v1-PERIODIC HOMOTOPY GROUPS OF SO(n)
MARTIN BENDERSKY AND DONALD M. DAVIS
Abstract.We compute the 2-primary v1-periodic homotopy groups
of the special orthogonal groups SO(n). The method is to calculate
the Bendersky-Thompson spectral sequence, a K*-based unstable
homotopy spectral sequence, of Spin(n). The E2-term is an Ext
group in a category of Adams modules. Most of the differentials
in the spectral sequence are determined by naturality from those
in the spheres.
The resulting groups consist of two main parts. One is sum-
mands whose order depends on the minimal exponent of 2 in sev-
eral sums of binomial coefficients times powers. The other is a sum
of roughly [log2(2n=3)] copies of Z=2.
As the spectral sequence converges to the v1-periodic homotopy
groups of the K-completion of a space, one important part of the
proof is that the natural map from Spin(n) to its K-completion
induces an isomorphism in v1-periodic homotopy groups.
1.Introduction
The p-primary v1-periodic homotopy groups of a topological space X, denoted
v-11ß*(X; p), are a localization of the portion of the actual homotopy groups d*
*etected
by K-theory. The study of these groups was first suggested in Mahowald's 1982
paper [32], although it was not until the 1991 paper [28] that a satisfactory d*
*efinition
appeared. In the 1989 paper [27], mod 2 v1-periodic homotopy groups of SO(n)
were computed for some small values of n, but the mod 2 groups do not contain
the information about higher 2-torsion in v-11ß*(X; 2). In 1988, the second au*
*thor
suggested to Mahowald that they try to compute v-11ß*(SO(n); 2), to which Mahow*
*ald
__________
Date: July 18, 2002.
1991 Mathematics Subject Classification. 55Q52,55T15,57T20.
Key words and phrases. homotopy groups, orthogonal groups, Adams spectral
sequences, K-theory.
Both authors thank the Johns Hopkins University JAMI program, which sup-
ported them during their work on this in Spring 2000. The second author thanks
the Reidler Foundation for support on this work during Summer 2002.
1
2 BENDERSKY AND DAVIS
wisely responded that it would be worthwhile to try the easier cases SU(n) and *
*Sp(n)
first.
It quickly became apparent that odd-primary groups were easier than 2-primary,
and the second author determined v-11ß*(SU(n); p) for odd primes p in 1989, pub*
*lished
in [23]. From these results, one easily reads off the groups v-11ß*(SO(n); p) f*
*or p odd.
In 1989, Mimura suggested to the second author that the computation of v-11ß*(X*
*; p)
for all compact simple Lie groups X and all primes p would be an interesting pr*
*oject.
Thanks to a new approach to odd primary v1-periodic homotopy groups introduced
in the 1999 Bousfield paper [15], the determination of v-11ß*(X; p) for all com*
*pact
simple Lie groups X and all odd primes p was completed in [22].
In this paper, we determine the long-sought groups v-11ß*(SO(n); 2). This lea*
*ves
v-11ß*(X; 2) for X the exceptional Lie groups E7 and E8 as the only cases remai*
*ning to
complete Mimura's challenge, with the E6 case having been completed very recent*
*ly
in [25] together with unpublished work of Bousfield.
Our method is to compute the v1-periodic Bendersky-Thompson spectral sequence
(BTSS) of Spin(n). This spectral sequence ([12]) is a K-based version of a spec*
*tral
sequence of Bousfield and Kan ([20]), and, for a collection of spaces which inc*
*ludes
Sn, Sn, and simply-connected finite H-spaces X for which H*(X; Q) is associati*
*ve,
converges to v-11ß*(X^), where X^ is the K-completion of X, which we will define
in Section 10. (From now on, all work will be 2-primary, and we write v-11ß*(X)
for v-11ß*(X; 2).) We say that X satisfies the Completion Telescope Property (C*
*TP)
if the natural map X ! X^ induces an isomorphism in v-11ß*(-). We will prove
in Theorem 2.13 that Spin(n) satisfies the CTP. Thus, since Spin(n) is the simp*
*ly-
connected cover of SO(n), a complete computation of the BTSS of Spin(n), includ*
*ing
differentials and extensions, yields v-11ß*(SO(n)).
A major advance in the understanding of the BTSS was made in [9], where it was
shown that for spaces X for which K*(X) is a nice exterior algebra, Es,t2(X) ca*
*n be
computed directly from the Adams module K*(X) as ExtsA(QK1(X; Z^2)= im(_2), K1(*
*St)).
The d3-differentials in the BTSS of Spin(n) are determined from the known behav*
*ior
of d3 in the BTSS of spheres, using naturality.
In this introductory section, we describe the result for SO(8a 1). The sim*
*ilar
results for SO(n) for other mod 8 congruences of n are described in Section 3, *
*which
PERIODIC HOMOTOPY GROUPS OF SO(N) 3
also includes discussion of the morphisms v-11ß*(SO(n)) ! v-11ß*(SO(n + 1)) and
some numerical examples.
The following numbers, which are closely related to the numbers of [8, 1.5], *
*play
an important role.
Definition 1.1. Let n 3 and let m be an odd integer satisfying m 2n.
n iX i j j o
eSp(m, n) = min (-1)k jkkm : j > 2n .
k
_ !
X n-1-kXi2n-1j X i 2n j
P1(m, n) = km i - 2 n-2-k-4t .
odd8k 1 i=0 t 0
< 2n-1 if n < (m + 1) + 3
P2(m, n) = :
T (m, n)=2 (m+1)+2if (m + 1) + 3, n
where T (m, n) =
X X i 2n j X X i 2n-1 j
(22n-1-3m+1+1) km n-2-k-4t-3.22n-2 km n-2-k-3t.
odd k 1 t 0 odd k 1 t 0
Our Pi(m, n) equals Ri(m, n - 1) of [8, 1.5]. The reason for the change is to*
* make
the formulas nicer for Spin(2n). The apparent difference between P2(m, n) here *
*and
R2(m, n - 1) of [8, 1.5] will be explained in Remark 3.2.
Let (-) denote the exponent of 2 in an integer.
Theorem 1.2. Let 2n + 1 = 8a 1. Let ` = [log2(4_3(n - 1))]. In the notation o*
*f 1.1,
let 8
>>G(2e1(4k-1)+e2(4k-1)) r = -3
>>> e (4k-1) e (4k-1)
>>>Z=2 1 Z=2 2 `Z2 r = -2
>>> 2`+2
>G(2`+4) r = 0
>>> e (4k+1)+5
>>>G(2 1 ) r = 1
>>> e1(4k+1)
>>>Z=2 Z=8 r = 2
:0 r = 3, 4
The G(-) when r = -3 has exactly ` summands. The G(-) groups when r = -1
and 0 are extensions of two Z2-vector spaces.
The reader may get a better feeling about where these groups come from and how
they are related to one another in Diagram 1.3, which pictures a stage of the B*
*TSS of
Spin(8a 1). As usual with charts of Adams spectral sequence type, position (t-s*
*, s)
depicts Es,tr(X) for appropriate r, differentials dr are homomorphisms from Es,*
*trto
Es+r,t+r-1r, and E*,*+i1is an associated graded for ßi(X) (here v-11ßi(X^)). A *
*small dot
represents an element of order 2, while a big o denotes a Z2-vector space of di*
*mension
`. We sometimes call elements in this vector space öl g-classes" because ` = [l*
*og2(-)].
Small labels D, 1, and 4a - 3 next to dots refer to names of elements which wil*
*l be
important later when we derive this chart. In position (2m, 1) with 2m = 8k 2*
*, we
have summands C1 = Z=2e1(m), while 8 represents Z=8 and C2 = Z=2e2(m). The lett*
*er
G in position (x, 2) denotes a group of same order as the neighboring group C1 *
* C2
or C1 Z=8 in position (x + 1, 1), but we don't know the group structure of G.
Lines of slope 1 connecting dots are the action of h1 on E2. This corresponds*
* to
the action of the Hopf map j on homotopy groups. We call these eta towers. This
action was defined in [9, 3.6], where it was shown that it acts bijectively on *
*groups
of filtration 2. This was shown in a slightly different context in [5]. Multi*
*ple lines
of slope 1 between big o's indicate nontrivial action of h1 on ` linearly indep*
*endent
elements. This carries the implication that G has at least ` summands, but we w*
*ill
show in 11.3 that G has exactly ` summands. Lines of slope -3 are d3-differenti*
*als,
__________
1If 8a + 1 = 9 and m 3 mod 4, there is an anomaly discussed in [8, 4.21]. In
this case, we have e1(m) = min( (m - 7) + 2, 8) and e2(m) = 3.
PERIODIC HOMOTOPY GROUPS OF SO(N) 5
which imply that the elements that they connect do not survive the spectral seq*
*uence.
If the chart does not depict h1x for an element x in filtration 2, it is beca*
*use the
omitted elements are involved in d3-differentials implied by the chart.
The lack of depicted h1-action on certain summands in filtration 1 carries no*
* impli-
cation about whether or not h1 is nonzero on their generators. The determinatio*
*n of
this requires careful analysis, which is stated in Proposition 1.4 and proved i*
*n Section
7. Indeed, the chart in Diagram 1.3 depicts the BTSS prior to the consideration*
* of
h1-action, d3-differentials, and extensions on the summands in position (8k + 2*
*, 1).
