SELF MAPS OF HP n VIA THE UNSTABLE ADAMS SPECTRAL SEQUENCE GUSTAVO GRANJA Abstract.We use obstruction theory based on the unstable Adams spectral sequence to construct self maps of finite quaternionic projective spaces* *. As a result, a conjecture of Feder and Gitler regarding the classification of* * self maps up to homology is proved in two new cases. 1.Introduction Let HPn denote n-dimensional quaternionic projective space and recall that H*(HPn; Z) = Z[u]=un+1 with |u| = 4. Given a self map f of HPn, the degree of f is the integer deg(f) such that f*(u) = deg(f)u. The homotopy classification of self maps of HP1 is well known: self maps are classified by their degrees and the allowable degrees are zero and the odd squa* *re integers [Ms ]. The situation for finite projective spaces is more complicated.* * It is not true in general that self maps are classified by their degrees [MR ], and e* *ven the set of possible degrees is unknown. We will write Rn = {deg(f) 2 Z: f : HPn ! HPn} for the set of possible degrees. Alternatively, Rn can be described as the set * *of self maps of S3 which admit an An structure (see [St]). There is a conjectural description of Rn due to Feder and Gitler. Consider the congruences 8 n-1Y < (2n)! ifn is even, (1) Cn : (k - i2) 0 mod i=0 : (2n)!_2ifn is odd. and define the sets of integers FG n= {k 2 Z | k is a solutionCof1, . .,.Cn}. In [FG ], Feder and Gitler computed the restrictions imposed by K-theory and Adams operations on the degrees of self maps of HPn: they are precisely that the degree should be in FG n. Therefore Rn FG n. Conjecture 1.1 (Feder and Gitler). Rn = FG n. This is trivial for n = 1. A proof for n = 2 is contained in [AC ], and for n* * = 3 in [MG2 ]. Feder and Gitler showed in [FG ] that the integers in FG 1 are preci* *sely the odd square integers and zero so the conjecture also holds for n = 1. In fac* *t, self maps of HP1 with these degrees were first constructed by Sullivan in [Su],* * and ____________ Key words and phrases. Self maps, quaternionic projective space, unstable Ad* *ams spectral sequence. Mathematics Subject Classification 2000: 55S35,55S36,55S37. Research supported in part by grants from FCT, FLAD and Funda,c~ao Gulbenkia* *n. 1 2 GUSTAVO GRANJA it was this which motivated Feder and Gitler's work. McGibbon has also proven that the conjecture holds stably, in a suitable sense (see [MG1 , Theorem 3.5]). The solutions to the congruences (1)are more easily described one prime at a time. It turns out (see Proposition 2.1) that the p-local integers which sat* *isfy C1, . .,.Cn consist of the p-adic squares (if p = 2 they must also be units or * *zero) together with all multiples of pe(p,n)for a certain function e described in (2)* *. The former can all be realized as degrees of self maps of the p-localization HP1(p)* *by work of Rector (see Proposition 2.2), so to prove the conjecture it is enough to bui* *ld self maps of HPn(p)with degree an arbitrary multiple of pe(p,n). The obstruction to extending a self map of HPn to HPn+1 is an element in ß4n+2S3 (see (5)). We will use some of the information available on ß*S3 to ana* *lyse this obstruction and produce self maps of HPn(p)with degree any multiple of pf(* *p,n) where f is a different function described in (11)(see Theorem 4.2). In general, e(p, n) < f(p, n) but for n 5 the two functions agree, and this implies our m* *ain result. Theorem 1.2. The Feder-Gitler conjecture holds for n 5. This is stronger evidence for the conjecture as the obstruction group ß4n-2S3 is detected by K-theory and Adams operations (i.e. by the e-invariant) only for n 3. For n 3 it is easy to produce self maps of