HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F MARTIN ARKOWITZ AND JEFFREY STROM 1 1. Introduction A map f : X -! Y is said to be detected by a collection F of topological spa* *ces if there is a space A 2 F such that the induced map f* : [A; X] -! [A; Y ] of homo* *topy sets is nontrivial. It is a standard technique in homotopy theory to use certai* *n simple collections F to detect essential homotopy classes. In studying the entire homo* *topy set [X; Y ] using this approach, one is led naturally to consider the set of ho* *motopy classes which are not detected by F, called the F-trivial homotopy classes. For example, if S is the collection of spheres, then f : X -! Y is detected by S p* *recisely when some induced homomorphism of homotopy groups ssk(f) : ssk(X) -! ssk(Y ) is nonzero. The S-trivial homotopy classes are those that induce zero on all homot* *opy groups. It is also important to determine induced maps on homotopy sets. For th* *is, one needs to understand composition of F-trivial homotopy classes. With this in mind, we study two basic questions in this paper for a fixed collection F: (1) * *What is the set of all F-trivial homotopy classes in [X; Y ]? and (2) In the special ca* *se X = Y , how do F-trivial homotopy classes behave under composition? We are particularly interested in the collections S of spheres, M of Moore spaces and of suspensio* *ns. We next briefly summarize the contents of this paper. We write ZF (X; Y ) for the F-trivial homotopy classes in [X; Y ] and set ZF (X) = ZF (X; X). After so* *me generalities on ZF (X; Y ), we observe in Section 2 that ZF (X) is a semigroup * *under composition. Its nilpotency, denoted tF (X), is a new numerical invariant of ho* *motopy type. For the collection of suspensions, we prove that t (X) dlog2(cat(X))e. * * In Section 3 we relate tF (X) to other numerical invariants for arbitrary collecti* *ons F. The F-killing length of X, denoted klF(X) (resp., the F-cone length of X, denot* *ed clF(X)), is the least number of steps needed to go from X to a contractible spa* *ce (resp., from a contractible space to X) by successively attaching cones on wedg* *es of spaces in F. We prove that tF (X) klF(X), and, if F is closed under suspension, that klF(X) clF(X). We also show that klF(X) behaves subadditively with respect to cofibrations. It is clear that for any X and Y , Z (X; Y ) ZM (X; Y ) ZS(X* *; Y ), and we ask in Section 4 if these containments can be strict. It is easy to find* * infinite complexes with strict containment. However, in Section 4 we solve the more diff* *icult problem of finding finite complexes with this property. From this, we deduce t* *hat containment can be strict for finite complexes when X = Y . The next two sectio* *ns are devoted to determining ZF (X) and tF (X) for certain classes of spaces. In * *Section ___________ 1Math. Subject Classifications: Primary: 55Q05 Secondary: 55P65, 55P45, 55M30 1 2 MARTIN ARKOWITZ AND JEFFREY STROM 5 we calculate ZF (X) and tF (X) for F = S; M and when X is any real or complex projective space, or is the quaternionic projective space HPn with n 4. In Sec* *tion 6 we consider t (Y ) for certain Lie groups Y . We show that 2 t (Y ) when Y = S* *U(n) or Sp(n) by proving that the groups [Y; Y ] are not abelian. In addition, we co* *mpute t (SO(n)) for n = 3 and 4. The paper concludes in Section 7 with a list of open problems. For the remainder of this section, we give our notation and terminology. All * *topo- logical spaces are based and connected, and have the based homotopy type of CW complexes. All maps and homotopies preserve base points. We do not usually di* *s- tinguish notationally between a map and its homotopy class. We let * denote the base point of a space or a space consisting of a single point. In addition to s* *tandard notation, we use for same homotopy type, 0 2 [X; Y ] for the constant homotopy class and id2 [X; X] for the identity homotopy class. For an abelian group G and an integer n 2, we let M(G; n) denote the Moore space of type (G; n), that is, the space with a single non-vanishing reduced ho* *mology group G in dimension n. If G is finitely generated, we also define M(G; 1) as a* * wedge of circles S1 and spaces obtained by attaching a 2-cell to S1 by a map of degre* *e m. The nthhomotopy group of X with coefficients in G is ssn(X; G) = [M(G; n); X]. A map f : X -! Y induces a homomorphism ssn(f; G) : ssn(X; G) -! ssn(Y ; G), and ss*(f; G) denotes the set of all such homomorphisms. If G = Z, we write ssn(X) * *and ssn(f) for the nthhomotopy group and induced map, respectively. We use unreduced Lusternik-Schnirelmann category of a space X, denoted cat(X). Thus cat(X) 2 if and only if X is a co-H-space. By an H-space, we mean a space with a homotopy-associative multiplication and homotopy inverse, i.e., a group-* *like space. For a positive integer n, the cyclic group of order n is denoted Z=n. If X is* * a space or an abelian group, we use the notation X(p)for the localization of X at the p* *rime p [12]. We let : X -! X(p)denote the natural map from X to its localization. A semigroup is a set S with an associative binary operation, denoted by juxta- position. We call S a pointed semigroup if there is an element 0 2 S such that x0 = 0x = 0 for each x 2 S. A pointed semigroup is nilpotent if there is an int* *eger n such that the product x1. .x.nis 0 whenever x1; : :;:xn 2 S. The least such int* *eger n is thenilpotency of S. If S is not nilpotent, then we say its nilpotency is 1. * *Finally, if x is a real number, then dxe denotes the least integer n x. 2. F-Trivial Homotopy Classes Let F be any collection of spaces. Definition 2.1. A homotopy class f : X -! Y is F-trivial if the induced map f* : [A; X] -! [A; Y ] is trivial for each A 2 F. We denote by ZF (X; Y ) the s* *ubset of [X; Y ] consisting of all F-trivial homotopy classes. We denote ZF (X; X) by