6 BENDERSKY AND DAVIS
Diagram 1.3. A stage of the BTSS of Spin(8a 1)
| | | | | |r | | |
| | | | - | |B | | |
| | | | BBB|BB | - | | |
| | | | BBB|BBBB | B BBBBB|BBB | |
| | | | BB|BBBB | BBBBBrB|BBB | |
_______________________________________________________________________|||*
*|||||||||||||||BBBBBBBBBBBBBBB
| | | | B|BBBBBr | B BBBB|rBBr | |
| | | - | |BBBBB | BBBBB|BB B | |
| | | | |BBBB- | B BBB|BBB B | |
4 | | | | |BBB | BBB|BBB | |
| | | | |BBBBBBr| B BB|BBBBBB | |
_______________________________________________________________________|||*
*|||||||||||||||BBBBBBBBBBBBBBBBB
| | | |r | BBBBBB| r B B|BBBBBBB|B |
| | - | | - | BBBBBB| r B |BBBBBBB|B |
| | | | | BBBBB| BB|BBBB |B |
3 | | | | r| BBBBB|BB B|BBBBBBB|BBB |
_______________________________________________________________________|||*
*|||||||||||||||BBBBBBBBBBBBBBBBB
| | 4a-|3r | | r BBB|BBB 4a-|3BBBrBB|BBBBB |
| | | | | r BB|B | BBBB B| B |
| G | | - | | BBB|B | B-BBBB|BB |
2 | | | | | BB|BBB | B| B |
| | | r | | G | | BB|BrB |
_______________________________________________________________________|||*
*|||||||||||||||BDDB
| | | |1r | | | |BBrBrB |
| | | | r | | | |1 D |
| |C C | | D | | C 8 | | |
s = 1 | | 1 2 | | | | 1 | | |
| | | | | | | | |
_______________________________________________________________________|||*
*||||||
t - s = 8k+ -3 -2 0 2 4
In this chart, the 1-line is from 3.1, the 2-line from 5.2, the eta towers fr*
*om 5.14,
and d3-differentials on them from 6.2. The fine tuning described in the follow*
*ing
result is proved in Section 7.
Proposition 1.4. The BTSS of Spin(8a 1) is as pictured in Diagram 1.3, with t*
*he
following modifications and clarifications.
(1)d3 is 0 on E1,8k+12;
(2)d3 : E1,8k+32! E4,8k+52is nonzero on both summands;
(3)there are nontrivial extensions (.2) from the two summands of
E1,8k+32to the two of E3,8k+52.
Theorem 1.2 follows immediately from 1.3 and 1.4. Some other extensions are rul*
*ed
out since 2j = 0 in ß*(-), while others are left undetermined as discussed in R*
*emark
3.9.
The rest of the paper is organized as follows. We begin in Section 2 by prov*
*ing
Theorem 2.13, that Spin(n) satisfies the CTP. This utilizes an analysis of the *
*BTSS
of the classifying spaces BSpin(n). In order to accomplish this, we utilize a g*
*eneral
PERIODIC HOMOTOPY GROUPS OF SO(N) 7
result, 2.2, about fibrations which induce a relatively injective extension seq*
*uence in
K-homology. The proof of this result is not needed in the rest of the paper, an*
*d is
presented in Section 10.
As suggested above, the form of v-11ß*(Spin(n)) depends on the mod 8 value of*
* n.
Results similar to those listed above for Spin(8a 1) will be collected in Sec*
*tion 3,
which will also give some explicit numbers. The next four sections go through t*
*he
details of the computation in the following order. In Section 4 we compute the *
*1-line,
by extending the methods of [8], which computed the 1-line for Spin(2n + 1). We
explain in 5.2 the simple reason why E2,4k+32(Spin(n)) has essentially the same*
* order
as E1,4k+32(Spin(n)). We compute the 1-line group explicitly as a direct sum of*
* two or
three summands; however, we do not know the group structure of the 2-line group*
*s.
As noted above, in filtration greater than 2, elements are arranged in what w*
*e call
eta towers. These are elements of the group ji(X), defined in 1.5.
Definition 1.5. The group ji(X) is defined to be the direct limit of the groups
Es,2s+2i+12(X),2 using h1 : Es,t2(X) ! Es+1,t+22(X).
The natural morphism Es,2s+2i+12(X) ! ji(X) is an isomorphism for s 3. In
Section 5, we compute the eta towers in Spin(n), and in Section 6 we compute the
d3-differentials on the eta towers. In Section 7, we compute the d3-differentia*
*l on the
1-line groups and the extensions in the BTSS.
In Section 8, we prove some combinatorial results needed earlier in the paper*
*. In
Section 9, we compare our results with those obtained by a J-homology approach
such as was employed in [26] and [27].
The method used to compute ExtA in Sections 5, 6, and 7 is that developed in *
*[9].
In Section 11, we describe an alternate way of computing ExtA, involving an exp*
*licit
small resolution. It has several advantages over the previous method: (a) it de*
*scribes
eta-towers in a way which does not involve an extension in a short exact sequen*
*ce
(compare 5.1 and 11.3); (b) it gives a different proof of a formula for h1-acti*
*on and
extends that formula to other situations (compare 7.2 with 11.5 and 11.18); and*
* (c) it
__________
2We use this notation here because eta towers for Spin(n) occur only when t-2s
is odd. If one is dealing with situations in which eta towers occur in both par*
*ities
of t - 2s, then defining ji(X) as the limit of Es,2s+i2(X) would be more sensib*
*le.
([25])
8 BENDERSKY AND DAVIS
gives a new interpretation of the 2-line groups, which shows exactly their numb*
*er of
summands, and lends hope to their complete calculation. (See discussion after 1*
*1.3).
2.The BTSS of BSpin(n) and the CTP
In this section, we prove, by induction on n, that Spin(n) satisfies the CTP *
*for all
n. In order to accomplish this, we consider also the BTSS and CTP for the class*
*ifying
spaces BSpin(n).
We begin by recalling an important definition.
Definition 2.1. Let A be a commutative ring with unit. Let S be the category of
coassociative cocommutative coalgebras over A which are free positively-graded *
*A-
modules of finite type. A relatively injective extension sequence is a sequenc*
*e of
maps in S
f g 00
C0 -! C -! C
such that
og is a split epimorphism of A-modules;
othe map f is the inclusion C C00A ! C;
oC is a relatively injective C00-comodule, which means that it is a
direct summand (over K*) of a C00-comodule of the form C00 N
for some K*-module N.([39, p.321])
This definition is that of [17, 2.1], modified to relatively injective comodu*
*les rather
than injective ones. The following result will be proved in Section 10.
j h
Theorem 2.2. Let F -! E -! B be a fibration. Suppose that, for X = F , E, and
B, the BTSS converges to v-11ß*(X^), and that the induced sequence of K*-coalge*
*bras
K*(F ) ! K*(E) ! K*(B)
is a relatively injective extension sequence. Then
^
(i)the induced maps of K-completions F ^! E^ -h! B^ form a
fibration,
(ii)there is an exact sequence
. .!.Es2(F ) ! Es2(E) ! Es2(B) -@! Es+12(F ) ! . .,.and
(iii)@ commutes with differentials in the BTSS.
PERIODIC HOMOTOPY GROUPS OF SO(N) 9
The following result is quite easy.
Proposition 2.3. The fibration Spin(2n - 1) ! Spin(2n) ! S2n-1 induces a rela-
tively injective extension sequence in K*(-).
Proof.Using [29] or [37], we easily see that the K*(-)-algebras of the fibratio*
*n are
polynomial algebras on indecomposables which form a short exact sequence. Indee*
*d,
in notation which will be prevalent in the last half of this paper, the generat*
*ors are
* p*
-i -
(2.4)
with i*(xj) = xj, i*(D ) = D, and p*(g) = D+ - D-. Dualizing this result (e.g.,*
* [2,
1.4]) says that K*(-) is a relatively injective extension sequence. ||
The following similar result is somewhat more delicate.
Proposition 2.5. There is a relatively injective extension sequence of coalgebr*
*as
K*(S2n) ! K*(BSpin(2n)) ! K*(BSpin(2n + 1))
induced by the fibration
S2n ! BSpin(2n) ! BSpin(2n + 1). (2.6)
Proof.We begin by showing that
* ffi 0 2n
R(Spin(2n + 1)) -i! R(Spin(2n)) -! K (S ) (2.7)
is a projective extension sequence. This means that i* is a split (over Z) mono*
*mor-
phism,
effi0 2n
R(Spin(2n)) im(i*)Z -! K (S )
is an isomorphism, and R(Spin(2n)) is a projective R(Spin(2n + 1))-module. (See
[36] for the analogue over a field.)
From [30] or [14], we have
R(Spin(2n + 1))= Z[æ2n+1, ~2æ2n+1, . .,.~n-1æ2n+1, ]
R(Spin(2n)) = Z[æ2n, ~2æ2n, . .,.~n-2æ2n, +, -].
For 1 j n - 2, i*(~jæ2n+1) = ~jæ2n+ ~j-1æ2n. Also, i*( ) = + + -, and
i*(~n-1æ2n+1) = + . - + ~n-2æ2n- ~n-3æ2n- ~n-5æ2n- . ...
10 BENDERSKY AND DAVIS
The morphism OE is the composite R(Spin(2n)) ! K0(BSpin(2n)) -j*!K0(S2n) with
j the map in (2.6), and satisfies OE(~iæ2n) = 0 and OE( ) = `, where K0(S2n) =
Z[`]=`2. After an obvious change of basis, (2.7) becomes
* ffi 2
Z[g01, . .,.g0n-1, ] -i! Z[g1, . .,.gn-2, +, -] -! Z[`]=`
with i*(g0i) = gi for i n - 2, i*(g0n-1) = + . -, OE(gi) = 0, and i*( ) and*
* OE( )
as above.
The first two properties of projective extension sequence are clearly satisfi*
*ed. To
see the projectivity, we observe that Z[{gi}, x, y] is a free Z[{gi}, xy, x + y*
*]-module on
1 and x. This can be achieved by noting that the change-of-basis matrix relating
(xy)n, x(xy)n-1(x+y), (xy)n-1(x+y)2, x(xy)n-2(x+y)3, (xy)n-2(x+y)4, . . .
to
xnyn, xn+1yn-1, xn-1yn+1, xn+2yn-2, xn-2yn+2, . . .
is triangular, and similarly in odd degree.
In [2, 1.2], the following result is proved.
Theorem 2.8. ([2]) If G is a connected compact Lie group, then K1(BG) = 0, and
there are natural isomorphisms
K0(BG) Hom(K0(BG), Z) Hom(R(G), Z),
where Hom(-, -) refers to continuous homomorphisms. Here R(G) has the I(G)-
adic topology, and Z is discrete.
