STABLE GEOMETRIC DIMENSION OF VECTOR BUNDLES OVER ODD-DIMENSIONAL REAL PROJECTIVE SPACES MARTIN BENDERSKY AND DONALD M. DAVIS Abstract.In [6], the geometric dimension of all stable vector bundles over real projective space Pn was determined if n is even and sufficiently large with respect to the order 2eof the bundle in gKO(Pn). Here we perform a similar determination when n is odd and e > 6. The work is more delicate since Pn does not admit a v1-map when n is odd. There are a few extreme cases which we are unable to settle precisely. 1.Statement of results The geometric dimension gd(`) of a stable vector bundle ` over a space X is t* *he smallest integer m such that ` is stably equivalent to an m-plane bundle. Equiv* *alently, gd(`) is the smallest m such that the classifying map X -`! BO factors through BO(m). The group gKO(P n) of equivalence classes of stable vector bundles over * *real projective space is a finite cyclic 2-group generated by the Hopf line bundle ,* *n. In [6], it was shown that, for sufficiently large even n, the geometric dimen* *sion of a stable vector bundle over P ndepends only on its order in gKO(P n) and the mod 8 value of n. For bundles of order 2e, this value, called sgd(n, e) or sgd(__n,* * e), where __nis the mod 8 residue of n, was completely determined; its approximate value * *is 2e. A key role in this analysis was played by KO-equivalences Pkn+8+8! Pkn, defined* * if n is even, k is odd, and n + 8 < 2k - 1. Such maps do not exist when n is odd, and so the methods and results are somewhat more complicated. The term "stable" geometric dimension (sgd) refers to the fact that the geometric dimension achie* *ves a stable value as n gets large within its congruence class. __________ Date: June 14, 2005. 1991 Mathematics Subject Classification. 55S40,55R50,55T15. Key words and phrases. geometric dimension, vector bundles, homotopy groups. We would like to thank Mark Mahowald for valuable conversations related to this work. 1 2 MARTIN BENDERSKY AND DONALD M. DAVIS An important role in [6] was played by the v1-periodic spectrum functor des* *cribed in [7, 7.2]. We are interested in the stable portion of [P n, BSO(m)], i.e. th* *e portion which persists under jm : BSO(m) ! BSO. To achieve this, we define the stable portion s[P n, BSO(m)] = [P n, BSO(m)]= ker(jm*), and similarly for spectral sequence groups that approximate these groups. The g* *roup s[P n, BSO(m)] is cyclic since it maps injectively to the cyclic group [P n, * *BSO]. In [6], we proved that, if n is even, sgd(n, e) m iff (s[P n, BSO(m)]) e. (1.1) Here and throughout, (-) denotes the exponent of 2 in an integer, and if C is a cyclic group, then (C) denotes (|C|). The backwards implication has a simple * *and natural proof ([6, 1.5]), while the forward implication was proved by noting th* *at all the requisite nonlifting results were already in the literature. For odd n, we determine (s[P n, BSO(m)]) completely in Theorem 1.2, provided m 12. We prove in 2.1 that the backwards implication of (1.1) holds when n is odd, except that here this sgd refers to stable bundles of order 2e over projec* *tive spaces of sufficiently large dimension n mod 2L, with L usually, but perhaps * *not always, equal to 3. We will observe in Theorem 1.3 that, in almost all cases, k* *nown nonlifting results of Section 3 imply the converse; i.e. (1.1) holds in almost * *all cases when n is odd. However, there are some rare cases in which our computation of (s[P n, BSO(m)]) suggests there should be an extra nonlifting result which we have been unable to establish. Most of our work is devoted to proving the following theorem. Theorem 1.2. If m = 8i + d 12, then (s[P n, BSO(m)])) = 4i + t, where t is given by the following table. The two entries indicated by asterisks must be de* *creased by 1 if (n + 1 - m) 1_2m - 2. | d || | ___________|0__1__2__3_4__5__6__7_| 1 ||00 1* 1 2 2 3 3 || n mod 8 3 ||00 1 2 3 3 3 3 || 5 ||00 1 1 2 2 3* 3 || __________7_||00__0__0_1__1__2__3_|| STABLE GEOMETRIC DIMENSION 3 Combining this with 2.1 for liftings, and using 3.1 and 3.2 for nonliftings, * *yields the following result, which is our main theorem. Theorem 1.3. Define ffi(__n, e) by the table | e mod 4 || | ______|0_1___2___3__| 1 ||00* 0 0 || __n3 | | |0 0 -1 -2 | 5 ||00 0 0* || ____7_||02___2___1__|| Let e 7. For sufficiently large n __nmod 8,1 the geometric dimension of sta* *ble vector bundles of order 2e over P nequals 2e + ffi(__n, e), except that entries* * indicated with an asterisk might be 1 greater than indicated if (n + 1 - 2e) e - 2. The idea of stable geometric dimension was first proposed in [10]. It was cla* *imed there that if e 75, then sgd(n, e) 2e + ffi(__n, e) with ffi(__n, e) as in * *Theorem 1.3, ignoring the asterisks. We do not contradict those results here. However, if th* *e exotic nonlifting results mentioned above can be proved, they would contradict this li* *fting result of [10], for certain extreme cases with n odd. This does not seem to be * *out of the question, for the sentence near the bottom of [10, p.60] which includes a commu* *tative diagram seems to lack justification, which could render that proof invalid. For even-dimensional projective spaces, we also obtained, in [6], results abo* *ut sta- ble geometric dimension for bundles of order 2ewhen e < 7. We could do that her* *e for odd-dimensional projective spaces, but the arguments are extremely delicate. Co* *nse- quently, we will defer these cases of small m and e to the future. 2. Proof of Theorem 1.2 In this section, we prove Theorem 1.2. We begin with a general result similar* * to [6, 1.6]. Proposition 2.1. Let n be odd and e a fixed positive integer. For each m, there exists an integer L such that if (s[P n, BSO(m)]) e then, for sufficiently * *large N satisfying N n mod 2L, the geometric dimension of any stable vector bundle* * of order 2e over P N is less than or equal to m. __________ 1If the asterisked entries are increased to 1, then n _nmod 8 must be modif* *ied to n _nmod 2e-2in these cases. 