ZF* * (X). HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 3 We study ZF (X; Y ) and ZF (X) for certain collections F. The following are s* *ome interesting examples. Examples. fi (a)S = {Sn fin 1}, the collection of spheres. In this case f 2 ZS(X; Y ) if a* *nd only if ss*(f) =f0.i (b)M = {M(Z=m; n) fim 0; n 1}, the collection of Moore spaces M(Z=m; n). Here f 2 ZM (X; Y ) if and only if ss*(f; G) = 0 for any finitely generated* * abelian group G. (c) = {A}, the collection of all suspensions. In this case f 2 Z (X; Y ) if and only if f* = 0 : [A; X] -! [A; Y ] for every space A. (d)P is the collection of all finite dimensional complexes. Then f 2 ZP (X; Y* * ) if and only if f : X -! Y is a phantom map [18]. In this paper our main interest is in the collections S, M and . In Section 7* * we will mention a few other collections. However, we begin with some simple, gene* *ral facts about arbitrary collections. Lemma 2.2. (a)If F F0, then ZF0(X; Y ) ZF (X; Y ) for any X andWY . (b)If Xff2 F for each ff in some index set, then ZF ( Xff; Y ) = 0 for every * *Y . (c)For any X, ZF (X) is a pointed semigroup under the binary operation of comp* *o- sition of homotopy classes, and with zero the constant homotopy class. __ Any map f : X -! Y gives rise to functions ef: X x Y -! X x Y and f : X _ Y -! X _ Y defined by the diagrams ef _f X x Y ____//_XOxOY and X _ Y ____//_XO_OY p1|| |i2| q1|| j2|| fflffl|f | fflffl|f | X _________//Y X ________//_Y: __ Clearly, if f 2 ZF (X; Y ), then ef2 ZF (X x Y ) and f 2 ZF (X _ Y ). The following lemma, whose proof is obvious, will be used frequently. Lemma 2.3. The functions : ZF (X; Y ) -! ZF (X x Y ) and OE : ZF (X; Y ) -! ZF (X _ Y ); __ defined by (f) = efand OE(f) = f are injective. Thus, ZF (X; Y ) 6= 0 implies * *that ZF (X x Y ) 6= 0 and ZF (X _ Y ) 6= 0 We conclude this section with some basic results about Z (X; Y ) and Z (X). Recall that a map f : X -! Y has essential category weight at least n, written E(f) n, if for every space A with cat(A) n, we have f* = 0 : [A; X] -! [A; Y ] ([29] and [24]). 4 MARTIN ARKOWITZ AND JEFFREY STROM Lemma 2.4. For any two spaces X and Y , Z (X; Y ) = {f | f 2 [X; Y ]; E(f) 2} = {f | f 2 [X; Y ]; f = 0}: Proof Let f 2 Z (X; Y ). If cat(A) 2 then the canonical map : A -! A has a section s. Thus the diagram f* [A; X]_______//_[A; Y ] * || *|| fflffl|f** fflffl| [A; X] ____//_[A; Y ] commutes, and so f* = s*f*** = 0. Since the reverse implication is trivial, th* *is establishes the first equality. Now assume that f* = 0 : [B; X] -! [B; Y ] for every space B. Taking B = X, we find that f O = 0 : X -! Y . Since this map is adjoint to f, we conclude that f = 0. Conversely, if f = 0, then f* = 0 : [B; X] -! [B; Y ], which means that f* = 0 : [B; X] -! [B; Y ]. This completes the proof. * *|| Remarks. (a)Since cat(A) 2 if and only if A is a co-H-space, a map f : X -! Y has E(f) 2 if and only if f* = 0 : [A; X] -! [A; Y ] for every co-H-space A. (b)By Lemma 2.4, we can regard the set Z (X; Y ) as the kernel of the looping function : [X; Y ] -! [X; Y ]. We see from (a) that ker = 0 if X is a co-H- space. The function has been extensively studied in special cases, e.g., w* *hen Y is an Eilenberg-MacLane space, then is just the cohomology suspension [3* *0, Chap.VII]. Proposition 2.5. Let X be a space of finite category, and let n log2(cat(X)).* * If f1; : :;:fn 2 Z (X), then f1O. .O.fn = 0. Thus the nilpotency of the semigroup * *Z (X) is at most dlog2(cat(X))e, the least integer greater than or equal to log2(cat(* *X)). Proof Since fi 2 Z (X), Lemma 2.4 shows that E(fi) 2. By the product formula for essential category weight [28,Thm.9], E(f1O. .O.fn) E(f1) . .E.(fn) 2n cat(X* *). From the definition of essential category weight, f1 O . .O.fn = 0. * * || Remark. We shall see later that the semigroup ZS(X) is nilpotent if X is a fin* *ite dimensional complex. It follows that this is true for ZM (X) and Z (X) (Remark * *(b) following Theorem 3.3). Definition 2.6. For any collection F of spaces and any space X, we define tF (X* *), the nilpotency of X mod F as follows: If X is contractible, set tF (X) = 0; Otherwise, tF (X) is the nilpotency of the semigroup ZF (X). HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 5 Thus, tF (X) = 1 if and only if X is not contractible and ZF (X) = 0. The set ZS(X) and the integer tS(X) were considered in [2], where they were written Z1 (X) and t1 (X). Since S M , we have 0 t (X) tM (X) tS(X) 1 for any space X. Since cat(A1 x . .x.Ar) r + 1 [15, Prop.2.3], we have the following result. Corollary 2.7. For any r spaces A1; : :;:Ar, t (A1 x . .x.Ar) log2(r + 1) : This paper is devoted to a study of the sets ZF (X; Y ), with emphasis on the nilpotency of spaces mod F for F = S; M and . 3. F-Killing Length and F-Cone Length Proposition 2.5 shows that dlog2(cat(X))e is an upper bound for t (X). In th* *is section, we obtain upper bounds on tF (X) for arbitrary collections F. We begin* * with the main definitions of this section. Definition 3.1. Let F be a collection of spaces and X a space. Suppose there is* * a sequence of cofibrations Li- ! Xi- ! Xi+1 for 0 i < m such that each Liis a wedge of spaces which belong to F. If X0 X * *and Xm *, then this is called an F-killing length decomposition of X with length m. If X0 * and Xm X, then this is an F-cone length decomposition with length m. Define the F-killing length and the F-cone length of X, denoted by klF(X) and clF(X), respectively, as follows. If X *, then klF(X) = 0; otherwise, klF(* *X) is the smallest integer m such that there exists an F-killing length decomposit* *ion of X with length m. The F-cone length of X is defined analogously. The main result of this section is that klF(X) is an upper bound for tF (X). * *We need a lemma. f g Lemma 3.2. If X -! Y -! Z is a sequence of spaces and maps, then there is a cofiber sequence of mapping cones Cf -! Cgf- ! Cg, where the maps are induced by f and g. The proof is elementary, and hence omitted. Theorem 3.3. If F is any collection of spaces and X is any space, then tF (X) klF(X): If F is closed under suspensions, then klF(X) clF(X). 