Note on desuspending the Adams map


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1 Math. Proc. Camb. Phil. Soc. (1986), 99, Printed in Qreat Britain Note on desuspending the Adams map BY F. R. COHEN AND J. A. NEISENDORFER University of Kentucky and Ohio State University (Received 9 May 1984; revised 17 May 1985) Let p denote a fixed prime and let P n (p r ) denote the cofibre of the degree p r map on consider n l).p 3 (p r ) and show that the map q in the cofibre sequence induces a split epimorphism on the ^primary component of 7T 2p S 3 if p > 2. That analogous maps q: P n (p r )>S n, n ^ 4, induce split epimorphisms on the jjprimary component of n n+2p _ 3 S n, p > 2, is shown in work of J. F. Adams [1]. It is the purpose of this note to document the above computation in the case n = 3 for the use of others. Recall that the generator a x of the ^primary component of n ip S 3,p > 2, has orders and thus extends to a map P 2p+1 (p)>s 3. The result of this note is that, if p is an odd prime, there is a choice of map A: P 2p+1 (p) > P 3 (p r ) such that qoa induces an epimorphism on the ^primary component of n 2p S 3. In [1], Adams constructed a map with n greater than 3 such that qoa induces an epimorphism on the ^primary components of Jr n+2p _ 3 S n. Thus the lift constructed here is a choice of desuspension of the Adams map. We give two proofs, one of which is elementary but longer. 1. The short proof Throughout this section, p is an odd prime. We wish to show PROPOSITION 11. There is a homotopy commutative diagram A S 2p where a^ is a generator for the pprimary component ofn 2p S 3. The proof of 11 occupies the rest of Section 1. Let S 2n+1 {p r } be the homotopytheoretic fibre of the degree p r map on S Zn+1. Note that the pinch map q: P 2^1^) > S 2n+1 factors thiough a map q': since p r q is null homotopic. Recall ([2], [6]) that there is a fibration sequence 00 fi«' G(n) x n * fc=l
2 60 F. R. COHEN AND J. A. NEISENDORFER where C(n) is the localization at p of the homotopytheoretic fibre of the double suspension E 2 : S 2n ~ l ~> Q 2 S 2n + 1 and V a P na (p r ) is a certain bouquet ([3], [6]). Furthermore, the factor 2 V a P Ua (P r ) * s a retract of QP 2n+1 (p r ) and the modp homology of Uk=i'S zpkn  1 {p r+1 } x QV<* P n "{p r ) injects into that of QP 3 (p r ) [4]. We will use this fibration when n = 1. Let y: P 2 v +1 (p) > S 3 denote an extension of a v Since y has orders, it is in the image of y': P 2 *> +1 (p) + S 3 {p r }. The adjoint y": P 2 P(p) » Q.S 3 {p r } represents an element in n 2p (Q.S 3 {p r ); Z/p) and all we need to show is that y" is in the image of an element in n 2p (QP 3 (p r ); Z/p). Hence, if 8 is the connecting homomorphism in the long exact modp homotopy sequence of the above fibration, we want 8(y") = 0. Notice that 7r 2 j,_ 1 (C(l); Z/p) is zero since C(l) has the same homotopy type as the 3fold loop space of the localization atp of the 3connected cover of S 3. Hence By" lies in 7T 2p _ 1 (S 2 P 1 {p r+1 2p _ 1 (Q\/ a P n '(p r );Z/p). Since By" maps to zero in nzpi(qp 3 i!p r );Z/p), the modp homology of S 2p ^{p r+l } x Q V a P na (p r ) injects into that of np 3 (p r ), and 7T 2p _ 1 (S 2 P 1 {p r + 1 }; Z/p) is isomorphic to H ip _ 1 {S 2 P 1 {p r + 1 }; Z/p), it follows that dy" lies in the second summand. Since the summand is a retract of n ip _ 1 {D.P z {p r ); Z/p), it follows that dy" = 0. The proposition follows. The modp homology H lt.(q,p 3 (p r ); Z/p) is the tensor algebra T[u,v] where u has degree 1, v has degree 2, and the rth homology Bockstein is given by f} T v = u. The diagram in 11 exists if and only if v v is in the image of the modp Hurewicz map $: n*(qp 3 (pr); Z/p) * H*(QP 3 (p>); Z/p). That these statements are equivalent follows from the following two facts: (i) The adjoint of a possible qfl is characterized by having modp Hurewicz image equal to i p where t generates H 2 (QS 3 ; Z/p) and v maps to i. (ii) In dimension 2p, the piimitives of T\u, v] are generated by v v and commutators of u and v. These commutators map to zero in H^{C1S 3 ; Z/p) and are in the image of the modp Hurewicz map. For more details on (i), see [4]. 2. The elementary proof Consider the map/: P 3 (p T )>K(Z/p r, 2) which induces an isomorphism on H 3 ( ; Z/p r ) together with the natural map : K(Z/p r, n) > K{Z, n+ 1) which induces an isomorphism on H n+1 ( ; Z/p r ). Thus there exists a homotopy commutative diagram where the rows (but not the columns) are fibrations: K(Z/p r,l)x