4 MARTIN BENDERSKY AND DONALD M. DAVIS Proof.From the definition of X in [11]2 as a periodic spectrum whose spaces are telescopes of L L +k2L L1X ! L1+2 X ! . .!. 1 X ! . .,. with L1 0 mod 2L for the 0thspace, it follows, using James periodicity, that L [P n, BSO(m)] colim[P n+k2L, BSO(m)]. k 1+k2 L Thus the hypothesis implies that the stable bundle of order 2e over P n+k2 lift* *s to BSO(m) if k is sufficiently large. || The informal claim that we made in Section 1 that L can usually be chosen to * *be 3 can be seen either from the fact that (s[P n, BSO(m)]) determined in 1.2 usu* *ally only depends on n mod 8, or by restricting to P n-1and using the result from [6* *] that geometric dimension over these even-dimensional projective spaces eventually on* *ly depends on the mod 8 value of n - 1. The way in which Proposition 2.1 will be u* *sed in the proof of Theorem 1.2 is to use known nonlifting results (3.1 and 3.2) to* * assert that (s[P n, BSO(m)]) < e for various values of the parameters. The proof of the following result occupies most of the rest of this section. Theorem 2.2. Let n be odd, m 12, and OEn,mdenote the restriction homomorphism s[P n, BSO(m)] ! s[P n-1, BSO(m)] between cyclic 2-groups. Then 8 <2 if n 1 mod 8 | ker(OEn,m)|=: 1 otherwise 8 >><2if n 1 mod 4 and n - m 0, 1, 2 mod 8 | coker(OEn,m)|=>2 if n 1 mod 4 and (n + 1 - m) m=2 - 2 >: 1 otherwise Theorem 1.2 follows directly from 2.2 and the following recapitulation of res* *ults of [6]. __________ 2called Tel1X there STABLE GEOMETRIC DIMENSION 5 Theorem 2.3. ([6, 1.7,1.8,1.10]) If n 6, 8 mod 8 and 8i + d 9, then 8 >>>-1 d = -1 >>< 0 d = 0, 1, 2, 3 (s[P n, BSO(8i + d)]) = 4i + > >>>1 d = 4, 5 >: 2 d = 6. If n 2, 4 mod 8 and 8i + d 9, then 8 >>>0d = 0, 1 >>< 1 d = 2 (s[P n, BSO(8i + d)]) = 4i + > >>>2d = 3 >: 3 d = 4, 5, 6, 7. The lengthy proof of Theorem 2.2 will occupy the remainder of this section. We let n = 2k + 1. Viewing s[P, BSO(m)] as jm* im([P, BSO(m)] -! [P, BSO], it is clear that the kernel of OE2k+1,min 2.2 equals the kernel of * 2k [P 2k+1, BSO] -i! [P , BSO]. The proof of 2.1 implies that this kernel equals that of L i* 2k+c2L colim[P 2k+1+c2, BSO] -! colim[P , BSO], which, by the calculation of gKO(P n) in [1], has order 2 if k 0 mod 4, and i* *s trivial otherwise. This establishes the kernel part of 2.2. The cokernel of OE2k+1,m(= si*) is much more delicate. It involves the exact * *sequence * 2k ff* -1 [P 2k+1, BSO(m)] i-![P , BSO(m)] -! v1 ss2k(BSO(m)), (2.4) where ff denotes the attaching map. The following proposition is elementary. Proposition 2.5. Let x 2 [P 2k, BSO(m)] satisfy jm*(x) 6= 0, so its equivalence class [x] is a nonzero element in s[P 2k, BSO(m)]. oIf ff*(x) = 0, then [x] 2 im(OE2k+1,m). oIf ff*(x) 6= 0 and there is no y 2 ker(jm*) such that ff*(y) = ff*(x), then [x] is a nonzero element of coker(OE2k+1,m). 6 MARTIN BENDERSKY AND DONALD M. DAVIS The main point here is the necessity of checking for y. The proof of the cokernel part of 2.2 varies depending on the mod 4 value of * *k and mod 8 value of m in (2.4). Case 1: k 2 mod 4, m -1, 0, 1 mod 8. Here v-11ss2k(BSO(m)) = 0 by [3, 1.2,3.4,3.6] and so by Proposition 2.5 OE2k+1,mis surjective in 2.2 in this cas* *e. Case 2: k 2 mod 4, m 3, 4, 5 mod 8. By x33, 8 <1 d = 3 (s[P 8`+5, BSO(8i + d)]) 4i + : 2 d = 4, 5. By Theorem 2.3, 8 <2 d = 3 (s[P 8`+4, BSO(8i + d)]) = 4i + : 3 d = 4, 5. Thus OE2k+1,min 2.2 must have nontrivial cokernel when m 3, 4, 5 mod 8 (and * *still k 2 mod 4). This cokernel can have order at most 2 because v-11ss2k(BSO(m)) = Z=2 if m 3, 5 mod 8 by [3, 3.10], while v-11ss2k(BSO(8i + 4)) Z2 Z2. Case 3: k 0 mod 4, m -1, 0, 1 mod 8. By x3, 8 <-1 d = -1 (s[P 8`+1, BSO(8i + d)]) 4i + : 0 d = 0, 1. By 2.3 8 <-1 d = -1 (s[P 8`, BSO(8i + d)]) = 4i + : 0 d = 0, 1. We have already proved ker(OE8`+1,m) = Z=2, and hence coker(OE8`+1,m) 6= 0. We * *must prove the order of this cokernel is only 2. By [3, 1.2,1.3,1.4], v-11ss8`-1(SO(m)) is an extension of two Z=2-vector spac* *es4, one in filtration 2 and the other in filtration 4. We will show that the filtration* *-4 elements are in the image of ff* in (2.4); they are hit not by the stable summand but ra* *ther by elements of order 2. This implies that the desired cokernel has order only 2. __________ 3As was remarked prior to Theorem 1.3, all the lower bounds of that theorem are immediate from 3.1 and 3.2, and by 2.1, all the non-asterisked " " parts of* * 1.2 follow from this. When we invoke one of these (sgd(-, -) -)-results, we will * *just say "By x3." 4This is the first time of many that we will utilize the isomorph* *ism v-11ssi(SO(m)) v-11ssi+1(BSO(m)). STABLE GEOMETRIC DIMENSION 7 The attaching map for the top cell of P 8`+1is j on the (8` - 1)-cell. By [6,* * (2.4)], [P 8`, BSO(m)] [P10-8`, BSO(m)] [M0(24`), BSO(m)]. Since, by [6, (2.6)], the stable summand of [M0(24`), BSO(m)] comes from the bottom cell of the Moore space, ff* in (2.4) is equivalent to ~*`: v-11ss-1(BSO(m)) ! v-11ss8`(BSO(m)), (2.6) where ~` is the element of highest Adams filtration in the (8` + 1)-stem, detec* *ted by P `h1 in the Adams spectral sequence. This is seen by observing that OE` 0 S8`-ff!P 8`-! P1-8` and ~` -1 deg 1 0 S8`- ! S -! P1-8` become equal in ss8`(P10-8`^ J) Z2 Z2, where each equals the element of high* *est filtration. Thus, since v-11ss*(P ) v-11ss*(P ^ J) for spectra P by [12], th* *e two composites become equal in v-11ss8`(P10-8`). Thus they are equal in v-11ss8`(BS* *O(m)). Here we have used the 2-local J-spectrum which is the fiber of _3 - 1 : bo ! 