As Anderson remarks on [2, p.5], the effect of this is given in the following*
* corollary.
Corollary 2.9. If {~i} is a set of irreducible representations of G and æi = ~i-
dim(~i), let æE = æe11. .æ.ekk2 R(G). Let OEE 2 Hom (R(G), Z) be dual to æE in *
*the
basis of æE's. Then K0(BG) is free abelian with basis {OEE}, and its coalgebra *
*structure
is given by
X
_(OEE) = OEF OEE-F .
F E
We apply Hom(-, Z) to the objects of (2.7), obtaining the sequence of 2.5, wh*
*ich
is 0 in odd gradings. This duality applied to a projective extension sequence y*
*ields
a relatively injective extension sequence. As described in Corollary 2.9, the *
*effect
PERIODIC HOMOTOPY GROUPS OF SO(N) 11
of Hom is to make the dualization act as if R(G) were finite dimensional. For t*
*he
tensor/cotensor criterion, we use [34, 3.2.2]. We remark that we need to use "r*
*ela-
tively injective" because these Z(p)-modules lack the divisibility to be inject*
*ive, but
projectivity does not have this problem. This completes the proof of Propositio*
*n 2.5.
||
The above proposition is relevant for Spin(n) because of the following result.
Proposition 2.10. i. There is an isomorphism of BTSS's
Es,tr(BSpin(n)) Es,t-1r(Spin(n)), r 2;
ii.the map of K-completions
Spin(n)^ ! (BSpin(n)^) (2.11)
induces an isomorphism in v-11ß*(-);
iii.Spin(n) satisfies the CTP if and only if BSpin(n) does.
Proof.We first establish the isomorphisms, for G = Spin(n),
Es,t2(BG) Exts,tG(K*BG) Exts,tU(P K*BG) Exts,t-1U(P K*G) Es,t-12(G).
Here G and U are the categories of unstable K*K-coalgebras and unstable K*K-
comodules discussed in Section 10 and in [12]. Our convention is to omit writin*
*g K*
as the first component of Ext groups.
The first isomorphism is [12, 4.3]. The fourth isomorphism is [12, 4.9], and*
* the
second follows similarly. To deduce the third isomorphism, first note that, for*
* X =
BG, the map X ! X induces K*-1( X) ! K*(X), which on primitives is an
isomorphism in U. The desired isomorphism then follows from the isomorphism
U(A[2n]) oeU(A[2n - 1]). (2.12)
Here U is the functor from free K*-modules to unstable -comodules defined in [*
*12,
x4], A[t] is the free K*-module on a generator of grading t, oe is suspension, *
*and (2.12)
follows from [12, 4.5].
To prove (i) for all r, we note that the isomorphism of E2-terms is induced b*
*y a map
of towers. To see this, we use the following natural map of augmented cosimplic*
*ial
spaces, where K(X) = 1 (K ^ 1 X). We take X = BSpin(n); however, the
argument works in much greater generality.
12 BENDERSKY AND DAVIS
- ___- ___-
X ___-K X ___-_K(K( X))___-_-K3 X___-_-_-. . .
| | | |
| | | |
|? |? - |? ___-_-3|? ___-_-
X ___- KX ___-_ K(KX) ___- K X ___-_-. . .
Applying ß*(-) and taking homology of the alternating sum to the first row yi*
*elds
E*,*2( X), and doing this to the second yields E*,*-12(X). The induced morphism*
* in
homology is the E2 isomorphism observed above. But these cosimplicial spaces gi*
*ve
rise, by filtering the Tot construction, to the towers that define the entire s*
*pectral se-
quence, and so the morphism induces a morphism of spectral sequences, which is *
*then
an isomorphism. See the first few pages of Section 10, or [12], for more detail*
*s regard-
ing the Tot construction and the BTSS. By [9, 5.1], the spectral sequences conv*
*erge,
respectively, to v-11ß*(Spin(n)^) and v-11ß*( (BSpin(n)^)), which is consequent*
*ly an
isomorphism.
Part iii follows from part ii and the commutative diagram
Spin(n) --'-! BSpin(n)
?? ??
'Spin?y 'BSpin?y
Spin(n)^---! (BSpin(n)^)
||
Now we can prove the main theorem of this section.
Theorem 2.13. For each n, the natural map Spin(n) ! Spin(n)^ induces an iso-
morphism in v-11ß*(-); i.e., Spin(n) satisfies the CTP.
Proof.It was proved in [9] that Sn satisfies the CTP. Since Spin(3) = S3, this *
*will
initiate the induction. The induction steps are immediate from Propositions 2.1*
*4 and
2.10.iii. ||
Proposition 2.14. a. If Spin(2n - 1) satisfies the CTP, then so does Spin(2n).
b. If BSpin(2n) satisfies the CTP, then so does BSpin(2n + 1).
Proof.It was proved in [9, 5.8,5.11,5.12] that if F ! E ! B induces a relatively
injective extension sequence in K*(-), and two of the spaces satisfy the CTP, t*
*hen
so does the third. The proposition then follows from 2.3 and 2.5. ||
PERIODIC HOMOTOPY GROUPS OF SO(N) 13
3. Listing of results
In this section, we state the results for the explicit form of the BTSS of Sp*
*in(N) for
the values of N not covered in Section 1. The proofs of these statements occupy*
* the
next four sections of the paper. Indeed, results for the 1-line are proved in S*
*ection
4, the 2-line in 5.2, the eta towers in 5.14, 5.16, and 5.22, and d3 on the eta*
* towers
in 6.2. Finally, d3 on the 1-line and the extensions are in Section 7, with ex*
*plicit
references there to the theorems of this section whose proofs are being complet*
*ed. At
the end of this section, we also describe homomorphisms induced by inclusion ma*
*ps,
and give explicit numerical examples.
We begin by recalling from [8] the determination of the 1-line groups of Spin*
*(N)
when N is odd, with a refinement established in Remark 3.2.
Theorem 3.1. ([8, 1.5], 3.2) If n 6, and m is odd, then
8
>< (m + 1) if (m + 1) < (n)
0(m + 1, n) = > (m + 1 - n) + 1 if (m + 1) = (n)
>:
(m + 1) + 1 if (m + 1) > (n)
and
R(m, 2n) = min(eSp(m, n), (P1(m, n)), (P20(m, n)), (P3(m, n))),
where eSp, P1, and P2 are as in 1.1, while
8
>><2P2(m, n)if (m + 1 - n) n - 3
P20(m, n)= > n if (m + 1) n - 3 and n even
>:
P2(m, n) otherwise, and
X i 2nj
P3(m, n) = 1_n km+1 n-k .
k odd
If n - 3 (m + 1), then
8
N. The BTSS is periodic, and *
*so
it suffices to specify the groups in this range. Thus, for example, the integer*
*s R(m, N)
of 3.1 and 3.3 are defined by the given formulas only for m > N.
We remark that it is true for dimensional reasons that E1,2m2(Spin(n)) = 0. A*
*lso,
for the omitted cases, since Spin(4) S3 x S3, its v1-periodic homotopy groups
PERIODIC HOMOTOPY GROUPS OF SO(N) 15
follow from [32] or [24, 4.2], while the 1-lines of Spin(6) and Spin(10) have a*
*rithmetic
anomalies and are covered in Proposition 4.33. Although they are quite rare, th*
*ere
are cases in which the value of R(m, N) is determined by eSp(m, N), and cases in
which the value of R(m, 2n) is determined by P3(m, n).
Now we describe the entire BTSS of Spin(N), divided into cases by the mod 8 v*
*alue
of N. We begin with the case N = 8a.
Theorem 3.4. The BTSS for Spin(8a) is the direct sum of the BTSS of S8a-1given
in Diagram 3.6 and the BTSS of Spin(8a - 1), as given in 1.3 and 1.4, except th*
*at if
s = 1 or 2, the short exact sequence
p* s,8k-1 8a-1
0 ! Es,8k-12(Spin(8a - 1)) -i*!Es,8k-12(Spin(8a)) -! E2 (S ) ! 0
(3.5)
is not always split. If s = 2, no claim is made about the structure of the grou*
*ps. If
s = 1, it splits if (k) < (a), but does not in the remaining cases:
oif (k) = (a) and (k - a) < 4a - 5, it is
p* (k-a)+4
0 ! Z=2e1 Z=2 (k)+4i*-!Z=2e1 Z=2 (k-a)+5 Z=2 (k)+3-! Z=2 ! 0
in which p* sends the second summand surjectively and the third
summand injectively, and i*(g2) = g3- 2 (k-a)- (k)+1g2; in par-
ticular, the initial summands for Spin(8a - 1) and Spin(8a) are
equal in this case; here we initiate a custom of letting gidenote
a generator of the ith summand;
oif (k - a) 4a - 5, it is
p* 4a-1
0 ! Z=2e1 Z=2 (k)+4i*-!Ze1+1 Z=24a-1 Z=2 (k)+3-! Z=2 ! 0
in which p* sends the first summand surjectively, the second
summand to multiples of 2, and the third summand injectively,
while i*(g1) = 2g1- g2 and i* sends g2 injectively to the second
summand plus surjectively to the third summand;
oif (a) < (k) < 4a - 5, then it is
p* (a)+4
0 ! Z=2e1 Z=2 (k)+4i*-!Z=2e1 Z=2 (k)+5 Z=2 (a)+3-! Z=2 ! 0
in which p* sends the second summand surjectively and the third
summand injectively, and i*(g2) = g3- 2g2;
16 BENDERSKY AND DAVIS
oif (k) 4a - 5, it is
p* (a)+4
0 ! Z=24a-1 Z=24a-1-i*!Z=24a-1 Z=24a-1 Z=2 (a)+3-! Z=2 ! 0
in which p* sends the first summand surjectively and the third
summand injectively.
Here we are using the following diagram for the BTSS of S8a-1, taken from [7,
p.488], in which C denotes Z=2min( (k-a)+4,4a-1), and 8 denotes Z=8.