3 Note on desuspending the Adams map 61 The main result of this note is given by the second statement in PROPOSITION 21. Letp be an odd prime. (a) There exists a map p: P 2p+1 (p) > X such that the composite jkp induces an isomorphism on H 2p ( ; Z/p). (b) There exists a homotopy commutative diagram where a x is a generator for the pprimary component of n 2p 8 3. Next consider M(p r ) = (V 0 *ij<zp2 pi+2i (P r )) V P 2p {p T+1 ). We shall show LEMMA 22. Ifp > 2, there exists a map A: M(p r )> X such that (i) A induces an integral homology isomorphism in degree less than 2p and (ii) A induces a split monomorphism on n i for i less than or equal to 2p. We need to know the integral cohomology of X in a range. LEMMA 23. Ifp > 2, then (Z/p r if i = 2k, 4 s? 2k < 2^2, if i = 2p + 1 or 2p + 2, and,0 for other i, 0 < i < 2p + 2. Furthermore, (jk)* is an isomorphism on H 2p+1 ( ; Z). Granting 22 and 23, we prove 21 (a). Let K denote the cofibre of A, the map which A 0 was given in 22. Consider the cofibre sequence M(p r ) > X * K. Let F denote the homotopytheoretic fibre of 6. The natural lift of A to F induces an integral homology isomorphism in dimensions < 2p + 2 by the Serre exact sequence. Thus by the long exact homotopy sequence for a fibration together with 22, there are split short exact sequences 0>7 for i < 2p. But H 2p (K; Z) is isomorphic to n 2p K by the Hurewicz theorem. Thus n 2p K is isomorphic to Z/p. Any choice of splitting for the epimorphism n 2p X > n 2p K induces a map p: S 2p > X which is of order p. Thus there is a choice of extension of p to p. pnp+i(p) _>. x. By 23, jkp induces an isomorphism on H 2p ( ; Z) and 21 (a) follows. To prove 21 (b), observe that a, x : S 2p » $ 3 <3> has nonzero Hurewicz image. This suffices. Observe that the above proof gives PROPOSITION 24. Ifp > 2, then M(p r ) if if i < 2p. We omit the proof of the following 2primary analogue: i = 2p, and
4 62 F. R. COHEN AND J. A. NEISENDORFER PROPOSITION 25. (i) n z P 3 (2 r ) ~ Z/2 r + 1, and fz/4 if r = 1, (ii) 7T 4 P 3 (2') S ' ' IZ/2 0Z/2»/ r> H*(X;Z) To compute H*(X; Z), we first give information on the cohomology algebra H*( Y; Z) by using the Serre spectral sequence in integral cohomology for the fibration 7 CP > Y > P 3 (p r ). The following result is stated without details of proof. LEMMA 31. (i) H 2i (Y; Z) ~ Z and f 0 i/ t < p, (H) jy^+^rjz)^ LZ/p t/ i=p; in # 2i (CP ; Z) if i $ 0(p),? in J?^(CP ;Z), j<2>; (iii) JAere are choices x i of generators for H 2i ( Y; Z) such that, ifi + k < p 2, then p r ~ 1 x i+k if either ior kis zero P r+1 %i+h if *i& + 0(p) ^ t + is O(JJ); (iv) j*: # 2»+ 1 (>S 3 <3>,Z) > /? 2 P +1 (y; Z) is an isomorphism. Using the information in 31 together with the integral cohomology Serre spectral sequence for the fibration S 1 > X > Y, we obtain ; Z) s: 'Z/p r if» = 2j and j^p1, 1 if i = 2p, if i = 2p + 1, and 1 or l/p Z/p T ~ x if i = 2p + 2. This is the content of Lemma Proof of 22 By construction, X is 2connected and n 3 X ~ n 3 P 3 (p r ) ^ Z/^) r. Consider the composite u given by S 1 > P 2 (p r ) > QP 3^) where ^ is the Freudenthal suspension. Notice that the modp reduction of the Hurewicz image of the Samelson product [u, u] is given by [u, u] in H*(Q.P z (j> r ); Z/p). Thus [u, u] in Z/p r is of order p r. Since [u, u] and [u, v] are the unique primitives in their respective degrees, we have shown that there is a map w: P 3 (p T ) * O.X such that [(Qh). w]^ gives an monomorphism on the module of primitives in dimensions 2 and 3. Let T denote the composite {Q.h).w. Thus T is a map from P 3 (p r ) to (XP 3 (# r ). Define ad*: (A P V)) A P 3 (p r ) * ilp 3 b r ), where A& denotes the Afold smash product and ad fc = [E[E[... [E,T]...] the {k+ 1) fold Samelson product. Since p is an odd prime, there exists a map >(AFV))APV) k
