STABLE GEOMETRIC DIMENSION OF VECTOR BUNDLES
OVER EVEN-DIMENSIONAL REAL PROJECTIVE SPACES
MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
Abstract.In 1981, Davis, Gitler, and Mahowald determined the
geometric dimension of stable vector bundles of order 2eover RP2n
if n is sufficiently large and e 75. In this paper, we use the
Bendersky-Davis computation of v-11ß*(SO(m)) to determine this
geometric dimension for all values of e (still provided that n is
sufficiently large). The same formula that worked for e 75 works
for e 6, but for e < 6 the formula is different due to anomalies
in the formula for v-11ß*(SO(m)) when m 10.
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. Many
papers (e.g., [1], [22], [23]) have been devoted to computing the geometric dim*
*ension
of multiples k,n of the Hopf bundles, in part because certain cases are equival*
*ent to
determining whether P ncan be immersed in a certain Euclidean space. (e.g., [11*
*])
ffi n
For n even, there exists a map P n+8-! P which induces a monomorphism in
KgO (-). This map may be obtained as a compression of multiplication by 16 of
the suspension spectra. Hence, as was observed in [12], the geometric dimension*
* of
vector bundles of order 2e over P nis a nonincreasing function of n for even n *
*in a
fixed congruence class mod 8, and so must achieve a stable value, which we call*
* the
stable geometric dimension sgd(__n, e). Here __n2 2Z=8 is the residue class of *
*n. Thus
we have the following result, which was proved in the first paragraph of [12].
__________
Date: September 22, 2003.
1991 Mathematics Subject Classification. 55S40,55R50,55T15.
Key words and phrases. geometric dimension, vector bundles, homotopy groups.
1
2 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
Proposition 1.1. For even __n2 Z=8 and e a positive integer, there is an integer
sgd(__n, e) which equals the geometric dimension of all bundles of order 2e in *
*gKO(P n)
for sufficiently large even n satisfying n __nmod 8.
Maps P n+8! P ninducing a monomorphism in gKO(-) do not exist when n is odd,
and so the situation for stable geometric dimension of bundles over odd-dimensi*
*onal
projective spaces is much more delicate, and will be discussed in a separate pa*
*per.
In [12] and [13], the value of sgd(__n, e) was determined for e 75.
Theorem 1.2. ([13]) Define ffi(__n, e) by the table
| e mod 4
|| |
________|0_1__2____3_ |
6, 80||2 2 1 ||
__n2, 4 | |
________0|_0_-1___-2_|
Then sgd(__n, e) 2e + ffi(__n, e), and if e 75, then sgd(__n, e) = 2e + ffi*
*(__n, e).
In fact, as follows from the fact that sgd is the limiting value of a nonincrea*
*sing
function but also is proved directly in [12], the geometric dimension of all ve*
*ctor
bundles of order 2e over all P nis 2e + ffi(__n, e).
Our first new result lowers the condition e 75 to e 6.
Theorem 1.3. If e 6, then sgd(__n, e) = 2e + ffi(__n, e).
If e 5, the value of sgd(__n, e) is sometimes greater than 2e + ffi(__n, e)*
*. It is given
in the following theorem, which is proved in Section 3. Thus we have determined*
* the
stable geometric dimension of all stable vector bundles over all even-dimension*
*al real
projective spaces.
Theorem 1.4. For e 5, sgd(__n, e) is given by the following table.
| e
|| |
_______|1_2__3__4___5__|
6, 85||6 7 11 12 ||
__n2, 4 | |
________5|_5_6__10__11_|
We remark that Adams ([1]) initiated the study of which bundles over RP nhave
specific small values of geometric dimension, and this topic has also been cons*
*idered
in [22, x3] and [17, x3], and in unpublished work of Lam and Randall.(e.g., [21*
*]).
Our new approach makes heavy use of the computation of v-11ß*(SO(m)) obtained
in [2]. We begin by indicating the relationship between this computation and sg*
*d.
STABLE GEOMETRIC DIMENSION 3
Let n be even. The maps OEk which are used in defining sgd can be factored as
(ffi0)kn
P n+8kcol-!P1n+8k+8k-!P . (1.5)
If (k) n=2,1 then James periodicity says that P1n+8k+8k' 8kP n, which, prec*
*eding
(OE0)k, yields a v1-map 8kP n! P n. Bousfield ([7, p.1251]) uses this v1-map t*
*o define
v-11ßi(Y ; P n) = colim[ 8kd+iP n, Y ]
d
for any space Y . With Y = BSO(m) and i = 0, this becomes colim[P1n+8kd+8kd, BS*
*O(m)].
This group contains one summand which is stable in the sense that it injects as*
* m
increases, along with some unstable summands. The stable summand corresponds
to multiples of the Hopf bundle ,, which comprise KgO(P n). We will denote this
stable summand by s in various contexts of v1-periodic homotopy groups or spect*
*ral
sequence groups which approximate them. It follows that
Proposition 1.6. If n is even, then sgd(__n, e) m if and only if, for n __n*
*mod 8
sufficiently large, the exponent of 2 of the cyclic group sv-11ß0(BSO(m); P n) *
*satisfies
(sv-11ß0(BSO(m); P n)) e.
This notation of (G) for the exponent of 2 in a cyclic group G will be adopted
throughout.
The situation when n 6, 8 mod 8 is particularly simple. We will prove the
following key result in Section 2.
Proposition 1.7. If n 6, 8 mod 8, then
sv-11ß0(BSO(m); P n) sv-11ß-2(SO(m)).
Note the simplification here_coefficients are no longer in a projective space*
*. The
groups v-11ß*(SO(m)) were computed in [2], where the following result was prove*
*d.
Theorem 1.8. If 8i + d 11, then
8
>>>-1 d = -1
>><
0 d = 0, 1, 2, 3
(sv-11ß-2(SO(8i + d))) = 4i + >
>>>1 d = 4, 5
>:
2 d = 6.
__________
1Slightly smaller values work, too.
4 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
Theorem 1.3 when __n= 6 or 8 is an immediate consequence of 1.6, 1.7, and 1.8.
