STABLE GEOMETRIC DIMENSION OF VECTOR BUNDLES
OVER ODDDIMENSIONAL 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
v1map 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 mplane 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 2group 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 KOequivalences 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 v1periodic 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 evendimensional 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
odddimensional 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 2e2in 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 n1and using the result from [6*
*] that
geometric dimension over these evendimensional 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 n1, BSO(m)]
between cyclic 2groups. 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 v11ss2k(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 v11ss2k(BSO(m)) =
Z=2 if m 3, 5 mod 8 by [3, 3.10], while v11ss2k(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], v11ss8`1(SO(m)) is an extension of two Z=2vector 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 nonasterisked " " 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
v11ssi(SO(m)) v11ssi+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)] [P108`, 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
~*`: v11ss1(BSO(m)) ! v11ss8`(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`! P18`
and
~` 1 deg 1 0
S8` ! S ! P18`
become equal in ss8`(P108`^ J) Z2 Z2, where each equals the element of high*
*est
filtration. Thus, since v11ss*(P ) v11ss*(P ^ J) for spectra P by [12], th*
*e two
composites become equal in v11ss8`(P108`). Thus they are equal in v11ss8`(BS*
*O(m)).
Here we have used the 2local Jspectrum which is the fiber of _3  1 : bo ! 4*
*bsp.
This spectrum played a key role in the early days of v1periodic homotopy theor*
*y,
especially in [12].
In the spectral sequence of [3] converging to v11ss*(SO(m)), elements in fil*
*tra
tion 2 occur in etatowers, 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 mod2 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 filtration4 elements y being ff*(x) with x an element of
filtration 3 follows from (2.7), since x is the element in an earlier etatower*
* 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 2adic coefficients from all K*
**()
groups, and will continue to do so. (ii) A is the category of 2adic stable Ad*
*ams
modules.([7]) (iii) We have replaced SO(m) by its double cover Spin(m). This do*
*es
not change v11ss*(), and indeed SO(m) = Spin(m). But for calculations such*
* as
(2.14), it is essential that the underlying space be simplyconnected. (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 24i2 (`i)D*
*+  x4i1.
Since restriction j#3to Spin(8i + 5) sends D+ to D and x4i1to x4i1, 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 v11ss8`1(SO(8i+*
*1))
and v11ss8`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
Z2vector 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 d2differentials, 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 Z2class 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 Z2class *
*labeled
x4i3. 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 d3differential in [P 8`, BSO(8i + 5)] not present in [P 8`, BSO(8i *
*+ 6)].
We can, however, consider an E2version 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 d4differential 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 C0summand to x4i1, 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 Kequivalent 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)] sv11ss02(SO(m)), (2.19)
where, by [6, (2.17)],
v11ss0n(X) [T n, (X)]. (2.20)
The analogue of (2.6) is that the morphism ff* in (2.4) is equivalent to
i*`: v11ss01(BSO(m)) ! v11ss8`+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`+4ff!P 8`+4! P18`
STABLE GEOMETRIC DIMENSION 15
and
i` 1 f 4
S8`+4! T ! P18`,
where f is, up to periodicity, a restriction of the map in [6, 2.8], become equ*
*al in
ss8`+4(P148`^ 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 v11ss8`+4(P14*
*8`),
and hence, following by any element g of [P148`, 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 mod2 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 Sdual. There is a
cofiber sequence
M6 ! D(MC(ei)) ! D(T 0).
In the chart below, the solid dots are from the M6 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 bp
r rA Ab rrr@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 v11ss*(Sn), this e*
*lement
supports a d3differential, but in that converging to v11ss0*(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 etatower 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 d3differential 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 etatowers 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 ss2k1(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 Kequivalences
__________
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
M1 ff! P18` ! P108` ! M0
x? x? x? x? (2.25)
=?? ?? ?? = ??
q 0 4`1 2 0 4` 0
M1 ! M (2 ) ! M (2 ) ! M
The top vertical maps are just the v1maps. 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[P128`, BSO(m)]ff![M1, BSO(m)]
?? ?? (2.26)
?y =?y
q* 1
sv11ss2(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 sv11ss1(BSO(m)) sv11ss2(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 v11ss8`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`1A! 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^2mod*
*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 = _33j. 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 2a1 CC (2.40)
22 2 0 A
22a1 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)22n4+ njj22n4j 8i 2`1iSi,j
j=2 i j1
with
j2X i j i ji
Si,j= (1)t 2j1t(2j  2t  1) jt2
t=0
given in [5, (4.20)]. The term (n  1)22n4is divisible by 2n+1 for n 5. The *
*other
terms are divisible by 22nj3with 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+2jj injjjX i2`j
(n + 1)22n3 22n+14j j  j2 8i i Si,j,
j 2 i j1
with Si,jas above, from [5, p.54]. The lead term (n + 1)22n3is divisible by 2n*
*+1 for
n 3. Other terms are divisible by 22nj2with 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 3generator 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 Dcomponent 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 = j1,1, the ,1component 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 ,1component of the Z=2ngroup
presented by (2.38) is then to have this Dcomponent of x3 be odd.