Diagram 3.6. BTSS of S8a-1
| | | | | | | | |
| | | | | | | | |
| | | | r | | | | |
| | | | C | | | | |
___________________________________________________________||||||||||*
*||||||||C
| | | | C | r | | | |
| | | | | | | | | |
| | | r | C | |r | | | |
4| | | | C | |B | | | |
___________________________________________________________||||||||||*
*||||||||BC|
| | | | r C| ||B | | | |
| | r | | r |C| B| |r | | |
3| | | | |C| B| | | | |
___________________________________________________________||||||||||*
*||||||||BC|||
| | | r | |C| |B || | | |
| | | r | |C8 r |B | | | |
2| C | | | | | B | | | |
___________________________________________________________||||||||||*
*||||||||B|
| | | | | | B|| | | |
| | | | r | | | | |
| | | | | | BB8 | | |
s = 1| | C | | | | | | |
___________________________________________________________|||||||||
t - s = 8k+ -2 0 2 4
Next we describe the BTSS for Spin(8a + 3) and Spin(8a + 5), except for the g*
*roup
structure of some 2-line groups. We begin with a picture of a certain stage of*
* the
BTSS, and then describe the result in a theorem.
PERIODIC HOMOTOPY GROUPS OF SO(N) 17
Diagram 3.7. A stage of the BTSS for Spin(8a + 4 1)
| |r | | r | | | | |
| | | | - | | | | |
| |B | | B BBB|BB | - | | |
| | B | | B BBB|BBBB | BBBB|BBBB | |
| | B | | B BB|BBBB | BBBBr|BBBB | |
_______________________________________________________________________*
*||||||||||||||||||BBBBBBBBBBBBBBBB
|Dr | B |r | B B|BBBBB | BBB|BBB | |
| | B |B - | B |BBBBB | BBB|BB r | |
| | B | B | B |BBBB- | BB|BBBB | |
4 | | | | |BBB | BB|BBBB | |
| | B | B | B |BBBBBBr| B|BBBBBB | |
_______________________________________________________________________*
*||||||||||||||||||BBBBBBBBBBBBBBBBBB
| |r B | B | B| BBBBBB| |BBBBBBBB|r |
| | - B| B | - B| BBBBBB| |r |BBBBBBB | |
| | B| B | |B BBBBB| | |BBBB B | |
3 | | |B B | r|B BBBBB|BB | |BBBBBBBB|B |
_______________________________________________________________________*
*||||||||||||||||||||BBBBBBBBBBBBBBBBBB
|Dr | |BBrD B | |BBr BBB|BBB| |DBBBBBBB|r |
| | | B| | r BB|B | | BBBB |B |
| G | | - | | BBB|B | | B-BBBB|BB |
2 | | | B| | BB|BBB| | |B |
| | | r |B | G | | | BB|rB |
_______________________________________________________________________*
*||||||||||||||||||||4a-14a-1BB
| | | |BBrD | | | | |DBr |
| | | | r | | | | | BBr |
| |C C | | 1 | | | | 1 |
s = 1 | | 1 2 | | | | C1 8| | |
| | | | | | | | |
_______________________________________________________________________*
*|||||||||
t - s = 8k+ -3 -2 0 2 4
Theorem 3.8. Let 2n + 1 = 8a + 4 1. The BTSS of Spin(2n + 1) is as depicted in
Diagram 3.7, with the following additions and interpretations.
oThe 1-line groups are as given in 3.1.
oA G in position (x, 2) represents an abelian group of the same
order as the group in position (x + 1, 1).
oAll elements x in filtration 2 are acted on freely by j in E2.
When the elements jix for i > 0 are not depicted, it means that
they support d3-differentials inferred from d3(x).
oThe big o's represent a vector space of dimension ` = [log2(4(n-
1)=3)] + ffiff(n-1),1, as specified in 5.14. Multiple lines indicate
d3-differentials or eta-actions acting bijectively on these vector
spaces. The groups G in position (8k - 1 2, 2) have exactly `
summands.
oIn addition to the differentials pictured, d3 on the generator
of the C1 summand in position (8k + 2, 1) is nonzero, while d3
from position (8k -2, 1) hits the class D. This differential is on
18 BENDERSKY AND DAVIS
the C2-summand if (k) + 3 < n; otherwise, C1 C2 Z=2n
and the differential is from C1. Other than these, there are no
more nonzero differentials.
oThere is a nontrivial extension (multiplication by 2) in dimen-
sion 8k - 2 from C2 to D.
Remark 3.9. We can make some other general statements about extensions in Di-
agram 3.7. For the summands x in the G in (8k - 3, 2) which are of order 2, the*
*re
must be an extension (.2) from (8k - 1, 2) to h21x in (8k - 1, 4). For summands*
* y in
(8k - 1, 2) which do not extend into (8k - 1, 4), there must be an extension fr*
*om G
in (8k + 1, 2) to h21y in (8k + 1, 4). These follow because the homotopy groups*
* of the
mod-2 Moore space imply that if ff 2 ßn(X) satisfies 2ff = 0, then j2ff is divi*
*sible by
2.
We easily read off the groups as follows.
Corollary 3.10. Let 2n + 1 = 8a + 4 1, ` = log2(4(n - 1)=3)] + ffiff(n-1),1, *
*and
let e1(m)and e2(m) be defined by the formulas of 1.2 unless m 3 mod 4 and n <
2 + (m + 1), in which case e1(m) = n - 1 and e2(m) = n + 1. Then
8
>>>G(2e1(4k-1)+e2(4k-1)) Z2r = -3
>>> e (4k-1) e (4k-1)
>>>Z=2 1 Z=2 2 `Z2 r = -2
>>> 2`+1
>(` + 2)Z2 r = 0
>>> e (4k+1)+4
>>>G(2 1 ) r = 1
>>> e1(4k+1)
>>>Z=2 Z=8 r = 2
:Z
2 Z2 r = 3, 4
The G(-) when r = -3 has exactly ` summands. The G(-) group when r = -1 is
an extension of two Z2-vector spaces.
The result for Spin(4a + 2) is given as follows.
Theorem 3.11. A chart for the BTSS of Spin(4a + 2) is as in Diagram 3.12. The
1-line groups are as given in Theorem 3.3. A group labeled G in position (x, 2)*
* has
the same order as the group in (x + 1, 1). The group labeled C0 in (4k - 1, 2)*
* is
cyclic of exponent 1 greater than that of the cyclic group C in (4k, 1). Each *
*big o
represents a Z2-vector space of dimension [log2(4_3(2a-3))]. The d3-differentia*
*l on the
generator of C1 in (8k + 2, 1) is nonzero if and only if R(4k + 1, 4a + 2) of 3*
*.3 equals
PERIODIC HOMOTOPY GROUPS OF SO(N) 19
R(4k + 1, 4a + 3) of 3.1. All other d3-differentials are 0, except those indica*
*ted in the
chart. The extension from C0 into (8k - 1, 4) is trivial.
Diagram 3.12.|The|BTSS of Spin(4a|+|2) | | | | |
| | | | - | | | | |
| | | | BBB|BB | - | | |
| | | | BBB|BBBB | BBBB|BBBB | |
| | | | BB|BBBB | BBBBr|BBBB | |
_______________________________________________________________________*
*||||||||||||||||||BBBBBBBBBBBBBB
| | | | B|BBBBB | BBBB|BB | |
| | | - | |BBBBB | BBB|BB r | |
| | | | B|BBB- | BBB|BB B | |
4 | | | | |BBB | BB|BBBB | |
| | | | |BBBBBBr| B|BBBBBB | |
_______________________________________________________________________*
*||||||||||||||||||BBBBBBBBBBBBBBB
| | | | | BBBBBB| |BBBBBBB|B |
| | - | | - | BBBBBB| |r |BBBBBBB| |
| | | | | BBBBB| | |BBBB B | |
3 | | | | 4a-r1| BBBBB|BB| |BBBBBBB|BB |
_______________________________________________________________________*
*||||||||||||||||||||BBBBBBBBBBBBBBB
| | | | | BBBB|BB| | BBBBBB|B |
| | | | | r BBB| | | BBBB |B |
| G | | - | | BBB|B | | B-BBB|BBB |
2 | | | | | BBB|BB| | |0B |
| | | C0 | | G | | | BC|BB |
_______________________________________________________________________*
*||||||||||||||||||||B
| | | | | | | | | B |
| | | | r | | | | | BBrC |
| |C C | | 1 | | | | 1 |
s = 1 | | 1 2 | | C | | C1 8| | |
| | | | | | | | |
_______________________________________________________________________*
*|||||||||
t - s = 8k+-3 -2 0 2 4
The groups which are the result of this chart are given in the following resu*
*lt.
Corollary 3.13. Let ` = [log2(4_3(2a - 3)], R(-, -) be as in 3.3,4
8
>>G(2e1(4k-1)+e2(4k-1)) r = -3
>>> e (4k-1) e (4k-1)
>>>Z=2 1 Z=2 2 `Z2 r = -2
>>> e3(4k)+1 2`
>Z=2e3(4k) (` + 2)Z2 r = 0
>>> e (4k+1)+5
>>>G(2 1 ) r = 1
>>> e1(4k+1)+1
>>>Z=2 Z=8 r = 2
:Z=2e3(4k+2) r = 3, 4
The G(-) when r = -3 has exactly ` summands. The G(-) group when r = -1 is
an extension of two Z2-vector spaces.
From Table 3.22 we can see, for example, that d3 is nonzero on E1,8k+32(Spin(*
*18))
iff k 2 mod 8 or k 259 mod 512 and that d3 6= 0 on E1,8k+32(Spin(22) iff k *
* 3
mod 27 or k 4 + 29 mod 212.
Next we describe the BTSS of Spin(8a + 4).