Indeed, for n 6, 8 mod 8, the smallest d such that
(sv-11ß0(BSO(8i + d); P n)) 4i + <0, 1, 2, 3>
is 8i + <0, 4, 6, 7>.
Proof of Proposition 1.8.Because of the mammoth nature of [2], we guide the rea*
*der
to the relevant results. Referring always to [2], the specific statements rega*
*rding
(sv-11ß-2(SO(8i + d))) are in 1.2 for d = 1, 3.10 for d = 4 1, and 3.13 for
d = 4 2. Specific statements are not made for d = 4 or 8, but only with relat*
*ion to
the case d - 1. In 3.4(last case) (resp. 3.14(last case)), it is shown that the*
* exponent
when d = 8 (resp. d = 4) is 1 greater than when d = 7 (resp. d = 3). ||
When __n= 2, 4, a similar program is followed but we must define and compute a
modified sort of v1-periodic homotopy group. In Section 2, we will utilize the *
*following
definition and prove Theorem 1.10, which, with 1.6, implies Theorem 1.3 when __*
*n= 2
or 4 just as in the previous case.
Definition 1.9. Let Mn+1(k) = Sn [k en+1 denote the usual Moore spectrum, and
Nn+1(k) = Mn+1 [''en+2[2en+3,
and define, for any space X and any integer i,
Le e
v-11ß0i(X) = colim[Ni+1+k2 (2 ), X].
k,e
The second part of this definition, analogous to the definition of v-11ß*(X) fi*
*rst given
Le i e i e i e+1
in [15], is made using v1-maps 2 N (2 ) ! N (2 ) and canonical maps N (2 ) !
Ni(2e), similarly to the situation for Moore spaces Mi(2e).
Theorem 1.10. (1) If n 2, 4 mod 8, then
sv-11ß0(BSO(m); P n) sv-11ß0-2(SO(m)).
STABLE GEOMETRIC DIMENSION 5
(2)If 8i + d 11, then
8
>>>0d = 0, 1
>><
1 d = 2
(sv-11ß0-2(SO(8i + d))) = 4i + >
>>>2d = 3
>:
3 d = 4, 5, 6, 7.
The requirement that e 6 in Theorem 1.3 is due to the condition 8i + d 11*
* in
1.8 and 1.10(2). In Section 3, we will prove the following result, which, with *
*1.6, 1.7,
and 1.10(1), implies Theorem 1.4.
Theorem 1.11. For 5 m 10,
8
>><1m = 5
(sv-11ß-2(SO(m))) = >2 m = 6
>:
3 m = 7, 8, 9, 10;
8
>><2m = 5
(sv-11ß0-2(SO(m)))= >3 m = 6, 7, 8, 9
>:
4 m = 10.
The stable summand is not in the image of v-11ß-2(SO(4)) ! v-11ß-2(SO(5)) or of
v-11ß0-2(SO(4)) ! v-11ß0-2(SO(5)).
When m = 5 or 6, a slight reinterpretation of (sv-11ß-2(SO(m))) is required.*
* In
this case, the stable classes do not form a direct summand, and by (sv-11ß-2(S*
*O(m)))
we mean (im(v-11ß-2(SO(m)) ! v-11ß-2(SO)), a definition that works for all m. A
similar interpretation is used for the primed groups.
2. Proof of main results
In this section we prove Proposition 1.7 and Theorem 1.10, which we have alre*
*ady
seen imply Theorem 1.3.
Let denote the v1-periodic spectrum functor described in [7, 7.2]. By [7, 7*
*.2(i)],
we have, if n is even,
v-11ß0(BSO(m); P n) [P n, BSO(m)] [P n, SO(m)]-1
v-11ß-1(SO(m); P n), (2.1)
or similarly with P nreplaced by another space with a v1-map. We will use the f*
*our
parts of (2.1) interchangeably.
6 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
Proof of Proposition 1.7.The proof utilizes the following result, which is part*
* of [14,
4.2]. Here and throughout, Mn(k) = Sn-1 [k en denotes a Moore spectrum.
Theorem 2.2. ([14]) For ffl = 0 and 1, and L sufficiently large, there is a K*-
L 4k-ffl 2L-2ffl
equivalence M2 (2 ) ! P2L+1-8k.
We also note the following elementary result.
ffi0 n
Proposition 2.3. A K*-equivalence Pbn+8+8-!Pb, with n even and b odd, as used
in (1.5), induces an isomorphism
ffi0*-1 n+8
v-11ß*(Y ; Pbn) -! v1 ß*(Y ; Pb+8)
for any space Y .
L n n -1 n
Proof.A K*-equivalence 2 Pb ! Pb used in defining v1 ß*(Y ; Pb) can be factored
as
L n n+8 ffi0 n
2 Pb ! Pb+8 -! Pb;
L n+8 n+8
thus OE0*is injective. Similarly a K*-equivalence 2 Pb+8 ! Pb+8 used in defini*
*ng
v-11ß*(Y ; Pbn+8+8) can be factored as
L n+8 2Lffi02Ln n+8
2 Pb+8 -! Pb ! Pb+8,
and so OE0*is surjective. ||
Thus
v-11ß*(Y ; P 8k-2ffl) v-11ß*(Y ; P1-2ffl-8k) v-11ß*(Y ; M0(24k-ffl)).
(2.4)
Here we use that a K*-equivalence induces an isomorphism in [-, Y ], since Y *
*is
K*-local, and also use the fact ([20, 3.7]) that the maps of 2.2 asymptotically*
* respect
the v1-maps of the two spaces.
With k sufficiently large, there is, by [15, 1.7], a split short exact sequen*
*ce
0 ! v-11ß-1(SO(m)) ! v-11ß-1(SO(m); M0(24k-ffl)) ! v-11ß-2(SO(m)) ! 0.
