* is onto so fi and hence ff are equivalences. Two choices were made in this proof: first we choose T from the factors of * *nG which had minimal connectivity. Then we choose G0from the factors of SnT which had mi* *nimal connectivity. However by choice, ßk+n(G0) ßk(T ) ßk+n(G). Since fi is an eq* *uivalence, these inclusions are equalities and G0is the only factor of SnT which is not k * *+ n connected and T is the only factor of nG which is not k connected. In fact, we have Corollary 2.3 In the correspondence G ! T between atomic k + n - 1-connected co* *-Hn spaces G and atomic k - 1-connected Hn spaces T each of the maps in the commuta* *tive diagram: ß`(T ) -h*-- ß`( nG) ? ? ffn?y ?y~= f* ß`+n(SnT ) --- ß`+n(G) is an isomorphism for ` 2k - 1. 6 Definition 2.4 We call a pair of connected atomic spaces (G, T ) a correspondin* *g pair if there are structure maps f, g, g0, h such that the composites: g n h T - ! G -! T f n g0 G - ! S T -! G are the identity. Proposition 2.5 Given a corresponding pair (G, T ) we may choose maps f, g, g0* *, h such that g and g0 are adjoint. Proof. Given f, g, g0, h, we will keep g, h and replace f with a map efso th* *at g* . ef= 1 where g* is the adjoint of g. Since the composite: f n g* G -! S T -! G induces an isomorphism in ßk+n and G is atomic, this composite is a homotopy eq* *uivalence. Now define ef= f O (g*f)-1. Clearly we could also prove this result retaining g* *0 and f and replacing h by a map ehwith the same effect. Athough our main focus will be on the case n = 1, at this point we will disc* *uss an example in the case n = 2. Example 2.6: 2S3<3> is an atomic H3 space, and the corresponding co-H3 spaces* * is P 2p+2= S2p+1[p'e2p+2; i.e., there are retractions: g 3 2p+2 h 2 3 2S3<3> ---! P ---! S <3> f 3 3 3 g0 2p+2 P 2p+2 ---! S S <3> ---! P . In particular, h = 2h0where h0: P 2p+2-! S3<3> is onto in homotopy. In fact, * *if p > 3, it can be seen that h0factors through S2p+1{p}. Proof: We first observe that S3<3> is 2p - 1 connected and ß2p(S3<3>) = Z=p. T* *hus we may construct a map P 2p+1-! S3<3> inducing an isomorphism in ß2p. We use the H* * space structure2 on S3<3> to extend this map to a map h0: P 2p+2-! S3<3>, and define* * h = 2h0. We construct g using a lifting H0of the loops on the Hopf invariant map 2S3 -H* *! 2S2p+1: 0 2 2p+1 2S3 -H! S {p} where S2p+1{p} is the fiber of the degree p map on S2p+1(see, for example, [G4,* * x4]). There is a natural map S2p+1{p} -L! P 2p+2obtained from the obvious fibrations, and * *these maps combine to define g: 0 2 2p+1 2L 3 2p+2 2S3<3> -! 2S3 -H! S {p} -! P ________________________________ 2h0is actually a loop map since S3<3> is a loop space. 7 all of these maps induce isomorphism in ß2p-2 and 2S3<3> = BW (1) is atomic. * *Since 2S3<3> is an H3 space and P 2p+2is a co-H3 space we have proven the correspond* *ence. Note that the corresponding map f an be defined as the composition: 3h 3 2 3 P 2p+2-! S3 3P 2p+2S-!S S <3> so f is a triple suspension and h is a triple loop map. One is tempted to generalize this. 2nS2n-1<2n + 1> is an H2n+1space, and on* *e seeks to understand the corresponding co-H2n+1 space G. Note that the transfer defines a* * map S2n+1Bnq -~!S2n+1<2n + 1> where Bnq is the nq skeleton of the p localization of B p. We can extend this t* *o a map h0: S2n+2Bnq -! S2n+1<2n + 1>. This gives a candidate for h: h = 2nh0: 2n+1S2n+2Bnq -! 2nS2n+1<2n + 1> and f = S2n+1f0 2n~ 2n 2n+1 f0 : SBnq -! 2nS2n+1Bnq -! S <2n + 1>. Constructing G is not easy and what we can see is that the composite: f 2n+1 2n 2n+1 S2n+2Bnq -! S S <2n + 1> -! G is a monomorphism in homology. G may, however, be somewhat larger. Another example is provided by Neisendorfers result [N ]: If P 2n+1(pr) = S2* *n [pre2n+1 and r 2, then 2P 2n+1(pr) contains as an atomic factor, the fiber D(n, p) of* * the map: 2S2n+1 -i! S2n-1 of degree pr constructed in [CMN ]. Thus D(n, r) $ P 2n+1(p* *r) is a correspondence of an atomic H2 space D(n, r) and an atomic co-H2 space P 2n+1(p* *r) for r 2. x3. Let us return to the case n = 1. Let (G, T ) be a corresponding pair, * *and choose structure maps (f, g, h) as in section 2 with g0 adjoint to g. These maps dete* *rmine an H space structure on T and a co-H space structure on G as follows: gxg h m : T x T -! G x G -! G -! T f g0_g0 n : G -! ST -! ST _ ST -! G _ G. In the pull back diagram: S G --u-! G _ G ? ? ? ? y ffl y p G ---! G x G 8 u is the composition S G -! S G _ S G -ffl_ffl!G _ G. Consequently the right i* *nverse corresponding to n is the composition: f sg : G -! ST -! S G. The case of the H space structure maps is more complicated. In the absence of h* *omotopy associativity, two non homotopic maps ~1, ~2 : ST - ! T can yield the same H s* *pace structure map m when composed with T x T -! ST x ST -! ST. However, one such choice is the composition: g0 h ST -! G -! T. At this point we introduce a öH pf fibration" sequence for T . This can be d* *one for any connected H space either with the classical Hopf construction: H(m) : T * T -! ST (Sugawara [S]) or via a construction of Dold-Lashof construction ([DL ], [G2 ]): Hm : T * T ' Em -! ST. These constructions are different3 and we will find the Dold-Lashof constructio* *n advanta- geous. Specifically we will use the following corollary of Propositions A1 and* * A2 of the appendix. Corollary 3.1 Suppose T is atomic. Then there is a 1-1 correspondence between f* *iber ho- motopy classes of fibrations: T --i-! E - H--! ST with i null homotopic and homotopy classes of maps m : T x T -! T such that: a) m(*, t) = t b) the map f : T -! T given by f(t) = m(t, *) is a homotopy equivalence. This correspondence is given by H = Hm . ________________________________ 3We wish to thank Yukata Hemmi and Norio Iwase for helpful e-mail notes at t* *his point. 9 Proof. The only part that does not follow immediately from A1 and A2 of the * *appendix is that if i is null homotopic, f is a homotopy equivalence. f is the compositi* *on: T -! ST -@! T where @ : ST -! T is from the fiber sequence. If i is null homotopic, @ has * *a right homotopy inverse. Hence f* : ßk(T ) -! ßk(T ) is onto. Now ßk is complete and* * finitely generated, and f* : ßk(T ) Z=pr ! ßk(T ) Z=pr is an isomorphism for each r.* * Hence f* is an isomorphism and f is a homotopy equivalence. We now choose an arbitrary H space structure m : T x T - ! T and a correspon* *ding fibration using (3.1). We will see later that some improvements can be made if * *we can choose a homotopy associative H space structure. Choose a map f : G -! ST which induce* *s an isomorphism in ßk and choose h as the composite: f @ G -! ST -! T, where @ comes from the fibration sequence. It may not be possible to choose g0a* *djoint to g so that hg = 1 and g0f = 1; however any choice of g which induces an isomorphis* *m in ßk-1 will yield that hg and g0f are homotopy equivalences. We now construct a commu* *tative diagram obtained by pulling back the fibration from (3.1) along the base. f g0 f G - --! ST ---! G ---! ST ? ? ? ? ? ? ? ? y @0y hy @y T - --! T ---! T ---! T ? ? ? ? (A) ?y ?y '0?y i?y R0 - --! Q ---! R ---! T * T ? ? ? ? i0?y ?y i?y Hm ?y f g0 f G - --! ST ---! G ---! ST By A1, Q is determined by the restriction of the action map T x T -! ST x T -a! T. 0 Since @0 = h O g0, the composition T -! ST -@! T is a homotopy equivalence,* * so Q ' T * T by Corollary 3.1. Since g0f is a homotopy equivalence R0' R and R is * *a retract of T * T . Since h has a right homotopy inverse, i0is null homotopic and we have Theorem 3.2 There is a homotopy equivalence G ' T x R W where R is a retract of T * T and hence a co-H space. Write R = Rffwith Rffat* *omic. Then 10 a) if G = SX, Rffis a retract of SX(i)for some i 2 b) if G is homotopy is associative, Sj-1Rffis a retract of G(j)for some j 2. We think of the exact sequence of spaces: * -! R -! G -@! T -! * as a minimal free presentation of T . Minimal since G is atomic and free since * *the homology of both R and G are tensor algebras. However @ is not an H map in general. Proof. The only parts needing attention are the last twoWassertions. If G = * *ST , R is a retract of T * T which in turn is a retract of SX * SX ' SX(i)^ X(j). Part * *b) follows from the following theorem of Theriault. Theorem 3.3 (Theriault [T ]) Let G be a simply connected homotopy co-associati* *ve co-H space. Then 1` S G ' G(k) k=1 where G(k) is a homotopy co-associative co-H space and Sk-1G(k) ' G(k). Proof of 3.2b. WeWwrite R as a retract of G * G ' G ^ (S G). It follows t* *hat R is a retract of G ^ G(k). Thus each Rffis a retract of G ^ G(k) for some* * k. But k 1 W G ^ G(k) is a retract of G ^ S G(k) ' G(`) ^ G(k); now ` 1 ` S G(k) ' G(k, j) j 1 with Sj-1G(k, j) ' G(k)(j). Consequently SjRffis a retract of G(`) ^ G(k)(j)and* * S`+kj-1Rff is a retract of S(k-1)jG(`)^ G(k)(j)' G(`+kj). There is another sense in which the H-space T is generated by G, namely: Proposition 3.4 The image of the homomorphism: f* : eHi+1(G) -! eHi+1(ST ) ' eHi(T ) generates the ring eH*(T ). As an example, consider T02n+1 P 2n+1, the atomic factor of [CMN ]. Then H*(* *T02n+1) is generated by u 2 H2n-1(T02n+1) and r 2 H2n(T02n+1) as a non-associative ring. Proof. Let R eH*(T ) be the subing generated by the image of f*. We firs* *t observe that the composition: fx1 a a0: G x T - --! ST x T ---! T defines a H*( G) module structure on eH*(T ). 11 Lemma 3.5 R is a H*( G) submodule. Proof. The map a0fits into a commutative diagram: 0 G x G x T -1xa--! G x T ? ? ? ? ymx1 y a0 0 G x T - a--! T which can be obtained from A3 of the appendix and the definition of a0. Thus (a0)*(ff1ff2 t) = (a0)*(ff1 (a0)*(ff2 t)) for ff1, ff2 2 H*( G) and t 2 H*(T ). Iterating we get (a0)*(ff1. .f.fk t) = (a0)*(ff1 (a0)*(a2, . .,.(a0)*(ffk t)* * . .).). It follows that it is sufficient to show that (a0)*(ff t) 2 R whenever t 2 R * *and ff is indecom- posable. Since indecomposable elements of H*( G) are in the image of the homomo* *rphism Hei(G) --*-! eHi(S G) ' eHi-1( G) and ~ Sg O f, any indecomposable element ff can be written as g*(r) for some * *r 2 R. Now we have a commutative diagram: T x T gxg. & gx1 G x G - 1xh--! G x T ? ? ? ? ym y a0 G - -h-! T where the commutative square is a restriction of the above square to G x G x * Gx GxT . Since the composition along the left and bottom is m, we have a*(g*(r* *) t) = m*(r t) = rt 2 R. This proves the lemma. We now use this lemma to prove the proposition. Let , 2 H*(T ) be an element* * of least degree that is not in R. , = h*(ff) for some ff 2 H*( G). ff cannot be indecomp* *osable since then it would be in the image of * and hence , would be in the image of f*. Th* *us ff = ff0iff00i with ff0iand ff00iof positive degree. In particular , = h*( ff0iff00i) = (a0)** *(ff0i h*(ff00i)). Since deg h*(ff00i) < deg,, h*(ff00i) 2 R and hence, by the lemma , 2 R. Proposition 3.6 If G is homotopy co-commutative, and co-associative each eleme* *nt in imf* is primitive (and hence H*(T ) is primitively generated). Conversely, if, in a* *ddition T is homotopy associative and homotopy commutative, each primitive element in H*(T )* * is a sum of prth the powers of elements in imf*. 12 This depends on the following lemma which has been noted by [T ] as a coroll* *ary of the results of Berstein [B ]. The proof in [B ] is not direct, and we offer here a * *direct proof. Lemma 3.7 If G is co-commutative co-associative co-H space, the image of * : eHi(G) -! eHi-1( G) consists of primitive elements. Proof. We first examine the case that G = SX for some X. Consider the compos* *ite: X : SX -S--! SX ^ X - W--!SX _ SX where is the diagonal and W is the Whitehead product. By definition, the Whi* *tehead product is '1 + '2 - '1 - '2 where '1, '2 : SX -! SX _ SX are the inclusions. * *X can be written as ('1+'2)-('2+'1) = OE-øOE where OE : SX -! SX _SX is the usual co-H s* *tructure map. In particular, the composite X ~ * iff the usual co-H structure is co-com* *mutative. Now consider the composite: G ffl_ffl G ---! S G ---! S G _ S G ---! G _ G. Since G is co-associative, is a co H map and hence (ffl _ ffl) ~G (ffl _ ffl)(OE G - øOE G ) ~ (ffl _ ffl)OE G - ø(ffl _ ffl)OE G ~ OEG - øOEG ~ *. However we further factor this as G ---! S G -S--! S( G ^ G) - W--!S G _ S G - ffl_ffl--!G _ G. The composite (ffl _ ffl)W is the inclusion of the fiber in the fibering S( G ^ G) ---! G _ G - --! G x G; since this inclusion has a null homotopic fiber, we conclude that the composite G ---! S G -S--! S( G ^ G) is null homotopic from which the result follows. Proof of 3.6. Since f : G ! ST is the composition G -! S G -Sh!ST1 the fir* *st part is immediate. However, if , 2 H*(T ) is primitive, so is g*(,) 2 H*( G). S* *ince H*( G) is a primitively generated tensor algebra, g*(,) is a sum of prth powers of com* *mutators (r 0) and prth powers of elements in the image of *. But if T is homotopy as* *sociative and homotopy commutative , = h*g*(,) is a sum of prth pwers of elements in imf*. We now derive a result from the following theorem of Theriault [T, Theorem 8* *.4]. 