4* *bsp. This spectrum played a key role in the early days of v1-periodic homotopy theor* *y, especially in [12]. In the spectral sequence of [3] converging to v-11ss*(SO(m)), elements in fil* *tra- tion 2 occur in eta-towers, with their Pontryagin duals described by elements* * in QK1(Spin(m))= im(_2), occurring with period 4. Dual to the composition (2.6) is v4`1s+1,t+2 # h#1 s,t # Es+1,t+2+8`2(Spin(m))# -! E2 (Spin(m)) -! E2 (Spin(m)) , (2.7) where v41is the isomorphism which shifts eta towers to elements with the same n* *ame, and h#1stays in the same eta tower. To see this, note that, with Y = Spin(m), * *if g 2 ssn(Y ), then g O ~`(= ~*`(g) in (2.6)) can be obtained as the composite ` n+2 ej n g S8`+n+1,! M8`+n+2(2) -A! M (2) -! S -! Y, (2.8) where A is an Adams map and ejan extension over the mod-2 Moore spectrum of j n Sn+1 -! S . Then (2.7) is dual to the horizontal composition in Diagram 2.9, wh* *ile (2.8) induces the composition around the top. The vertical maps @ are Bockstein homomorphisms for .2. 8 MARTIN BENDERSKY AND DONALD M. DAVIS Diagram 2.9. Diagram involving Bockstein and h1 s,s+n+2 v4`1 s,s+n+2+8` E2 (Y ; Z2) _____- E2 (Y ; Z2) | | ` | | ej* | | |@ |@ h1 s+1,s+n+2|? v4`1 s+1,s+n+2+8`|? Es,s+n2(Y )____- E2 (Y ) _____- E2 (Y ) Now the claim about filtration-4 elements y being ff*(x) with x an element of filtration 3 follows from (2.7), since x is the element in an earlier eta-tower* * with the same name as y. This completes the proof of Case 3. For the remaining cases, we will need the following result, where Q(-) denote* *s the indecomposables. Theorem 2.10. For any positive integers n and m, there is a spectral sequence Er(n, m) converging to [P n, SO(m)]* with Es,t2(n, m) = ExtsA(K*( Spin(m)), K*( tP n)).(2.11) If n is even, then Es,2r2(n, m) = 0, and if n is also sufficiently large, there* * is a short exact sequence 0 ! ExtsA(QK1Spin(m)= im(_2), K1S2r+1) ! Es,2r+12(n, m) -ffi!Exts+1 1 2 1 2r+1 A (QK Spin(m)= im(_ ), K S ) ! 0. (2.12) If n is odd and sufficiently large, there is a split short exact sequence q* s,t i* s,t 0 ! Exts,n+tA(QK*(Spin(m))= im(_2)) -! E2 (n, m) -! E2 (n - 1, m) ! 0. (2.13) Several remarks are in order here. (i) We omit 2-adic coefficients from all K* **(-)- groups, and will continue to do so. (ii) A is the category of 2-adic stable Ad* *ams modules.([7]) (iii) We have replaced SO(m) by its double cover Spin(m). This do* *es not change v-11ss*(-), and indeed SO(m) = Spin(m). But for calculations such* * as (2.14), it is essential that the underlying space be simply-connected. (iv) Beg* *inning with (2.13), we will often abbreviate ExtsA(M, K*St) as Exts,tA(M). (v). The sp* *litting of (2.13) is just claimed for E2, not necessarily for the entire spectral seque* *nce. STABLE GEOMETRIC DIMENSION 9 Proof.By [7, 7.2], the spectrum SO(m) is K=2*-local, and so the existence of t* *he spectral sequence follows from [7, 10.4].5 By [8, 9.1], there is an isomorphism* * in A 8 <0 i = 0 Ki( Spin(m)) : QK1(Spin(m))= im(_2) i = 1. (2.14) By [1], if n is even, then 8 4i - 2, while if (` - i) 4i - 2, it is Z=25+ (`-i)generated by 24i-2- (`-i)D* *+ - x4i-1. Since restriction j#3to Spin(8i + 5) sends D+ to D and x4i-1to x4i-1, we deduce* * that j#3maps onto D if and only if (` - i) 4i - 2, establishing the claim in 2.2 * *about coker(OE8`+5,8i+6), one of the asterisk cases in 1.2 and 1.3. Case 6: k 0 mod 4, m 2 mod 8. The argument is similar to that of Case 5, although it has one additional complication. We use a diagram of exact seque* *nces STABLE GEOMETRIC DIMENSION 13 analogous to that of Case 5, with dimensions of projective spaces and indices of BSO(-) decreased by 4. By [6, 1.7,1.8], sj2 is an isomorphism of Z=24i. Using * *x3, (s[P 8`+1, BSO(8i + 1)]) < 4i + 1. As we showed at the beginning of the proof * *of 2.2, ker(OE8`+1,8i+1) = Z=2, and hence OE8`+1,8i+1cannot be surjective. What complicates the argument as compared to Case 5 is that v-11ss8`-1(SO(8i+* *1)) and v-11ss8`-1(SO(8i + 2)) are larger than the corresponding groups that appear* *ed in Case 5. These groups are taken from [3, 1.3,3.12]. Both of these groups have a * *large Z2-vector space in filtration 4, which maps isomorphically under j3. It is not * *an issue as possible image of ff*1on the stable summand because, as in Case 3, it is in * *the image under ff*1from a similar sum of Z2's. From the point of view of the spect* *ral sequence of 2.10, they are already hit by d2-differentials, and so we don't hav* *e to worry about whether they are hit by d4's. What is more of a worry is that E2,8`+11(Spin(8i + 1)) and E2,8`+11(Spin(8i +* * 2)) have, in addition to, respectively, the Z2-class D and the larger cyclic summan* *d C0 that they had in Case 5, also a summand L, which is the sum of many Z2's and ma* *ps isomorphically under j3, while the first group also has an additional Z2-class * *labeled x4i-3. The summand L is depicted by the big dots in [3, 1.3,3.12] and has dimen* *sion [log2(4_3(4i - 1))]. We will show that ff*1sends the generator of the stable su* *mmand to just the class D. The analysis of whether D hits the element of order 2 in * *C0 proceeds exactly as in Case 5. We obtain that j3 sends D nontrivially, and hen* *ce coker(OE8`+1,8i+2) = Z=2, if and only if (` - i) 4i - 4, which translates to* * the claim of the theorem in this case, the other asterisk case in 1.2 and 1.3. It remains to verify the claim about ff*1, which is done by applying Pontryag* *in duality. By (2.6) and (2.7), ff#1is determined by h#1 1,-1 # E2,12(Spin(8i + 1))# -! E2 (Spin(8i + 1)) . That this sends only the class D nontrivially to the stable summand is proved e* *xactly as in the two paragraphs of [6] which appear shortly after Diagram 2.24 of that paper. The first of the two paragraphs begins "In order to show that d3(g1) = 0* *." In summary, a presentation of E1,-12(Spin(8i + 1))# is given, and, for each basis * *element b of E2,12(Spin(8i+1))#, (h1)#(b) is interpreted as an element in that presente* *d group, and it is observed that only (h1)#(D) is nonzero. 