(2.5)
The stable summand sv-11ßi(Y ; SO(m)) may be defined to be v-11ßi(Y ; SO(m)) mo*
*d-
ulo the kernel of the stabilization v-11ßi(Y ; SO(m)) ! v-11ßi(Y ; SO). Similar*
*ly to [6,
STABLE GEOMETRIC DIMENSION 7
1.9], v-11ß-1(SO) = 0. Thus (2.5) induces an isomorphism
sv-11ß-1(SO(m); M0(24k-ffl)) ! sv-11ß-2(SO(m)).(2.6)
With (2.1) and (2.4), this yields the desired conclusion of Proposition 1.7. ||
Next we prove Theorem 1.10(1). One viewpoint for the relevance of Nn+1(k) in-
volves a comparison of charts of KO*(-) computed, for example, by the method of
[12, p.41] or [14, p.133] as v-11ko*(-). In Diagram 2.7, the left side is a ch*
*art of
KO*(P 8k-2ffl) with ffl = 0 or 1 and main groups of order 24k-ffl, while the ri*
*ght side is
KO*(P 8k+2ffi) with ffi = 1 or 2 and larger (middle) groups of order 24k+ffi+1.*
* A chart
for KO*(M(2n)) is given by the left side of Diagram 2.7 with main groups of ord*
*er
2n, while a chart for KO*(N(2n)) is given by the right side of the diagram with*
* the
larger groups of order 2n+2. Here we have not listed a superscript for M(-) or *
*N(-)
since the effect of the superscript is just to translate the chart horizontally*
*. These
charts are not necessary for the proof; they merely form one way of understandi*
*ng
the need for resorting to Nn+1(k). The charts for M(2n) match nicely with those*
* of
P 8k-<0,2>, but must be modified to those of N(2n) to match with P 8k+<2,4>.
____________________________Diagram_2.7.||| |
| | |_____________________________|
| | | |
|KO*(P|8k-<0,2>) orKO*(M(2n)) || |KO*(P|8k+<2,4>) orKO*(N(2n)) ||
| ||r | | |r |
| |r |r qq | | |r |r qq |
| r|r | q | | r|r | q |
| r |r | | | r |r | |
| | | | | | | |
| |r | | | | |r | | |
| |r | | | | |r | | |
| | | | | | | | | |
| | | | | | | | | |
| | | |r r | | | | |r |
| | | |rr | | q | | |r |
| | | |r | | qq | |r r |
| | | | | | | |
| qq | |r | | | |rr |
| q |r r | | |r |r |
| |rr | | |r |
| | | | |
| |r | | |
_________________________________|| |_____________________________|
Proposition 2.8. For sufficiently large L, there exist K*-equivalences
4kL 4k f1 2
N2 (2 ) -! P1-8kand
4k+1L 4k+1 f2 4
N2 (2 ) -! P1-8k.
8 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
Proof.In [14, 4.2], a K*-equivalence M0(24k) -f! P10-8kwas constructed. Let J =
v-11J denote the periodic J-spectrum. The chart for J*(P12-8k) in the range -2
* 5 is given in Diagram 2.9.
Diagram 2.9.
|rr
r |r
rr |r
|r r|||@
|rr |@||
|||rr |@|
|| | |
|| |@|
|| |@|
|| |@|
|| | |
|| |@|
|| |@|
|| |@| r
|| | |
|| |@|r
||r |||rrr@
_________________|||rr|rr
|r |r
|r
-1 3
Let M1 = S0 [2e1. Consider the composite
~'' -1 j 0 4k f 0 i 2
M1 -! S -! M (2 ) -! P1-8k-! P1-8k^ J .
(2.10)
The cofiber of j O ~jis N0(24k). Note that J*(P110-8k) has a chart like that of*
* Diagram
2.9 with the same top and extending 4 units lower. The commutative diagram indu*
*ced
by the inclusion P12-8k! P110-8k
~''* 1 2
J-1(P12-8k)---! [M , P1-8k^ J ]
?? ??
24?y =?y
~''* 1 10
J-1(P110-8k)---![M , P1-8k^ J ]
implies that its top morphism is 0. Thus the composite (2.10) is trivial, and h*
*ence
~f
the extension N0(24k) -! P12-8k^ J of i O f exists.
4kL
By [24], P12-8k^ J is the telescope v-11P12-8kover v1-maps of P12-2-8k-24kLas*
* L !
1. Thus the map ~ffactors through a map f1 whose (24kL)-suspension is as in the
statement of the proposition. This f1 is a K*-equivalence by the Five Lemma app*
*lied
STABLE GEOMETRIC DIMENSION 9
to
4kL 4k 24kL 4k 2+24kL
M2 (2 )---! N (2 )---! M (2)
?? ? ?
?y ??y ??y
P10-8k ---! P12-8k ---! M2(2).
In [14, 4.2], a K*-equivalence M0(24k+1) ! P-08k-1is constructed. As in the
4k+1L04k+1 f02 2
proof of the first part, this yields a K*-equivalence N2 (2 ) -! P-8k-1.*
* By
[16, 3.1], there is a filtration-3 K*-equivalence P-28k-1h1-!P--48k-7. The fil*
*tration-
4 K*-equivalences used in the definition of stable geometric dimension yield a *
*K*-
4k+1 -24k+1 4 4k+1
equivalence P--48k-7h2-!P14-2-8k-24k+1' P1-8k. The 2 -fold suspension*
* of
h2O h1O f02is our desired K*-equivalence f2. ||
Thus for ffi = 1, 2, we have
sv-11ß0(BSO(m); P 8k+2ffi) sv-11ß0(BSO(m); N0(24k+ffi-1)).
(2.11)
Here again we use from [20] that K*-equivalences such as those in 2.8 must asym*
*p-
totically commute with v1-maps. Similarly to (2.5), for k sufficiently large, t*
*here is a
split short exact sequence
0 ! v-11ß-1(SO(m)) ! v-11ß-1(SO(m); N0(24k+ffi-1)) ! v-11ß0-2(SO(m)) ! 0,
(2.12)
which, similarly to (2.6), induces an isomorphism
sv-11ß-1(SO(m); N0(24k+ffi-1)) sv-11ß0-2(SO(m)).
(2.13)
We will expand slightly upon the proof of (2.12) following Definition 2.14. The*
*orem
1.10(1) is an immediate consequence of (2.11), (2.1), and (2.13). ||
We expand 1.9 to include another related spectrum.
