Associativity in two-cell complexes Brayton Gray Dept. of Mathematics, Statistics and Computer Science (M/C 249) University of Illinois at Chicago 851 South Morgan Street Chicago, IL 60607-7045 brayton@uic.edu 1. The purpose of this paper is to examine the smash powers of a two-cell comp* *lex P = S2m-1 [ e2n when localized at primes grater than 3. We have two applicat* *ions in mind. We intend to introduce Samelson and Whitehead products into the homot* *opy groups with coefficient in P given by ssk(X; P ) = [k-2nP; X] for k 2n: Neisen* *dorfer did this in the case that n = m [N], and we generalize his results here. This * *will be applied in a forthcoming paper that generalizes the splitting theorems of [CMN]* * for P . The second purpose is to enhance the spectrum P0 = -2m+1 P = S0 [ e2n-2m+1 with the structure of a homotopy associative, homotopy commutative ring spectrum. T* *hus, in particular, any homotopy associative homotopy commutative ring spectrum can * *have "coefficients" introduced by smashing with -2m+1 P and retain its ring structur* *e. Finally we will examine the Hurevicz homomorphism in this context. The essence of these results lies in constructing a favorable splitting: P ^ P ' 2m P _ 2n+1P ; that such splittings exist when localized away from 2 has been known for some t* *ime ([G2], [K]); here we find that a careful choice of splitting allows us to handle the a* *ssociativity and commutativity of the resultant maps. Throughout this paper we will assume t* *hat all spaces are localized at a prime p > 2. From 2.5 onwards, we will assume that p * *> 3. 2. In this section we construct the required splittings. Recall that S1 localiz* *ed at p is not a co-H space. However, if P is localized at p, P = S1 ^ P is still p local and * *a co-H space where S1 is the unlocalized circle. Lemma 2.1. P is a co-commutative co-H space. 1 Proof: We will show that for any X the group structure defined on [P; X] is in * *fact Abelian. Our argument is strengthening the obvious generalization of [N; 9.7]. * *We consider two maps f; g : P ! X and compare their products fg and gf: P ----! P x P -fxg---!X x X ----! X P ----! P x P -fxg---!X x X ---o-! X x X ----! X where o is the twist map. The two composites from P x P to X agree on P _ P . We take their difference and factor it over P ^ P . It follows that the differenc* *e between fg and gf factors: P ----! P ^ P ----! X: Our task is complete when we show that : P -! (P ^ P ) is null-homotopic. By cellular approximation, the map compresses through S2m-1 ^ S2m-1 uniquely * *if n > m, and this map factors uniquely over S2n. In other words, there is a uniq* *ue map fl : S2n -! S4m-2 so that the diagram: P ----! P ^ P ? x j?y ??i^i S2n --fl--!S2m-1 ^ S2m-1 where i : S2m-1 -! P and j : P ! S2n are the maps from the cofibration sequenc* *e. Let o : P ^ P - ! P ^ P be the twist map. Since o = , ofl ~ fl; i.e., (-1) O f* *l ~ fl. Consequently fl : S2n+1 -! S4m-1 has order 2 and hence is null-homotopic. The * *case n = m is actually easier. The map fl may not be unique; however 2n 4n - 2 onl* *y if n = 1. In this case the degree of fl is determined by the cup product structur* *e in H*P and this is zero localized away from 2. We now state the splitting result. The existence of the maps and * goes ba* *ck to [G2]. Theorem 2.2. Localized at a prime p > 2, there are unique co-H maps: : P ^ P ----! 