HPn with degree in FG n+1 which do not extend one stage. Theorem 1.2 is proved by inductively building ma* *ps of high Adams filtration thus ensuring their obstructions to extension are dete* *cted by the e-invariant. Organization of the paper. In section 2 we describe the p-local solutions to the congruences (1) and quote results of Rector which reduce the Feder-Gitler conjecture to producing maps with degree any multiple of pe(p,n). In section 3,* * we gather the necessary information on the unstable Adams spectral sequence for S3. In section 4 we show that the degree of a self map of HPn is in FG n+1if and on* *ly if the Adams e-invariant of the obstruction to extension vanishes. The results quo* *ted in section 3 then allow us to prove Theorem 1.2. Acknowledgments. The author thanks Mike Hopkins for his guidance and useful suggestions, and also Maxence Cuvilliez for a useful discussion. 2.The local version of the conjecture We will write |k|p = sup{l :pl|k} for the p-adic valuation of k and [x] for t* *he least integer x. In this section we reduce the Feder-Gitler conjecture to construct* *ing self maps of the localizations HPn(p)of degree k for all k with |k|p e(p, n) * *for a certain function e(p, n). First note that, by celullar approximation, the inclusion of HPn in HP1 induc* *es a bijection [HPn, HPn] ! [HPn, HP1 ]. As HP1(0)= K(Q, 4), it follows easily from Sullivan's arithmetic square that in order to produce a self map of HPn of degr* *ee k, it suffices to do so after localizing at each prime p (or, in fact, at each * *prime p 2n - 1). SELF MAPS OF HPn VIA THE UNSTABLE ADAMS SPECTRAL SEQUENCE 3 We will write Rn,p= {deg(f) 2 Z(p)|f : HPn(p)! HPn(p)} and FG n,pfor the set of solutions in Z(p)of the congruences (1). Let Dp denote the set defined by æ Dp = {k{2kZ(2)|2kZ= 0 ork 12mod 8} ifp =^2, (p)| k = u for someui2fZp}p > 2. It will soon emerge that Dp = R1,p = FG 1,p. Note that D2 consists of the 2-adic unit squares in Z(2)together with 0. We define the function 8 ><0 ifn = 1, (2) e(p, n) = >#{k n | k = pj ork = (p+1_2)pj-1 for somej i1}fp > 2, :1 + 2[log 2(n)] otherwise. Proposition 2.1. Let p be a prime and n < 1. Then FG n,p= Dp [ pe(p,n)Z(p). Proof.The sets Dp are the closure of R1 = {0, 1, 9, 25, . .}.in Z(p)with the p- adic topology. Since the solution set of each congruence is closed we conclude * *that Dp FG n,pfor every n, p. We will conclude the proof in the case p = 2 (the ca* *se when p is odd is similar). If k 2 Z(2)is a solution of C2 then either k 1 mod 8, or k = 23l for some l 2 Z(2)so it suffices to consider solutions k with |k|2 3. For these, since |(4m+2)!=2|2 = |(4m)!|2 it suffices to consider the congruences C2m. So let |k|* *2 3. Factoring out units in Z(2)we see that k is a solution of C2m if and only if m-1Y (k - (2i)2) 0 mod 2|(4m)!|2. i=0 Writing k = 4l and noting that |(4m)!|2 = 2m + |(2m)!|2, this becomes m-1Y (l - i2) 0 mod 2|(2m)!|2 i=0 so if k = 4l 2 FG 2m,2then l 2 FG m,2. Since l 62 Zx(2), assuming the result is true for all n < 2d and letting 2d 2m < 2d+1, it follows by induction that R2m,2- D2 22d+1Z(2). It remains to check that for 2d 2m < 2d+1 we have k 2 FG 2m,2as long as |k|2 2d + 1. That is, for j m and |k|2 2d + 1, fifi2j-1 fi fifiY(k - i2)fifi |(4j)!| fii=0 2 fifi 2 2j-1X |k|2 + 2|i|2 2j + j + . .+.[j=2d-1] i=1 |k|2 + 2(j + . .+.[j=2d-1]) - 2|2j|22j + j + . .+.[j=2d-1] |k|2 + j + . .+.[j=2d-1] 2j + 2|2j|2. The last inequality is easy to check if we write 2j = 2rv with v 2 Zx(2)and not* *ice that r d and 2d-r v < 2d-r+1. 