13 Theorem 3.8 (Theriault) If G is a homotopy co-commutative and co-associative * *co-H space and (l, p) = 1, there is a decomposition: G = Ulx Fl induced by a map OEl: Ul- ! G, where the homology of Ulis the subalgebra of th* *e homology of G generated by the commutators of length l. Corollary 3.9 For some H space structure on T , the image in homology of ß : * *R ! G contains all commutators of length l where (l, p) = 1, for l > 1. Proof. Ulis homotopy equivalent to a product of some of the factors of G. * *Since Ul is lk - 1 connected, T is not among them. These factors lie in R, and we repla* *ce them by Ul, for each l with (l, p) = 1, and take T to be the fiber of this new map. Th* *is new T will still be the bottom atomic factor of G, but the map from G to T and hence the* * H space structure may be different. (See example 3.13 below.) G is a generator for T in another sense as well: Proposition 3.10 Let X be any H space and _i: T ! X be H maps for i = 1, 2. Th* *en a) _1 ~ _2 () (S_1)f ~ (S_2)f f S_i G -! ST -! SX. __ __ b) If, in addition, G = SK and f = Sf for f : K ! T , then __ __ _1 ~ _2 iff_1f ~ _2f. Proof. From the commutative diagram: f S_i G -! ST - ! SX h& #~ # _i T - ! X __ we see that if (S_1)f ~ (S_2)f,__1h ~ _2h so _1 ~ _1hg ~ _2hg ~ _2. If f = Sf , (S_1)f ~ (S_2)f iff S(_1f) ~ S(_2f). Since X is an H space, this is true iff _1* *f ~ _2f. There are numerous examples in the literature of spaces K with this property* * ([CMN ], [G1 ], [Se]). Write PX (t) for the Poincar'e series for X. Proposition 3.11 PR = PTPG - (t + 1)(PT - 1). 14 Proof. Since G = R x T , we have ______PT______ 1 = ______________ 1 - t-1(PR - 1) 1 - t-1(PG - 1) which implies the result. W Corollary 3.12 If G = SX and G ^ G ~= SnffG, then R ' G ^ W where W is a wed* *ge ff2A of spheres and t(PT - 1) PW = PT - _________+ 1. PG - 1 Q Furthermore if Tn corresponds to SnX for eachWn 1, SX ' Tnifor some sequen* *ce ni. Proof. The hypothesis implies that R = Rffwhere each Rffis a suspension of* * G by 3.2. Thus R ' G ^ W where W is a wedge of spheres. The formula for PW follow* *s from 3.11. Applying the Hilton-Milnor theorem and induction decomposes the loop spac* *e of the wedge into a product. In the next example we see that the map ß does depend on the H space structu* *re. Example 3.13 Let T = S2p-3{p}, the fiberWof the degree p map on S2p-3. We wil* *l write P n=WSn-1 [p'en. Then G = P 2p-2, and R = Rffwhere each Rff= P nffby 3.12. In* * fact, R = P 2p-1+k(2p-4). k 1 One way of obtaining this decomposition is given in [CMN ], where* * the maps P 2p-1+k(2p-4)-! P 2p-2are given by iterated Whitehead products. In particular * *ß is stably inessential. On the other hand, let B = (B p)(p). There is a fibration sequence p 2p-3 B -! S2p-3- ! S - ! B. Since B ' S2p-3{p}, this determines another H space structure on S2p-3{p}. The* *re is a pull back diagram W i S2p-3{p}- --! P 2p-1+k(2p-4)---!P 2p-2 ? k 1 ? ? '?y ?y ?y B - --! P B ---! B from which we can see that the composition ` P 2p-1+k(2p-4)-! P 2p-2-! B k 1 15 is null homotopic. In particular, with k = 1, the restirction of ß to P 4p-5is * *the attaching map for the next two cells of B. But the 4p - 4 skeleton of B is P 2p-2[ CP 4p-* *5and the Steenrod operation P1 is non zero. Thus P 4p-5-! P 2p-2is stably essential. In * *particular, the Whitehead product map P 4p-5-! P 2p-2does not factor through R and the comm* *utator P 4p-6-! P 2p-2-! S2p-3{p} is non trivial. We now consider the special case when there is a homotopy associative multip* *lication on T . Proposition 3.14 Suppose there is a homotopy associative multiplication on T .