Nearly Trivial Homotopy Classes Between Finite Complexes Martin Arkowitz and Jeffrey Strom Abstract We construct examples of essential maps of finite complexes f :* * X -! Y which are trivial of order n. This latter condition implies that for a* *ny space K with cone length n, the induced map f* = 0 : [K, X] -! [K, Y ]. The main result establishes a connection between the skeleta of the infinite * *dimen- sional domains of essential phantom maps and the finite dimensional domai* *ns of maps which are trivial of order n. In particular, there are essential * *maps f : 2i(CPt=S2) -! M(Z=ps, 2l + 3) which are trivial of order n. 2000 MSC: Primary 55P99; Secondary 55M30, 55P60 Keywords: cone length, killing length, weak category, phantom maps. 1 Introduction Let f : X -! Y be a map and consider the class K(f) of all spaces K such that f O h is homotopic to the constant map * for every map h : K -! X. The larger * *the class K(f), the more nearly trivial we consider the map f to be. We are interes* *ted in finding essential maps f : X -! Y such that the class K(f) is large. For examp* *le, if K(f) contains all finite dimensional complexes, then f is a phantom map. Howeve* *r, if f is an essential phantom map, then the domain X must be infinite dimensiona* *l. In this note we study analogs of phantom maps between finite complexes - that i* *s, we study essential maps f : X -! Y of finite complexes for which K(f) contain* *s a large class of finite dimensional spaces. A natural first step is to search for maps f : X -! Y for which K(f) contai* *ns Sn for each n 0, or equivalently, such that ß*(f) = 0 : ß*(X) -! ß*(Y ). We say * *that such a map is trivial of order at least 1. There are numerous examples of maps* * of this kind - for example, the canonical quotient map X x X -! X ^ X. To extend this notion, we will define what it means for a map to be trivial of order at l* *east n. A (spherical) cone length decomposition of length n for a connected space K * *is a sequence of cofibrations Si-! Li-! Li+1, 0 i < n, where Si is a wedge of spheres, L0 * (i.e., L0 is contractible) and Ln K. * *The (spherical) cone length of K, denoted cl(K), is defined as follows: if K is con* *tractible, set cl(K) = 0; otherwise, cl(K) is the smallest integer n such that there exist* *s a cone length decomposition of K with length n. If instead we require that L0 K, Ln * * * and that each Sibe connected, then we have a (spherical) killing length decompo* *sition 1 of length n. The (spherical) killing length of K, written kl(K), is defined ana* *logously. It is shown in [2] that kl(K) cl(K) for any space K. Furthermore, any cellul* *ar decomposition of an n-dimensional complex K is a cone length decomposition of length at most n. Therefore, if K is an n-dimensional complex, then kl(K) cl(* *K) n. These definitions suggest two numerical homotopy invariants of maps f : X -!* * Y . We write Tc(f) n if f O h ' * for every map h : K -! X with cl(K) n. Simil* *arly, we write Tk(f) n if f O h ' * for every map h : K -! X with kl(K) n, and say that f is trivial of order at least n. This includes the particular case me* *ntioned above: f is trivial of order at least 1 in this latter sense if and only if ß** *(f) = 0. Since kl(K) cl(K) for all K, it follows that Tk(f) Tc(f). Moreover, if f is* * trivial of order at least n, then f O h ' * for any h : K -! X with K an n-dimensional complex. It follows that if X is an n-dimensional complex, then no essential m* *ap f : X -! Y can be trivial of order at least n. Killing length is also related to the weak category of a space. The reduced * *(n+1)- fold diagonal map z__n+1_factors"_______- dn+1 : X -! X ^ X ^ . .