Definition 2.14. Let T n= Sn [''en+2[2en+3.
The reason for the choice of names of these spaces is "next letter of alphabe*
*t." The
space T nhas appeared in other guises as variations on a sphere. In [8, 10.7], *
*it was
called C, and its K*-localization was shown in [8, 10.6] to be the only other K*
**-local
spectrum to have the same K*(-)-groups as SK . The spectrum bsp, which was used
10 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
in many papers of the second and third authors (e.g. [12, p.41], [14, p.127], [*
*16, p.41])
involving the J-spectrum, equals T 0^ bo.
The split short exact sequence (2.12) is induced from cofiber sequences
eL-1 2eL 4k+ffi-1 2eL24k+ffi-1
T 2 ! N (2 ) ! S - ! , (2.15)
where k is large enough that SO(m) has H-space exponent 24k+ffi-1, and e and L *
*are
large. This induces a split short exact sequence
eL-1 4k+ffi-1 2eL-2
0 ! ß2eL-1(SO(m)) ! [N2 (2 ), SO(m)] ! [T , SO(m)] ! 0,
and, similarly to [15, 2.6], there is a direct system of these split short exac*
*t sequences
with respect to increasing 2eL, the direct limit of which is (2.12). See also *
*(2.17),
which suggests that (2.12) can be obtained by applying [-, BSO(m)] to (2.15).
We will use the following spectral sequence to compute v-11ß0*(X), which was *
*defined
in 1.9.
Proposition 2.16. If X is an odd sphere or simply-connected compact Lie group,
there is a spectral sequence converging to v-11ß0*(X) with E2-term
eEs,t2 ExtsA(QK1(X; Z^2)= im(_2), K1(St; Z^2)).
Note that the E2-term is isomorphic to that of the spectral sequence of [4] c*
*on-
verging to v-11ß*(X). We will call it eE2(X) when it is the initial term of the*
* spectral
sequence converging to v-11ß0*(X).
Proof of Proposition 2.16.We begin by mimicking the proof of [7, 7.5]. With D
denoting S-duality, there are isomorphisms
v-11ß0*(X) lim[N1(2k), (X)]* colimß*(DN1(2k) ^ (X))
k k
ß*(D(T 0) ^ M0(Z=21 ) ^ (X))
[T 0, (X)]*. (2.17)
Here we have used that applying ^M0(Z=21 ) to the torsion spectra S0 [2 e1 and
(X) leaves them unchanged.
By [7, 10.4],2 there is a spectral sequence converging to [T 0, (X)]* with
Es,t2 ExtsA(K*( (X); Z^2), K*(T t; Z^2)).
__________
2Although many results of [7] only work when p is odd, this one also works wh*
*en
p = 2.
STABLE GEOMETRIC DIMENSION 11
Bousfield ([9]) has proved that for X as in this theorem, there is an isomorphi*
*sm in
A 8
<0 i = 0
Ki( X) :
QK1(X)= im(_2) i = 1,
where Q(-) denotes the indecomposables. This is the 2-primary analogue of [7, 9*
*.2].
The proof of the proposition is completed by noting that there is an isomorphis*
*m in
A, K*(St; Z^2) K*(T t; Z^2). Indeed, the morphism in K*(-; Z^2) induced by t*
*he
inclusion St,! T tis a monomorphism onto multiples of 2. ||
The next result gives the primed v1-periodic homotopy groups of odd spheres. *
*The
conclusion is that the d3-differentials between the eta-towers in the spectral *
*sequence
for v-11ß0*(S2n+1) are the opposite of the way they are in the spectral sequenc*
*e for
v-11ß*(S2n+1). Here n can be even or odd.
Theorem 2.18. The spectral sequence of 2.16 converging to v-11ß0*(S2n+1) is as *
*pic-
tured in Diagram 2.19. Here 8 means Z=8, while C is Z=2min(n,4+ (k+1)). We do n*
*ot
picture many portions of eta-towers which are involved in nontrivial d3's. The *
*dotted
differential when n 1, 2 is nonzero unless (k + 1) + 4 > n, in which case d3*
* = 0
and the extension in v-11ß02n+8k+7(S2n+1) occurs. The action of h1 on the gener*
*ator of
C in position (2n + 8k + 8, 1) is nontrivial, but the class which it hits depen*
*ds upon
whether or not (k + 1) + 4 > n.
12 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
Diagram 2.19.
| | | | | | | | |
| | | | | | | | r |
| | | | | | r | | |
-1 0 2n+1 | | | | | | BB | | B |
v1 ß*(S ) __________________________________________________|||||||||||*
*|||||||BrB
n 0, 3 mod 4 | r | | | | r | B | pp | B |
| BB | | | | | B | ppr| B | r
__________________________________________________|||||||||||*
*|||||||CpppBBBB
| B | | | r | | r |BppC|p B| B
| B| | | | | r|Bpp |C pr|B BB
__________________________________________________|||||||||Cp*
*ppB
| B| | | | r | |Bp | p |BBr
| |B |8 | | | |BC |C pp|
| |B | | | r| | r |C pp|
__________________________________________________|||||||||Cpp
| |B | | | | | | Cpp|
| |Br | | | | | | |
| | | | 8 | | r | | CC |
__________________________________________________|||||||||
i = 2n + 8k+ 1 2 3 4 5 6 7 8
| | | | | | | | |
| | | | | | | | |
| | | | | | | | | r
| r | | r | | | | | | B
__________________________________________________|||||||||||*
*|||||||BBBBB
r | B | | B | | | | |B r | BB
BB | B | pr| B| | | r|pprp|B r |
-1 0 2n+1 __________________________________________________|||||||||||*
*|||||||BpppBpppppBBBBB
v1 ß*(S ) B | B| pp| |B | B | p|pppr|B B|
n 1, 2 mod 4 B| r |Bpp| |B | Brr| p|pppr|pB B|B
__________________________________________________|||||||||Bp*
*pppppBB
B| |Bpp| | B | | p|pppp| |
|B | | | | | C |pppp| B |
r |B |B8 | | BBr|r | r | ppp|p BB|r
__________________________________________________|||||||||pp*
*pp
|B | | | | | | ppp|p |
| Br | | | | | | p | |
| | | 8 | | r | | pC| |
__________________________________________________|||||||||
i = 2n + 8k+ 2 3 4 5 6 7 8 9
Proof.We begin by using a J-homology approach to determine v-11ß0*(S2n+1). These
methods were developed in [24], and described quite thoroughly in [10, x3,x4,x5*
*]. We
assume that the reader has some familiarity with those methods. For a reader who
has no such expertise, an alternate proof is given after this one.