6 MARTIN ARKOWITZ AND JEFFREY STROM Proof Assume that klF(X) = m > 0 with F-killing length decomposition fi pi Li- ! Xi- ! Xi+1 for 0 i < m. Let g0; : :;:gm-1 2 ZF (X) and consider the following diagram, wi* *th dashed arrows to be inductively defined below: f0 g0 g1 g2 gm-2 gm-1 L0 _____//_X0 X ____//_X____//_X___//_._._.//_X____//X;;v<< z >>_ ">> p0|| vv z z __ "" f1 fflffl|g00vvzz _ _ "" L1 ________//X1 zzg0 _ " z 1 _ " p1|| zz __ "" f2 fflffl|z _ _ " L2 ________//X2 _ "" _ _ g0m-2" p2|| _ "" fflffl| _ " g0m-1 .. _ _ "" . _ " _ " pm-2 || __ " fm-1 fflffl|_ " " Lm-1 ______//Xm-1 "" " pm-1 || "" fflffl|" Xm : Since L0 is a wedge of members of F and g0 2 ZF (X), we have g0 O f0 = 0 by Lemma 2.2(b). Thus there is a map g00: X1- ! X extending g0. The same argument inductively defines g0ifor each i, and shows gm-1 . .g.1g0 = g0m-1O(pm-1 . .p.1* *p0). Now gm-1 . .g.1g0 = 0 since Xm *. This proves the first assertion. Next we let m = clF(X), and show that klF(X) m. Let fi pi Li- ! Xi- ! Xi+1 for 0 i < m be an F-cone length decomposition of X. Set hi= (pm-1 pm-2 . .p.i+1) O pi: Xi_______//_Xm X for i < m and hm = id. Since hi= hi+1O pi, Lemma 3.2 yields cofiber sequences Cpi-! Chi-! Chi+1; for 0 i < m. This is a killing length decomposition of X. To see this, observe* * that Cpi Li, which is a wedge of spaces in F because F is closed under suspension. Furthermore, h0 : X0 * -! X, so Ch0 Xm X. Finally, Chm 0 because hm = id: X -! X. || HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 7 Remarks. (a)The notion of cone length has been extensively studied. The version in Defi* *nition 3.1 is similar to the one given by Cornea in [6] (see (c) below). It is pre* *cisely the same as the definition of F-Cat given by Sheerer and Tanre [25]. The F-cone length clF(X) can be regarded as the minimum number of steps needed to build the space X up from a contractible space by attaching cones on wedges of sp* *aces in F. The notion of F-killing length is new and also appears in [2] for the* * case F = S. It can be regarded as the minimum number of steps needed to destroy X (i.e. go from X to a contractible space) by attaching cones on wedges of spaces in F. We note that Theorem 3.3 appears in [2, Thm. 3.4] for the case F = S. For the collection S, it was shown in [2, Ex.6.8] that the inequalit* *ies in Theorem 3.3 can be strict. (b)A space need not have a finite F killing length or F-cone length decomposit* *ion. For example, kl (CP1 ) = 1 because all 2n-fold cup products vanish in a spa* *ce X with kl (X) n. However, if X is a finite dimensional complex, then the process of attaching i-cells to the (i-1)-skeleton provides X with a S-cone* * length decomposition. Thus in this case, klS(X) clS(X) dim(X). Since S M , it follows that kl (X) klM(X) klS(X) and cl (X) clM(X) clS(X), and so dim(X) is an upper bound for all of these integers. If X is a 1-conn* *ected finite dimensional complex, then a better upper bound for clM(X) is the num* *ber of nontrivial positive-dimensional integral homology groups of X. This can * *be seen by taking a homology decomposition of X [11, Chap.8]. (c)It follows from work of Cornea [6] that the cone length of a space X, denot* *ed cl(X), can be defined exactly like the -cone length cl (X) above, except th* *at one does not require L0 2 . It follows immediately that cl(X) cl (X). (d)The inequality klF(X) clF(X) also follows from work of Sheerer and Tanre since the function klF satisfies the axioms for F-Cat [25, Thm.2]. We conclude this section by giving a few properties of killing length. j q Theorem 3.4. If F is any collection of spaces and X -! Y -! Z is a cofiber * *se- quence, then klF(Y ) klF(X) + klF(Z): Proof Write klF(X) = m and klF(Z) = n. Let fi Li- ! Xi- ! Xi+1 for 0 i < m be a F-killing length decompositiongoffX. Set g0 = j : X0 X -! Y and define Y1 by the cofibration L0- ! Y -! Y1. 0By0Lemma 3.2, there is an aux* *illiary cofibration Cf0_____//Cg0f0___//Cg0 || || || || || || || g1 || || X1 ______//Y1_____//_Z 8 MARTIN ARKOWITZ AND JEFFREY STROM which defines g1. We proceed by induction: given gi: Xi- ! Yi, let Yi+1be the c* *ofiber of the map gifi: L0- ! Yi and use Lemma 3.2 to construct an auxilliary cofibrat* *ion Cfi _____//Cgifi__//_Cgi || || || || || || || gi+1 || || Xi+1 _____//Yi+1____//_Z which defines gi+1. This defines cofiber sequences of the form Lj- ! Yj- ! Yj+1* * with 0 j < m. Since Xm *, the (m + 1)stcofiber sequence, Xm -! Ym -! Z, shows that Ym Z. Now adjoin the n cofiber sequences of a minimal F-killing length decomposition of Z to the first m cofiber sequences to obtain an F-killing leng* *th decomposition for Y with length n + m. || Finally, we obtain an upper bound for kl (X) and hence an upper bound for t (* *X). This provides a useful complement to Proposition 2.5 when cat(X) is not known. Proposition 3.5. Let X be an N-dimensional complex which is (n - 1)-connected for some n 1. Then ss N + 1 kl (X) log2 ______ : n Proof We argue by induction on log2 N+1_n . If log2 N+1_n = 1, then N 2n - 1. It is well known that this implies that X is a suspension, which means that kl (X) = * *1. Now suppose log2 N+1_n = r and the result is known for all smaller values. Let* * Xk denote the k-skeleton of X, and consider the cofiber sequence X2n-1- ! X -! X=X2n-1: By Theorem 3.4, kl (X) kl (X2n-1) + kl (X=X2n-1). The inductive hypothesis applies to X2n-1 and to X=X2n-1, so kl (X) 1 + (r - 1) = r. || 4. Distinguishing ZF for Different F We have a chain of pointed sets Z (X; Y ) ZM (X; Y ) ZS(X; Y ): Simple examples show that each of these containments can be strict. There are n* *on- trivial phantom maps CP1 - ! S4 [18]. Each of these lies in ZM (CP1 ; S4) becau* *se M P (see Examples in Section 2), but not in Z (CP1 ; S4), by Lemma 