^.X = X(n+1) is the composite of the diagonal X -! Xn+1 with the projection Xn+1 -! X(n+1) onto the smash product. We say wcat(K) n if dn+1 ' *, i.e., if dn+1 is homoto* *pic to *. We have the following commutative diagram for any map h : Sn -! X _______h_______// Sn X d2|| |d2| fflffl|h^h fflffl| Sn ^ Sn _________//_X ^ X. Since wcat(Sn) = 1 for each n 1, it follows that Tc(d2) Tk(d2) 1. We will* * show how this example can be greatly generalized. The inequalities wcat(K) cat(K) cl(K) [6] and kl(K) cl(K) do not direc* *tly imply any relation between killing length and weak category, but our first resu* *lt shows that such a relation does exist. Proposition 1 If K is a space with kl(K) = m, then wcat(K) < 2m . This allows us to produce a large family of examples of essential maps betwe* *en finite complexes which are trivial of order at least n. n) Example 2 Let X be a finite complex with wcat(X) 2n. Then d2n : X -! X(2 is essential and it follows from the naturality of the reduced diagonal that it* * is trivial n of order at least n. Since wcat(CPt) = t, we can take X = CP2 . 2 In these examples, f : X -! Y is an essential map with Tk(f) n, and wcat(* *X) 2n. Furthermore, if f : X -! Y is essential and Tk(f) n, then we must have kl(X) > n. In view of Proposition 1, it is reasonable ask whether wcat(X) 2n whenever X is a finite complex that is the domain of an essential map which is trivial of order at least n. In the special case n = 1, the answer is no: in * *[3] we constructed essential maps f : 2(CPt=S2) -! S5 which are trivial of order at l* *east 1, and wcat( 2(CPt=S2)) = 1 because it is a suspension. There remains the quest* *ion: is there an upper bound on the triviality of an essential map of finite complex* *es whose domain is a suspension? In this note we show that the answer to this more general question is again * *no. We refine and expand the examples of [3] by constructing, for each n, maps f wi* *th Tk(f) n and domain a suspension of a finite complex. Since every map with Tk * * n is trivial of order at least 1, this provides many more examples of the type co* *nsidered in [3]. These new examples are closely related to phantom maps. Our main result (The- orem 3) forges a link between the infinite dimensional domains of phantom maps and finite complexes which are domains of maps with Tk n. In fact, these fin* *ite complexes are quite common - they lurk among the skeleta of most familiar infin* *ite dimensional spaces. Let M(G, m) denote the Moore space with homology G in dimension m. We write Xk for the k-skeleton of the CW complex X and X(p)for the p-localization of the space X. Theorem 3 Let X be a 1-connected CW complex of finite type, let p > 3 be a pr* *ime and let n 1. Assume that there is an essential phantom map 2lX -! S2(k+l)+* *1(p) where k, l 1. Then, for each i < l, there are positive integers t = t(n), s * *= s(n) and a map f : Xt=X2k- ! M(Z=ps, 2k + 1) such that 2if : 2i(Xt=X2k) -! M(Z=ps, 2(k +i)+1) is essential and trivial of * *order at least n. To show that the theorem is not vacuous, and to give the desired examples, we r* *ecall the following basic result from the theory of phantom maps (see [19, Thm. D] and [10, Thms. 5.2 and 5.4]). Let bYpdenote the Sullivan p-completion of the space * *Y [16], and write Ph(X, Y ) [X, Y ] for the set of phantom maps from X to Y . Theorem Let X be a 1-connected CW complex of finite type. If the pointed mappi* *ng space map *(X, bS2(k+l)+1p) is weakly contractible, then Ph(X, S2(k+l)+1(p)) = [X, S2(k+l)+1(p)] ~=H2(k+l)(X; R). 