Let Ui = Si-3[2ei-2[''ei. Note that T iand U-i are S-dual. The map 2n+1S2n+1!
QP 2nof [10, 3.3] induces an isomorphism in v-11ß0*(-). Thus
v-11ß0i(S2n+1) v-11[T i, 1 2n+1P 2n]
v-11ßi(U2n+1^ P 2n)
v-11Ji(U2n+1^ P 2n).
STABLE GEOMETRIC DIMENSION 13
Arguing similarly to [10, p.1011], there is a short exact sequence of A1-modules
0 ! H*U5 ! A1==A0 ! Z2 ! 0,
and hence isomorphisms
Exts,tA1(H*(U5 ^ X), Z2) -! Exts+1,tA1(H*X, Z2)
for s > 1 and any space X. Here A1 is the subalgebra of the mod 2 Steenrod alge*
*bra
generated by Sq1and Sq2, which is relevant since the E2-term of the Adams spect*
*ral
sequence converging to ß*(X ^ bo) is ExtA1(H*X, Z2). Inverting v1, we conclude
v-11ßi(U0 ^ X ^ bo) v-11ßi+4(X ^ bo).
Thus, since v-11J is the fiber of _3-1 : v-11bo ! v-11bo, the chart for v-11J*(*
*U0^X) is
like that of v-11J*(X) pushed back by 4, but the differentials between adjacent*
* towers
(corresponding to _3-1) of U0^X are the same as those in X in the same dimensio*
*n.
We obtain charts for v-11ß0*(S2n+1) as in Diagram 2.20. Here the differential*
* between
the second pair of towers in either box is d (4k+4). The height (number of dots*
*) of the
towers in the left box is n. The height of the smaller (left) towers in the rig*
*ht box is
n - 1, while that of the larger towers is n + 1.
| | | |
Diagram_2.20._v-11ß0i(S2n+1)__|||| _________________________________||||
| | | |
|n 0, 3 mod 4 | |n 1, 2 mod 4 | |
| |r | | |r |
| r|r | | r| |
| r|r| | | r ||r |
| r r|r| | | r r || |
| | | | | | | || |
| |rr | | | | |rr || |
| |r|r | | | | |r|r || |
| |@| | | | | |@| || |
| | | | | | | | | || |
| |@| | | | | |@| || |
| |@|| | | | | |@|| || |
|| |@| | || q || || |@| ||| q ||
| |||@ | | qq | | |||@ || qq |
| |@| ||| | | |@| ||| |
| |@|| | || | | |@|| || |
|| q |||@ |||r || || q |||@ ||||r ||
| qq |@| r| | | qq |@| ||rr |
| |@|rr |r | | |@| ||rr |
| | | | | | | | |
| |r|rr@ | | |r|r@ |rr |
| |rr | | |r |r |
| |r | | |r |
| | | |
|i = 2n + 8k+ 4 8 | |i = 2n + 8k+ 4 8 |
______________________________|| _________________________________||
14 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
By Proposition 2.16, the E2-term in 2.19 is the same as that for v-11ß*(S2n+1*
*) as
given, for example, in [3, p.488]. The d3-differentials in 2.19 are the only w*
*ay of
inserting them to yield groups which agree with v-11ß0*(S2n+1) as given in 2.20*
*. ||
Now we easily deduce the following key result.
Proposition 2.21. The spectral sequence of (2.16) for v-11ß0*(Spin(m)) has Ee2as
given in [2, 1.3,3.4,3.7,3.12,3.14] but with d3-differentials between eta-tower*
*s the op-
posite of those given there.
Proof.As described in [2, x5], an eta-tower is a family of Z2 elements related *
*by
h1 : eEs,t2! eEs+1,t+22, beginning in filtration 1, 2, or 3. If x is an eta-tow*
*er, then there
is an eta-tower with the same name appearing every 4 (horizontal) dimensions, a*
*nd
either all those congruent mod 8 to x support d3-differentials hitting the othe*
*rs, or
else all those congruent mod 8 to x are hit by d3-differentials from the others*
*. In [2],
it was shown that all these d3's in Spin(m) could be determined by naturality f*
*rom
those in the odd spheres. Since we saw in 2.18 that the d3's in the spectral se*
*quence
for v-11ß0*(S2n+1) are opposite of those in the spectral sequence for v-11ß*(S2*
*n+1), we
can deduce that the same happens for Spin(m). ||
Now we give an alternate proof of Theorem 2.18 which does not involve J-chart
technology. This argument can probably be used to prove Proposition 2.21 concur-
rently with 2.18.
Alternate proof of Theorem 2.18.Let t be odd, and let Mt(j) = St [''et+2. The
obvious cofibration induces a short exact sequence in A
0 ! K*(St+2) ! K*(Mt(j)) ! K*(St) ! 0,
and hence, for any A-object N, an exact sequence
Exts,t+2A(N) ! ExtsA(N, K*Mt(j)) ! Exts,tA(N) -h1!Exts+1,t+2A(N).
If, as is the case when N = K*( S2n+1) or K*( Spin(n)), Exts,tA(N) -h1!Exts+1,*
*t+2A(N)
is an isomorphism for s > 2, then ExtsA(N, K*Mt(j)) = 0 for s > 2.