* * Then in diagram (A), @ = @0: ST ! T is an H map, h is an H map, the fiberings Q -! ST * *and R -! ST are fiber homotopy equivalent, g can be chosen so that hg ~ 1 and g0f ~* * 1 and ß is the composition g0 R - --! T * T- Hm--!ST - --! G. Proof. We show that there are structure maps (f, g, h) so that the composit* *ion Sg0 h ST -! G -! T factors the given map ~ : ST -! T extending the multiplicatio* *n. To do this, we choose any structure maps (f0, g0, h0). We begin with the compositi* *on: g0 f0 ~ e : T---! G - --! ST ---! T e induces an isomorphism in ßk, so it is a homotopy equivalence. Define g = g0* *(e-1) and h = ~( f0). Then hg = 1. Now the composition: f0 g0 e0: G---! ST ---! G is a homotopy equivalence where g0is the adjoint of g. Let f = f0(e0)-1. Then g* *0f ~ 1. To show that these structure maps determine ~, define __~as the indicated composit* *e: __~: ST -!g0 G -f0! ST -~! T. * *fi Since ~ is an H map, __~is as well. To see that ~ ~ __~we need only calculate _* *_~fiT= ~( f0)g = hg ~ 1. By A3 of the appendix, @ ~ ~ and @0 = h( g0) ~ ~. The fibering Q -! S* *T is determined by the restruction of the action map: 0 T x T -! ST x T -a! T a0is the composite: fx1 g0x1 a ST x T - ! G x T - ! ST x T -! T. Since T is homotopy associative, the formula in A3 implies that a is the compos* *ite ~x1 ST x T ---! T x T ---! T ; 16 combining these we see that a0is the composite: fx1 hx1 m ST x T ---! G x T ---! T x T - --! T and restricting to T x T is thus m. x4. Given the strong duality involved here, it is natural to ask whether a d* *ual discussion can be obtained. This would require a dual to the Hopf fibering. The following * *Conjecture is due to Ganea [Ga ]. Conjecture 4.1 Given a co-H space G there is a cofibration sequence: G -ff!X -! G -! S G. This is certainly the case when G is a suspension, for if G = SA we can take* * X = SA=A. This is false in general. Example 4.2 Let ff1 : S2p -! S3 be the first element of order p with p > 3 and G = S3 [ffe2p+1. Harper [H ] has shown that ff is a co-H map and hence G is a c* *o-H space. Suppose such a space X does exist. H*( G) through dimension 2p has classes u, u* *2, . .,.up and v with |u| = 2 and |v| = 2p. In p-local cohomology, the class up- pv transg* *resses to the class in dimension 2p + 1 in the base. Since p > 3, the classes u2 and up-2 are* * in the image of oe*. Hence up is in the image of oe*. This is impossible since, mod p, up tr* *ansgresses. The requirement that p > 3 is essential in this example. f 2n+1 Proposition 4.3 Suppose S2Y - ! S is a co-H map localized at 3 and n 1, * *and set G = S2n+1[f CS2Y . Then there is a Hopf co-fibration: G --ff-!X ---! G ---! S G. The proof of 4.3 depends on the following Lemma 4.4 Let bS2n= S2n[ e4n[ . .[.e2n(p-1) (S2n)1 . Then the loops on the p* *rojection map: Sb2n ---! S2n(p-1) is null homotopic. Proof. Let K = S2n-1 [wn e2np-2where wn is the first element in the kernel * *of the double suspension localized at p. Then there is a loop map SK -! Sb2nwhich h* *as a right homotopy inverse [G1 ], so it suffices to show that the composite: SK ---! Sb2n ---! S2n(p-1) 17 is null homotopic. We will show that the composite SK -! Sb2n -! S2n(p-1)is n* *ull homotopic. The degree k map on S2n induces a map [k] : Sb2n- ! Sb2nand there a* *re commutative diagrams: SK ---! bS2n---! S2n(p-1) ? ? ? ? ? ? y ffi y[k] ykp-1 SK ---! bS2n---! S2n(p-1) S2n ---! SK ---! S2np-1 ? ? ? ? ? ? p y k yffi yk S2n ---! SK ---! S2np-1. Since S2wn ~ 0, the composite SK -! bS2n-! S2n(p-1)factors uniquely over S2n* *p-1: SK ---! bS2n ? ? ? ? y y S2np-1--ffi-!S2n(p-1) and these diagrams, together with the uniqueness imply that kp-1ffi = ffikp. Si* *nce ffi must be a suspension, we conclude that (kp - kp-1)ffi = 0. Let k = -1 to get -2ffi = 0 * *or ffi = 0. Proof of 4.3. According to Harper [H ], the Hopf invariant of f is the co-H * *deviation and hence is null homotopic. Therefore we can factor: f0 2n+1 SY - ! S & " f~ Sb2n= S2n [[',']e4n. Consequently (S2n+1[f CS2Y ) contains as a subcomplex bS2n[_fCSY . Then ffi2n (S2n+1[f CS2Y ) S __ contains S4n[feCSY where efis the projection of f onto S4n. However efis null h* *omotopic ffi since j : Sb2n-! S4n is nullfhomotopic.fThusif(S2n+1[ffCS2Yi) S2n contains * *(S4n_ S2Y ). Let X = (S2n+1[fCS2Y ) S2n S2Y and oe be the projection (S2n+1[fCS2Y )* * -! X. In homology oe* is onto and it's kernel consists of all tensors of length 1.