2m P * : 2n+1P ----! P ^ P such that a) O (i ^ 1) ~ 1 b) (j ^ 1) O * ~ 1 c) O ~ * 2 d) O o ~ - e)o O * ~ where o : P ^ P -! P ^ P is the twisting map. The proof will reply on constructing self maps of P ^P which are idempotent* *s as self maps. This is stronger than knowing that their induced homomorphisms are idempo* *tent, but the proof is of the same flavor (see [C]). Given an idempotent self map e : X -! X, we construct a telescope; i.e., th* *e homo- topy colimit X ---e-! X ---e-! X ----! . . . which we write as T (e) or T (X; e) and denote byie: : X -! T (e) the inclus* *ion of the first factor. Lemma 2.3. There is a 1-1 correspondence between homotopy classes of maps g : T* * (e) -! Y and homotopy classes of maps h : X -! Y such that he ~ h given by composition* * with ie. Proof: We first observe that [T (e); Y ] is the inverse limit of the diagram: * e* . . .----! [X; Y ]--e--! [X; Y ]----! [X; Y ]; i.e., there are no phantom maps. This follows since the Mittag-Leffler conditio* *n holds since e2 = e (see [G1]). However, given a sequence of maps gi : X -! Y with gie ~ gi-* *1, we calculate gi~ gi+1e = gi+1e2 ~ gie ~ gi-1. So gi= g for each i. Corollary 2.4. Suppose W is another space. Then T (X ^ W; e ^ 1) ' T (X; e) ^ W* * . Proof: We use 2.3 and adjointness. By 2.3, we can construct a unique map je : T (e) - ! X such that jeie ~ e a* *nd ieje ~ 1. It is also easy to see that if X is a co-H space and e : X -! X is a * *co-H map, then X(e) is a co-H space and ie; je are co-H maps. Proof of 2.2: Since P is a co-commutative co-H space by 2.1, P ^ P is also. Thus 2 ^ 1 : P ^ P - ! P ^ P is a co-H map and a homotopy equivalence. Let us write 1_ 1_ 2 : P ^ P -! P ^ P for the homotopy inverse. 2 is thus a co-H map. Now it is ea* *sy to see that the sum and difference of two co-H maps is again a co-H map since P ^ * *P is co- associative and co-commutative. Consequently e+ = 1_2O(1+o) and e- = 1_2O(1-o) * *are both co-H maps. Thus composition is distributive over addition and we get e+ O * *e+ = e+ , e+ O e- = *, e- O e- = e- , and 1 = e+ + e- . Write X = T (e ), i = ee , j * *= je . 3 Then j O i ~ e , i O j ~ 1, and i O j ~ * since e O e ~ *. Since 1 = e* *+ + e- , we have a splitting P ^ P ' X+ _ X- . Now write ae; oe for the compositions: 2m P -i^1---!P ^ P --i---! X- X+ --j+--!P ^ P -j^1---!2n+1P respectively. These are co-H maps since e are co-H maps. They are easily see* *n to be homotopy equivalences. So we define = ae-1i- and * = j+ O oe-1 ; the statement* *s in 2.2 now easy to verify. From this point forward we will assume that all spaces are localized at a p* *rime p > 3. Theorem 2.5. There are homotopy commutative diagrams: 2n* P ^ P ^ P -^1---! 2m P ^ P 4n+1P ----! 2n+1P ^ P ?? ? ? ? y1^ ?y2m-1 ?y2n* ?y1^* 2m-1 *^1 2m P ^ P -----! 