14 MARTIN BENDERSKY AND DONALD M. DAVIS Case 7: k 0 mod 4, m 6 mod 8. Let k = 4` and m = 8i + 6. This time the diagram of the sort used in Case 5 does not quite work because j2 is not surjec* *tive, due to a d3-differential in [P 8`, BSO(8i + 5)] not present in [P 8`, BSO(8i * *+ 6)]. We can, however, consider an E2-version of the diagram, where ff*1and ff*2are, * *after dualizing, given by (2.7). The diagram below addresses what amounts to the d2- differential on sE0,-12(8` + 1, 8i + 6). The d4-differential on this summand i* *s then eliminated similarly to Cases 3, 4, and 6. v4`1h#12,8`+1 # sE0,-12(8`, 8i + 5)#---!sE1,-12(Spin(8i + 5))#---E2 (Spin(8i + 5)) x? x? x? ?? j#2?? j#3?? v4`1h#12,8`+1 # sE0,-12(8`, 8i + 6)#---!sE1,-12(Spin(8i + 6))#---E2 (Spin(8i + 6)) As in Case 6, the v4`1h#1on Spin(8i + 5) sends only D nontrivially, and j#3se* *nds the generator of the C0-summand to x4i-1, since ((8` + 1) - (8i + 5)) = 2. Thus v4* *`1h#1 on Spin(8i + 6) is 0, and hence OE8`+1,8i+6is surjective. Case 8: k 2 mod 4, m 2 mod 8. Let k = 4` + 2. The argument is similar to that of Case 7, but is complicated by P 8`+4not being K-equivalent to a Moore spectrum. Let, as in [6, 2.14], T n= Sn [j en+2[2en+3. From [6, (2.11),(2.13)], we have s[P 8`+4, BSO(m)] sv-11ss0-2(SO(m)), (2.19) where, by [6, (2.17)], v-11ss0n(X) [T n, (X)]. (2.20) The analogue of (2.6) is that the morphism ff* in (2.4) is equivalent to i*`: v-11ss0-1(BSO(m)) ! v-11ss8`+4(BSO(m)), where i` : S8`+5! T 0is the element of highest filtration (4` + 2) in its stem * *in the Adams spectral sequence of T 0. It is j~`on the top cell. The reason for this i* *s similar to the discussion between (2.6) and (2.7). In this case, both OE` 4 S8`+4-ff!P 8`+4-! P1-8` STABLE GEOMETRIC DIMENSION 15 and i` -1 f 4 S8`+4-! T -! P1-8`, where f is, up to periodicity, a restriction of the map in [6, 2.8], become equ* *al in ss8`+4(P14-8`^ J) Z2 Z2, where each is the element of highest filtration. No* *te that f has Adams filtration -1. Thus the two composites are equal in v-11ss8`+4(P14-* *8`), and hence, following by any element g of [P14-8`, BSO(m)] [P 8`+4, BSO(m)], ff*(g) = i*`(gOf) in ss8`+4( BSO(m)). Note that f induces the isomorphism obtai* *ned from (2.19) and (2.20). ei 0 n Let M6 -! T be an extension of i. Here M is the mod-2 Moore spectrum with top cell in dimension n. We claim that ei*: K0(T 0) ! K0(M6) (2.21) is the nontrivial morphism from Z^2to Z=2. One way to see this is to obtain ku** *(D(ei)) from ko*(D(ei)) by using bu = bo [j 2bo. Here D denotes the S-dual. There is a cofiber sequence M-6 ! D(MC(ei)) ! D(T 0). In the chart below, the solid dots are from the M-6 and the circles from D(T 0)* *. The differential in the ko*-chart is due to the j2 connection. It implies the diffe* *rential in the ku*-chart, which is the asserted homomorphism (2.21). Diagram 2.22. ko*(D(MC(ei))) and ku*(D(MC(ei))) e |6 e |6 ko*(D(MC(i))) | ku*(D(MC(i))) | r | r r | rr A b| r r rr@A b|p r rA| Ab rr|r@rb|@Abp _______________rbA _______________brr@b -7 0 -7 0 From e.g. [4, p.488] or [3, 3.6,3.16], Ext1,n+6A(P K1(Sn)) Z=2. We will nam* *e the nonzero class v21h1. In the spectral sequence converging to v-11ss*(Sn), this e* *lement supports a d3-differential, but in that converging to v-11ss0*(Sn), it survives* * to a ho- motopy class, which is the class i discussed above. (See [6, 2.18].) We obtai* *n the following analogue of Diagram 2.9. 16 MARTIN BENDERSKY AND DONALD M. DAVIS Diagram 2.23. Diagram involving Bockstein and v21h1 s,s+n+6 v4`1 s,s+n+6+8` E2 (Y ; Z2) _____- E2 (Y ; Z2) | | ` | | ei* | | |@ |@ v21h1 s+1,s+n+6|? v4`1 s+1,s+n+6+8`|? Es,s+n2(Y )____- E2 (Y ) _____- E2 (Y ) Here Y could be any space, but we use Y = Spin(m). The point of the diagram is that the composition around the top is ff*, while the composition on the bot* *tom sends an eta-tower to one with the same name. The claim about (2.21) was needed to establish commutativity of the triangle. Now that we have related ff* to v4`+21h1, we obtain the following analogue of* * the diagram in Case 7. v4`+21h#12,8`+5 # sE0,-12(8` + 4, 8i +-1)#--!sE1,-12(Spin(8i + 1))# ----E2 (Spin(8i + 1)) x? x? x? ?? j#2?? j#3?? v4`+21h#12,8`+5 # sE0,-12(8` + 4, 8i +-2)#--!sE1,-12(Spin(8i + 2))# ----E2 (Spin(8i + 2)) The same argument as in Case 7 now implies d2 = 0 : sE0,-12(8` + 5, 8i + 2) ! E2,02(8` + 5, 8i + 2). The d3-differential on sE0,-13(8`+5, 8i+2) is as it was on sE0,-13(8`+4, 8i+2),* * which was shown to be 0 in [6].7 That d4 = 0 on sE0,-14(8` + 5, 8i + 2) is seen as in* * most of the previous cases, using Diagram 2.23 to assert that the target was already hi* *t by d2 applied to eta-towers with the same name. Case 9: k 3 mod 4, m 6 2 mod 4, and m 12. We decompose ff* in (2.4) as * 2k+1 i* -1 [P 2k, BSO(m)] -eff![M , BSO(m)] -! v1 ss2k-1(SO(m)), (2.24) where Mn = Mn(2), and effis the attaching map for the top two cells of P 2k+2. * *Let k = 4` - 1. There is a commutative diagram in which rows are cofiber sequences * *and columns are K-equivalences __________ 7It was done in the paragraph of [6] near the end of Section 2, which begins * *"We prove now that d3= 0 on eE1,-12(Spin(8i + 2))." STABLE GEOMETRIC DIMENSION 17 M8`-1 --eff-! P 8`-2 ---! P 8` ---! M8` ?? ?? ?? ?? A`?y ?y ?y ?y 0 -2 M-1 --ff-! P1-8` ---! P10-8` ---! M0 x? x? x? x? (2.25) =?? ?? ?? = ?? q 0 4`-1 2 0 4` 0 M-1 ---! M (2 ) ---! M (2 ) ---! M The top vertical maps are just the v1-maps. The middle square on the bottom is * *from [6, 2.2], which was