Now we consider the cofiber sequence
q t+3
St+2- ff!Mt(j) -i! T t-! S ,
STABLE GEOMETRIC DIMENSION 15
where ff is a coextension of 2, i the inclusion, and q the collapse. It induces*
* a short
exact sequence in A
0 ! K*(T t) ! K*(Mt(j)) ! K*(St+2) ! 0
and hence an exact sequence
ExtsA(N, K*T t) ! ExtsA(N, K*Mt(j)) ! Exts,t+2A(N) -ffi!Exts+1A(N, K*T t).
With N as above, since ExtsA(N, K*Mt(j)) = 0 for s > 2, ffi induces an isomorph*
*ism
of eta-towers. Note that since K*T t K*St in A, this ffi can be considered to *
*be a
morphism
h-11 s,t-2
Exts,t+2A(N) ! Exts+1,tA(N) -! ExtA (N),
since h1 is an isomorphism. As noted in the proof of 2.21, names of eta-towers *
*have
period 4 in t. Thus this ffi maps a set of eta towers to a set of eta towers wi*
*th the
same name. It can be shown, using the Small Complex of [2, x11], that this ffi *
*sends
an eta tower to the one with the same name, at least if N = K*( S2n+1). Since t*
*he
proof is somewhat involved and this is only an alternate proof, it is omitted h*
*ere.
Finally we note that this ffi commutes with d3-differentials since it is indu*
*ced by the
map q. Thus the d3-differential on eta towers in eEs,t-22of the spectral sequen*
*ce for
v-11ß0*(S2n+1) agree with d3 on Es,t+22of the spectral sequence for v-11ß*(S2n+*
*1). The
conclusion is that E2 is the same for the two spectral sequences, but d3 on eta*
*-towers
is opposite. For d3 on the 1-line, more delicate analysis is required, which wi*
*ll be the
focus of the next proposition. ||
We close this section by proving Theorem 1.10(2), the determination of the re*
*quired
sv-11ß0-2(SO(m)). This is accomplished using the spectral sequence of 2.16, and
follows from the following result.
Proposition 2.22. In the spectral sequence of 2.168with X = Spin(m),
< 0 m 0, 1, 2 mod 4
oIf 4a m 4a+3, then (sEe1,-12) = 2a+:
1 m 3 mod 4.
oThere is a nontrivial extension (.2) from eE1,-11to eE3,11if m 7.
od3 : E1,-13! E4,13is nonzero if and only if m 0, 1 mod 8.
16 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
Proof.We use the observation after 2.16 that eE2is isomorphic to the E2-term of*
* the
spectral sequence converging to v-11ß*(X). From [2, 3.1], (sE1,-12(Spin(2n+1))*
*) = n,
while from [2, 3.3]
8
>>:2
0 otherwise.
The extensions must be nontrivial by a form of Bott periodicity. A similar situ*
*ation
is discussed in [6, 1.19].
In [2, x7], d3 from the 1-line of the spectral sequence converging to v-11ß*(*
*Spin(m))
was determined by noting that d3(x) = y iff d3(h1x) = h1y. Since d3 from the 2-*
*line
had already been computed, it sufficed to compute h1x. Methods for computing h1
from the 1-line were developed in [2, 7.2,7.9]. The same methods work here in t*
*he
spectral sequence converging to v-11ß0*(Spin(m)). The biggest difference is th*
*at, as
shown in 2.21, the d3's from the 2-line here are opposite of the way they were *
*in [2].
We focus here on the cases where we must show d3 = 0 on sEe1,-12(Spin(m)). The
nonzero d3's when m 0 or 1 are implied by 1.2, 1.6, and 1.10, since these re*
*sults
imply that (sv-11ß0-2(SO(8i+d))) is equal to or less than the value claimed in*
* 1.10(2)
STABLE GEOMETRIC DIMENSION 17
in these cases, and, given the first two parts of 2.22, the only way to make th*
*e group
this small is with a nonzero d3. The method of the proof which follows can also*
* be
used to obtain these nonzero d3's.
We begin with the spectral sequence for v-11ß0*(Spin(8i + 3)). The E2-term eq*
*uals
that of [2, Diagram 3.7]. In Diagram 2.23, we present the relevant portion, wit*
*h the
d3-differentials which apply to v-11ß0*(Spin(8i + 3)).
Diagram 2.23. Part of the spectral|sequence|for|v-11ß0*(Spin(8i|+ 3))
____________________||||||||
| |Dr | |
| | yr4|i-1 |
s = 5 | | BBBB|BBBBB|BBBBB
____________________||||||||BBBBBBB
|rD | BBB|BBB |
| y r| BBB|BB |
| 4i|-1 BB|B |
| | BB|BBBBB|
____________________||||||||BBBBBBB
| | B|BBBBB|BB
| | |BBBBB|B
| | |BBBBB|BB
____________________||||||||BBBBBBB
| | |rDBBB|BBBB
| | | ByBB|BBBrB4i-1
| | | |
____________________||||||||
| | | |
| | | |
| | | |
s = 1 | |C1 C2| |
____________________||||
-3 -2 -1
The dual group (Ee2,12)# has basis {D, x4i-1} [ BC[2i, 4i], where
BC[2i, 4i] = {xj : 2i j 4i and j - 2 (j)+1< 2i}.
The set BC[2i, 4i] has [log2(16i=3)] + ffiff(i),1elements and is represented by*
* the big o
in Diagram 2.23. We use the same names for elements of the dual basis. By [2, 3*
*.7]
and Proposition 2.21, all basis elements of eE2,12except D support nonzero d3 i*
*n the
spectral sequence for v-11ß0*(Spin(8i + 3)). By [2, 7.9], D is a summand of h1(*
*g1) in
the case at hand; we will see why this is true in the next paragraph.
In order to show that d3(g1) = 0, we must show that the basis elements of eE2*
*,12other
than D are not summands of h1(g1). We adopt the dual point of view as explained*
* in
the proof of [2, 7.9]. In the notation of that proof, we are in the first case *
*considered
18 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
there_4` + 3 = 8k - 1 with >> n. Since n which we have been using in this pap*
*er
to denote the dimension of a projective space is not relevant to this propositi*
*on, we
are free here to use n as it was used in the proof of [2, 7.9], namely 8i + 3 =*
* 2n + 1
so n = 4i + 1. The four relations described there which yield (Ee1,8k-12)# are *
*A12n,1,
A22n,1- 2n+1 , A32n,1- 2n , and u2n,1+ 2 with u odd.3 In fact, A1 is even by
the discussion following [2, 8.1], and A2 is even by [2, 3.2]. Hence in the Z=2*
*n Z=2n
group presented, it is only the last relation whose division by 2 lowers the or*
*der of
the first (,1) summand.