3 The mapping space condition holds for any infinite loop space or for any sus* *pension of such a space [11, Thm. 2]. Thus Theorem 3 applies, for example, when X = CP1* * , and we conclude that there are essential maps 2l(CPt=S2) -! M (Z=ps, 2l + 3) which are trivial of order at least n. Other examples can be obtained by apply* *ing Theorem 3 to the classifying space BG, where G is any simply connected Lie group [10, Thm. 5.6]. In particular, it follows that for any k, l, m, n 1 there are* * essential maps f : BU(m)t=BU(m)2k- ! S2k+1 such that 2if is essential and trivial of ord* *er at least n for each i < l. For information about the cone length and killing length of spaces in a more general context, we refer the reader to [5, 2, 3]. The invariants Tc and Tk int* *roduced in this paper are closely related to the the essential category weight of a map* * (also known as the strict category weight), as studied in [15, 13]. In fact, a map f * *: X -! Y has essential category weight at least n, written E(f) n, if f O h ' * for an* *y map h : K -! X with cat(K) n. Since cat(K) cl(K), E(f) is a lower bound for Tc(f). We would like to thank Don Stanley for the statement and the proof of Lemma * *4. 2 Proofs In this section, we prove Proposition 1 and Theorem 3. In x2.1 we establish Pro* *po- sition 1. We then give some definitions and lemmas which are used in x2.3 to pr* *ove Theorem 3. We only consider based spaces of the homotopy type of 1-connected CW com- plexes. We use localization techniques, and write ~ : X -! X(p)for the natural* * map from X to its localization at the prime p. We refer to [9] for the standard pro* *perties of localization. 2.1 Proof of Proposition 1 We proceed by induction. If kl(K) = 0, then K is contractible and the result fo* *llows. Assume that the result is known for all spaces with killing length less than m * *and j k that kl(K) = m. Write S -! K -! L for the first step in a minimal killing leng* *th decomposition for K, so kl(L) = m - 1. Since d2 O j ' * : S -! K ^ K, there is a map ffi : L -! K ^ K such that d2 ' ffi O k. Thus we have the homotopy commutat* *ive 4 diagram d2 d2m-1 m K _________//_OOKO^_K_____________//K(2O)OO OOO OOOO ffi| || (2m-1) kOOOOOO | |ffi O''|______d2m-1'*___//_ m-1 L L(2 ). m-* *1) Since kl(L) = m - 1, the inductive hypothesis shows that d2m-1 ' * : L -! L(2 * * , m) and therefore d2m = d2m-1d2 ' * : K -! K(2 . * *|| 2.2 Lemmas We begin with a lemma which will allow us to modify a given killing length deco* *m- position. Lemma 4 [14] Let j0 k0 S0 _________//_L0________//L1 j1 k1 S1 _________//_L1________//L2, be two cofibrations in which S0 and S1 are wedges of spheres, S0 is (n - 1)-con* *nected, L0 and S1 are n-connected and L2 is (n + 1)-connected. Then there is another pa* *ir of cofibrations __ _j0 _k0 __ S 0__________//L0________//_L1 __ _j1 __ _k1 S 1__________//L1________//_L2 __ __ __ __ __ where S 0and S 1are wedges of spheres, S 0is n-connected and S 1and L 1are both (n + 1)-connected. Proof Write S0 = T _ U where T is the subwedge consisting of all n-spheres in S0 and * *U is the complementary subwedge. Since j0|T ' *, L1 ' T _ C where C is the cofiber * *of the map j0|U. Notice that C is n-connected. Write S1 = V _ W , where V is the subwedge of all (n + 1)-spheres and W is t* *he complementary subwedge. Applying Hn+1 to the second cofibration, we obtain the exact sequence Hn+1(V )_________//Hn+1( T _ C)________//Hn+1(L2) = 0. 5 Thus j = j1|V : V -! T _C is surjective on Hn+1 and hence on ßn+1. Let b1, . * *.,.bk 2 ßn+1( T ) be the standard generators, and choose a1, . .,.ak 2 ßn+1(V ) such th* *at j*(ai) = bi for each i. Then the map s = (a1, . .,.ak) : T -! V satisfies j O* * s = i T, the inclusion of T into T _C. The long homology exact sequence of the cofibra* *tion _j __________________________________________________* *_____________________________________________________________________@ ""_________________________________________________* *___________ T ____s____//V_______//_A, _ where j = p T O j and p T projects T _ C onto T , induces