originally from [11]. The construction in [11] implies comm* *utativity of the lower right square. If this cofiber sequence is pushed one space farthe* *r, a commutative square is obtained which is the suspension of the lower left square. Hence the lower left square commutes. Thus we obtain a commutative diagram * s[P 8`-2, BSO(m)]-eff--![M8`-1, BSO(m)] x? x? ?? ?? *0 s[P1-2-8`, BSO(m)]ff---![M-1, BSO(m)] ?? ?? (2.26) ?y =?y q* -1 sv-11ss-2(SO(m))- --! [M , BSO(m)], where q is the collapse map. In the bottom row, s[M0(24`-1), BSO(m)] has been replaced by sv-11ss-1(BSO(m)) sv-11ss-2(SO(m)) because ` can be taken to be arbitrarily large, and so the maps from the top cell of the Moore space are eph* *emeral. When the eff*in the top row is followed by i* into v-11ss8`-3(SO(m)) to yield (* *2.24), we obtain from the diagram something agreeing up to isomorphisms with that obtained by applying s[-, BSO(m)] to the composite ` -1 q -1 S8`-2,! M8`-1-A! M -! S . (2.27) By [2], this composite is the element of order 2 in the stable image of J in * *the (8`- 1)-stem; however, we will compute it using (2.27) rather than this imJ descript* *ion. 18 MARTIN BENDERSKY AND DONALD M. DAVIS We will show that the composite ae2 1,-1 A` 1,8`-1 sE1,-12(Spin(m))-!q*E2 (Spin(m); Z2) -! E2 (Spin(m); Z2) @-! E2,8`-1(Spin(m)) (2.28) i* 2 is 0.8 Noting that E4,8`+11(Spin(m)) = 0 (2.29) by [3, 1.3,3.6,3.7], Theorem 2.2 follows in this case. We show that the Pontryagin dual of (2.28) is 0. Let C0 -d1!C1 -d2!C2 be the sequence of free Z(2)-modules associated to the sequence of free Z^2-mod* *ules in [3, 11.9]. Thus C0 = F , C1 = F F F , and C2 = F F F F , where F i* *s a free Z(2)-module on [m=2] generators. The transpose of the matrix of d1 is (0 2 4`-1), (2.30) and the transpose of the matrix of d2 is 0 1 -2 2 4`-1 0 B@ 0 0 0 C 4`-1A , (2.31) 0 0 0 - 2 and then the homology at Cs is Exts,8`-1A(P K1(Spin(m)= im(_2)). Here 2 (resp. j) is the matrix of _2 (resp. _3 - 3j) on P K1(Spin(m)). We are using here that for a rationally acyclic complex of finitely generated free Z(2)-modules, the i* *nclusion induces an isomorphism H*(-; Z(2)) ! H*(-; Z^2). In the remainder of this proof* *, we will write Z when we really mean Z(2). As observed in [3, proof of 11.3], Es,8`-12(Spin(m))# is the homology at C*s-* *1of the chain complex C* given by d*1 * d*2 * C*0- C1 - C2, (2.32) where C*s= Hom (Cs, Z) and the matrices of d*1and d*2are those of (2.30) and (2* *.31). The shift from s to s - 1 is due to the short exact sequence 0 ! Z ! Q ! Q=Z ! 0. __________ 8Note that ae2 and @ are parts of different Bockstein exact sequences, and so* * it is not automatic that the composite is 0. STABLE GEOMETRIC DIMENSION 19 Note that Es,4`-1 # * 2 (Spin(m); Z=2) is the homology at Cs=2 of the mod 2 reduction of (2.32), and ae#2: E1,8`-12(Spin(m); Z=2)# ! E1,8`-12(Spin(m))# is the boundary homomorphism ffi in the exact sequence of homology groups induc* *ed by the short exact sequence of chain complexes 0 ! C* -2! C* ! C*=2 ! 0. (2.33) To see this, note that the commutative diagram 0 ---! Z --2-! Z ---! Z=2 ---! 0 ?? ?? ?? 1?y 1_2?y i?y 0 ---! Z ---! Q ---! Q=Z ---! 0 induces a commutative diagram H1(C*=2) --ffi-!H0(C*) ?? ?? ae*2?y =?y H1(C* Q=Z) ---! H0(C*), from which the agreement of ffi and ae*2is immediate. The composite which we wish to show is 0 (dual to (2.28)) may now be identifi* *ed as ae2* * = * ffi * H1(C*(4`-1)) -! H1(C(4`-1)=2) -! H1(C(-1)=2) -! sH0(C(-1)). (2.34) Here the parenthesized subscript of C* is the subscript of , and C*=2 means the mod 2 reduction of C*. The identity map in the middle is due to the subscript n* *ot mattering mod 2, and the fact that A*is the identity homomorphism of K*(M). Sin* *ce, for the same parenthesized subscript, im(ae*2) = ker(ffi), we are reduced to pr* *oving ffi` * * ffi0 * ker(H1(C*=2) -! H0(C(4`-1))) ker(H1(C =2) -! sH0(C(-1))). (2.35) We will need the following result, culled from [3]. Theorem 2.36. Suppose m 12. 20 MARTIN BENDERSKY AND DONALD M. DAVIS oIf m = 2n + 1, then 8 (`) + 4 (2.37) with e > n. The group is presented by a matrix 0 1 2A1 0 B@u A2 nC 22 2 A , (2.38) u32n 2v where ui is odd, Ai > n, and v = min( (`) + 4, 2n + 1). The columns of this matrix correspond to generators ,1 and D of P K1(Spin(m)) under the isomorphism H0(C*(4`-1)) E1,8`-12(Spin(m))# P K1(Spin(m))=(_2, `4`-1), (2.39) where `j = _3-3j. The first row of (2.38) is due to a combina- tion of relations of the form _2(,i) and `4`-1(,i), while the sec- ond row is a combination of such relations together with _2(D) (with coefficient 1), and the third row is a combination of such relations together with 1.`4`-1(D). The first summand of (2.37) is the stable summand; it corresponds to the first (,1) column of (2.38). oIf m = 4a, then 8 2a and e3 e2 < 2a. The group is presented by a matrix 0 A 1 2 1 0 0 BB 0 2M -2M C B@u A2 2a-1 CC (2.40) 22 2 0 A 22a-1 u32v1 u42v2 with ui odd, Ai > 2a, M = min(2a - 1, (2` - a) + 3), v1 = min 0( (a) + 2, (`) + 4), and v2 = (a) + 2. Here min0(A, B) = min (A, B) unless A = B, in which case it is greater than either. STABLE GEOMETRIC DIMENSION 21 Under the isomorphisms of (2.39), the columns of (2.40) corre- spond to generators ,1, D+, and D-, and of the rows (relations) only the last one involves an odd multiple of `4`-1(D). Proof.For the first part, we use [3, 3.1,3.2] and [5, 3.18]. The proof of [5, * *3.15] explains how the rows of the presentation matrix are obtained, while [5, x4] de* *rives the inequalities for the exponents in those relations. Actually, [5, 3.18] only* * proves Ai n. The stronger result needed here follows by a more careful analysis of * *the proof of [5, x4]. It follows from [5, 3.18], refined to say that eSp(4` + 1, n)* * > n + 1 and the coefficients of ,1 in [5, (3.19)] and [5, (3.20)] are divisible by 2n+1. By [3, 8.1], eSp(-, n) is divisible by (2n + 1)!, which is divisible by 2n+1 * *for n 2. The divisibility of [5, (3.20)] is proved using its representation as n=2Xi j X i j (n - 1)22n-4+ n-jj22n-4j 8i 2`-1iSi,j j=2 i j-1 with j-2X i j i ji Si,j= (-1)t 2j-1t(2j - 2t - 1) j-t2 t=0 given in [5, (4.20)]. The term (n - 1)22n-4is divisible by 2n+1 for n 5. The * *other terms are divisible by 22n-j-3with 2 j n=2, which will be sufficiently divi* *sible except when (n, j) is (6,3). In this case, the additional divisibility is prov* *ided by S2,3= 30. The divisibility of [5, (3.19)] is proved similarly using its representation * *as X iin+2-jj in-jjjX i2`j (n + 1)22n-3 22n+1-4j j - j-2 8i i Si,j, j 2 i j-1 with Si,jas above, from [5, p.54]. The lead term (n + 1)22n-3is divisible by 2n* *+1 for n 3. Other terms are divisible by 22n-j-2with 2 j n=2, which is divisible* * by 2n+1. For the second part, we use [3, 3.3] and its proof in [3, x4]. The classes ,i* *, D, D+, and D- in P K1(Spin(m)) are as in [5, 3.10] and [3, 4.1], but do not play a maj* *or role in this paper. || We remark that the condition m 12 is necessary for the divisibilities of th* *e entries of the matrices to hold. 22 MARTIN BENDERSKY AND DONALD M. DAVIS By the definition of ffi using (2.33), if x = (x1, x2, x3) 2 C*1=2 is a cycle* * representing an element of H1(C*(4`-1)=2), then ffi(x) = 1_2_2(x2) + 1_2`4`-1(x3), (2.41) viewed as an element in the group presented by one of the matrices of 2.36. He* *re xi 2 F *or F=2*. We write ffi0 and ffi` for the boundaries ffi associated to C** *(-1)and C*(4`-1), respectively. Note that the relations ,j = j4`-1,1 are used to bring* * these elements into the 2- or 3-generator form of 2.36. This relation is a consequenc* *e of [5, 3.9], which says that modding out by _j- j4`-1for j = 3 and -1 also accomplishes modding out by _j- j4`-1for other odd j. The matrix (2.38) implies that when m = 2n + 1, sH0(C*(-1)) is isomorphic to Z=2n generated by ,1, since v = 2n + 1 in this case, and that in (2.41) with ` * *= 0, ffi0(x1, x2, x3) 6= 0 2 sH0(C*(-1)) if and only if the D-component of x3 is odd* *. This key point may warrant some explanation. The interpretation of the rows of (2.38) gi* *ven after (2.39) implies that when _2(x2) or `-1(x3) are written in terms of ,1 and* * D, using ,j = j-1,1, the ,1-component of each will be divisible by 2n+1 unless the* * D- component of x3 is odd, and when these are multiplied by 1/2, as they are in (2* *.41), the only way to obtain a nonzero component in the ,1-component of the Z=2n-group presented by (2.38) is then to have this D-component of x3 be odd. If the D-component of x3 is odd, then ffi`(x1, x2, x3) 6= 0 2 H0(C*(4`-1)), (2.42) since it is 1_2times the last row of (2.38) plus perhaps 1_2times the other row* *s. Such a vector is easily seen to be nonzero in the group presented by (2.38), regardles* *s of the value of v. This establishes the contrapositive of (2.35). The same argument applies when m = 4a, using the matrix (2.40). The previous paragraph carries through verbatim, with n replaced by 2a - 1. Case 10: k 3 mod 4, m 2 mod 4. The method of Case 9 does not apply here, since _-1 6= -1 in P K1(Spin(m)) when m 2 mod 4. However the result he* *re follows by naturality from Case 9. Let k = 4`+3 and m = 4j +2. The morphism sE0,-12(8`+7, 4j +1) ! sE0,-12(8`+ 7, 4j + 2) is bijective by [3, 3.3]. As we have just seen that d2 = 0 on the f* *ormer, it must also be 0 on the latter. Note that d3 on sE0,-13(8` + 7, 4j + 2) equals* * d3 on STABLE GEOMETRIC DIMENSION 23 sE0,-1 3 (8` + 6, 4j + 2), by the general form of the spectral sequence, and this e* *quals d3 on E1,-13(Spin(4j + 2)) by the paragraph after Diagram 2.16 beginning "By the proof." By [3, 3.12], this is zero. As there is nothing for d4 to hit by (2.29)* *9, we deduce that the generator of E0,-12(2k + 1, m) is an infinite cycle in this case, esta* *blishing Theorem 2.2 in this case. Case 11: k 1 mod 4, m 6 2 mod 4, m 12. Let k = 4` + 1. Similarly to (2.25), we have, using [6, 2.8], a commutative diagram in which rows are cofibr* *ations and columns are K-equivalences. M8`+3 --eff-! P 8`+2 - --! P 8`+4 ?? ? ? ?y ??y ??y M3 ---! P12-8` - --! P14-8` x? x x ?? ??? ??? 4`+1L 24`+1L 4` 2 24`+1L 4`+1 2 F ---! N (2 )---! N (2 ) where Nn(k) = Mn(k) [jen+1[2en+2, the map labeled 2 has degree 2 on the bottom 4`+1L cell, and 2 F is the stable fiber of this map. Thus F = M-1 [j M1 [2M2, and, with T n= Sn [j en+2[2en+3 as in Case 8, there is a cofiber sequence T -2! F ! T -1-2! T -1. (2.43) Similarly to (2.26), we obtain a commutative diagram, using [6, (2.13)] * s[P 8`+2, BSO(m)]--eff-![M8`+3, BSO(m)] x? x? ?? ?? s[P12-8`, BSO(m)]---! [M3, BSO(m)] ?? ?? ?y ?y 4`+1L sv-11ss024`+1L-2(SO(m))---![ 2 F, BSO(m)]. __________ 9which also holds when m 2 mod 4 24 MARTIN BENDERSKY AND DONALD M. DAVIS Since ` is large, the 24`+1Lmay be omitted by periodicity, and so ff* in (2.* *4) is obtained as the composite * -1 sv-11ss0-2(SO(m)) ! [M3, BSO(m)] -! [M8`+3, BSO(m)] -i! v1 ss8`+1(SO(m)). (2.44) This can be considered as the d2- and d4-differentials in the spectral sequence* * de- scribed prior to Case 4. Recall from [6, 2.16] that the E2-term for v-11ss0*(-)* * equals that for v-11ss*(-). The cofibration (2.43) yields a short exact sequence 0 ! K-1(T -1) -2! K-1(T -1) ! K-1(F ) ! 0 which is 0 ! Z^2-2! Z^2! Z=2 ! 