If the Dcomponent 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 Kequivalences.
M8`+3 eff! P 8`+2  ! P 8`+4
?? ? ?
?y ??y ??y
M3 ! P128`  ! P148`
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 = M1 [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 12! 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[P128`, BSO(m)]! [M3, BSO(m)]
?? ??
?y ?y
4`+1L
sv11ss024`+1L2(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
sv11ss02(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 d4differentials in the spectral sequence*
* de
scribed prior to Case 4. Recall from [6, 2.16] that the E2term for v11ss0*()*
* equals
that for v11ss*().
The cofibration (2.43) yields a short exact sequence
0 ! K1(T 1) 2! K1(T 1) ! K1(F ) ! 0
which is
0 ! Z^22! Z^2! Z=2 ! 0.
Thus (2.44) is, at the E2level, 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 Kequivalence 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 Dcomponent 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`+2ff!P 8`+2! P128`! v11P128`' v11N0(24`).
(2.47)
The class of this composite is divisible by 4 in v11ss4`+2(N0(24`)) v11ss4`*
*+2(P 8`+2).
Call it 4fl.
To see this divisibility, we use that ff goes to 0 in v11ss8`+2(P 8`+4), sin*
*ce it is
an attaching map. Diagram 2.48, which is similar to those of [12, pp 945], dep*
*icts
v11ss*(P 8`+2) ! v11ss*(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. v11ss*(P 8`+2) ! v11ss*(P 8`+4) near * = 8` + 2
r____________________ r
r r rr
rrr r rr
rrr@ r rr@
r rr@ r rr@
rr@ rr@
rr@r rr@
P 8`+2rr@r P 8`+4 rr@r
rr@rg rr@r
rr rr@r
r_____________________ rr
r
6 6
This chart also depicts v11ss*(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 v11T 1iT!v11N0(24`) induces in ss8`+2() an injection Z=8 ! Z=8 Z*
*=2.10
Let g denote the generator of v11ss2(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`+3i! M8`+3 A! M3 ! v1 T 1
?? ?? ?? ??
2?y 2?y 2?y =?y
` 2g 1 (2.49)
S8`+3i! M8`+3(4) A!M3(4) ! v1 T 1
__________
10v11T1 can be defined to be T1 ^ v11J.
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 Z2vector 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
E2version 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+ffl4k+8bffiu24b+ffl1
P 4k+tx P u2 ! P .
This is done by showing that _3  1 applied to the dual class in
4b+ffl1
ko2(P24kt1^ Pu24b+ffl+4k8b+ffi1^ P u2(3.3))
is nonzero. This class is called the axial class.
4b+ffl1
Lemma 3.4. Let X = P24kt1^ Pu24b+ffl+4k8b+ffi1. Then ko*(X ^ P u2 ) c*
*on
tains summands
4b+ffl1 u24b+ffl2
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
A1module 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 A1modules
0 ! 5Z2 ! N ! N0 ! 0.
If M is an A1module 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+ffl2
Here we have used that, except in its bottom few cells, the A1module H*P u2
4b+ffl5
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+ffl2will not alter the Ext gro*
*ups in the
region of interest. Note that H*X is A0free except in the case where t = 3 = f*
*fi, in
which case it is a direct sum of an A0free 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
ko1(P24kt1^ P4k8b+ffi1).
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
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[2]________, On the groups J(X), IV, Topology 5 (1966) 2171.
[3]M. Bendersky and D. M. Davis, The v1periodic homotopy groups of SO(n),
Memoirs Amer Math Soc 815 (2004).
[4]________, 2primary v1periodic homotopy groups of SU(n), Amer Jour Math
114 (1991) 529544.
[5]________, The 1line of the Ktheory BousfieldKan spectral sequence for
Spin(2n + 1), Contemp Math AMS 279 (2001) 3756.
[6]M. Bendersky, D. M. Davis, and M. Mahowald, Stable geometric dimension of
vector bundles over evendimensional real projective spaces, to appear in T*
*rans
Amer Math Soc. http://www.lehigh.edu/~dmd1/sgd2.html
[7]A. K. Bousfield, The Ktheory localization and v1periodic homotopy groups *
*of
finite Hspaces, Topology 38 (1999) 12391264.
[8]________, On the 2primary v1periodic homotopy groups of spaces, Topology
44 (2005) 381413.
[9]D. M. Davis, Generalized homology and the generalized vector field problem,
Quar Jour Math Oxford 25 (1974) 169193.
[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)
3961.
[11]D. M. Davis and M. Mahowald, Homotopy groups of some mapping telescopes,
Annals of Math Studies 113 (1987) 126151.
[12]M. Mahowald, The image of J in the EHP sequence, Annals of Math 116
(1982) 65112.
Hunter College, CUNY, NY, NY 10021
Email address: mbenders@shiva.hunter.cuny.edu
Lehigh University, Bethlehem, PA 18015
Email address: dmd1@lehigh.edu