the split short exa* *ct sequence _j __*________________________________________________* *_____________________________________________________________ ________________________________________________________* *_____________________________________________________________________@ zz_______________________________________________________* *__________________________________________________ 0____//_Hn+1( T_)_s*___//_Hn+1(V_)______//_Hn+1(A)___//_0. Thus A has the homotopy type of a wedge of (n + 1)-spheres and there is a homot* *opy equivalence (s, t) : T _ A -! V for some map t : A -! V . With these identifications, the following diagram commutes j1 V _OWO ____________//_ T _ C | || (s,t)_id| || | (i T,g) || T _ A _ W __________// T _ C, where g : A _ W -! T _ C is some map. Thus we have a square of cofibrations T __________________ T_____________//* i|| |i|T || fflffl| (i T,g) fflffl| fflffl| T _ (A _ W ) _________//_ T _ C_________//L2 | | | | | | fflffl| pCOg fflffl| fflffl| A _ W ________________//C____________//_L2. We have now constructed the following pair of cofibrations j0|U U ____________//L0__________//C pCOg A _ W _________//_C_________//L2. It remains to move the A term in the second cofibration to the first cofibratio* *n. By definition, U is n-connected so the map L0- ! C is surjective in ßn+1. Since A * *is a wedge of (n + 1)-spheres, the map pC O g|A : A -! C lifts through l : A -! L0. 6 __ _ Finally, the desired_pair of cofibrations_is obtained_as follows:_let S0 = U* * _ A, j0= (j0|U, l) and let L1 be the cofiber of j0; let S1 = W and let j1be the comp* *osite pCOg|W __ _ W -! C ,! L1. It is a simple matter to verify that the cofiber of j1 is homot* *opy equivalent to L2. || Proposition 5 Let c 2 and let K be a (c-1)-connected CW complex with kl(K) = m. Then (a)K has a minimal killing length decomposition in which each Si and Li is (c* * - 1 + i)-connected; (b) for each q 0, kl(Kq) m + 2; (c)for each q 0, kl(K=Kq) 2m + 2. Proof ji ki Let Si-! Li-! Li+1, 0 i < m, be a minimal killing length decomposition for K - that is, L0 K, Lm *, and Si is a wedge of spheres for each i. We observe that, to prove (a), it is enough to show that K has a minimal kil* *ling length decomposition in which S0 is (c - 1)-connected and Siand Liare c-connect* *ed for i > 0. Then (a) follows on applying this to the resulting L1 and its minima* *l killing length decomposition of length m-1, and then applying it to the resulting L2 an* *d its minimal killing length decomposition of length m-2, and so on. Let k be the gre* *atest integer for which S0 is (k - 1)-connected and Siand Liare k-connected for i > 0* *. We want to show there is a killing length decomposition of K with k c. Assume th* *at k < c. Let i denote the greatest index for which Si or Li is not (k + 1)-connec* *ted. If i > 0 then we have a pair of cofibrations Si-1__________//Li-1_________//_Li Si____________//Li_________//_Li+1 in which Si-1is (k - 1)-connected, Si and Li-1are k-connected, and Li+1is (k + * *1)- connected. By Lemma 4, these cofibrations can be replaced by the cofibrations __ __ Si-1__________//Li-1________//Li __ __ Si___________//_Li_________//_L2 __ __ __ in which Si-1is k-connected and Li and Siare (k + 1)-connected. Continuing in t* *his way, we eventually obtain a spherical killing length decomposition in which eac* *h Si 7 and Li, i > 0, is (k + 1)-connected. Applying Lemma 4 to the first two cofibrat* *ions in this decomposition shows that there is a killing length decomposition of K w* *ith S0 k-connected and Si and Li (k + 1)-connected for all i > 0. This shows that t* *here is a minimal killing length decomposition for K in which S0 is (c - 1)-connecte* *d and Si and Li are c-connected for_i > 0. Now we prove (b). Let_Si_be the subwedge_of_Si consisting_of the spheres wi* *th dimension at most q, let L0 = Kq and_let j0: S0- ! L 0be a_lift of j0|_S0, whic* *h exists by cellular approximation. Define L1 to be the cofiber of j0. Thus, we have a d* *iagram of cofibration sequences __ __ __ S 0__________//L0________//_L1 | | | | | | fflffl| fflffl| fflffl| S0 __________//L0________//_L1 __ In this diagram, L1 is a subcomplex of L1_which_contains the_q-skeleton_of_L1. * *By cel- lular approximation, we may lift j1|_S1: S1- !