4m-1 P 2n+1P ^ P ----! P ^ P ^ P where an obvious shuffle of suspension coordinates is implicit in 1 ^ and 1 ^ * **. To prove this we need a further lemma. Suppose e; f : X -! X, ef = fe and e* *2 = e. Then by 2.3, f induces a unique map T (e) -f*!T (e) such that the diagram: T (e)--f*--!T (e) x? x ? ie ??ie X ---f-! X commutes up to homotopy. Furthermore, if f is also an idempotent, so are f* and* * ef. Lemma 2.6. There is a homotopy commutative diagram: X --ief--! T (X; ef) ? x ie?y ??OE T (X; e)--if*--!T T (X; e); f* where OE is a homotopy equivalence. Proof: We easily construct OE and its inverse using 2,3. For example, there is * *a unique map : T (X; e) -! T (X; ef) with ie ~ ief and then . f* ~ by uniqueness. Sim* *ilarly, we have 4 Lemma 2.7. There is a homotopy commutative diagram: T (X; e) --je--! X x x jf*?? ??jef OE T T (X; e); f*---- T (X; ef) where OE is a homotopy equivalence. This follows similarly to 2.6 using Lemma 2.8. There is a 1-1 correspondence between homotopy classes of maps g : Y* * ! T (e) and homotopy classes of maps h : Y ! X with eh ~ h given by composition w* *ith je. Proof: Let H(g) = jeg and G(h) = ieh. Then H G(h) = eh ~ h an G H(g) = g. Proof of 2.5. We construct idempotent self maps of P ^ P ^ P by using the actio* *n of 3. Let a = (12) and b = (123); then ab = (23) and aba = b2. Define e1 = 1_2O (1* * - a), e2 = 1_2O (1 - ab), and f = 1_3O (1 + b + b2). Here we interpret the elements o* *f 3 as self maps of P ^ P ^ P . As before, these are idempotents as is E = e1f = fe1 = e2f * *= fe2. Thus 2.6 applied twice gives a homotopy commutative diagram: ie2 P ^ P ^ P ----! T (e2) ? ? ie1?y OE2if*?y T (e1) ----! T (E) OE1if* where OE1 and OE2 are equivalences. By 2.4, T (e1) ' T (e- )^P ' 2m P ^P and T (e2) ' P ^T (e- ) ' P ^2m P ' 2m* * P ^ P . Furthermore, under these equivalences ie1 corresponds to ^ 1 and ie2 corre* *sponds to 1 ^ . To understand if* we must understand the effect of f* on T (e1) and T (e* *2). On T (e1), f* is determined by the composition T (e1) -j1!P ^ P ^ P -f!P ^ P ^ P -i1!T (e1) if we combine this with the equivalence: 2m P ^ P ' T (e- ) ^ P ' T (e1) we obtain: 2m P ^ P -i^1-!P ^ P ^ P -e1!P ^ P ^ P -f!P ^ P ^ P -i1!T (e1): 5 However fe1 = -fe1ab = -fe1(23) so we have fe1 O (1 ^ o) = -fe1. Consequently t* *he composition 2m P ^ P - ! T (e1) -! T (E) factors through 2m-1 : 2m P ^ P - ! 2m-1 X- . The same considerations apply to f* on T (e2) and we obtain a homoto* *py commutative diagram: 2m-1 P ^ P ^ P - 1^---!2m P ^ P -----! 2m-1 P ?? ? ? y^1 ?y ?y2 2m-1 2m P ^ P -----! 4m-1 P ----! T (E) 1 An easy calculation for H* T (E) shows that 1 and 2 are homotopy equivalences.* * Re- stricting to S2m-1 ^ P ^ S2m-1 gives 1 ~ 2. Composing with -1 ~ -12gives the result. The case of * is similar, using 2.7 and replacing e1 and e2 with e01= 1* *_2O (1 + a), e02= 1_2O (1 + ab). 3. In this section we will apply the pairings of section 2 to produce the Same* *lson and Whitehead products as well as the ring spectra structure. We begin with some no* *tation. Let P k= k-2nP for k 2n and Pi = i-2m+1 P for i 2m - 1. We can now define maps i;jand *k;`as follows. Let i;j= (-1)j+1ci;jwhere ci;jis the composite: i+j-4m+1 Pi^ Pj = i-2m+1 P ^ j-2m+1 P ' i+j-4m+2 P ^ P --------! i+j-2m+1 P = Pi+j for i + j 4m - 1. Let *k;`be the composite: k+`-4n-1* pk+` = k+`-2nP ---------! k+`-4nP ^ P ' k-2nP ^ `-2nP ' P k^ P ` for k + ` 