5 Note on desuspending the Adams map 63 which induces an isomorphism on H 2k+S { ; 1/p) [5]. Notice that adj;.t + (t>') = [i>[i;[... [vu]...] in ff»(qp 8 (p'); 1/p) = T[tt,»] where t/ is the top class in H 2k+3 (P 2k + 3 (p r ); 1/p). By ([2], sections 3 and 4) the elements {ad* (v) (u), ytf r ad* (v) (u)\ 0 < k < p  2} are nonzero and algebraically independent. Since Q.X is the homotopytheoretic fibre of a map ClP 3 (p r ) > K(l/p r, 1), the maps &d k.i lift to QX. Thus, we obtain maps and the multiplicative extension of e to 8: A P 3+2i (p r ) > 2 Q(e):nS( By the methods of [4], we show LEMMA 41. There exists a map V 0sy<p2 a: QX > Q2( V such that o(qe) is homotopic to the identity. Furthermore, the map e induces an integral homology isomorphism in degrees < 2p 2. Holding the proof of 41 in abeyance, we first consider the cofibre D of the map e. Notice that there is a split short exact sequence 0 > " 2p i( V P 4+2 '(2> r )) ^ *,_! Z > *,_! D > Since Z) is (2j> 2)connected, there is a map /i: P 2p (p r+1 ) > X which induces an isomorphism on H 2p _ x ( ; Z) by Lemma 23 and the above remarks. Thus the composite A given by P i+2i (p r ))vp 2p (p r+1 ) > XvX > X 3 induces an isomorphism on integral homology in degrees < 2p l. That A induces a split monomorphism on np i < 2p, follows from the HiltonMilnor theorem [7] and the splitting ev/i fold ^^2 P iy^^2p>( V P 4+2 V)) H 2p _ s Y, where the righthand map is given by the splitting in 41 and the Hurewicz homomorphism. That A induces an integral homology isomorphism in degrees less than p follows from 41 and 23. Thus 22 follows. We finish this section by supplying a proof of 41. Since the elements ad* (v)(u) and fi T &d h (v)(u), 0 ^ k ^ p 2, are algebraically independent, (Oh)* (Q.e)+ is a monomorphism with Bockstein acyclic image. As in proposition 17 of [4] there is a map rjr: I,QP 3 {p r ) * SQS( V P 3+2i (p r )) K3
6 64 F. R. COHEN AND J. A. NEISENDORFER such that ^.S(QA.Oe) is homotopic to the identity. Thus by [4], there is a map a: QP 3 (p r )+ ii2(v 0^<p3P 3+2 'b r )) such that a(qh.q.e) is homotopic to the identity. The map or of 41 is given by a. Qh. Finally, it suffices to show that e induces an integral homology isomorphism in degrees < 2p 2. If F denotes the homotopytheoretic fibre of e, it suffices to check that F is (2p 3)connected by inspection of the integral homology Serre spectral sequence. Since F is simplyconnected, we will consider the Serre spectral sequence in modp homology for the fibration Oft Q( V P 4+2 '(i^)) ttx > F. It suffices to show that ( lh)+ is an isomorphism in degrees < 2p 2. Evidently (O.h)^ is a monomorphism. We compare the Poincar6 seiies for the modp homology of ^(V 0^j^P3 pi+2i (P r )) and D.X. Since the fibration OP 3 (p r ) + K(Z/p r, 1) is orientable and induces an epimorphism in modp homology in degrees through 2p 1, it follows that the Poincar6 series for the modp homology of 2X in degrees ^ 2p 2 is given by %, where Thus in degrees < 2p 2, 1t * ~ 1 / t 2 ' If 6 is the Poincare' series for the modp homology of O( V P i+21 (P r )), then (KK3 6 = In degrees ^ 2p 2, 6 is given by in this range and 41 follows. 1,.,. 1t )/())  which is1tt* Both authors were partially supported by National Science Foundation grants. REFERENCES [1] J. F. ADAMS. On the groups J(X), IV. Topology 5 (1966), [2] F. R. COHEN, J. C. MOORE and J. A. NEISENDORFER. Torsion in homotopy groups. Ann. of Math. 109 (1979), [3] F. R. COHEN, J. C. MOORE and J. A. NEISENDORFER. The double suspension andpprimary components of the homotopy groups of spheres. Ann. of Math. 110 (1979), [4] F. R. COHEN, J. C. MOORE and J. A. NEISENDORFER. Exponents in homotopy theory, to appear in the Conference Proceedings in honor of J. C. Moore's 60th birthday. [5] J. A. NEISENDORFER. Primary homotopy theory. Memoirs of the A.M.S., 232, L6] J. A. NEISENDORFER. 3 primary exponents. Math. Proc. Cambridge Philos. Soc. 90 (1981), [7] G. W. WHITEHEAD. Elements of Homotopy Theory. Graduate Texts in Mathematics (Springer Verlag, 1978).
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