4 GUSTAVO GRANJA As e(p, n) tends to 1 with n, this means that FG 1,p = Dp. The following result1 of Rector [Re ] implies that in fact we have R1,p = FG 1,p = Dp and therefore, to prove the conjecture it suffices to construct self maps with degr* *ee in pe(p,n)Z(p). Proposition 2.2 (Rector). If k 2 Dp then there is a self map of HP1(p)of degree k. Remark 2.3. In view of the above, the Feder-Gitler conjecture implies the follo* *w- ing: If a self map of S3(2)admits an A2k-structure then it admits an An-structu* *re for all n < 2k+1 (not necessarily extending the A2k-structure). Similarly for p* * odd. It would be interesting to know whether this holds more generally. A related question is whether a 2-local H-space admitting an A2k-structure ne* *c- essarily admits an An structure for n < 2k+1. In this regard, note that a group* *like multiplication on a rational space always admits a loop structure [Sch, Corolla* *ry 2]. 3. The unstable Adams spectral sequence for HP1 Let U denote the category of unstable modules over the mod p Steenrod algebra. If M 2 U, U(M) denotes the free unstable algebra generated by M and for P 2 U a free module, we write K(P ) for the product of mod p Eilenberg-Maclane spaces s* *uch that H*(K(P ); Fp) = U(P ). Recall from [HM2 ] that if X is a simply connected space such that H*(X; Fp) = U(M) for some M 2 U of finite type, then given a free resolution 0 - M - P0 - P1 - P2 - . . . of M with Ps of finite type and 0 in degree s + 1, one can construct a tower * *of principal fibrations over X, called an Adams resolution for X (3) ... p2|| fflffl|j2 X{2} _______//K( 2P2) p1|| fflffl|j1 X{1} _______//_K( P1) p0|| fflffl|j0 X{0} = X ______//K(P0) where denotes the algebraic loop functor (the left adjoint to suspension). The corresponding homotopy spectral sequence is the Massey-Peterson version of the unstable Adams spectral sequence. For a simply connected space of finite type t* *his spectral sequence converges strongly and Es,t2(X) = Exts,tU(M, Fp) ) ßt-sX bZp. The Adams filtration of a map f : W ! X is the largest s such that f factors through X{s}. ____________ 1See also [JMO ] for a more up to date account of self maps of p-completed c* *lassifying spaces. SELF MAPS OF HPn VIA THE UNSTABLE ADAMS SPECTRAL SEQUENCE 5 Notice that H*(S3; Fp) = U( 3Fp) and H*(HP1 ; Fp) = U(M) where M is such that M = 3Fp. One easily checks that the first derived functor 1M = 0 hence looping an Adams resolution for HP1 yields an Adams resolution for S3. This in turn implies that the adjunction isomorphim (4) ßkS3 ' ßk HP1 ' ßk+1HP1 lifts to an isomorphism of spectral sequences Es,tr(S3) = Es,t+1r(HP1 ). We will now collect some well known information about this spectral sequence. Figures 1 and 2 describe the portion of the E2 term of the spectral sequence for HP1 along the vanishing line (i.e. the v1-periodic part). As usual, vertical l* *ines represent multiplication by p and the slanted lines composition with j if the p* *rime is 2 and with ff1 if the prime is odd. As usual, for p an odd prime, we set q = 2(* *p-1). | q s s = 4k+ |6_ a q |6_a 3 | 5 | | 2 |_ a a|q 4 |_|a q |_ a q q ____ 3 |_|a a q 1 |_ a q| |_|a a a| 0 | 2 | |j -1 |_ a 1 |_|aa -2 |_ 0 |___________________-|||||||||a | 4 5 6 7 8 91011 t - s ____________________-|||||||| t - s = 8k+4 5 6 7 89 1011 Figure 1. E(s,t)2(HP1 ) for p = 2. s ||6 s = k+ ||6 3 |_a |_ q a |_|a q a 1 | i i 2 | | ff1i i 0 |_ q ai 1 |_|a i ai -1 |_ 0 |___________________-|||||||||aii | 4 3 + q 2 + 2q t - s ____________________-|||||||| t - s = qk+ 2 3 2+q Figure 2. E(s,t)2(HP1 ) for p odd. Theorem 3.1 (Adams, Mahowald, Miller, Harper-Miller). Consider Figures 1,2. (a) Above the classes shown and the dotted lines in the columns where no classes appear, the E2 term vanishes. (b) The classes in dimensions qk +3 for p odd and the classes in dimension 8k +7 with filtration 4k + 2 for p = 2 correspond to elements of ß*S3 which are stable and detected by the (stable, real) e-invariant. 