4n + 1. Write Ii: Si -! Pi and Jk : P k-! Sk for the appropriate sus* *pensions of the maps i; j from section 2. Then we have Theorem 3.1. Suppose i + j 4m - 1 and k + ` > 4n. Then there are maps i;j: Pi^ Pj -! Pi+j *k;`: P k+`-! P k^ P ` such that: a)i;jO o = (-1)ijj;i 6 b) i;jO (Ii^ 1) ~ 1 c)i;j+kO (1 ^ j;k) ~ i+j;kO (i;j^ 1) d) o O *k;`~ (-1)k`*`;k e)(Jk ^ 1) O *k;`~ 1 f)(1 ^ *k;`) O *j;k+`~ (*j;k^ 1) O *j+k;` Corollary 3.2. Let fflss2k(S0). Then the spectrum S0 [ e2k+1 has a unique ho* *motopy commutative homotopy associative ring spectra structure when localized at a pri* *me p > 3. Proof: This follows from 3.1 a; b and c. We now use the maps *k;`to construct Samelson and Whitehead produces in hom* *o- topy with coefficients in P . We follow the methods of [N] which we generalize* * to this case and make mild improvements due to the uniqueness of our structure maps. I* *n [N; section 9], the author defines a space C to be co-abelian if the diagonal map C* * -! C ^ C is null-homotopic. This guarantees that [C,G] is an Abelian group whenever G is* * a group like space (a homotopy associative H space with homotopy inverse). However, as * *we have seen (2.1), it suffices to assume that : C -! C ^ C is null homotopic, and thi* *s will suffice for all applications. Definition 3.3. A space C is quasi-co-Abelian if : C -! C ^ C is null homotopi* *c. Following [N], we define external Samelson products as follows. Let ff : C* *1 -! G, fi : C2 -! G. Then : C1 ^ C2 -! G is given by the composition C1 ^ C2 ff^fi---!G ^ G -c!G where c is an extension over the smash product of the group commutator map (a; * *b) -! aba-1b-1 defined in the homotopy category. Then [N, 9.10], in the quasi-co-Abel* *ian context yields: Proposition 3.4. Suppose C1; C2 and C3 are quasi-co-Abelian. Let ff; ff0 2 [C* *1; G], fi; fi02 [C2; G], and fl 2 [C3; G]. Then the following formulas hold: i) = -o* ii) = + ff ff ff iii)ff; + b* fi; + (b2)* fl; = 0 7 where b = (123). In particular, this applies when Ci = P kfor k 2n. We now apply this to co* *nstruct internal Samelson products. Let ssk(X; P ) = [P k; X]. This is an Abelian group* * if k > 2n or X is a group like space. If G is a group like space, ff 2 ssk(G; P ), fi 2 s* *s`(G; P ) we define [ff; fi] to be the composition: *k;` k ` P k+`--! P ^ P ---! G: Strictly speaking, this is not defined if k = ` = 2n. The suspension of the com* *position is defined, and we can recover [ff; fi] using the retraction G ! G given by the H * *space structure and adjointness. Thus we have the analogue of [N;10.1]. Proposition 3.5. Let ff; ff02 ssk(G; P ), fi; fi02 ss`(G; P ) and fl 2 ssq(G; P* * ). Then: i)[ff; fi] = (-1)k`+1[fi; ff] ii)[ff + ff0; fi] = [ff; fi] + [ff0; fi] iii)ff; [fi; fl] = [ff; fi]; fl + (-1)k` fi; [ff; fl] Whitehead products can be defined, as usual, using adjointness. The reader * *can easily construct the definition and state the analogous result to 3.5 for Whitehead pr* *oducts. The map Jk : P k-! Sk induces a homomorphism of Lie algebras: ss*(G) -! ss*(G; P ): Consequently, there is a Samelson product: ssk(G) ss`(G; P ) -! ssk+`(G; P ) expressing the Lie module structure of ss*(G; P ) over ss*(G). However, this S* *amelson product can be defined for all k 1, ` 2n directly and one easily proves Proposition 3.6. Suppose ff 2 ssk(G), fi 2 ss`(G), fl 2 ssp(G; P ) and ffi 2 ss* *q(G; P ) with k; ` 1 and p; q 2n. Then the following identities hold: a) ff; [fi; fl] = [ff; fi]; fl + (-1)k` fi; [ff; fl] b) fi; [fl; ffi] = [fi; fl]; ffi + (-1)`p fl; [fi; ffi] . 