* * Let C be the cofiber of oe. Then H*(C) -! H*(S G) consists of the suspension of the tensors * *of length 1 and hence the composite C -! S G -ffl!G is a homotopy equivalence and we are done. 18 x5. We need to say something about naturality. Given corresponding pairs (G1* *, T1), (G2, T2) with structure maps (f1, g1, h1) and (f2, g2, h2), a map from (G1, T1) to (G2, * *T2) should be a pair of maps (OE, _) so that certain diagrams commute. In many situations we * *may begin with an H map _ : T1 -! T2 and it is not possible to find an appropriate co-H m* *ap OE. For example, _ = fl where fl : S2n+1- ! S2m+1 is not a co-H map. The following is the strongest result that seems reasonable, and we offer th* *is as a definition of a map from (G1, T1) to (G2, T2). Proposition 5.1 Suppose _ : T1 ! T2 is an H map and OE : G1 ! G2 is a co-H map* *. Then the following are equivalent: f1 S_ g02 a) OE is the composition: G1 -! ST1 -! ST2 -! G2 b) The following square commutes: ffi G1 ---! G2 ? ? h1?y ?yh2 _ T1 ---! T2 g1 ffi h2 c)_ is the composition: T1 -! G1 -! G2 -! T2 d) The following square commutes: ffi G1 ---! G2 ? ? f1?y ?yf2 Sffi ST1 ---! ST2. Proof. To see that a) implies b) note that ~i = hiO g0i. So hi ~ ~iO fi;* * thus _ O h1 ~ _ O ~1 O f1 ~ ~2 O S_ O f1 ~ h2 O g02O S_ O f1 = h2 O OE. To se* *e that b) g1 implies c), compose the square with the map T1 -! G1. The last two parts are s* *imilar to the first two parts. On the other hand if T2 is homotopy associative b) implies that _ is an H ma* *p, and if G1 is homotopy co-associative d) implies that OE is a co-H map. x6. In this section we discuss a further refinement in the determination of* * R in case that T has a homotopy associative H-space structure and p > 2. The results are * *based on the following well known result. Lemma 6.1 Suppose p > 2 andføi: X ! X is a map such that ø2 ~ 1. Then SX ' X+* * _X- where eH*(X ) = {, 2 bH*(SX)fiø*(,) = ,}. 19 Proof. Let e : SX -! SX be given by e = ø 1 and e e X = lim{SX -! SX -! SX -! . .}.. -! Then the composition SX - ! SX _ SX - ! X+ _ X- induces homology isomorphisms between simply connected spaces. We will apply this lemma in two cases. We first consider the transposition * *map ø : T ^ T - ! T ^ T , from which we write T * T ' R+ _ R-. The second application * *deals with the inverse map for a homotopy associative H-space. For any connected H-sp* *ace, it is standard to construct left and right slicing maps: T x T ---! T (a, b)---! a=b (a, b)---! a\b by choosing homotopy inverses for the maps: T x T - --! T x T (a, b)- --! (m(a, b), b) (a, b)- --! (a, m(a, b)). These maps define operations on the set of homotopy classes [(X, *), (T1e)] whi* *ch satisfy the identities: ffi (ff . fi) fi=ff (ff=fi) . fi=ff ff\(fffi)= fi ff(ff\fi)= fi. In case T is homotopy associative one can see that e=ff = ff\e and write ff-1 f* *or this homotopy class; then we have (ff-1)-1 = ff and (fffi)-1 = fi-1ff-1. Write fl : T -! T fo* *r the inverse of the identify map. Then fl2 = 1, so we have ST ' T+ _ T-. If , 2 eH*(T ) is primitive, fl*(,) = -,. Since T is k - 1 connected, all cl* *asses in eHk(T ) are primitive. Consequently, the composition: f f0 : G---! ST ---! T- ---! ST is an isomorphism in dimension k. Now choose g0, h0 so that g00f0 ~ 1 and h0g0 * *~ 1. We then apply the modification of 3.8 to obtain a triple (f, g, h) with ~ ~ h( g0)* *, and observe that f : G -! ST factors through T-. 20 Proposition 6.2 The diagram: f G -! ST & # -Sfl f ST. commutes up to homotopy. fi Proof. It suffices to show that Sfl(T ) T and (-Sfl)fiT-= 1. This follows* * from the commutative ladders: 1-fl 1-fl ST - --! ST ---! ST ---! . . . ? ? ? -Sfl?y ?y1 ?y1 1-fl 1-fl ST - --! ST ---! ST ---! . . . 1+fl 1+fl ST - --! ST ---! ST ---! . . . ? ? ? -Sfl?y ?y-1 ?y-Sfl 1+fl 1+fl ST - --! ST ---! ST ---! . ... Theorem 6.3 The following diagram commutes up to homotopy T * T- --! ST ? ? -S(fi)?y -Sfl?y T * T- --! ST. Corollary 6.4 R is a retract of R- T * T . Proof of 6.4. The diagram: T * T % ? ? R ?y-S(fi) & T * T commutes after projection to ST by 6.3. However the inclusion of the fiber T -* * ! T * T is null homotopic, so this diagram actually commutes up to homotopy. As in the * *previous case, this implies that the projection onto R+ is null homotopic. 21 Proof of 6.3. We begin by considering the equivalence Em ' T * T from A2 of* * the appendix (where A = F = T and ` = m). From this we see that corresponding to ß1* * in the push out diagram for Em is the map (a, b) -! a=b in the push out diagram for T * ** T . In other words, the map Em -! ST corresponds to the map of push out diagrams: T -i1 T?