0. Thus (2.44) is, at the E2-level, given by ae2 1,3 1,8`+3 @ 2,8`+3 sE1,-12(Spin(m)) -! E2 (Spin(m); Z=2) -! E2 (Spin(m); Z=2) -! E2 (Spin* *(m)), (2.45) similarly to (2.28). We can justify the ae2 between distinct bigradings in two * *ways. (a) Exts,tA(-; Z=2) has period 4 in t; (b) The morphism is induced by F ! T -1,* * and there is a K-equivalence F ! M3. Hence, by the same argument used in Case 9 to go from (2.28) to (2.35), showi* *ng that d2 = 0 on sE0,-12(8` + 3, m) is equivalent to proving ffi0` * * ffi0 * ker(H1(C*=2) -! H0(C(4`+1))) ker(H1(C =2) -! sH0(C(-1))). (2.46) Here ffi0`(x1, x2, x3) = 1_2_2(x2) + 1_2`4`+1(x3). The proof that (2.46) holds is similar to that of Case 9, except that the mat* *rix, using _3 - 34`+1instead of _3 - 34`-1has a slightly different form. The matrix* * is described in Lemma 2.50 when m is odd. One must prove, analogous to (2.42), that if the D-component of x3 is odd, then ffi0`(x1, x2, x3) 6= 0 2 H0(C*(4`+1)* *). This is easier than in Case 9 because of the 23 in the last row of (2.51). As before, t* *he last row is characterized by being the relation due to `4`+1(D) plus other terms. He* *nce ffi0`(x1, x2, x3) will involve 1=2 times the last row of (2.51), which, because* * of the 23 is certainly nonzero in the group presented by (2.51). STABLE GEOMETRIC DIMENSION 25 Finally, we must show d4 = 0 on sE0,-1 4 (8` + 3, m). The composite (2.44) may be viewed as applying [-, BSO(m)] to S8`+2-ff!P 8`+2! P12-8`! v-11P12-8`' v-11N0(24`). (2.47) The class of this composite is divisible by 4 in v-11ss4`+2(N0(24`)) v-11ss4`* *+2(P 8`+2). Call it 4fl. To see this divisibility, we use that ff goes to 0 in v-11ss8`+2(P 8`+4), sin* *ce it is an attaching map. Diagram 2.48, which is similar to those of [12, pp 94-5], dep* *icts v-11ss*(P 8`+2) ! v-11ss*(P 8`+4) near * = 8`+2. The group where * = 8`+2 is in* *dicated with an arrow, and the nonzero element in the kernel of this homomorphism is ci* *rcled. Diagram 2.48. v-11ss*(P 8`+2) ! v-11ss*(P 8`+4) near * = 8` + 2 |r____________________- ||r r |r r|r rr||r r r||r rr|||r@ r r|||r@ r r||r@ r r|||r@ r|||r@ r||r@ r|||r@r r|||r@ P 8`+2r||r@r P 8`+4 r|||r@r r|||r@rg r||r@r r|r r|||r@r r|_____________________- r|r r| |6 |6 This chart also depicts v-11ss*(N0(24`)), and the circled element equals the * *composite (2.47) (since the ff is nontrivial, because Sq4 is nonzero in its mapping cone)* *. The inclusion v-11T -1-iT!v-11N0(24`) induces in ss8`+2(-) an injection Z=8 ! Z=8 Z* *=2.10 Let g denote the generator of v-11ss-2(T -1), and let 2eg denote an extension* * of 2eg over an appropriate Moore spectrum. Then (2.47) equals the top row of the commutative diagram (2.49) followed by iT. ` 4g -1 S8`+3--i-! M8`+3 -A--! M3 ---! v1 T -1 ?? ?? ?? ?? 2?y 2?y 2?y =?y ` 2g -1 (2.49) S8`+3--i-! M8`+3(4) -A--!M3(4) ---! v1 T -1 __________ 10v-11T-1 can be defined to be T-1 ^ v-11J. 26 MARTIN BENDERSKY AND DONALD M. DAVIS Here 2 : M8`+3! M8`+3(4) from the mod 2 Moore spectrum to the mod 4 Moore spectrum has degree 2 on the bottom cell and degree 1 on the top cell. Since E3,8`+42(Spin(m)) and E4,8`+52(Spin(m)) are Z2-vector spaces, and there* * can be no extension from filtration 2 to filtration 3 by naturality, the only way t* *hat ff* in (2.44) could hit an element in filtration 4 is if fl* hits an element of ord* *er 4 in filtration 2, and there is a nontrivial extension. We will show that (2fl)* can* *not be nonzero in filtration 2. Since ff*(= (4fl)*) is given by applying [-, BSO(m)] to the top composite in (2.49), then (2fl)* is given by applying [-, BSO(m)] to the bottom composite. * *The E2-version of this bottom composite is just like (2.45) with Z=2 replaced by Z=* *4. Thus showing that (2fl)* is 0 in filtration 2 is equivalent to proving the anal* *ogue of (2.46) with C*=2 replaced by C*=4. We need the following lemma. Lemma 2.50. The matrix, analogous to (2.38) in the interpretations of its rows * *and columns, which presents H0(C*(4`+1)) for Spin(2n + 1) with n > 5 is 0 1 2A1 0 B@u A2 nC 22 2 A (2.51) u32n 23 with uiodd and Ai n + 1. This is proved similarly to 2.36. It differs in that it involves 4` + 1 rath* *er than 4` - 1. It is just [5, 3.18] with a lower bound for some exponents being 1 larg* *er than was proved in [5]. As we don't need this refinement here, we will not present * *the details of the proof, which are extremely similar to those of 2.36. Now the analogue of (2.46) with 4 instead of 2 is proved by the same method used for 2. Now we have that ffi0(x1, x2, x3) 6= 0 2 sH0(C*(-1)) if and only if* * the D- component of x3 is not divisible by 4. Here we need that Ai n + 1 in (2.38) wh* *en ` = 0, which was proved in 2.36. In this case, ffi0`(x1, x2, x3) is nonzero in * *H0(C*(4`+1)) because it is 1_4or 1_2times the last row of (2.51) plus 1_4times multiples of * *the other rows. This will be nonzero because of the 23 in the second column. This completes the argument (for Case 11) when m is odd. If m = 4a a similar argument works. A matrix of the same general form as (2.40) presents H0(C*(4`+1* *)). Its rows and columns have analogous interpretations. As in the case m odd, the STABLE GEOMETRIC DIMENSION 27 key point is a 23 which occurs in the last row, second column. This is due to * *the (3m+1 - 1)-factor in [3, (4.27)]. The m of that paper is our 4` + 1. This 23 wi* *ll cause (2.46) to hold, and with the 2 replaced by a 4, just as it did when m is odd. Case 12: k 1 mod 4, m 2 mod 4. Similarly to Case 10, the method of Case 11 does not apply because the chain complex used there required _-1 = -1. Again, we can make the required deductions by naturality. The morphism sE0,-12(* *8`+ 3, 4j + 1) ! sE0,-12(8` + 3, 4j + 2) is bijective by [3, 3.3]. If j is odd, the* * generator of E0,-12(8` + 3, 4j + 1) is a permanent cycle by Case 11, and hence so is its * *image. Now let j be even. The same naturality argument shows that d2 = 0 on sE0,-12(8`* * + 3, 4j + 2). That d3 = 0 is proved by the method of Case 10, using that d3 = 0 on Ee1,-13(Spin(4j + 2)) by [6, 2.23]. Finally we consider d4. We cannot use natur* *ality from E4(8` + 3, 4j + 1) because it had a nonzero d3 by [6, 2.23]. Instead we us* *e the argument in Case 11, that the attaching map ff equals 4fl. We use naturality fr* *om E2(8` + 3, 4j + 1) to see that (2fl)* must be zero in filtration 2, and deduce * *as in Case 11 that ff* is 0 in filtration 4. 