Theorem 3.14. The BTSS for Spin(8a + 4) is the direct sum of the BTSS of S8a+3
given in Diagram 3.16 and the BTSS of Spin(8a + 3) as described in Theorem 3.8
except that the short exact sequence
p* s,4`-1 8a+3
0 ! Es,4`-12(Spin(8a + 3)) -i*!Es,4`-12(Spin(8a + 4)) -! E2 (S ) ! 0
(3.15)
is not always split when s = 1 and 2. No claim is made about the group structure
when s = 2. If s = 1 and ` = 2k + 1, then
oif (k - a) 4a - 4, the sequence is
0 ! Z=2e1 Z=8 ! Z=2e1 Z=2 (k-a)+5 Z=4 ! Z=2 (k-a)+4! 0
with the Z=8 mapping injectively to the second summand and
surjectively to the third, while these summands map, respec-
tively, surjectively and injectively to the Z=2 (k-a)+4. Note that
the two Z=2e1summands are of equal order. The d3-differential
from E1,8k+32(Spin(8a + 4)) is nonzero on only the first sum-
mand.
PERIODIC HOMOTOPY GROUPS OF SO(N) 21
oif (k - a) > 4a - 4, the sequence is
0 ! Z=2e1 Z=8 ! Z=2e1+1 Z=24a+1 Z=4 ! Z=24a+1! 0
with i* sending the first generator to (2, 1, 0) and the second to
(0, 24a-2, 1), while p* sends the the three generators, respectively,
to 1, -2, and 24a-1. The d3-differential from E1,8k+32(Spin(8a+
4)) is nonzero on just the first and second summands.
If s = 1 and ` = 2k, then
oif (k) 4a - 4, the sequence is
0 ! Z=2e1 Z=2 (k)+4! Z=2e1 Z=2 (k)+5 Z=4 ! Z=8 ! 0
with the Z=2 (k)+4mapping injectively to the second summand
and surjectively to the third, while these summands map, respec-
tively, surjectively and injectively to the Z=8. The d3-differential
from E1,8k-12(Spin(8a + 4)) is nonzero on just the second and
third summands.
oif (k) > 4a - 4, the sequence is
0 ! Z=24a+1 Z=24a+1! Z=24a+2 Z=24a+1 Z=4 ! Z=8 ! 0
with i* sending the first generator to (2, 0, 1) and the second
summand bijectively to the second summand, while p* sends the
first summand surjectively and the third summand injectively.
The d3-differential from E1,8k-12(Spin(8a+4)) is nonzero on just
the first and third summands.
Here we are using the following diagram for the BTSS of S8a+3, taken from [7,
p.488], in which C0 denotes Z=2min( (k-a)+4,4a+1), and 8 denotes Z=8. The dott*
*ed
d3-differential is present iff 4a + 1 (k - a) + 4.
22 BENDERSKY AND DAVIS
Diagram 3.16. BTSS|of|S8a+3| | | | | | |
| | | | r | | | | |
| | | | | | | | |
| | | | B | | | | |
___________________________________________________________|||||||*
*|||||||||||BB
| | | | | | | | |
| | | r | B | | | | |
| | | | B| r | | | |
4 | r | | | B| |. | | | |
___________________________________________________________|||||||*
*|||||||||||BBB|..
| B | r | | |B ||..| | | |
| B | | | r |B | . | | | r |
3 | B| r|| | |B | .|. | | |
___________________________________________________________|||||||*
*|||||||||||B|B||..
| 8 |B ||| | | BBC0 |. | | |
| |B | | | | |.. | | |
2 | r |B | | r | | |. | r | |
___________________________________________________________|||||||*
*|||||||||||B|..
| | B ||| | | | .. | | |
| | B| | | | | . | | |
r | | | | | | C0 | | |
s = 1| | B8 | | | | | | |
___________________________________________________________|||||||*
*||
t - s = 8k+ -2 0 2 4
The morphisms in v-11ß*(-) induced by inclusion maps Spin(n) ! Spin(n+1) can,
for the most part, be determined from the above charts together with knowledge *
*of
the induced morphism of 1-line groups and eta towers. For the 1-line groups, t*
*he
results are presented in Proposition 3.17. These are easily read off using the *
*names
of generators described in Section 4. Some details of the proof will be given i*
*n that
section.
Proposition 3.17. Let E2m(N) = E1,2m+12(Spin(N)), and consider the sequence
E2m(4a+1) -i1!E2m(4a+2) -i2!E2m(4a+3) -i3!E2m(4a+4) -i4!E2m(4a+5),
beginning with E2m(11).
a. Let m be even. All groups have a summand Z2 generated by ,1, which maps
across. In addition, there are summands
Z2 -i1!Z=2 -i2!Z2 -i3!Z2 Z2 -i4!Z2
satisfying i1 is injective, i2 = 0, i3 hits the sum of the generators, and i4 s*
*ends both
generators nontrivially.
b. Let m be odd. Each group E2m(4a + d) has an initial summand, the orders of
which increase with d. These map injectively to one another, plus possibly also*
* to a
summand in the second component. In addition, there are summands as described
below.
PERIODIC HOMOTOPY GROUPS OF SO(N) 23
If 2a < (m + 1) + 1, then the summands are
Z=22a-i1!Z=22a-i2!Z=22a+1-i3!Z=22a+1 Z=2e -i4!Z=22a+2,
(3.18)
satisfying i1 is bijective, i2 hits multiples of 4, i3 is bijective into the fi*
*rst summand,
i4 sends the first summand to multiples of 4, and the second summand injectivel*
*y.
If 2a > (m + 1) + 1, then, with = (m + 1) + 2, the summands are
Z=2 -i1!Z=2 -i2!Z=2 -i3!Z=2 +ffl Z=2e -i4!Z=2 ,
where ffl and e can be determined from Theorem 3.3. Then i1 is bijective, i2 h*
*its
multiples of 2, the first component of i3 is injective, and i4 sends the second*
* summand
injectively. If (m + 1) < (2a + 2), then ffl = 0, e = , and i4 is multiplica*
*tion by
2 on the first summand. If (m + 1) (2a + 2), then ffl > 0, e = - 1, the s*
*econd
component of i3 is surjective, and i4 is multiplication by 2 on the first summa*
*nd.
Some more detailed information is given in Theorems 3.4 and 3.14.
If 2a = (m + 1) + 1, the groups and morphisms are as in (3.18) except that t*
*he
last group is Z=22a+1and i4 sends the first summand to multiples of 2.
The anomalies in the 1-lines of Spin(6), Spin(9), and Spin(10) cause the foll*
*owing
changes in the morphisms involving them. We will sketch portions of the proof *
*in
Section 4 along with the sketch for Proposition 3.17.
Proposition 3.19. Let m = 2k + 1 with k odd, and E2m(n) = E1,2m+12(Spin(n)).
Then E2m(5) ! E2m(6) and E2m(9) ! E2m(10) are bijective, and
j1 j2 j3 j4
E2m(6) -! E2m(7) -! E2m(8) -! E2m(9) -! E2m(11)
is 8
j1 j2 j3 1 j4 < Z32 Z16 k 1 (4)
Z16-! Z8 Z8 -! Z8 Z8 Z8 -! Z8 Z=2 -! :
Z=2 2 Z32 k 3 (4)
where 1 = min(8, (k - 3) + 3) and 2 = min(8, (k - 19) + 3). We have j1(g) =
2g1 + g2, j2(g1) = g1, j2(g2) = g2 + g3, j3(g1) = g1, and j3 sends the second a*
*nd
third summands injectively into the second summand plus an even component in the
first summand. If k 1 mod 4, then j4(g1) = 4g1 + 4g2 and j4(g2) = 4g1 + 2g2. *
*If
k 3 mod 4, j4 sends the first summand injectively into both summands, while t*
*he
24 BENDERSKY AND DAVIS
component of j4 from the second summand to the first (resp. second) summand has
kernel consisting of elements of order 2 (resp. 8).
The eta towers which occur in big blocks (big o) in the BTSS charts of this s*
*ection
are described most explicitly in Table 6.1 using (5.7) and (5.8). We tabulate i*
*n Table
3.20 for a range of values of n the integers j for which xj 2 ji(Spin(2n + 1)) =
ji(Spin(2n + 2)). These depend on just the parity of i. The BTSS charts show th*
*at
elements in filtration 2, 3, and 4 in jev(Spin(N)) (resp. jod(Spin(N))) surviv*
*e to
elements of v-11ß8k+r(Spin(N)) for -1 r 1 (resp. -3 r -1).
_n_||jev(Spin(2n_+_1))jod(Spin(2n_+_1))
11 || 5, 6, 8 8, 9, 10
12 || 6, 7, 8 8, 10, 11
13 || 6, 7, 8, 12 8, 10, 11, 12
14 || 7, 8, 10, 12 8, 10, 12, 13
15 | 7, 8, 10, 12 8, 12, 13, 14
Table 3.20. |
16 || 8, 9, 10, 12 8, 12, 14, 15
17 ||8, 9, 10, 12, 168, 12, 14, 15, 16
18 || 9, 10, 12, 16 12, 14, 16, 17
19 || 9, 10, 12, 16 12, 16, 17, 18
20 || 10, 11, 12, 16 12, 16, 18, 19
21 || 10, 11, 12, 16 16, 18, 19, 20
From 5.14, we can easily verify8the following proposition.
<2j + 3 if j = 2ior 3 . 2i
Proposition 3.21. Let b(j) = : . Then
4j + 5 - 2 (j)+3otherwise
xj 2 jev(Spin(N)) iffb(j) N 4j + 4
and 8
<4j + 4 if j = 2i
xj 2 jod(Spin(N)) iff2j + 3 N :
2j + 2 + 2 (j)+2otherwise.
Explicit values of 1-line exponents for Spin(11) and Spin(13) were presented *
*in [8,
1.6]. There was one mistake in that table. The formula for e1(2n + 1) when n = 5
and m 7 mod 8 should have been min(8, (m - 39) + 2). The cause of this mista*
*ke
was just overlooking one of the two numbers whose minimum equalled e1.
In 3.1 and 3.3, we give formulas for the largest exponent, R(m, N), in E1,2m+*
*12(Spin(N))
when m is odd. In Table 3.22, we give explicit values for R(2k + 1, 2n - 1) for
PERIODIC HOMOTOPY GROUPS OF SO(N) 25
8 n 13. We have verified that in this range R(m, 2n) = R(m, 2n - 1) when n *
*is
odd. It is proved in 3.4 and 3.14 that if n is even, then R(m, 2n) = R(m, 2n-1)*
* unless
(m + 1 - n) n - 3 or (m + 1) n - 3, in which case R(m, 2n) = R(m, 2n - 1)*
* + 1.