2 ____________ 2See (8)below for the definition of the e-invariant. 6 GUSTAVO GRANJA Proof.Recall from [Ma1 ] and [HM1 ] that for each prime there is a bigraded com* *plex ( (3), d) such that Es,t2(S3) = Hs,t-3 (3) There is a short exact sequence of complexes 0 -! (1) -! (3) -! W (1) -! 0 which induces a split short exact sequence on homology. (1) is a complex with 0 differential which corresponds to the Z-tower in t - s = 3. Let M denote the E1-term of the stable Adams spectral sequence for the mod p Moore spectrum given by the -algebra. The main results of [Ma1 ] and [HM1 ] state that there is a map of complexes W (1) -! M inducing an isomorphism on homology for t - s < 5s - 16 for p = 2 and t - s < (p + 1)qs - (p + 2)q for p odd. This reduces (a) to a statement about the stable E2-term of the Moore spectrum except in low dimensions where it is easily checked directly by comput* *ing the homology of the complex (3). The E2-term of the Moore spectrum is computed in the required range in [Mi ] for p odd and in [Ma3 ] for p = 2. It agrees with the description given in Figu* *res 1 and 2. This proves (a). Let Mn denote the mod p Moore space with top cell in dimension n and i : Sn-1 -! Mn denote the inclusion of the bottom cell. Let A : Mn+r -! Mn with r = 8 if p = 2 and r = q if p > 2 be a map inducing an isomorphism in K-theory (see [Ad ]). Suppose first that p is odd. Let ~ffdenote an extension of a generator ff 2 ß* *2pS3 to M2p+1. Then by [Ad , Proposition 12.7], ffk = ~ffO Ak-1O i 2 ßqk+2S3 survive* *s to a stable class which is detected by the e-invariant. Since ffk has Adams filtra* *tion at least k it must be represented in E2(S3) by the class in bidegree (t - s, s)* * = (qk + 2, k). Next let p = 2. Let ~~denote an extension to M13 of the generator ~ of ß12S3. Then ~k = ~~O Ak-1 O i 2 ß8k+4S3 has Adams filtration at least 4k + 1 (the Adams filtration of the Adams map is 4 since we are in the stable range). It follows * *that the Toda bracket < ~k, 2, j >2 ß8k+7S3 has Adams filtration at least 4k + 2. [A* *d , Proposition 12.18] shows that this bracket survives to a stable class of order 4 detected by the e-invariant so the bracket is represented on E2(S3) by the clas* *s in bidegree (t - s, s) = (8k + 6, 4k + 2). This completes the proof of (b). 4. Construction of maps Let : S4n+3 ! HPn denote the Hopf map. The cofiber sequence S4n+3 -! HPn ! HPn+1 shows that the obstruction to extending a map f : HPn ! HP1 to HPn+1 is (5) o(f) := [f O ] 2 ß4n+3HP1 . Alternatively, one can regard the obstruction as an element in ß4n+3HPn+1, in which case, it is given by the formula (6) o(f) := [f O ] - deg(f)n+1[ ] 2 ß4n+3HPn+1 as a simple cohomology calculation shows. Note that, as S3 ! HP1 factors through S4, the isomorphism (4)shows that any element of ß*HP1 (in particular the obstruction to extension) factors uniqu* *ely SELF MAPS OF HPn VIA THE UNSTABLE ADAMS SPECTRAL SEQUENCE 7 through the bottom cell of HP1 . We will still write o(f) for the corresponding element in ß4n+3S4. We will write KO(X), K(X), KSp(X) for the reduced K-groups of X. Using the Atiyah-Hirzebruch spectral sequence, one easily checks that K0(HPn) = Z[x]=xn+1, and KO0(HPn) = Z[y, z]= < yn+1, z - 4y2 > where x and y are the classes repre- sented by the reduced tautological bundle over HPn. The complexification map c : K(HPn) ! KO(HPn) sends x to 2y. The forgetful map r : K(HPn)_! KO(HPn) sends yzk to x2k+1 and zk to 2x2k. Moreover KSp(HPn) = KO 4(HPn) is a free abelian group of rank n and the image of the forgetful map KSp(HPn) ! K(HPn) consists of those integer combinations of the xi where the coefficient of xi is* * even when i is even. In [FG ], Feder and Gitler show that the Adams operations on K(HPn) are given by the formula 2(l2 - 1) . .