2.2(b). For the other containment, the Bockstein applied to the fundamental cohomology clas* *s of M(Z=p; n) [3] corresponds to a map f : M(Z=p; n) -! K(Z=p; n + 1). If p is an o* *dd prime, then ssn+1(M(Z=p; n)) = 0 [3, pp.268-69], so f is in ZS(M(Z=p; n); K(Z=p* *; n+ 1)). Since it is essential, f cannot lie in ZM (M(Z=p; n); K(Z=p; n + 1)). In these examples either the domain or the target is an infinite CW complex. Thus they leave open the possibility that if X and Y are finite complexes, all * *of the HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 9 pointed sets above are the same. We will give examples which show that, even f* *or finite complexes, these inclusions can be strict. These examples are more diffi* *cult to find and verify. They are inspired by an example (due to Fred Cohen) from [8]. Recall that if p is an odd prime, then S2n+1(p)is an H-space [1]. Moreover, i* *f f is in the abelian group f 2 [2X; S2n+1] then the order of O f 2 [2X; S2n+1(p)] is ei* *ther infinite or a power of p. Lemma 4.1. Let X be a finite complex and let h : X -! S2n+1 be a map such th* *at for some odd prime p, O 2h is nonzero and has finite order divisible p. Then t* *here is an s > 0 such that the composite X -h! S2n+1- i!M(Z=ps; 2n + 1) is essential. Proof Consider the diagram ______________//_2n+1 S2n+1 S(p) | | ps|| ps| h fflffl| fflffl|2n+1 X _________//_MMMS2n+1_________//_S(p) MMM MMM i| j| iOhMMM&&Mfflffl|| fflffl|| M(Z=ps; 2n + 1) __=_//_M(Z=ps; 2n + 1) in which the vertical sequences are cofibrations and ps denotes the map with de* *gree ps. If i O h = 0, then j O O h = 0. It can be shown that ( O h) lifts through * *the map ps : S2n+2(p)-!S2n+2(p). Suspending once more, we obtain a lift given by the da* *shed line in the diagram lS2n+3(p)55 ll l ll l ps|| l l l fflffl| 2n+3 2X _2h__//S2n+3___//_S(p) : Since X is a finite complex, the torsion in [2X; S2n+3(p)] is p-torsion and has* * an ex- ponent e. Since S2n+3(p)is an H-space, the map ps induces multiplication by ps* * on [2X; S2n+3(p)]. If s e, then the image of ps : [2X; S2n+3(p)] -! [2X; S2n+3(p* *)] cannot contain any nontrivial torsion. But O 2h is nonzero and has finite order. Ther* *efore the lift cannot exist, and so i O h 6= 0. * * || 10 MARTIN ARKOWITZ AND JEFFREY STROM Theorem 4.2. Let X be a finite complex, let p be an odd prime and let g : X -* *! S2n+1 be an essential map. (a)Assume that ss2n+1(X) is a finite group, and that O 2g is nonzero with fin* *ite order divisible by pn+1. Then there is an s > 0 such that the composite ________l___________________________________________* *______________________________________________________________________@ _______________________________**___________________________* *______________________________________________________________________@ X __g_//_S2n+1pn_//_S2n+1i__//M(Z=ps; 2n + 1) is essential and ss*(l) = 0. (b)Assume that ssk(X) = 0 for k = 2n and 2n + 1, and that O 2g is nonzero with finite order divisible by p2n+1. Then there is an s > 0 such that the compo* *site f ____________________________________________________* *______________________________________________________________________@ _______________________________**___________________________* *______________________________________________________________________@ X __g_//_S2n+1p2n//_S2n+1i__//M(Z=ps; 2n + 1) is essential, and ss*(f; G) = 0 for any finitely generated abelian group G. Proof In part (a), the composition O 2(pn O g) has finite order divisible by p. Ther* *efore Lemma 4.1 shows that l = i O pn O g is essential if s is large enough. Similarl* *y, if s is large enough, the map f in part (b) is essential. From now on, we assume that s* * has been so chosen. We use the commutative diagram g pk i s X ____//_CS2n+1__//_S2n+1___//M(Z=p ; 2n + 1) CC | | CCC | | |= Og C!!Cfflffl| fflffl| fflffl|| pk 2n+1j S2n+1(p)__//_S(p)___//_M(Z=ps; 2n + 1): We take k = n in part (a) and k = 2n in part (b). Proof of (a) Since M(Z=ps; 2n + 1) is p-local, there is only p-torsion to cons* *ider. By results of Cohen, Moore and Neisendorfer [5, Cor. 3.1], the p-torsion in ss** *(S2n+1(p)) has exponent n. Since S2n+1(p)is an H-space, pn : S2n+1(p)-!S2n+1(p)annihilate* *s all p- torsion in homotopy groups. Thus ss*(l) can be nonzero only in dimension 2n + * *1. But ss2n+1(g) is a homomorphism from a finite group to Z, so ss*(l) = 0. Proof of (b) It suffices to show that ssm (f; G) = 0 for any cyclic group G; b* *y part (a) we need only consider G = Z=pr. For each r 1 and each m 0, there is the exact coefficient sequence [11, Chap.5] 0 -! Ext (Z=pr; ssm+1 (Y )) -! ssm (Y ; Z=pr) -! Hom (Z=pr; ssm (Y )) -!* * 0: Let Y = S2n+1(p). Since the p-torsion in ss*(S2n+1(p)) has exponent n [5], the * *exact sequence shows that the p-torsion in ssm (S2n+1(p); Z=pr) has exponent at most 2n if m 6* *= 2n. Thus the map p2n : S2n+1(p)-!S2n+1(p)induces 0 on the mthhomotopy groups with coeffi* *cients in any finite abelian group if m 6= 2n. Taking Y = X in the coefficient seque* *nce, HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 11 we have ss2n(X; Z=pr) = 0. Therefore ss*(f; G) = 0 for any finitely generated a* *belian group G. || We apply this theorem to construct examples of finite complexes which disting* *uish the various ZF . Our first example shows that ZM (X) can be different from ZS(X) even when X is a finite complex. Using the coefficient exact sequence for homotopy groups, we * *find that [M(Z=pr; 2n); S2n+1] = ss2n(S2n+1; Z=pr) ~=Z=pr for each r; this is a stable group. Therefore, if r > n, there are essential m* *aps g : M(Z=pr; 2n) -! S2n+1 with finite order divisible by pn+1. Applying part (a* *) of Theorem 4.2, we have the following example. Example. Let r > n > 1. For p an odd prime and s large enough, there are essential maps l : M(Z=pr; 2n) -! M(Z=ps; 2n + 1) such that ss*(l) = 0. Therefore by Lemma 2.3, r s ZS M(Z=p ; 2n) _ M(Z=p ; 2n + 1) 6= 0 while, of course, r