The fourth relation here is due to (_3 - 34k-1)( ), the third to _2( ), and t*
*he
first two to _2 and _3 - 34k-1acting on various xj. It was observed in the proo*
*f of
[2, 7.9] that dividing the fourth relation by 2 corresponds to modding (Ee1,8k-*
*12)# by
h#1(D). Modding (Ee1,8k-12)# by h#1(b) for other elements b in the basis of (Ee*
*2,8k+12)#
corresponds to dividing other relations _2( ), _2(x), or (_3-34k-1)(x) by 2. Si*
*nce it
is only dividing the fourth relation by 2 that lowers the order of the fourth s*
*ummand,
P 1,8k-1#
we deduce that the first component of h#1(ff0D + ffixi) in (Ee2 ) equals f*
*f0
times the element of order 2, or dually that h1(g1) = D. This implies d3(g1) = *
*0 since
d3(D) = 0.
We prove now that d3 = 0 : Ee1,-12(Spin(8i + 2)) ! Ee4,12(Spin(8i + 2)). By *
*[2,
3.3], eE1,-12(Spin(8i + 1)) ! eE1,-12(Spin(8i + 2)) is bijective. By the proof *
*of [2, 3.11],
Ee4,12(Spin(8i+2)) ! eE4,12(Spin(8i+3)) is injective.4 By [2, 3.1], eE1,-12(Spi*
*n(8i+1))
Z=24i Z=24iand eE1,-12(Spin(8i+3)) Z=24i+1 Z=24i+1. Let x 2 eE1,-12(Spin(8i+2*
*)).
Then i*(x) = 2y 2 eE1,-12(Spin(8i + 3)). Hence i*(d3(x)) is divisible by 2, and*
* hence is
0, since it lies in a Z2-vector space. The injectivity of i* on eE4,12implies t*
*hat d3(x) = 0.
Next we consider Spin(8i + 4). From [2, 6.1], we see that eE4,12(Spin(8i + 4)*
*) has
basis dual to
{x4i-1, D+} [ BC[2i, 4i] [ {(D+ - D-)s, (D+ - D-)u},
where the two classes (D+ - D-) map nontrivially to eE4,12(S8i+3). By Diagram 2*
*.19,
d3 acts injectively on eE4,12(S8i+3), and hence it does also on the classes (D+*
* - D-).
__________
3We use to denote elements of K1(Spin(8i + 3)) instead of the D that was
used in [2] to avoid confusion with the element D of (E2,8k-12)#. Also note tha*
*t k
of [2, 7.9] is 0 here.
4The proof there deals with E4,8k+52but applies also to E4,8k+12.
STABLE GEOMETRIC DIMENSION 19
The element D+ also supports a nonzero d3 from Ee4,1
3 (Spin(8i + 4)). This is true
because of 2.21 and the fact that in [2, 3.7], the element D in position (8k - *
*3, 4) did
not support a nonzero d3 in the spectral sequence for v-11ß*(Spin(8i + 3)). Thu*
*s the
only elements that d3(g1) might hit are dual to x4i-1or BC[2i, 4i]. By the argu*
*ment
used above in the case of Spin(8i+3), h#1does not send the corresponding elemen*
*ts of
Ee2,12(Spin(8i+4))# to the element of order 2 in eE1,-12(Spin(8i+4))# because d*
*ividing
the corresponding relations by 2 will not lower the order of the first (,1) sum*
*mand.
Thus d3(g1) = 0 on the stable summand of eE1,-12(Spin(8i+4)). That the same is *
*true
in Spin(8i + 5) and Spin(8i + 6) follows by naturality, since
Ee1,-12(Spin(8i + 4)) ! eE1,-12(Spin(8i + 5)) ! eE1,-12(Spin(8i + 6))
send the first summand bijectively. ||
3.Proof of results for SO(m) when m 10
In this section, we prove Theorem 1.11, which we showed in Section 1 implies
Theorem 1.4. Regarding the comment after 1.11 about the meaning of the so-called
stable summand s, we will make the following distinction: as described there, s*
* will
refer to stable classes, those which map nontrivially to SO, while f will refer*
* to the
"first" summand, the summand generated by the stable classes, which may contain
multiples of the stable class which are not stable, inasmuch as they become 0 u*
*pon
stabilization. It is the case that s and f are equal in Spin(m) for m 7, as w*
*e shall
see.
By [2, 3.19], for m = 5, 6, 7, 8, 9, and 10, fE1,-12(Spin(m)) fEe1,-12(Spin*
*(m)) is
given by
Z=16 -! Z=16 -2! Z=8 -! Z=8 -! Z=8 -! Z=8.
(3.1)
In the spectral sequence converging to v-11ß*(Spin(m)), when m = 7 the extension
(.2) from fE1,-11to E3,11is trivial by [2, 1.4], as is d3 : fE1,-13! E4,14. The*
* same is
true for m = 8, 9, and 10 by naturality.
In the spectral sequence converging to v-11ß*(Spin(6)), differentials and ext*
*ensions
from E1,-1rare trivial by [2, 3.11]. Indeed, there is nothing for d3to hit, and*
* extensions
are ruled out by 2j = 0. In the spectral sequence converging to v-11ß*(Spin(5))*
*, there
20 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
is a nontrivial d3-differential and a nontrivial extension from E-1,1r, as can *
*be seen
by comparison with [19, 1.7], using that Spin(5) = Sp(2). Thus fv-11ß-2(Spin(m)*
*) is,
for m = 5, 6, 7, 8, 9, and 10, given by
Z=16 -2! Z=16 -2! Z=8 -! Z=8 -! Z=8 -! Z=8.