(.l2 - (n - 1)2) (7) _l(x) = l2x + . .+.2l______________________(2n)!xn. and this in turn determines the action of the Adams operations on KO(HPn) (reca* *ll that both c and r commute with Adams operations). Let A be Adams' category of abelian groups with Adams operations [Ad , Section 6]. Recall from [Ad ] that the (real) e-invariant is a group homomorphism (8) Z -e! ExtA(KO(X), KO(Sj+1)) where Z = {ff 2 ßj(X)|KO(ff) = 0}. e(ff) is the Yoneda class of the extension 0 ! KO(Sj+1) ! KO(X [ffej+1) ! KO(X) ! 0. If X is a sphere there is a natural identification of the target of e with a su* *bgroup of Q=Z. The stable e-invariant of fi 2 ßk+lSk is defined by representing fi as* * a class fi0 2 ß8m+lS8m for some m and then setting es(fi) = e(fi0) 2 Q=Z. This is independent of the choice of m. Proposition 4.1. Let f be a self map of HPn and let o(f) 2 ß4n+3S4 denote its obstruction to extension. Then deg(f) 2 Rn+1 if and only if the stable e-invari* *ant of o(f) vanishes. Proof.Let k = deg(f). Feder and Gitler show in [FG ] that k 2 Rn+1 if and only * *if there is a ring endomorphism OE of K0(HPn+1) commuting with Adams operations, such that OE(x) is in the image of the forgetful map KSp(HPn) ! K(HPn). In turn, this is equivalent to the existence of a map _ : KO(HPn+1) ! KO(HPn+* *1) in A such that the following diagram in A commutes (9) 0 _____//KO(S4n+4)____//_KO(HPn+1)_____//ØKO(HPn)____//0 |kn+1| Ø_Ø f*|| fflffl| fflffl fflffl| 0 _____//KO(S4n+4)____//_KO(HPn+1)_____//KO(HPn)_____//0 and _(y) is in the image of the forgetful map KSp ! KO. In fact, a ring endomorphim of K0(HPn+1) commuting with Adams operations determines a unique ring endomorphism _ of KO0(HPn+1) commuting with Adams operations and this will necessarily make (9)commute. Conversely, such a _ is determined by its value _(y) since by (7), _l(y) generate KO0(HPn+1) over Q. Commutativity of (9)then implies that _(y) is determined by deg(f) and it then 8 GUSTAVO GRANJA follows that _(y) must correspond to the unique (see [FG ]) endomorphism OE of K0(HPn+1) Q commuting with Adams operations such that OE(x) = deg(f)x+. ... Consider the extension E in A which is (9) as an extension of groups but where the Adams operations are replaced with ~_l= l2_l. Because KO(HPn) is torsion free, the existence of the map of extensions (9) is equivalent to the existence* * of the corresponding self map of E. Cupping with the generator of KO(S4) yields a map in ExtA (10) E -! (0 ! KO(S4n+4) ! KO( 4HPn+1) ! KO( 4HPn) ! 0) which is termwise an injection of abelian groups. The condition that OE(x) lie * *in the image of the forgetful map from KSp is easily seen to be equivalent to the cond* *ition that the self map of E extends over the monomorphism (10). We conclude that k 2 Rn+1 iff the following self map exists in ExtA 0 ____//_KO(S4n+8)____//KO( 4HPn+1)Ø_____//KO( 4HPn) ____//_0 kn+1|| ØØ |4f*| fflffl| fflffl fflffl| 0 ____//_KO(S4n+8)____//KO( 4HPn+1) _____//KO( 4HPn) ____//_0 or equivalently, writing 4 : S4n+3 ! 