s ZM M(Z=p ; 2n) _ M(Z=p ; 2n + 1) = 0 by Lemma 2.2 (b). It can be shown that any s r will suffice in this example. Freyd's generating hypothesis [9] is the conjecture that no stably nontrivial* * map between finite complexes can induce zero on stable homotopy groups. The map l in this example is stably nontrivial, but our argument does not show that lar* *ge suspensions of l induce zero on homotopy groups; the difficulty is that after t* *wo suspensions, l factors through pn : S2n+3(p)-!S2n+3(p), which need not annihila* *te all p-torsion. i 2n+1 j Our second example is a map f : 2n-2 CPp =S2 -! M(Z=ps; 2n + 1) which we use to show that ZM (X) can be different from Z (X) when X is a finite compl* *ex. We need some preliminary results to show that Theorem 4.2 applies to this situa* *tion. Lemma 4.3. Let f : n+1CPm -! Sn+3. The degree of f|n+1S2 is divisible by lcm(1; : :;:m), the least common multiple of 1; : :;:m. Proof We may assume that f is in the stable range. If f|n+1S2 has degree d, then f n+3 n+1 m n+1CPm -! S ,! CP : has degree d in Hn+3(n+1CPm ) and is trivial in all other dimensions. According* * to McGibbon [18, Thm.3.4], d is divisible by lcm(1; : :;:m). * * || 12 MARTIN ARKOWITZ AND JEFFREY STROM Proposition 4.4. The image of the n-fold suspension map t 2 3 n pt 2 n+3 n : [CPp =S ; S ] -! [ (CP =S ); S ] contains elements of order pt for every n 1 and t 1. Proof Write m = pt and examine the commutative diagram [CPm ; S3] _________//_[S2; S3]_________//[CPm =S2; S3] |n| n|| |n| fflffl| fflffl| fflffl| [n+1CPm ; Sn+3] _____//[n+1S2; Sn+3]____//[n(CPm =S2); Sn+3] |*| *|| |*| fflffl| fflffl| fflffl| [n+1CPm ; Sn+3(p)]___//[n+1S2; Sn+3(p)]//_[n(CPm =S2); Sn+3(p)]: To show that the image of * O n : CPm =S2; S3] -! [n(CPm =S2); Sn+3(p)] contains elements of order pt, we modify the above diagram as follows: the image and cok* *er- nel of [CPm ; S3] -! [S2; S3] ~= Z are kZ and Z=k, respectively, for some integ* *er k; similarly for [n+1CPm ; Sn+3] -! [n+1S2; Sn+3] and [CPm ; S3(p)] -! [S2; S3(* *p)]. Thus we have a commutative diagram with exact rows kZ _______//Z_____//Z=k | | | |~ | | |= | fflffl| fflffl| fflffl| lZ _______//Z_____//Z=l || || || fflffl| fflffl| fflffl| lpZ(p)____//Z(p)___//Z=lp for some integers k, l and lp, where lp is the largest power of p which divides* * l. Lemma 4.3 shows that lp is divisible by pt. The composite Z=k -! Z=l -! Z=lp is surje* *ctive, and this completes the proof. * *|| It follows from Proposition 4.4 that part (b) of Theorem 4.2 applies to the s* *pace 2n+1 2 2n-2(CPp =S ) for each n > 1, and so we obtain our second example. Example. For each odd prime p and each n 1, there is an s > 0 such that there are essential maps i 2n+1 j f : 2n-2 CPp =S2 -! M(Z=ps; 2n + 1) which induce zero on homotopy groups with coefficients. Therefore, i i 2n+1 j j ZM 2n-2 CPp =S2 _ M (Z=ps; 2n + 1) 6= 0 HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 13 while, of course, i i 2n+1 j j Z 2n-2 CPp =S2 _ M (Z=ps; 2n + 1) = 0: The map f can be chosen to be stably nontrivial. As in the previous example, the suspensions of f might not be trivial on homotopy groups with coefficients. 2n+1 2 r s Finally, let A = 2n-2(CPp =S ), B = M(Z=p ; 2n) and C = M(Z=p ; 2n + 1) for s large. Then Z (A _ B _ C) < ZM (A _ B _ C) < ZS(A _ B _ C); so both of these inequalities can be strict for a single finite complex. 5. Projective Spaces We show that for projective spaces FPn with F = R; C or H, Z (FPn) = ZM (FPn) = ZS(FPn); and we completely determine these sets for F = R and C and all n. We also deter* *mine tS(HPn), for n 4. 5.1. General Facts. We first prove some general results that will be applied la* *ter. W Proposition 5.1. If X Snff, then ZS(X; Y ) = ZM (X; Y ) = Z (X; Y ) for any space Y . Proof f Let f 2 ZS(X;WY ). The map f is adjoint to the composition X -! X -! Y . Since X Snff, f O = 0, and so f = 0. Thus f 2 Z (X; Y ). || By Lemma 2.4, the condition ZS(X; Y ) = Z (X; Y ) is equivalent to the condit* *ion that if f : X -! Y induces zero on homotopy groups, then f = 0. W Proposition 5.1 applies to X = Sn+1 because, according to James [13], Sn+1 1 nk+1 n+1 k=1S . Of course ZS(S ; Y ) = 0. Since (A x B) A _ B _ (A ^ B) for any A and B [11, 11.10], James's result allows us to apply Proposition 5.1 * *to any space whose loop space splits as a finite type product of spheres and loop spac* *es on spheres. Moreover, if X and X0 both satisfy the hypotheses of Proposition 5.1, * *then so does X x X0. For F = R; C or H, let d = 1; 2 or 4, respectively. For each n 1 there is a homotopy equivalence FPn Sd-1 x S(n+1)d-1[10]. Corollary 5.2. For F = R; C or H and each n 1, ZS(FPn) = ZM (FPn) = Z (FPn). 14 MARTIN ARKOWITZ AND JEFFREY STROM Another corollary of Proposition 5.1 applies to intermediate wedges of sphere* *s. For spaces X1; X2; : :;:Xn, the elements (x1; : :;:xk) 2 X1 x . .x.Xk with at least* * j coordinates equal to the base point form a subspace Tj(X1; : :;:Xk) X1x . .x.X* *k. Porter has shown [23, Thm. 2] that Tj(Sn1; : :;:Snk) has the homotopy type of a product of loop spaces of spheres for each 0 j k. Our previous discussion establishes the following. Corollary 5.3. For any n1; : :;:nk 1 and any 0 j k, ZS(Tj(Sn1; : :;:Snk)) = ZM (Tj(Sn1; : :;:Snk)) = Z (Tj(Sn1; : :;:Snk)): Remarks. (a)Taking j = 0 in Corollary 5.3, we deduce from Corollary 2.7 that tS(Sn1 x . .x.Snk) log2(k + 1) : This reproves [2, Prop.6.2] by a different method. (b)It is proved in [2, Prop. 6.5] that for any positive integer n, there is a * *finite product of spheres X with tS(X) = n. By Corollary 5.3, the same is true for t (X) and tM (X). Thus the integers tF (X) for F = S; M or and any X take on all positive integer values. Finally, we observe that the splitting of FPn gives a useful criterion for de* *ciding when a map f : FPn -! Y lies in ZS(FPn; Y ). Proposition 5.4. Let i be the inclusion Sd = FP1 ,! FPn, and let p : S(n+1)d-1* *-! FPn be the Hopf fiber map. Then the map (i; p) : Sd _ S(n+1)d-1-! FPn induces a sur* *jec- tion on homotopy groups. Therefore, a map f : FPn -! Y satisfies ss*(f) = 0 if * *and only if f O i = 0 and f O p = 0. 