4. In this section we will discuss the Bockstein homomorphism defined in homoto* *py with coefficients in P . Let oe = 2n - 2m + 1. Then we construct fi : ssk(X; P ) -! ssk-oe(X; P ) which is defined when k 2n + oe = 4n - 2n + 1 and is a homomorphism when k > 4n - 2m + 1 or X is group like. Let Bk = Ik-oeJk-oe: P k-oe-! P k. Then fi is d* *efined by fi(f) = (-1)kf O Bk for f : P k-! X. 8 Lemma 4.1. *B4n+1 ' (i ^ 1) + (1 ^ i) O o : S2m ^ P -! P ^ P . Proof: The right hand side is invariant under o, so it factors as * O OE where* * OE : P 2n+2m -! P 4m+1; applying (i ^ 1) to the equation gives OE ~ B4n+1. Proposition 4.2. If u 2 ssk(X; P ) and v 2 ss`(X; P ) fi = + (-1* *)k. Proof: Suspending and keeping track of signs, we get *k;`O Bk+` = (1 ^ I`) O o * *+ (-1)`O Ik ^ 1 : P k+`-oe-! P k^ P `and hence *k;`O Bk+` = (1 ^ B`) O *k;`-oe+ (-1)`(Bk ^ 1) O *k-oe;` from which the result follows. 5. Let E be a commutative associative ring spectrum. Let P0 = 1-2m P = S0 [ * *eoe. Then by 3.2, EP = E ^ P0 is also a commutative, associative ring spectrum. We d* *efine EPr(X) = ssr(X ^ EP ) = eEr+2m-1(P ^ X). In this section we discuss the relati* *onship between ssr(X; P ) and gEPr(X). Lemma 5.1. There is a long exact sequence: . .-.!eEr-oe+1(X) -*!Eer(X) -i!gEPr(X) -@!eEr-oe(X) -! . . . where *(x) = 0x where 02 Eoe-1is the composition Soe-1-! S0 -! E. Corollary 5.2. Suppose 0= *(1) = 0 2 Eoe-1. Then EP*(X) ~=E*(X)^(eoe) as E*(X) modules where ^(eoe) is the p-local exterior algebra on one generator eoeof dim* *ension oe. Proof: Since 0= 0, * = 0 and we have a short exact sequence: 0 -! eEr(X) -i!gEPr(X) -@!eEr-oe(X) -! 0 let eoebe such that @eoe= 1. Since EP* is a module over E*, we can define a rig* *ht inverse to @ by the action of eE*(X) on eoe. Consequently gEPr(X) ~=eEr(X) eoeeEr-oe(X* *). Since oe is odd, e2oe= 0 and we are done. We now define a Hurewicz homomorphism. OE : ssr(X; P ) -! gEPr(X) for r 2n* *. Let fflr be the composition: *2n;oe r Sr+2m-1 -Ir+2m-1----!P r+2n---! P ^ P : Let er = oer+2m-1 (1) 2 Eer+2m-1(Sr+2m-1 ), and ir = fflr*(er) 2 Eer+2m-1(P ^ P* * r) = gEPr(P r). We then define OE by OE(f) = f*(ir) for f : P r-! X. Clearly OE is w* *ell defined and a homomorphism for r > 2n. Define fi : gEPk(X) -! gEPk-oe(X) to be the composition i@ from 5.1. 9 Proposition 5.3. fi OE(f) = -OE fi(f) for f : P k-! X with k 2n + oe. Proof: This is a simple diagram chase using the definitions here and in section* * 4. It now follows that OE is a homomorphism of Bockstein spectral sequences. N* *ote that if n > m, OE is nilpotent so both the E-homology and homotopy Bockstein spectral s* *equences converge after a finite number of items (depending only on OE). In case 0= *(1) = 0, we can introduce a Hurewicz homomorphism: h : ssk(X; P ) -! eEk(X) as follows. By 5.2 OE(f) = h(f) + eoeh0(f). Since fieoe= 1, we get