Table_3.22._R(2k_+_1,_2n_-_1)_________________________________________
| n | k 15 mod 16 ||n | k 0 mod 4 |
|___|______________________________||__|______________________________|||||||
|| 8 m|ax(7,|11 - (k + 1)) |8|||min(13,||(k - 4) + 10) ||
|| 9 m|ax(8,|13 - (k + 1)) |9|||min(14,||(k - 4) + 10)1012 ||
||10 |max(9,|15 - (k + 1)) |1|0m|in(25,||(k|- 8 - 210- 2 ) + 12) ||
||11 |max(10,|17 - (k + 1)) |1|1m|in(26,||(k|- 8 - 2 ) + 13) ||
||12 |max(11,|19 - (k + 1)) |1|2m|in(26,||(k|- 8) + 17) ||
|_13_|max(12,_21_-__(k_+_1))________|1|3m|in(26,__(k_-_8)_+_17)_______|||||||
|____|____________________________|_|__|_____________________________|_||||||
|_n_|____k___1__mod_8______________||n_|____k___5__mod_8______________|||||||
|| 8 1|1| ||8|m|in(15,||(k - 5) + 9) ||
|| 9 1|1| ||9|m|in(15,||(k - 5) + 9) ||
||10 |13| ||10m|in(16,||(k|- 21) + 11) | |
||11 |min(24,| (k - 9) + 15)10 |11||16||| ||
||12 |min(27,| (k - 9 - 210) +115)1 |12|||17|| | |
|_13_|min(28,__(k_-_9_-_2__-_2__)_+_16)||131|8________________________|||||||
|____|____________________________|_|__|_____________________________|_||||||
|_n_|____k___2__mod_8______________||n_|____k___6__mod_8______________|||||||
|| 8 1|2| ||8|m|in(18,||(k - 6) + 10)9 ||
|| 9 1|2| ||9|m|in(20,||(k - 6 - 28) + 10) ||
||10 |14| ||10m|in(21,||(k|- 6 - 2 ) + 12) ||
||11 |15| ||11m|in(21,||(k|- 6) + 13) | |
||12 |min(29,| (k - 10) + 18) |12||21||| ||
|_13_|min(29,__(k_-_10)_+_18)_______|13||21___________________________|||||||
|____|____________________________|_|__|_____________________________|_||||||
|_n_|____k___3__mod_8______________||n_|____k___7__mod_16_____________|||||||
|| 8 9|| ||8|||8| 10 | |
|| 9 1|0| ||9|m|in(18,||(k - 7 - 29 ) + 7) ||
||10 |11| ||10m|in(19,||(k|- 7 - 28) + 9) ||
||11 |14| ||11m|in(20,||(k|- 7 - 27) + 11) ||
||12 |16| 14 ||12m|in(21,||(k|- 7 - 2 )6+ 13) ||
|_13_|min(30,__(k_-_11_-_2__)_+_16)_|13_||min(22,__(k_-_7_-_2_)_+_15)_ |
26 BENDERSKY AND DAVIS
4.The 1-line of Spin(2n)
In this section we prove Theorem 3.3 regarding E12(Spin(2n)). We begin with t*
*he
following adaptation of [8, 3.4,3.10] to Spin(2n).
Proposition 4.1. The abelian group of indecomposables QK1(Spin(2n)) has gener-
ators ,i, i 1, D+, and D-, and relations Tn-1, Tn, Rn+1, . .,.R2n-1, Sj, j *
*2n
defined by:
X X i 2nj
Tn-1 : 2n-1(D+ + D-) + (-1)k n-j-k,k;
j oddk
j 1
X i 2nj X X i 2nj
Tn : (-1)k n-k ,k + 2 (-1)j (-1)k n-j-k,k;
k j 1 k
X i 2nj X i 2n j
Rj : (-1)k j-k ,k - (-1)k 2n-j-k,k;
k k
X ijj
Sj : (-1)k k ,k.
k
Adams operations satisfy _t,k = ,ktfor t > 0, _-1,k = -,k, and
_t(D+ - D-) = tn-1(D+ - D-). (4.2)
Each relation with subscript j expresses ,j in terms of ,iwith i < j and D . *
*Thus
QK1(Spin(2n)) is a free abelian group with basis ,1, . .,.,n-2, D+, D-. Formula*
*s for
_t(D ) will be obtained in the proof of 4.9.
Proof of Proposition 4.1.Naylor ([37, p.151]) describes the use of Hodgkin's th*
*eorem
([29]) and the representation ring R(Spin(2n)) to determine QK1(Spin(2n)). Unfo*
*r-
tunately, his description contains many typographical errors. Husemoller ([30, *
*p.189])
has correct versions of the results about R(Spin(2n)), and proofs.
Similarly to [8, 3.1], the morphism j* : R(SU(2n)) ! R(Spin(2n)) satisfies
j*(~i) = j*(~2n-i), (4.3)
where ~iis the ith exterior power of the canonical representation. Then R(Spin(*
*2n))
has fundamental representations j*~1, . .,.j*~n-2, +, - with relations
+ - = j*~n-1+ j*~n-3+ j*~n-5+ . . .(4.4)
+ + + - - = j*~n + 2(j*~n-2+ j*~n-4+ . .)..(4.5)
PERIODIC HOMOTOPY GROUPS OF SO(N) 27
Hodgkin associates to each fundamental representation ` of G a primitive elem*
*ent
fi(`) of K-1(G). We denote by efi(`) the element of K1(G) which corresponds to *
*this
under Bott periodicity. Then efi(` ø) = dim(`)fei(ø)+dim (ø)fei(`). In QK1(Spin*
*(2n)),
we let D = efi( ) and Bi= efi(j*~i). We obtain from (4.4) and (4.5),
2n-1(D+ + D-) = Bn-1+ Bn-3+ Bn-5+ . . . (4.6)
2n(D+ + D-) = Bn + 2(Bn-2+ Bn-4+ . .).. (4.7)
Under the isomorphism QK1(SU(2n)) fK0(CP 2n-1), let ,0icorrespond to ,i- 1,
with , the Hopf bundle, and let ,i= j*(,0i) 2 QK1(Spin(2n)). From [8, 3.2], we *
*have
P k+1i2nj
Bj = (-1) j-k,k. Now Tn-1 follows from (4.6), Tn is obtained from (4.7) and
(4.6), Rifollows from (4.3), and Sj is a consequence of (, - 1)j = 0 in fK0(CP *
*2n-1)
for j 2n.
The formula for _t,k follows from _t, = ,t in fK0(CP 2n-1). The formula for _*
*-1,k
follows from [8, 3.17]. The formula for _t(D+ - D-) follows from the short exa*
*ct
sequence
p* 1 i* 1
0 ! QK1(S2n-1) -! QK (Spin(2n)) -! QK (Spin(2n - 1)) ! 0
(4.8)
with p*(gen) = D+ - D-. ||
From Proposition 4.1 and [8, 3.18], we deduce the following result.
28 BENDERSKY AND DAVIS
Proposition 4.9. The Pontryagin dual of the abelian group E1,2m+1
2 (Spin(2n)) is
generated by ,1, D+, and D- subject to the following relations:
iX ijjm j
kk ,1, j 2n; (4.10)
ik odd
X ii 2nj i2njj m j
j-k - j+k k ,1, n + 1 j 2n - 1; (4.11)
ik odd i jj
1_X m+1 2n
n k n-k ,1; (4.12)
k odd
(1 + (-1)m ),1; (4.13)
i X i j j
2n-1(D+ + D-) - n2n-j-kkm ,1; (4.14)
kjoddodd 1
2n-1(D+ - D-); (4.15)
(3n-1- 3m )(D+ - D-); (4.16)
iX X i jj
km n-2n2-k-4t,1- 2n-1D+; (4.17)
k odd t 0
i X X i jj
- 2n-1 km n2n-1-2-k-3t,1+ (1_6(22n-1+ 1 + 3n) - 3m )D+
k odd t 0
+1_6(22n-1+ 1 - 3n)D-; (4.18)
(1 + (-1)m )D if n even; (4.19)
D+ + (-1)m D- if n odd. (4.20)
Proof.By [8, 1.1], E1,2m+12(Spin(2n))# is the quotient of QK1(Spin(2n)) by the *
*image
of _2 and _r- rm for all odd r, although it suffices to consider just r = -1 an*
*d 3. As
in [8, 3.15], _2,m = ,2m allows us to remove all ,ev, and _r,1 = ,r allows us t*
*o equate
all ,r with r odd to rm ,1. This reduces the generating set to ,1, D+, and D-, *
*and
the relations of Proposition 4.1 become (4.10), (4.11), (4.12), and (4.14). For*
* (4.12),
we first obtain
iX ii j X i jjj
km 2nn-k+ 2 (-1)j n2n-j-k,1,
k odd j 1
i j i j i j
but then when each 2niin this expression is expressed as 2n-1i+ 2n-1i-1, all*
* terms
in the alternating sum cancel out except
X `i2n-1j i2n-1j' X i 2nj
km n-k - n-k-1 = 1_n km+1 n-k .
k k
PERIODIC HOMOTOPY GROUPS OF SO(N) 29
The relation (4.13) is from _-1,1, and (4.15) and (4.16) are from _t(D+ - D-) as
given in 4.1.
P 1
Suppose that we know _t(D) = bj,j+ cD in QK (Spin(2n - 1)). We claim that
it follows that
X
_t(D+ + D-) = 2 bj,j+ c(D+ + D-)
in QK1(Spin(2n)). By adding and subtracting this with (4.2), we obtain
X
_t(D ) = bj,j+ 1_2(c + tn-1)D + 1_2(c - tn-1)D .
(4.21)
Using this, (4.17) and (4.18) follow from [8, (3.20),(3.21)].