5.2. Complex Projective Spaces. Next we show that certain skeleta X of Eilenber* *g- MacLane spaces have the property that ZS(X) = 0. We apply this to CPn and nCP2 for each n. Let G be a finitely generated abelian group. Give the Eilenberg-MacLane space K(G; n) with n 2 a homology decomposition [11, Chap. 8] and denote the mth section by K(G; n)m . Thus K(G; n) is filtered * K(G; n)n . . .K(G; n)m . . .K(G; n) and there are cofiber sequences M(Hm+1 (K(G; n)); m) -! K(G; n)m -! K(G; n)m+1* * . Theorem 5.5. If the group Hm (K(G; n)) is torsion free and Hm+1 (K(G; n)) = 0, then ZS(K(G; n)m ) = 0. Proof We write X = K(G; n)m . Then Hk(K(G; n); X) = 0 for k m + 1. By Whitehead's theorem [30, Thm. 7.13], the induced map ssk(X) -! ssk(K(G; n)) is an isomorphi* *sm HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 15 for k m. Since Hm (K(G; n)) is torsion free, X has dimension at most m, and so X has a CW decomposition _ Sn = Xn Xn+1 . . .Xm X: For f 2 ZS(X) we prove by induction on k that f factors through X=Xk for each k m. The first step is trivial since ssn(f) = 0 implies f|Xn = 0. Inductively,* * assume that f factors through X=Xk with n k < m. There is a cofibration _ Sk+1 Xk+1=Xk -! X=Xk -! X=Xk+1: Since n < k + 1 m, it follows that ssk+1(X) ~=ssk+1(K(G; n)) = 0, so f extends* * to X=Xk+1. Taking k = m, we find f = 0. || Remark. Clearly, ssk(K(G; n)m ) = 0 for n < k < m. The hypotheses in Theorem 5.5 are needed to conclude further that ssm (K(G; n)m ) = 0. As an application of Theorem 5.5, we have the following calculations. Theorem 5.6. (a)ZF (CPn) = 0 for each n 1 and each F = ; M or S. (b)ZF (nCP2) = 0 for each n 1 and each F = ; M or S. Proof By Proposition 5.1 it suffices to consider the case F = S. Since CP1 = K(Z; 2) * *and the CPn are the sections of a homology decomposition of CP1 , part (a) follows * *from Theorem 5.5. Recall from [7] that for n 2 ae Z if k = n or n + 2 Hk(K(Z; n)) = 0 if k = n + 1 or n + 3. Since Sq2 is nontrivial on Hn(K(Z; n); Z=2), we have K(Z; n)n+2 n-2CP2. Thus Theorem 5.5 applies to n-2CP2. || This theorem immediately shows that tF (CPn) = tF (nCP2) = 1 for F = ; M or S and each n 1. 5.3. Real Projective Spaces. In this subsection we completely calculate ZS(RPn). By the Hopf-Whitney theorem [30, Cor. 6.19], [RP2n; S2n] ~=H2n(RP2n) ~=Z=2. The unique nontrivial map q : RP2n- ! S2n is theqquotientpmap obtained by factoring* * out RP2n-1. Let f2n denote the composite RP2n- ! S2n -! RP2n where p is the univers* *al covering map. Theorem 5.7. For F = ; M or S and each n 1, (a)ZF (RP2n-1) = 0 (b)ZF (RP2n) = {0; f2n}. 16 MARTIN ARKOWITZ AND JEFFREY STROM Proof Let f : RPm -! RPm with ss1(f) = 0. Because ssk(RPm ) = 0 for 1 < k < m, an argument similar to the proof of Theorem 5.5 shows that f must factor through q : RPm -! Sm . For m > 1, any map Sm -! RPm lifts through p : Sm -! RPm . Thus there is a map g : Sm -! Sm of degree d which makes the following diagram commute p f Sm G____//_GRPm___//_RPmOO GGG q| p| qOpGG##Gfflffl|| | g | Sm _____//_Sm : First let m = 2n - 1. We may assume n > 1. The composite q O p : S2n-1- ! S2n* *-1 is known to have degree 2. Since ssi(p) is an isomorphism for i > 1, f O p repr* *esents 2d 2 Z ~=ss2n-1(RP2n-1). If f 2 ZS(RP2n-1), then d must be 0, and so f = 0. This proves (a). Now take m = 2n. The composite q O p : S2n -! S2n is trivial because it is ze* *ro on homology. Therefore f O p = 0, and since ss1(f) = 0, Proposition 5.4 shows t* *hat f 2 ZS(RP2n). Since RP2n is connected, there is a bijection ~= fi p* : [RP2n; S2n]________//f fif 2 [RP2n; RP2n]; ss1(f) = 0 = ZS(RP2n): Since [RP2n; S2n] = {0; q} as noted above, ZS(RP2n) = {0; f2n}, where f2n = p O* * q. || Remark. This argument actually shows that, if ssk(Y ) = 0 for 1 < k < 2n+1, th* *ere is a bijection between ZS(RP2n+1; Y ) and the set of elements ff 2 ss2n+1(Y ) s* *uch that 2ff = 0. Corollary 5.8. For each n 1, (a)tF (RP2n-1) = 1 for F = ; M and S. (b)tF (RP2n) = 2 for F = ; M and S. Proof It suffices to prove part (b) for F = S. Since ZS(RP2n) 6= 0, tS(RP2n) 2. The only possibly nonzero product in this semigroup is f2nO f2n. But this is zero b* *ecause ZS(RP2n) is nilpotent by Theorem 3.3. || 5.4. Quaternionic Projective Spaces. The quaternionic projective spaces are not skeleta of Eilenberg-MacLane spaces, and it is much more difficult to compute t* *heir nilpotency. Let f 2 [HPn+1; HPn+1], and assume that f is cellular. Then f|HPn : HPn -! HPn and the homotopy class f|HPn is well defined. Lemma 5.9. If f 2 ZS(HPn+1), then f|HPn 2 ZS(HPn). HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 17 Proof Let f 2 ZS(HPn+1) and let g = f|HPn. Consider the diagram _ __ h___ _//4n+3 S4n+3 S p || |p| i fflffl| g fflffl| S4 _____//_HPn___________//HPnJ JJJ j || j|| JlJJJ fflffl| f fflffl|JJ$$m HPn+1 __________//HPn+1 ____//_HP1 q || |q| fflffl|h fflffl| S4n+4 _ _ _ _ _ //_S4n+4: p n l 1 where i; j; m and l are inclusions. Since S4n+3- ! HP -! HP can be regarded * *as a fibration and l O (g O p) = m O (f O (j O p)) = 0, it follows that g O p lift* *s to the map h. Since f 2 ZS(HPn+1), f induces zero on H4(HPn+1), and hence is zero in cohomology. Therefore h is zero in cohomology and hence is trivial. Thus h = 0,* * so g O p = 0. Also, g O i = 0, so g 2 ZS(HPn) by Proposition 5.4. * * || Next we indicate how we will apply Lemma 5.9. If ZS(HPn) = 0 and f 2 ZS(HPn+1), then f|HPn = 0, so f factors through q : HPn+1 -! S4n+4. By Proposi- tion 5.4, if i : S4 ,! HPn+1, then ss4n+4(i) is surjective, so f factors as in * *the diagram f m 1 HPn+1 _____//HPn+1____//_HPOO::t tt q || |i| tttt fflffl|g | ttt l S4n+4 ______//_S4: By cellular approximation, f is essential if and only if m O f is essential. T* *he map l O g : S4n+4- ! HP1 is adjoint to a map g0 : S4n+3- ! S3, which in turn is ad* *joint to l O g0. By cellular approximation again, i O g = i O g0, so we may assume th* *at g is in the image of the suspension : ss4n+3(S3) -! ss4n+4(S4). The proof of our main result about quaternionic projective spaces requires so* *me detailed information about homotopy groups of spheres. Since we refer to Toda's* * book [30] for this information, we use his notation here. For example, jk : Sk+1 -! * *Sk and k : Sk+3 -! Sk are suspensions of the Hopf fiber maps. Theorem 5.10. (a)ZF (HPn) = 0 for F = S; M or and n = 1; 2 and 3 (b)ZF (HP4) 6= 0 for F = S; M or . 