h0(f) = fi* *OE(f) = -OE fi(f) = -h fi(f) = eoeh0 fi(f) . Consequently h0(f) = -h fi(f) and we ha* *ve: OE(f) = h(f) - eoeh fi(f) : This definition depends on the choice of eoewith @eoe= 1. However in the case o* *f ordinary Z(p)homology, eoeis unique. We now examine the multiplicative structure. We first define an external p* *roduct pairing in homotopy: ^_: ssk(X; P ) ss`(Y ; P ) -! ssk+`(X ^ Y : P ) by defining f ^_g to be the homotopy class of the composite: *k;` k ` f^g P k+`--! P ^ P --! X ^ Y: With this definition, the Samelson product is given by = c*(ff ' fi) w* *here c : G ^ G -! G. is the commutator map for section 3. Clearly E ^ P0 is a commutative and associative ring spectrum. consequently* * there is an external pairing in homology: ^_: gEPk(X) gEP`(Y ) -! gEPk+`(X ^ Y ): Recalling the definition in [G3; section 2.3], we observe that this pairing is * *represented by (-1)r times the composition: gEP)k(X) gEP`(Y ) =Eek+2m-1(X ^ P ) eE`+2m-1(Y ^ P ) _^_ -! eEk+`+4m-2(X ^ P ^ Y ^ P ) oe-!eE 1 k+`+4m-1(X ^ P ^ Y ^ P ^ S ) o-!E k+`+4m-1 X ^ Y ^ (P ^ P ) (1^)*----!eE 2n+2m-1 k+`+4m-1(X ^ Y ^ P ) ~=eEk+`+2m-1(X ^ Y ^ P ) = eEPk+`(X ^ Y ): Proposition 5.4. OE(ff ^_fi) = OE(ff) ^_OEfi) Proof: This is a straightforward diagram chase, using two applications of the f* *ollowing. 10 Lemma 5.6. *a+b-oe;cO a;b+c' (a;b^ 1) O (1 ^ *b;c) as homotopy classes in [* *P a^ P b+c; P a+b-oe^ P c]. Proof: It suffices to show that both composites are homotopic when precompos* *ed with both (Ia ^ 1) and *a;b+c. In the later case both composites are actually nul* *l homotopic. Theorem 5.7. In a group like space OE = OE(ff); OE(fi) Proof: OE = OE c*(ff^_fi) = c*OE(ff^_fi) = c* OE(ff)^_OE(fi) : Now although E*(X) is not a coalgebra in general, it is still possible t* *o define primitive elements - elements 2 E*(X) such that *() = (i)*() + (i2)*() where ; i1; i2* * : X -! X x X are the diagonal and the two axial injections respectively. Clearly ev* *ery element in E*(P k) is primitive, and hence OE(ff) and OE(fi) are primitive. Now OE(* *ff) x_OE(fi) 2 Ek+`(G x G). Let : G x G -! G ^ G. So * OE(ff) x_OE(fi) = OE(ff) ^_OE(fi).* * Consequently c* OE(ff) ^_OE(fi) = c** OEOE(ff) x_OE(fi) C is represented by the composite G x G -x--! G x G x G x G -1xox1---!G x G x G x G -1x1xx-----!G x G x G x * *G -! G where is the inverse map. If is primitive, *() = - so one obtains c** OE(f* *f) x_OE(fi) = OE(ff)OE(fi) - (-1)|ff| |fi|OE(fi)OE(ff) = OE(ff); OE(fi) . One can also easily check that if 0= 0, h = h(ff); h(fi) . References [C]F. Cohen, Splitting certain spaces via self maps, Ill. J. Math. 20 (1975* *), 336-347. [CMN] F.R. Cohen, J.C. Moore, and J.A. Neisendorfer, Torsion in homotopy group* *s, Ann of Math. 109 (1979), 121-168. [G1] B. Gray, Operations and a problem of Heller, thesis, University of Chica* *go (1965). [G2] B. Gray, Operations on two-cell complexes. Proceedings of the Conference* * on Algebraic Topology, UICC (1968), 61-68. [G3] B. Gray, Homotopy Theory: An introduction to algebraic topology, Academi* *c Press, 1975. [K]W. Komornicki, Multiplication in two-cell complexes, thesis UIC, 1979. [N]J.A. Neisendorfer, Primary Homotopy Theory Memoirs AMS #232 (1980). 11