The above "claim" follows from the short exact sequence (4.8) in which i*(D )*
* = D,
once we observe that D+-D- cannot appear in _t(D++D-). This follows by working
in R(Spin(2n)). Recall that _t in QK1(G) corresponds to (-1)t+1~t in I=I2, where
I is the augmentation ideal of the representation ring. The representation rin*
*g of
the maximal torus is Z[ff11=2, . .,.ffn1=2], and here all the ~i(æ)'s and + + *
* - are
invariant under ff 7! ff-1, while + - - is not. (See [30].) So the exterior p*
*owers
of these invariant classes will also be invariant, and hence cannot contain + *
*- -
as a summand.
Finally, (4.19) follows from _-1 = -1 in QK1(Spin(2n)) if n is even, which fo*
*llows
from (4.8) and [8, 3.17], and (4.20) follows from _-1(D ) = -D in QK1(Spin(2n))
if n is odd, which is a consequence of (4.21) with c = -1. ||
Now we can prove Theorem 3.3.
Proof of Theorem 3.3.We begin by observing that all coefficients in relations (*
*4.10)
through (4.19) are even, with the single exception of (4.17) when n = 3, which
accounts for the anomaly for Spin(6) mentioned following Theorem 3.3. Although*
*ij
sometimes a bit of argument is required, these all follow from the facts that *
*abis
i j i j P i j
even if a is even and b odd, that 2a2b ab mod 2, that i ni= 2n, and that
inj i nj
i = n-i. P P i *
* j
For example, the coefficient of ,1 in (4.17) is congruent to k oddt 0 n-22n*
*-k-4t.
If n is even, then all terms are even. If n = 2a + 1, then writing k = 2b + 1,
P P i 2a+1 j
the sum is congruent to b 0 t 0ca,b,twith ca,b,t= a-1-b-2t. If a is even, t*
*hen
30 BENDERSKY AND DAVIS
ca,2b0,t ca,2b0+1,t, and hence the sum is even. If a is odd, then ca,2b0,t ca*
*,2b0-1,tfor
b0 > 0, c2d+1,0,d-2A c2d+1,0,d-2A-1,iandjc2d+1,0,0is even unless d = 0. This *
*is the
anomaly for Spin(6)_that 2ddis even unless d = 0.
Now the case m even and n even of 3.3, that E1,2m+12(Spin(2n)) Z2 Z2 Z2,
with generators ,1, D+, and D-, is immediate, since (4.13) and (4.19) give these
relations, and there are no relations involving odd multiples of these generato*
*rs.
The case m even and n odd, that E1,2m+12(Spin(2n)) Z2 Z=2min(n-1, (m+1-n)+2)
with generators ,1 and D+, is also easy. We use here, and throughout, that
8
<1 if e odd
(3e- 1) = :
(e) + 2 if e even.
The Z2 comes from (4.13). Replace 2,1 by 0, and use (4.20) to replace D- by -D+.
The other relations reduce to 2n-1D+ from (4.17) and -3n-1(3m+1-n - 1)D+ from
(4.18).
When m is odd, (4.13) gives no information, and so the analysis becomes more
complicated. From (4.10), we have 2eSp(m,n),1 = 0. We shall prove in Lemma 8.3 *
*that
(4.11) gives no additional information. For Spin(2n + 1), we proved the analog*
*ous
result in [8, 3.6] using topology (two ways of computing E1,2m+12(Sp(n))). For *
*Spin(2n)
it seems that we must resort to combinatorics.
We also need the following estimate, which we prove at the beginning of Secti*
*on 8.
Lemma 4.22. The expressions of Theorem 3.3 satisfy, if m is odd and n 6,
min(eSp(m, n), (P1(m, n)), (P3(m, n)) n.
We now obtain 3.3 in the case when m is odd and n is odd. The generators will
be ,1 and D+ + c,1 for appropriate c. The relations (4.10) and (4.11) give 2eSp*
*(m,n),1,
as just explained, while (4.12) gives 2 (P3(m,n)),1. The relation (4.20) allow*
*s us to
replace D- by D+. The other relations become
2nD+ - Y1,1 (4.23)
Y2,1- 2n-1D+ (4.24)
-3 . 2n-1Y3,1+ (22n-1+ 1 - 3m+1)D+, (4.25)
P
where Y1, Y2, and Y3 refer to the sums (just the -part) in (4.14), (4.17), an*
*d (4.18).
Replacing (4.23) by (4.23)+2(4.24) yields P1(m, n),1.
PERIODIC HOMOTOPY GROUPS OF SO(N) 31
It was proved in [8, 3.18] that5
(Y2) n forn 6. (4.26)
Thus the smallest 2-exponent in (4.24) and (4.25) is M := min(n - 1, (m + 1) +*
* 2),
and by 4.22, this is smaller than the exponents which have occurred earlier in *
*the
analysis. The summand with generator D+ + c,1 is obtained by dividing whichever
of (4.24) or (4.25) has the smallest exponent by 2M , so that its coefficient o*
*f D+ is
odd. Subtracting an appropriate multiple of this relation from the other yields*
* the
final relation, P2(m, n),1.
Finally, we consider the case m odd and n even. We obtain 2eSp(m,n),1 and
2 (P3(m,n)),1 as in the previous case. Let M = min(n - 1, (m + 1 - n) + 2), an*
*d let
Y1, Y2, and Y3 be as above. By (8.2), (Y1) n, while Y3 is odd by 8.11. We al*
*so use
(4.26). The relations are
2n-1(D+ + D-) - Y1,1, (4.27)
2M (D+ - D-), (4.28)
Y2,1- 2n-1D+, (4.29)
-3 . 2n-1Y3,1+ (22n-2+ 1_2(3n - 1) - (3m+1 - 1))D+
+(22n-2- 1_2(3n - 1))D-. (4.30)
Replace (4.27) by (4.27)+2n-1-M (4.28)+2(4.29) to get P1(m, n),1. Now we divide
into subcases.
If (m + 1) < (n), the smallest 2-exponent in any coefficient is (m + 1) + *
*2 in
(4.28). This gives a summand Z=2 (m+1)+2generated by D+ -D-. Use this to replace
2 (m+1)+2D- by 2 (m+1)+2D+ in (4.30), which becomes
- 3 . 2n-1Y3,1+ (22n-1- 3m+1 + 1)D+. (4.31)
We get a second Z=2 (m+1)+2summand from (4.31)=2 (m+1)+2. Add a multiple of
(4.31) to (4.29) to get the final relation, P2(m, n),1. The generators of the r*
*espective
summands in 3.3 are ,1, D+ + c,1, and D+ - D-.
If (m + 1) (n), the smallest 2-exponent in any coefficient is (n) + 1 in*
* D- (or
D+) in (4.30). We get a summand Z=2 (n)+1with generator (4.30)=2 (n)+1. Replace
__________
5There it was stated and proved that (Y2) n - 1, but the same argument
establishes this stronger result, which we will need.
32 BENDERSKY AND DAVIS
(4.28) by
(22n-2- 1_2(3n - 1))2- (n)-1(4.28) + 2M- (n)-1(4.30),
obtaining
- 3 . 2n-1+M- (n)-1Y3,1+ 2M- (n)-1(22n-1- 3m+1 + 1)D+.
(4.32)
The smallest 2-exponent in the remaining coefficients ((4.29) or (4.32)) is
8
(n),
with the (-) + 3 coming from D+ in (4.32), and n - 1 coming from D+ in (4.29).
We get a summand of this 2-exponent generated by the relevant relation divided *
*by
its 2-power. Add a multiple of this relation to the other to get P20(m, n),1. *
* The
respective generators in 3.3 in this case are ,1, D+ + c,1, and D- + uD+ + c0,1.
That R(m, 2n) = n - 1 when n - 1 (m + 1) + 2 and n is odd follows as in
Remark 3.2. If n is even and (m + 1) n - 3, then M = (n) + 2 and in the abo*
*ve
argument the relation P20(m, n) has its 2-divisibility determined by 3 . 2nY3, *
*which
equals a unit times 2n by 8.11. ||
Here is a sketch of proof that was postponed in Section 3.
Proof of Proposition 3.17.These groups are Pontryagin dual to the groups that we
j0 j1
computed as quotients of the K-groups. For Spin(2b) -! Spin(2b + 1) -! Spin(2*
*b +
2), we have j*1(D ) = D and j*0(D) = D+ + D-. We focus on the hardest case, i4
when 2a > (m + 1) (n). Here n = 2a + 2.
We have = (m + 1) + 2, e = (n) + 1, and ffl = (m + 1 - n) + 1 - (n). The
image of the generator of Z=2 under i#4is
D+ + D- + c,1 2 Z=2 +ffl Z=2e,
where the respective generators of these summands are g1 = D+ + c0,1 and g2 =
D+ -D- +2ffD+ +c00,1 with ff = (22n-1-3m+1 +1)=(-22n-1+3n-1). All coefficients
of ,1 are sufficiently 2-divisible that they can be ignored. We obtain
i#4(gen) = -g2+ (2ff + 2)g1.
PERIODIC HOMOTOPY GROUPS OF SO(N) 33
Dualizing, this says that the second summand injects under i4, while the kernel*
* of
the morphism from the first summand has 2-exponent 1 + (ff + 1) = 2 + (m + 1 -
n) - (n) = ffl + 1 and hence this morphism maps onto multiples of 2. ||
The cases of Spin(6) and Spin(10) were omitted from 3.3 because of arithmetic
anomalies. We handle those cases now.
Proposition 4.33.
8
>Z=8 Z=2min( (m-5)+2,6)if m 1 mod 4
>: min( (m-7)+2,8)
Z=8 Z=2 if m 3 mod 4.
Proof.We first consider Spin(6). Proposition 4.9 is still valid. As remarked pr*
*eviously,
the subsequent argument fails because the coefficient of ,1in (4.17) is odd whe*
*n n = 3.
Use (4.17) to replace ,1 by 4D+, and use (4.20) to replace D- by (-1)m+1D+. The
smallest resulting relation on D+ is 8D+ from (4.13) if m is even, while if m i*
*s odd,
we have 16D+ from (4.14) and 1_3(3m+1 - 1 + 16)D+ from (4.18), yielding the res*
*ult.