* T -i2! T ? ?ff y * - T - ! * where oe(a, b) = a=b. This is, in fact, the classical Hopf construction on the * *map oe: H(ff) T * T- --! ST ; here we use the reduced join X*Y , which is the quotient of XxIxY under the ide* *ntifications: (x, 0, y)~ (x, 0, y0) (x, 1, y)~ (x0, 1, y) (*, t, *)~ *, and the reduced suspension. We introduce maps ff : S(X x Y ) -! X * Y and fi : * *X * Y - ! S(X ^ Y ) whose composite is homotopic to the quotient map S(X x Y ) -! S(X ^ Y* * ). We define ff by 8 < (x, 1 - 3s, *)0 s 1=3 ff(s, x, y) = (x, 3s - 1, y)1=3 s 2=3 : (*, 3 - 3s, y)2=3 s 1 and fi(x, t, y) = (t, x, y). fi collapses the subspace (*, t, y) [ (x, t, *) to* * a point. Since the join is reduced, this subspace is contractible. Consequently ff has a right homotopy* * inverse, and we can study H(oe) by considering H(oe)ff: 8 < (x, 1 - 3t) 0 t 1=3 H(oe)(ff(t, x, y)) = (x=y, 3t - 1)1=3 t 2=3 : (*=y, 3 - 3t)2=3 t 1 Thus H(oe)ff = -S(ß1) + S(m(ß1, flß2)) - S(flß2) here we use associativity to write *=y = fl(y) and x=y = x . fl(y). Now observe* * that (-S(fl)) . H(oe)ff = S(flß2) - S(flm(ß1, flß2)) + Sß2. Note that flm(ß1, flß2) = m(ß2, flß1), so H(oe)ff(-Sø) = (-S(ß2) + S(m(ß2, flß1)) - S(flß1))(-1) = S(flß1) - Sm(ß2, flß1) + S(flß2) 22 which completes the proof. Finally we observe that in case p = 2 it is possible to have R = T * T . For* * an example let T = RP 1 and note that G = SRP 1 is atomic, even as a module over {Sq1, Sq2}. I* *t follows that R = RP 1 * RP 1. x7. In this section we will show that in many cases, an associative H space * *structure on T is not possible. The model for this is the case G = P 2n+1(pr) of [CMN ]* *. In this case Neisenderfer has pointed out that the space T02n+1which is a retract of P* * 2n+1(pr) does not admit an associative H space structure. The key fact here is that ther* *e is a class v 2 H2n+1(P 2n+1(pr)) with fi(r)(v) 6= 0. We will consider generalized Bockste* *in homol- ogy operations fi* defined and natural on some full subcategory Cfiof the categ* *ory of CW complexes (for example, spaces on which fi(1), . .f.i(r-1)are all zero, and hen* *ce fi(r)is well defined homology operation). We assume that Cfiis closed under finite products* * colimits and retracts. Consequently it is closed under suspension and the James construc* *tion. The operations we consider include both the Bockstein operations and the Mil* *nor oper- ations Qi as well as possible higher order Milnor operations. We assume a stabl* *e homology operation: fi : Hi(X) -! Hi-2d-1(X) defined and natural for X 2 Cfi. We assume fi2 = 0, fi(oe(x)) = oe(fi(x)) wher* *e oe is the homology suspensions, and fi(x x y) = fi(x) x y + (-1)|x|x x fi(y). Note that if T 2 Cfi, ST 2 Cfiand hence G 2 Cfi. Conversely, if G 2 Cfiand G is* * homotopy associative and homotopy commutative, G 2 Cfiby 3.3. (Here we see the fact th* *at fi is stable so fi can be defined on X iff it can be defined on SX.) Consequently T 2* * Cfi. Proposition 7.1 Suppose G admits a homotopy co-associative and co-commutative * *struc- ture, and H*(G) is fi-acyclic. Suppose that T admits a homotopy associative and* * commutative structure. Then 0 = fi : H2n+1(G) -! H2n-2d(G). One may apply this result, for example, to the space _(1)n = P n-2p+1[A CP n-1 n 2p + 4 where A : P n-1-! P n-2p+1is the Adams map. We apply Q0 in case n is odd and Q1 in case n is even to see that the corresponding space T (v1)n-1 does not carry * *a homotopy associative and homotopy commutative H space structure. Proof: Suppose v 2 H2n+1(G) is such that fiv06= 0. Let v = f*(v0) 2 H2n(T ) * *and u = fiv. By 3.6, u and v are both primitive. Since T is homotopy associative and commuta* *tive, u2 = 0 23 while uvi 6= 0 for i < p by induction and applying * : H*(T ) -! H*(T ) H*(T* * ). Also fi(uvp-1) = -u2vp-1 = 0. Since H*(G) is fi-acyclic H*( G) and H*(T ) are also f* *i acyclic. Consequently there is a class w 2 H2np(T ) with fiw = uvp-1. Define e!2 H*(T ) * * H*(T ) by p-1X` ' 1 p i p-i e!= ! 