4HPn for the attaching map of the top cell of HPn+1, iff the following equality holds in ExtA: e( 4 ) O 4f* = kn+1 O e( 4 ) where O denotes the Yoneda product. Since the e-invariant sends compositions of maps to Yoneda products this is the same as e( 4(f O )) = e(kn+1 4 ) and, since e is a group homomorphism, by (6)this is equivalent to e( 4o(f)) = 0 in ExtA(KO( 4HP n), KO(S4n+8)). Since o(f) factors through the inclusion of the bottom cell of HPn, letting X* * = 4HPn [o(f)e4n+8, we have a map of extensions 0 ____//_KO(S4n+8)__________//_KO(X)___________//KO( 4HPn) ____//_0 |1| || i*|| fflffl| fflffl| fflffl| 0 ____//_KO(S4n+8)____//_KO(S8 [ 4o(f)e4n+8)____//_KO(S8)______//_0 where i denotes the inclusion of the bottom cell, so we have the equality es(o(f)) O i* = e( 4o(f)) The long exact sequence in ExtA induced by * 0 -! KO( 4(HPn=HP1)) -! KO( 4HPn) -i!KO(S8) -! 0 together with the easily checked fact that Hom A(KO( 4(HPn=HP1)), KO(S4n+8)) = 0 show that composition with i* is injective which concludes the proof. We are now ready to complete the proof of our main result. Consider the funct* *ion 8 >>4 _2-h3 i forp = 2, n > 5 >:- 1 - 2n-2_ p-1 forp 3, n > 5. Theorem 1.2 is now an immediate consequence of the following result. SELF MAPS OF HPn VIA THE UNSTABLE ADAMS SPECTRAL SEQUENCE 9 Theorem 4.2. pf(p,n)Z(p) Rn,p. Proof.Pick an Adams resolution for HP1(p)constructed from a minimal resolution * *so that all the d1-differentials vanish. Then the Adams covers X{s} are 3-connecte* *d, ß4(X{s}) = Z(p)and the projection maps X{s} ! HP1 induce multiplication by ps on ß4. Thus to prove the theorem, it is enough to provide a map HPn ! X{f(p, n)} of degree k on the bottom cell for an arbitrary k 2 Z(p). This will * *be obtained by extending a map HP1 ! X{f(p, n)} of degree k. Consider first the case when p = 2. If n is even, then for k < n - 1 the obstruction to extension of the map HPk ! X{f(2, n)} to HPk+1 lies in the group ß4k+3X{f(2, n)}. By the vanishing line of Theorem 3.1(a) this group is 0 so the map extends. Thus we have a map HPn-1 ! X{f(2, n)}. Since the Hopf map S4n-1 ! HPn-1 has Adams filtration 1, the composite S4n-1 ! X{f(2, n)} factors through X{f(2, n) + 1}. Let fl : S4n-1 ! X{f(2, n) + 1} denote this obstruction class. By Theorem 3.1(b), its image in ß4n-1HP1(2)is detected by the e-invariant (it is in the summand of order 4 corresponding to one of the top lightning flas* *hes). Since the map ß4n-1X{f(2, n) + 1} ! HP1(2)is injective (because there are no nonzero differentials E4n,k-r+1r! E4n-1,krfor k > f(2, n)), by Proposition 4.1 * *we must have fl = 0. Therefore our map extends to give a map HPn ! X{f(p, n)} of degree k. Moreover, this map extends one more stage because the obstruction to extension lies above the vanishing line of Theorem 3.1(a). This completes the proof for p = 2. For p odd, the reasoning is identical, except that for p = 3 and n = 5 we have to use the fact that ß19S3(3)is cyclic (see [To]) and hence detected by the e-invariant. Remark 4.3. For p odd, Theorem 4.2 can be improved in a small range of values of n using the following observations: (a) For n < (2p+1)(p-1)_2all classes in ß4n-2(S3(p)) are detected by the e-inva* *riant, therefore the conjecture holds at odd primes for these values of n. This fo* *llows from the results of Mahowald and Harper-Miller (cf. proof of Theorem 3.1) together with calculations with the -algebra in low degrees. (b) Let g(p, n) = sup{k : pkZ(p) Rn,p}. It is not hard to show that if f, g are self maps of HPn, the obstruction to extension of the composite is o(fg) = deg(g)no(f) + deg(f)o(g). Since (by Proposition 2.2) there is a self map of HPn(p)of degree p2, Selick's exponent theorem [Se] implies that g(p, n + 1) 2 + g(p, n). References [Ad] J. F. Adams, On the groups J(X).IV, Topology 5 (1966) 21-71. [AC] M. Arkowitz and C. R. Curjel, On maps of H-spaces, Topology 6 (1967) 137-1* *48. [CM] E. Curtis and M. Mahowald, The unstable Adams spectral sequence for S3, Co* *ntemp. 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