_L1 to_a map_j1: S1- ! L 1. Conti* *nuing in this way, we obtain_cofibration sequences Si- ! Li- ! L i+1for each_0_ i < * *m in which each inclusion_Li- ! Li is a (q - 1)-equivalence. Since Lm *, Lm is (q * *- 1)- connected. Now Lm is (q + 1)-dimensional by construction, which shows that Lm h* *as killing length at most 2. Append the two cofibrations of a killing length decom* *posi- tion of Lm to the previously constructed sequence of m cofibrations to obtain a* * killing length decomposition for Kq with length m + 2. Finally, we prove (c). The cofibration K -! K=Kq- ! Kq yields the inequali* *ty kl(K=Kq) kl(K) + kl( Kq) by [3, Thm. 3.4]. Since kl( Kq) kl(Kq) the result follows from part (b). || If p > 3 is an odd prime, then S2k+1(p)is a homotopy commutative, homotopy associative H-space [1]. Hence [K, S2k+1(p)] is a p-local abelian group for an* *y finite complex K. Our next lemma provides an upper bound on the exponent of this group, and may be interesting in its own right. Lemma 6 Let K be a (2k + 1)-connected finite complex (k 1) with kl(K) = m * *and let p > 3 be a prime. Then [K, S2k+1(p)] is a finite abelian group with exponen* *t dividing pmk. Proof We work by induction on kl(K). If kl(K) = 0, then K is contractible so the conc* *lusion is obvious. Now assume the result is known for any space with killing length le* *ss than m. By Lemma 4 we may find a minimal killing length decomposition for K in which 8 W n all terms are (2k + 1)-connected. If S i- ! K -! L is the first step in suc* *h a decomposition, then kl(L) < m. From the exact sequence W n 2k+1 2k+1 2k+1 [ S i, S(p)o]o_____[K, S(p) ]oo______[L, S(p) ] we see that [K, S2k+1(p)] is a finite group with exponent at most the product o* *f the W n 2k+1 2k+1 exponents of [ S i, S(p) ] and [L, S(p) ]. By the inductive hypothesis applied* * to L and a result of Cohen, Moore and Neisendorer [7], this product is at most pkpk(* *m-1)= pmk. || The following well-known lemma will be used in the proof of Lemma 8. j Lemma 7 Let A -i!B -! C be a cofibration and let f : X -! B be any map. If j O f ' *, then there is a map s : X -! A such that i O s ' f. Proof Since j O f ' *, the composite factors through CX, the cone on X. Thus we may construct a homotopy commutative ladder of cofibrations _____id___// X __________//CX__________// X X f|| || |s| ||f fflffl|j fflffl| fflffl| i fflffl| B ___________//C__________//_ A_________// B. || Armed with this lemma, we give a criterion which guarantees that certain maps f : X -! S2k+1 remain essential when composed with the standard inclusion map 's : S2k+1 ,! M(Z=ps, 2k + 1). Lemma 8 Let X be a finite complex and let h : X -! S2k+1 be a map such that* * for some odd prime p, ~ O 2h 2 [ 2X, S2k+3(p)] is nontrivial and has finite order * *divisible by p. Then the composite 's s X ____h____//S2k+1_______//M(Z=p , 2k + 1) is essential for sufficiently large s. 9 Proof Consider the diagram ______~_______//_2k+1 S2k+1 S(p) | | ps|| ps| h fflffl| ~ fflffl|2k+1 X _________//_MMS2k+1__________//_S(p) MM MMM | | 'sOhMMMMM's| ('s)(p)| M&& fflffl| fflffl| M(Z=ps, 2k + 1) _=__//_M(Z=ps, 2k + 1) in which the vertical sequences are cofibrations and ps denotes the map with de* *gree ps. If 's O h ' *, then ('s)(p)O ~ O h ' *, and so (~ O h) lifts through the * *map ps : S2k+2(p)-!S2k+2(p)by Lemma 7. Suspending once more, we obtain the lift ind* *icated by the dashed line in the diagram gg3S2k+3(p)3 g g g gg g g g |ps| gg g g fflffl| g __________//2k+3 2X _____2h__//_S2k+3 ~ S(p) . The torsion subgroup of [ 2X, S2k+3(p)] is a finite abelian p-group, and so it * *has an exponent pe. If s e, then ~ O 2h ' *, which is a contradiction, and so 's O* * h is essential. || 2.3 Proof of Theorem 3 Let G denote the tower {[ ( 2lXt), S2(k+l)+1(p)]}. Since Ph( 2lX, S2(k+l)+1(p)* *) is natu- rally isomorphic to lim1G, the tower G cannot be Mittag-Leffler [4]. This means that the index of Im([ ( 2lXt), S2(k+l)+1(p)]) [ ( 2lX2k), S2(k+l)+1(p)] (whi* *ch is finite by [12, Prop. 0]) is unbounded as t increases. Let r be the rank of the group [ ( 2lX2k), S2(k+l)+1(p)], let T be its torsion subgroup, and write ` ' At= Im [ 2l+1Xt, S2(k+l)+1(p)] -! [ 2l+1X2k, S2(k+l)+1(p)] and ` ' Zt= Im [ 2l+1X2k, S2(k+l)+1(p)] -! [ 2l(Xt=X2k), S2(k+l)+1(p)] . Choose t large enough that the index of At [ 2l+1X2k, S2(k+l)+1(p)] is divisi* *ble by pr((2n+2)(k+l)+1)|T |. Then the quotient [ 2l+1X2k, S2(k+l)+1(p)]=AtT is an ab* *elian group 10 which is generated by a set of at most r elements and which has order divisible* * by pr((2n+2)(k+l)+1). Since there is a surjection of finite groups Zt~= [ 2l+1X2k, S2(k+l)+1(p)]=At- ! [ 2l+1X2k, S2(k+l)+1(p)]=AtT it follows that Zt also contains elements of order divisible by p(2n+2)(k+l)+1. The commutativity of the diagram [ Xt, S2k+1(p)]_______//[ X2k, S2k+1(p)]_____//_[Xt=X2k, S2k+1(p)] | | | | 2l ~=| 2l |2l fflffl| fflffl| fflffl| [ 2l+1Xt, S2(k+l)+1(p)]//_[ 2l+1X2k, S2(k+l)+1(p)]//_[ 2l(Xt=X2k), S2(k+l)* *+1(p)] clearly shows that Zt Im( 2l) [ 2l(Xt=X2k), S2(k+l)+1(p)]. Thus there is a * *map g : Xt=X2k- ! S2k+1(p)such that 2lg has finite order divisible by p(2n+2)(k+l)* *+1. Notice that g itself also must have finite order since it is an element of the finite * *group Zt, and so 2ig has finite order divisible by p(2n+2)(k+l)+1for 0 i l. Since the composition ~O 2l(p(2n+2)(k+l)Og) has finite order divisible by p,* * Lemma 8, applied to 2(l-1)g, shows that 's O p(2n+2)(k+l)O 2(l-1)g is essential if * *s is large enough. Fix such an s and define f = 's O p(2n+2)(k+l)O g. Thus 2if is essenti* *al for each 0 i < l. Finally, we demonstrate that, for 0 i < l - 1, the essential map 2if is t* *rivial of order at least n. Let kl(K) n and let h : K -! _2i(Xt=X2k)_be any map. Si* *nce 2i(Xt=X2k) is 2(k + i)-connected, h factors through h : K=K2(k+i)-! 2iXt=X2k. Since 2ig has finite order, the induced homomorphism ß2(k+i)+1( 2ig) = 0. Ther* *efore __ 2(k+i)+1 2ig O h can be extended to a map eh: K=K2(k+i)+1-! S(p) as in the diagram ________2if_____________________________* *_____________________________________________________________________@ _________________________________________________* *_____________________________________________________________________@ ____2ig__________________________________((_________* *______________________________________________________2(k+i)+1p(2n+2)@ K ______h_//o2i(Xt=X2k)___//S(p)77_____________//_S(p) __'s//_M oo p77p __66____________________* *__________________ | _hoooo eh ppp _____________________________* *_____________________________________________________________________@ | oooo pppp ____________________________________* *_____________________________________________________________________@ fflffl|oo pp _______________*_______________________________* *_____________________________________________________________________@ K=K2(k+i) ____//_K=K2(k+i)+1__________________________________________ where we have abbreviated M = M(Z=ps, 2(k + i) + 1). 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