1 + __ v v + 1 !. i=1p i Then p-1X` ' p - 1 i p-i-1 i p-i-1 *(uvp-1) = {uv v + v uv } i=1 i = fi(e!). Consequently fi( *(!) - e!) = 0. Claim: if z 2 H*(T ) H*(T ) is a cycle its does not contain the term v v* *p-1. Proof: write for the subspace spanned by x. Then H2n(T ) = A H2n-2d-1(T )= B and we can arrange A and B so that fi(a) 2 B for each a 2 A. Likewise write H2n(p-1)(T )= C H2n(p-1)-2d-1(T )= D with fi(c) 2 D for all c 2 C. Write z = ziwith zi2 Hi(T ) Hm-i(T ). If fiz =* * 0, it follows that (fi 1)zi6= (1 fi)zi-1= 0 for all i and hence (fi fi)zi= 0 for all i. Now H2n(T ) H2n(p-1)(T ) ~= CA A C. Applying fi fi we obtain an element of D B B D. Since fi fi(v vp-1) = <-u uvp-2>, the term v vp-1 is not present in any* * cycle. It is present in e!, so it must also be present in *(!). Since T is primitively gene* *rated by 3.6, each term in ! is a product of primitives; now X *(!1. .!.s) = cfffi!ff !fi where cfffiare coefficients and !ff, !fiare products of the !i in such a way th* *at !ff!fi= !1. .!.s. Consequently if *(!) contains the term v vp-1, it must come from a* * term vp in !. But *(vp) = vp 1 + 1 vp so this is impossible. 24 Appendix In this section we collect some general facts about Hopf map constructions. We* * wish to thank Yukata Hemmi and Norio Iwase for some helpful e-mail conversations. Proposition A1 There is a 1-1 correspondence between homotopy classes of maps* * ` : A x F -! F with `(*, f) = f and fiber homotopy classes of fibrations: F -! E -! SA where ` is the restriction to A x F of the action map SA x F -a! F defined by the homotopy lifting property. Proof. For each such ` : A x F -! F we define a quasifibering: E` = F [` (CA) x F -! SA where the subspace A x F (CA) x F is identified with F via `. Both the cone a* *nd the suspension are reduced. Thus construction is due to Dold and Lashof [DL ] and i* *t is shown in [G2 ] that each Hurewicz fibering of the form considered here is homotopy eq* *uivalent to such_a construction. Given a_homotopy `t : A x F - ! F with `t(*, f) = f we co* *nstruct ` : A x (F x I) -! F x I by `(a, f, t) = (`t(a, f), t). From this we construct * *a quasi fibering F x I -! E_`-! SA and the inclusion of E`0and E`1into E_`are clearly homotopy equivalences. To recover the map ` : A x F -! F from an arbitrary Hurewicz fibering we fir* *st discuss the action map a : B x F -! F defined for each Hurewicz fibering F -! E -! B. This is constructed in a standard way by choosing a lifting in the diagram: B x F x 0 -i! E L # % # ß B x F x I -H! B where i(w, f, o) = f and H(w, f, t) = w(t). Then L(w, f, 1) 2 F = ß-1(*) so by * *restriction L defines the action map: fi a = Lfi BxFx1: B x F -! F. 25 For each map _ : X -! B we can, by restriction, construct a lifting L0in the d* *iagram: iO(ffix1) X x F?x 0 -! ?E ? L0 ? ? % ? ß y y HO(ffix1) X x F x I -! B. We now assert that given any two choices L00and L01of liftings, the associated * *ä ction maps" a0, a1 : X x F -! F (defined by restricting to X x F x 1) are homotopic. To see this we define a map I : X x F x (I x 0 [ 0 x I [ 1 x I) [ * x F x I x I -! E by applying L0fflon X x F x ffl x I (for ffl = 0, 1) and projecting X x F x I x* * 0 [ * x F x I x I onto F E. Define J : X x F x I x I -! B by J(x, f, s, t) = _(x)(t). We then e* *xtend I to a homotopy covering J, and this homotopy, when restricted to X x F x I x 1 is a* * homotopy between a0 and a1. It follows that in the case of a fibering F -! E -! SA 0 we have constructed a well defined homotopy class of maps SAxF -`! F with `0(** *, f) = f. By restriction to A x F we obtain a class `. We only need to see that we can ch* *oose ` as a lifting for E`: A x F?x 0 -'! E`? ? ? ? L% ?ß y y A x F x I -H! SA with '(a, f) = f 2 F E`, H(a, f, t) = (a, t) 2 SA and L(a, f, t) = ((a, t), f* *) 2 CA x F . Here CA = A x I=A x 0 [ * x I and L(a, f, 1) = `(a, f) 2 F E. Proposition A2 If the composition A x * A x F -`! F is a homotopy equivale* *nce, i : F -! E` is null homotopic and E` ' F * F . fi fi Proof. Since `fiAx*is a homotopy equivalence, i factors through `fiAx*up to * *homotopy and hence it factors through (CA) x *, so i is null homotopic. E` is the reduce* *d homotopy push out of the diagram: F -` A x F -i2!F. 26 We use the commutative diagram: Ffl- ` A x?F -i2! Ffl fl ? fl fl ? fl fl y fl F -i1 F x F -i2! F where (a, f) = (`(a, f), f). By hypothesis, is a homotopy equivalence so E` * *is homotopy equivalent to the reduced homotopy push out of F -i1 F x F -i2!F which is the reduced join F * F . Proposition A3 Replacing SA by the James construction A1 , the action map a : A1 x F -! 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