Since the multiples of 4 in v-11ß-2(Spin(6)) and multiples of 2 in v-11ß-2(Spin*
*(5))
stabilize to 0, they are not included in s, as discussed at the beginning of th*
*is section,
and so we obtain the first part of Theorem 1.11.
Since Spin(4) S3xS3, we deduce 4v-11ß*(Spin(4)) = 0, and so v-11ß-2(Spin(4)*
*) !
v-11ß-2(Spin(5)) cannot hit an element which stabilizes nontrivially, since 8 t*
*imes such
an element is nonzero in v-11ß-2(Spin(5)).
We will show that in the spectral sequence eEr(Spin(m)), whose eE1,-12is give*
*n in
(3.1), there is a nontrivial extension from eE1,-11for m = 7, 8, 9, and 10 (but*
* not when
m = 5 or 6), and a nonzero d3-differential from eE1,-13when m = 5, 7, 8, and 9 *
*(but
not when m = 6 and 10). From this, it is immediate that fv-11ß0-2(Spin(m)) is, *
*for
m = 5, 6, 7, 8, 9, and 10, given by
Z=8 -2! Z=16 -1! Z=8 -! Z=8 -! Z=8 -2! Z=16.
Since the multiples of 8 in v-11ß-2(Spin(6)) and multiples of 4 in v-11ß-2(Spin*
*(5))
stabilize to 0, they are not included in s, and so we obtain the second part of*
* Theorem
1.11.
First we show that d3 6= 0 on sE1,-13(Spin(m)) when m = 9. This implies d3 6=*
* 0
when m = 7 and 8, too. By the argument after [2, 7.2], h1 : Ee1,-12(Spin(m)) !
Ee2,12(Spin(m)) is injective. Since d3 on eta-towers of eE3(Spin(m)) is opposit*
*e to that
on E3(Spin(m)), we deduce from [2, 1.3] that d3 acts injectively on Ee2,1(Spin(*
*9)).
Naturality of h1 now implies that d3 acts injectively on generators of eE1,-13(*
*Spin(9)).
Next we deduce the nonzero extension (and give another proof of the nonzero d*
*3)
on eE1,-1r(Spin(7)) by a comparison of charts for v-11ß0*(G2) deduced from [18,*
* p.666]
and [5, Fig.2,p.1276]. The relevance of G2 is the 2-primary decomposition Spin(*
*7) '
G2x S7. This extension result was already alluded to in the previous section, w*
*here
it was noted that this is the beginning of a range of nontrivial extensions, an*
*d Spin
is at the end, and either one of them can be used to deduce the whole batch, so*
* that
this proof is not really essential to our proof.
STABLE GEOMETRIC DIMENSION 21
The aforementioned charts are for v-11ß*(G2), and so must be modified as in t*
*he
proof of 2.18 to give v-11ß0*(G2). The J-chart approach of [18] must be shifte*
*d by
4 dimensions with higher differentials staying the same in a fixed dimension. T*
*hus
v-11ß0*(G2) near * = -2 is as in Diagram 3.2.
Diagram 3.2. v-11ß0*(G2) ___________||
| r |
| | |
| r r||
| | |
| r r r||
| |@| |
| r r|r||@
| |@| |
| r r|r||@
| |@| |
| r|r||@
| | |
| r| |
| | |
| r| |
| -2 |
___________||
From the point of view of 2.16, we have the same E2 as in [5], but d3 on eta-*
*towers
is reversed. The chart near t - s = -2 is given in Diagram 3.3.
Diagram 3.3. v-11ß0*(G2) | | | |
__________________||||||||BB
| | |B |
| rrp|BBB|rBr|BBB
__________________||||||||BBBBpp
| p|rrB|BBB|p
__________________||||||||BBBBBBppppp
| 8 |ppp|rBr|BBpp
__________________||||||||BBppppp
| | 8 | |
| | | |
__________________||||
-2
By [2, after 7.2], h1 : Ee1,-12! Ee2,12is injective, and since d3 acts inject*
*ively on
Ee2,13, it must also act injectively on eE1,-13. The extension from eE1,-11must*
* be non-
trivial to give the Z=8 group deduced from the first approach. The extensions f*
*rom
Ee1,-11(Spin(m)) for 8 m 10 are then deduced by naturality as explained in *
*the
previous section.
Similarly to the proof in the previous section for Spin(8i + 2) with i > 1, we
deduce that d3 = 0 from sEe1,-13(Spin(10)). Indeed, eE4,12(Spin(10)) ! eE4,13(S*
*pin(11))
22 MARTIN BENDERSKY, DONALD M. DAVIS, AND MARK MAHOWALD
is injective, but sEe1,-1 e1,-1
2 (Spin(10)) ! E2 (Spin(11)) maps onto elements divisibl*
*e by
8.
The groups v-11ß0*(Spin(5)) = v-11ß0*(Sp(2)) can be obtained similarly to the*
* J-chart
determination of v-11ß*(Sp(2)) in [19]. To obtain v-11ß0*(Sp(2)), [19, Fig.2.1]*
* should
be shifted by 4 dimensions, and d1-differentials inserted from the new 8k +2 to*
* 8k +1.
But these differentials are not needed for our purposes. Since v-11ß0-2(S3) = 0*
* and
v-11ß0-2(S7) = Z=8, the exact sequence of the fibration S3 ! Sp(2) ! S7 implies
that v-11ß0-2(Sp(2)) is at most Z=8. Thus the Z=16 in fEe1,-12(Spin(5)) must su*
*pport
a nonzero d3 and cannot extend.
There can be no extension from Ee1,-11(Spin(6)) by naturality. Finally d3 is*
* 0 on
fEe1,-13(Spin(6)) since its image in Spin(7) consists of multiples of 2, but th*
*e target
classes eE4,13map injectively from Spin(6) to Spin(7) by [2, 6.1].
References
[1]J. F. Adams, Geometric dimension of vector bundles over RPn, Proc Int Conf
on Prospects in Math, Kyoto (1973) 1-14.
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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
Northwestern University, Evanston, IL 60208
E-mail address: mark@math.northwestern.edu