18 MARTIN ARKOWITZ AND JEFFREY STROM Proof First HP1 = S4, so ZS(HP1) = 0. If f 2 ZS(HP2), then there is a commutative diagram f m 1 HP2 _____//HP2____//_HPOO;;v vv q || |i| vvv fflffl|g | vvv l S8 _____//_S4;;w ww 5 || www fflffl|j4www S5 in which the vertical sequence is a cofibration. If g = 0, then f = 0, so we m* *ay assume that g 6= 0. We know that ss8(HP1 ) ~=ss7(S3) ~=Z=2, generated by j3O4 [* *29, p.43-44]. Thus we can take g = j4 O 5. Since 5 O q = 0, we conclude that g O q * *= 0, so f = 0. This shows that ZF (HP2) = 0. The proof that ZS(HP3) = 0 is similar. Let f 2 ZS(HP3) and apply Lemma 5.9 to get a similar factorization. The resulting map g : S12- ! S4 is either ffl3* * or 0 [29, Thm. 7.1]. If g = ffl3, then results of [14, (2.20a)] and [29, Thm. 7.4] s* *how that f O p 6= 0. Thus g = 0 and so f = 0. For part (b), we make use of the diagram preceding Theorem 5.10 and the fact * *that g can be taken to be a suspension map. If f 2 ZS(HP4), then we have p f m 1 S19 ____//_EHP4___//_HP4___//_HPOO;; EE | | vvv EEE q| |i vvvv E""Efflffl|g| vv l S16 _____//_S4: According to Toda [30], ss16(S4) = (ss15(S3)) ~=Z=2 Z=2. By [14, (2.20a)], q O* * p is 4162 ss19(S16). Then g O (416) = 4g O 16 because 16 is a suspension. Since 4g =* * 0, any map HP4- ! HP4 which factors through q lies in ZS(HP4). Marcum and Randall show in [17] that the map (i O 0) O 7) O q : HP4__________//HP4 is essential, where 0 2 ss6(S3) generates the 2-torsion and 7 2 ss16(S7) genera* *tes a Z=2 summand [29, Thm.7.2]. Thus ZS(HP4) 6= 0, and so tS(HP4) 2. || As before, we obtain the nilpotency. Corollary 5.11. (a)tF (HP1) = tF (HP2) = tF (HP3) = 1 for F = S; M or (b)tF (HP4) = 2 for F = S; M or . Proof It suffices to prove that tS(HP4) 2. Suppose f; g 2 ZS(HPn). The proof of Theo* *rem 5.10 shows that f factors through S16. Now g O f = 0 because g 2 ZS(HP4). * *|| HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 19 6. H-Spaces In this section we study the nilpotency of H-spaces Y mod . We make calculati* *ons for specific Lie groups such as SU(n) and Sp(n) and show that Z is nontrivial * *in these cases. If Y is an H-space, the Samelson product of ff 2 ssm (Y ) and fi 2* * ssn(Y ) is written 2 ssn+m (Y ) [30, Chap.X]. We first give a few general results which are needed later. Lemma 6.1. If Y is an H-space and 6= 0 for some ff 2 ssm (Y ) and fi* * 2 ssn(Y ), then [Sm x Sn; Y ] is not abelian. Proof The quotient map q : Sm x Sn -! Sm ^ Sn Sm+n induces a monomorphism q* : [Sm ^ Sn; Y ] -! [Sm x Sn; Y ] such that q* = [ffp1; fip2], the commuta* *tor of ffp1 and ffp2. || It is well known that if an H-space Y is a finite complex, then it has the s* *ame rational homotopy type as a product of spheres S2n1-1x . .x.S2nr-1with n1 . . . nr. If p is an odd prime such that 2n -1 2n -1 2n -1 2n -1 Y(p) S 1 x . .x.S r (p) S(p1) x . .x.S(pr) ; then p is called a regular prime for Y . If Y is a simply-connected compact L* *ie group, then p is regular for Y if and only if p nr [16, Sec.9-2]. We need a second product decomposition for p-localized Lie groups. By [20, Se* *c.2] there are fibrations S2k+1- ! Bk(p) -! S2k+2p-1for k = 1; 2; : :.:An odd prime * *p is called quasi-regular for the H-space Y if ! Y Y Y(p) S2ni-1x Bmj(p) : i j (p) 6.1. The Groups SU(n) and Sp(n). We apply the notions of regular and quasi- regular primes to the Lie group SU(n), which has the rational homotopy type of S3 x S5 x . .x.S2n-1, and to the Lie group Sp(n), which has the rational homoto* *py type of S3 x S7 x . .x.S4n-1. It is well known [20, Thm.4.2] that if p is an od* *d prime then (a)p is regular for SU(n) if and only if p n; p is quasi-regular for SU(n) if* * and only if p > n_2 (b)p is regular for Sp(n) if and only if p 2n; p is quasi-regular for Sp(n) i* *f and only if p > n. It is also known [4, Thm. 1] that if n 3 and r + s + 1 = n, there are generato* *rs ff 2 ss2r+1(SU(n)) ~=Z, fi 2 ss2s+1(SU(n)) ~=Z and fl 2 ss2n(SU(n)) ~=Z=n! such* * that = r!s!fl. If p is an odd prime and ff02 ss2r+1(SU(n)(p)), fi0 2 ss2s+1* *(SU(n)(p)) and fl0 2 ss2n(SU(n)(p)) are the images of ff, fi and fl under the localization* * homo- morphism * : ss*(SU(n)) -! ss*(SU(n)(p)) ~=ss*(SU(n))(p), then = r!s!fl02 ss2n(SU(n))(p)~=Z=n! Z(p): 20 MARTIN ARKOWITZ AND JEFFREY STROM Now we prove the main result of this section. Theorem 6.2. The groups (a)[SU(n); SU(n)] for n 5 and (b)[Sp(n); Sp(n)] for n 2 are not abelian. Proof Consider SU(n) for n 5 and let p be the largest prime such that n_2< p < n. If n 12, then it follows from Bertrand's postulate [26,p.137] that there are two * *primes p and q such that n_2< q < p < n. This implies that 2n + 6 < 4p. For 5 n < 12, and n 6= 5; 7; 11, it is easily verified that 2n + 6 < 4p. Assume that n 5 and that n 6= 5; 7 or 11. Since p > n_2, it follows that p * *is quasi-regular for SU(n). Since 2n + 6 < 4p, the spheres S2n-2p+3 and S2p-3 both appear in the resulting product decomposition. Thus we have 2n-2p+3 2p-3 2p-1 SU(n)(p) B1(p) x . .x.Bn-p(p) x S x . .x.S x S (p): Assume [SU(n); SU(n)] is abelian. Then [SU(n)(p); SU(n)(p)] is abelian, and the* *refore [S2n-2p+3x S2p-3; SU(n)(p)] is abelian. There are ff02 ss2n-2p+3(SU(n)(p)), fi02 ss2p-3(SU(n)(p)) and fl02 ss2n(SU(n)* *(p)) so that = (n - p + 1)!