3.Nonlifting results In [9], the following result was proven. Theorem 3.1. If u is odd and 24b+ffl> 4k + t, then gd(u24b+ffl,4k+t) 4k - 8b + d, where d is given in the following table. | ffl || | _____|0__1___2____3_ | 1 ||0 -2 -2 -4 || t 2 ||2 2 0 -4 || 3 ||2 2 0 -4 || ___4_||4_2___2____0_ || Several more nonlifting results could have been obtained by the same method. * *The author of [9] did not give careful enough consideration to Pbtwith t 1 mod 4 * *or b 2 mod 4. We sketch a proof of the following result. Theorems 3.1 and 3.2 to* *gether provide all the nonlifting results in Theorem 1.3, and those of [6, 1.1(2)]. 28 MARTIN BENDERSKY AND DONALD M. DAVIS Theorem 3.2. If u is odd and 24b+ffl> 4k + t, then gd(u24b+ffl,4k+t) 4k - 8b + ffi if (ffl, t, ffi) = (0, 2, 3), (0, 3, 3), (1, 4, 3), (1, 1, 0), or (0, 1, 2). Proof.We must show there does not exist an axial map 4b+ffl-4k+8b-ffiu24b+ffl-1 P 4k+tx P u2 ! P . This is done by showing that _3 - 1 applied to the dual class in 4b+ffl-1 ko-2(P--24k-t-1^ P-u24b+ffl+4k-8b+ffi-1^ P u2(3.3)) is nonzero. This class is called the axial class. 4b+ffl-1 Lemma 3.4. Let X = P--24k-t-1^ P-u24b+ffl+4k-8b+ffi-1. Then ko*(X ^ P u2 ) c* *on- tains summands 4b+ffl-1 u24b+ffl-2 ko*(X ^ Su2 ) ko*(X ^ P ). The upper edge of the second of these summands extends one filtration higher th* *an that of the first. Proof.Let A1 denote the subalgebra of the mod 2 Steenrod algebra generated by Sq1 and Sq2. We use that the Adams spectral sequence converging to ko*(X) has E2 = ExtA1(H*X). (We omit writing Z2 in the second variable.) Let N denote the A1-module with classes in grading 0, 2, 3, and 5 with Sq2Sq1Sq2 6= 0, and let N* *0 be defined by the short exact sequence of A1-modules 0 ! 5Z2 ! N ! N0 ! 0. If M is an A1-module which is free as a module over the subalgebra A0 generated* * by Sq1, then ExtA1(M N) = 0 in filtration > 0, and hence, for s > 0, we have Ext s,tA1(M 4Z2) Exts,t+1A1(M 5Z2) -! Exts+1,t+1A1(M N0). (3.5) The first of these groups can correspond roughly to the first summand of the le* *mma, and the last to the other summand, after adjoining many copies of ExtA1(M N). The filtration shift in (3.5) yields the conclusion of the lemma. 4* *b+ffl-2 Here we have used that, except in its bottom few cells, the A1-module H*P u2 4b+ffl-5 is built by short exact sequences from many copies of iN and one of u2 N0.* * A STABLE GEOMETRIC DIMENSION 29 deviation due to the bottom few cells of P u24b+ffl-2will not alter the Ext gro* *ups in the region of interest. Note that H*X is A0-free except in the case where t = 3 = f* *fi, in which case it is a direct sum of an A0-free summand and one that is inconsequen* *tial here. || Using some suspension isomorphisms, the part of (3.3) corresponding to the fi* *rst summand in 3.4 is ko-1(P--24k-t-1^ P4k-8b+ffi-1). The subscript of one P is odd11and the other 2 mod 4. The P4`+2is built from copies of N, which, after tensoring with the other P , give no Ext in positive * *filtration, together with , which changes bo to bu. Thus the chart for * *the portion of 3.4 due to the top cell is given by the diagram below, with the bott* *om class in dimension -8b + ffi - t - 2. Diagram 3.6. |r | |r |r | | |r |r | | |r |r | | |r |r | | r |r |r | | | r r| |r |r | | | | r r| r| . . . |r |r | | | | __r____r____r____r____________________________||||r All of our cases12have ffi - t = 1 - 2ffl. Thus the chart starts in -8b - 2f* *fl - 1, and its top element in dimension -1 is in filtration 4b + ffl. The summand of (* *3.3) corresponding to the second summand of 3.4 has top element in filtration 4b + f* *fl + 1. According to the third case of Table 12 of [9], the axial class has a compone* *nt 2 . u24b+fflin this second summand, i.e. at height 4b + ffl + 1, and so is nonz* *ero. || __________ 11except for the case (0, 3, 3), which is equivalent to (0, 2, 3) plus an add* *itional split summand 12with the exception noted in the previous footnote 30 MARTIN BENDERSKY AND DONALD M. DAVIS References [1]J. F. Adams, Vector fields on spheres, Annals of Math 75 (1962) 603-632. [2]________, On the groups J(X), IV, Topology 5 (1966) 21-71. [3]M. Bendersky and D. M. Davis, The v1-periodic homotopy groups of SO(n), Memoirs Amer Math Soc 815 (2004). [4]________, 2-primary v1-periodic homotopy groups of SU(n), Amer Jour Math 114 (1991) 529-544. [5]________, The 1-line of the K-theory Bousfield-Kan spectral sequence for Spin(2n + 1), Contemp Math AMS 279 (2001) 37-56. [6]M. Bendersky, D. M. Davis, and M. Mahowald, Stable geometric dimension of vector bundles over even-dimensional real projective spaces, to appear in T* *rans Amer Math Soc. http://www.lehigh.edu/~dmd1/sgd2.html [7]A. K. Bousfield, The K-theory localization and v1-periodic homotopy groups * *of finite H-spaces, Topology 38 (1999) 1239-1264. [8]________, On the 2-primary v1-periodic homotopy groups of spaces, Topology 44 (2005) 381-413. [9]D. M. Davis, Generalized homology and the generalized vector field problem, Quar Jour Math Oxford 25 (1974) 169-193. [10]D. M. Davis, S. Gitler, and M. Mahowald, The stable geometric dimension of vector bundles over real projective spaces, Trans Amer Math Soc 268 (1981) 39-61. [11]D. M. Davis and M. Mahowald, Homotopy groups of some mapping telescopes, Annals of Math Studies 113 (1987) 126-151. [12]M. Mahowald, The image of J in the EHP sequence, Annals of Math 116 (1982) 65-112. Hunter College, CUNY, NY, NY 10021 E-mail address: mbenders@shiva.hunter.cuny.edu Lehigh University, Bethlehem, PA 18015 E-mail address: dmd1@lehigh.edu