(p - 2)!fl0 in ss2n(SU(n)(p)) ~= Z=n! Z(p)~= Z=p. Since fl0 is a generator of Z=p, we have 6= 0. By Lemma 6.1, [S2n-2p+3x S2p-3; SU(n)(p)] is not abelian, and* * so [SU(n); SU(n)] is not abelian. It remains to prove that [SU(n); SU(n)] is not abelian for n = 5; 7; 11. The * *argu- ment we now give applies to SU(p) for any prime p 5. Notice that p is regular * *for SU(p), so it suffices to show that [S3 x S2p-3; SU(p)(p)] is nonabelian. Since* * p is a regular prime for SU(p), we choose generators ff 2 ss3(SU(p)), fi 2 ss2p-3(SU(p* *)) and fl 2 ss2p(SU(p)) so that = (p - 2)!fl06= 0 2 Z=p! Z(p)~=Z=p: Therefore [S3 x S2p-3; SU(p)(p)] is nonabelian by Lemma 6.1. The proof that [Sp(n); Sp(n)] is not abelian for n 2 is analogous: one uses * *Bott's result for Samelson products in ss*(Sp(n)) [4, Thm. 2] together with a quasi-re* *gular decomposition for Sp(n). We omit the details. * *|| Corollary 6.3. (a)For n 5, Z (SU(n)) 6= 0, and 2 t (SU(n)) dlog2(n)e. (b)For n 2, Z (Sp(n)) 6= 0, and 2 t (Sp(n)) d2 log2(n + 1)e. Proof For an H-space Y , a commutator in [X; Y ] is an element of Z (X; Y ) [28, Thm.* * 7]. Thus, if [Y; Y ] is nonabelian, 2 t (Y ). The upper bound for t (SU(n)) follo* *ws HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 21 from Proposition 2.5 since Singhof has shown that cat(SU(n)) = n [28]. The upper bound on t (Sp(n)) follows from Proposition 3.5. * *|| Schweitzer [25,Ex.4.4] has shown that cat(Sp(2)) = 4, so it follows from Prop* *osition 2.5 that t (Sp(2)) = 2. 6.2. Some Low Dimensional Lie Groups. Here we consider the Lie groups SU(3), SU(4), SO(3) and SO(4) and make estimates of t by either quoting known results or by ad hoc methods. We first deal with SU(3) and SU(4). Proposition 6.4. The groups [SU(3); SU(3)] and [SU(4); SU(4)] are not abelian. Proof For the group SU(3) this follows from results of Ooshima [21, Thm. 1.2] For SU(* *4), observe that the prime 5 is regular for both SU(4) and Sp(2), so SU(4)(5) (S3 x S5 x S7)(5) and Sp(2)(5) (S3 x S7)(5): If [SU(4); SU(4)] is abelian, then so is [SU(4)(5); SU(4)(5)] ~=[S3 x S5 x S7; * *SU(4)(5)], and thus [S3 x S7; SU(4)(5)] is abelian. If ff02 ss3(Sp(2)(5)) and fi02 ss7(Sp(2)(5)) are the images of generators of * *ss3(Sp(2)) ~= Z and ss7(Sp(2)) ~= Z then it follows from [4] that 6= 0 2 ss10(Sp(2* *)(5)) ~= Z=5! Z(5)~=Z=5. Now we relate SU(4) to Sp(2) via the fibration Sp(2) -i!SU(4) -! S5. The exact homotopy sequence of a fibration shows that ss10(i) is an isomorphism after loc* *alizing at any odd prime. Since i is an H-map, = i* 6= 0 2 ss10(SU(4)(5)): Thus [S3 x S7; SU(4)(5)] is not abelian, so [SU(4); SU(4)] cannot be abelian. * * || Corollary 6.5. (a)Z (SU(3)) 6= 0, and t (SU(3)) = 2 (b)Z (SU(4)) 6= 0, and t (SU(4)) = 2. Proof Since the groups [SU(n); SU(n)] are not abelian for n = 3 and 4, t (SU(3)) and t (SU(4)) are at least 2. But cat(SU(n)) = n by [28], so the reverse inequalit* *ies follow from Proposition 2.5. * *|| Next we investigate the nilpotence of SO(3) and SO(4). This provides us with examples of non-simply-connected Lie groups. Proposition 6.6. Z (SO(3)) = 0 and Z (SO(4)) 6= 0. Proof Since SO(3) is homeomorphic to RP3, the first assertion follows from Theorem 5.* *7. For the second assertion, recall that SO(4) is homeomorphic to S3 x SO(3). For notational convenience, we write X = SO(3) and Y = S3. We show that Z (XxY ) 6= 22 MARTIN ARKOWITZ AND JEFFREY STROM 0. Let j : X _ Y -! X x Y be the inclusion and q : X x Y -! X ^ Y be the quot* *ient map. Consider q* : [X ^ Y; X x Y ] -! [X x Y; X x Y ]: Notice that Im(q*) Z (X x Y ) because q 2 Z (X x Y; X ^ Y ), so q induces a function q** : [X ^ Y; X x Y ] -! Z (X x Y ). Consider the exact sequence of gr* *oups j* q* [(X x Y ); X x Y ] -! [(X _ Y ); X x Y ] -! [X ^ Y; X x Y ] -! [X x Y; X x Y ]: Since j* has a left inverse, ker(q*) = 0. Thus q** is one-one, so it suffices t* *o show that [X ^ Y; X x Y ] ~= [3SO(3); SO(3)] [3SO(3); S3] is nonzero. This follows from [31, Cor.2.12], where it is shown that [3SO(3); S3] ~=Z=4 Z=12. * *|| Corollary 6.7. t (SO(3)) = 1 and t (SO(4)) = 2. Proof We only have to show that t (SO(4)) 2. The remark following Theorem 5.7 shows that if f 2 Z (X x Y ) then f|X_Y = 0, so q** is onto. Thus f factors through* * a sphere, so we can proceed as in the proof of Corollary 5.11. * * || 6.3. The Group E (Y ). We conclude the section by relating Z (Y ) to a certain group of homotopy equivalences of Y . For any space X, let E (X) [X; X] be the group of homotopy equivalences f : X -! X such that f = id. This group has been studied by Felix and Murillo [8] and by Pavesic [22]. We note that if Y is an H* *-space, then the function : Z (Y ) -! E (Y ) defined by (g) = id+ g is a bijection of pointed sets. In general does not pre* *serve the binary operation in Z (Y ) and E (Y ). Thus E (Y ) is nontrivial whenever Z* * (Y ) is nontrivial. Proposition 6.8. The groups E (Y ) are nontrivial in the following cases: Y = SU(n), n 3; Y = Sp(n), n 2; and Y = SO(4). The groups E (Y ) are triv- ial in the following cases: Y = SU(2), Sp(1), SO(2) and SO(3). 7. Problems In this brief section we list, in no particular order, a number of problems w* *hich extend the previous results or which have been suggested by this material. 1.Calculate tF (X) for F = S; M or and various spaces X. In particular, what is t (HPn) for n > 4, and t (Y ) for compact Lie groups Y not considered in Section 6? 2.Find general conditions on a space X such that Z (X; Y ) = ZS(X; Y ). One s* *uch was given in Section 5. Is Z (Y ) = ZS(Y ) if Y is a compact simply-connect* *ed Lie group without homological torsion, such as SU(n) or Sp(n)? 3.Find lower bounds for tF (X) in the cases F = S; M or in terms of other numerical invariants of homotopy type. HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 23 4.With F = S; M or , characterize those spaces X such that ZF (X) = 0. 5.What is the relation between kl (X) and dlog2(cat(X))e? 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