For Spin(10), the case m even follows as in Theorem 3.3. The anomaly is the s*
*ame
as occurred for Spin(9) in [8, 4.21]. Here it occurs as (4.26), where Y2 = 45. *
*Since
(4.20) makes D+ = D-, the relations become the same as in Spin(9). ||
We close the section with another postponed proof.
Proof of Proposition 3.19.We compute the duals of the morphisms, using 4.33, 3.*
*1,
and 3.3 for the groups. We use ,1 and D as the generators of E2m(7)#. They map *
*to
,1 and D+ + D- in E2m(6)#, which is generated by D+ and has relations ,1 = 4D+
and D- = D+.
With u = (3m - 3)=8, we use ,1 and D + (8 + 3u),1 as generators of E2m(11)# a*
*nd
j#4 #
(6+u),1+2D and D as generators of E2m(9)#. The morphism E2m(11)# -! E2m(9)
is easily determined using j#4(D) = 2D. At the end, we need (u + 1) = 1. ||
34 BENDERSKY AND DAVIS
5.Eta towers
In this section, we compute the groups Es,t2(Spin(n)) for s > 2 (and s = 2 if*
* s + n
is even). These are called eta-towers because of the isomorphism, Es,t2h1-!Es+1*
*,t+22,
which on classes which survive to homotopy is related to composition with the H*
*opf
'' t-s
map St-s+1-! S . Notation for the eta-towers was initiated in 1.5. We begin *
*with
the closely-related computation for Sp(n).
We first recall the following key results from [9, 1.1,3.1].
Theorem 5.1. ([9]) a. If X is a simply-connected finite H-space with H*(X; Q)
associative, then the E2 term of its BTSS satisfies
8
2, and
0 ! coker(`|M2) ! Exts,2b+1A(M)# ! ker(`|M=2) ! 0
if s + b is even and s > 1.
These hypotheses apply to all cases considered here except Spin(4a + 2), where
_-1 6= -1; in (5.17) we will present a modified version of 5.1b which will appl*
*y in
that case.
The following result from [9, 3.1] is also useful.
Proposition 5.2. If M is a finite stable 2-adic Adams module with _-1 = -1 and
n is odd, there is a split short exact sequence
0 ! coker(_3 - 1|M=2) ! Ext2,2n+1A(M)# ! ker((_3 - 3n)|M) ! 0.
Also, |ker((_3 - 3n)|M)| = | Ext1,2n+1A(M)|.
An analogue when it is not true that _-1 = -1 is given in [9, 3.10], and is s*
*imilar
to (5.17).
Because of the following elementary proposition, the functors of Theorem 5.1.b
depend only on QK1(X; Z=2).
PERIODIC HOMOTOPY GROUPS OF SO(N) 35
Proposition 5.3. Let Q be a torsion-free 2-adic Adams module with _2 injective
(viz. Q = QK1(X; Z^2) with X as above), and let M = Q= im(_2). There is an
isomorphism of stable Adams modules
1_2 2
2_ : ker(_ |Q=2) ! M2.
Proof.Let K = ker(_2|Q=2), and apply the Snake Lemma to the commutative dia-
gram
0 ---! K
?? ?
?y ??y
0 ---! Q --2-! Q ---! Q=2 ---! 0
?? ?? ??
_2?y _2?y _2?y
0 ---! Q --2-! Q ---! Q=2 ---! 0
?? ?
?y ??y
M --2-! M
||
We have the following result for QK1(Sp(n); Z=2).
Proposition 5.4. Let ,i= ,i- 1 be generators for QK1(Sp(n); Z=2) as used in [8,
3.4]. Let
X iij
xi= j,i-2j.
j 0
Then {x1, . .,.xn} is a basis for QK1(Sp(n); Z=2) which satisfies
oIf i : Sp(n - 1) ! Sp(n) denotes the inclusion map, then
i*(xn) = 0 and i*(xj) = xj for j < n.
o_2(xi) = x2i, and is 0 if 2i > n.
P iij
o_3(xi) = j 0j xi+2j.
Proof.Since it was shown in [8, 3.4] that {,1, . .,.,n} forms a basis, it is im*
*mediate
that {x1, . .,.xn} does as well. To prove the result about i*(xn), we observe f*
*rom [8,
3.4] that
X ` i2n-1j i2n-1j' X i2nj
i*(,n) = n-k + n-1-k ,k = n-k,k.
k0 j>0 j>0
Since _2(,i) = ,2i, we have
X iij X i2ij X i2ij
_2(xi) = j,2i-4j= 2j,2i-4j= k ,2i-2k= x2i.
j 0 j 0 k 0
Here we have used several elementary facts about binomial coefficients mod 2. T*
*he
proof of the _3 formula involves a more elaborate combinatorial argument, which*
* is
relegated to Proposition 8.6. ||
Remark 5.5. Because of the elaborate computer-dependent proof of the _3-formula,
some may prefer the following simpler argument. This argument just proves
_3(xi) xi+ xi+2 (i)+1mod i + 2 (i)+1>,
but that is all that we really need.
First, by using _2_3 = _3_2 and the formula for _2(xi), it suffices to prove *
*that if
i is odd, then _3(xn) = xn+ xn+2 in QK1(Sp(n + 2)=Sp(n - 1)); Z=2). The definit*
*ion
of xiand formula for _3(,i) easily imply that _3xican only involve xj with j *
*i mod
2. Thus, if the claimed formula is not true, then _3-1 = 0 in QK1(Sp(n+2)=Sp(n-
1)); Z=2). It was shown in [11, 2.7] that _3 - 1 6= 0 in ku*(Qn+2n) Z=2, henc*
*e in
P K*(Sp(n + 2)=Sp(n - 1); Z=2), and hence in QK1(Sp(n + 2)=Sp(n - 1); Z=2) by
duality.
Using 5.4, we easily derive the following description of the eta-towers for S*
*p(n).
Proposition 5.6. For X = Sp(n), the split SES's of 5.1b become
0 ! ! Es,2b+12(Sp(n))# ! K[[n_2] + 1, n] ! 0
if s + b is odd, and
0 ! C[[n_2] + 1, n] ! Es,2b+12(Sp(n))# ! ! 0
if s + b is even. Here n* is the largest odd integer satisfying n* n,
K[a, b] = b>, (5.7)
PERIODIC HOMOTOPY GROUPS OF SO(N) 37
and
C[a, b] = . (5.8)
Proof.With M = QK1(Sp(n); Z=2)= im(_2) in 5.1b, we have, using 5.4, M=2 = and `(xi) xi+2mod H (which we will use to mean mod terms
with larger subscripts). Thus coker(`|M=2) and ker(`|M=2) = .
Let Sn = {i : [n_2] + 1 i n} and, for e 0, let Sn(e) = {i 2 Sn : (i)*
* = e}.
By 5.3, M2 . Let M2(e) = . Then ` induces
automorphisms of each M2(e) given by `(x2eu) x2e(u+2)mod H, with u odd, and
so, similarly to the previous paragraph
ker(`|M2)
and
coker(`|M2) .
||
This result is, for all intents and purposes, dual to [11, 1.8]. We illustra*
*te with
the case n = 10, which is depicted in [11, 1.13]. Columns 2 and 4 (resp. 3) in *
*[11,
1.13] correspond to Es,2b+12(Sp(10)) with s + b odd (resp. even), as can be see*
*n by
comparison with [11, 1.7]. The boundary pattern from u to (u - 2)0in [11, 1.13]*
* is
dual to our formula for ` mod H in M=2, and the elements 1 and 90 that survive
correspond to our and . Note that the computations should be dual, si*
*nce
[11] is depicting E2, while we are computing E#2.
The comparison of our `|M2 with the boundaries in [11, 1.13] is complicated s*
*lightly
by different ways of filtering elements in the two approaches. The two approac*
*hes
would agree if the elements in 5u and 60in [11, 1.13] were interchanged, and al*
*so the
elements 7u and 100. If this change were made, then the bottom part of columns 2
38 BENDERSKY AND DAVIS
and 3 of [11, 1.13] would be
60u 10u
70u 9u
80u 6u
90u 7u
100u 8u,
which is dual to our `|M2.
As in [11, 1.14], we conclude that the number of eta-towers in Es,2b+12(Sp(n)*
*) is
1 + [log2(4n=3)], for either parity of s + b.
Now we perform a similar analysis for Spin(2n + 1).
Proposition 5.9. QK1(Spin(2n + 1); Z=2) has basis {x1, . .,.xn-1, D}, with _2(x*
*i)
and _3(xi) as in QK1(Sp(n - 1); Z=2), _2(D) = xn-1, and _3(D) = D.
Proof.By [8, pf of 3.1], there is an Adams-module morphism
ffi 1
QK1(Sp(n)) -! QK (Spin(2n + 1)),
and by [8, 3.10] QK1(Spin(2n + 1)) has integral basis {,1, . .,.,n-1, D}, where*
* ,i=
OE(,i) and
n-1X n-kXi j
2n+1D = (-1)n+1OE(,n) + (-1)k+1,k 2n+1j.
k=1 j=0 (5.10)
Reduce mod 2 and change to the basis {xi} of 5.4. The mod 2 reduction of (5.10)*
* is
X inj
0 = OE(,n) + ,n-2j j;
j>0
P n 1
i.e., OE(xn) = 0. Thus if, for i < n, _k(xi) = j=1ffjxj in QK (Sp(n); Z=2), w*
*here
P n-1
coefficients ffj are as in 5.4, then, applying OE, we obtain _k(xi) = j=1ffjx*
*j in
QK1(Spin(2n + 1); Z=2), which is exactly the formula in QK1(Sp(n - 1); Z=2).
By [8, (3.20)] we obtain the integral formula
X X i 2n+2 j
_2(D) = (-1)k,k n-1-k-4t+ 2nD. (5.11)
k ! Es,2b+12(Spin(2n + 1))# ! K[[n_2], n - 1] ! 0
if s + b is odd, and
0 ! C[[n_2], n - 1] ! Es,2b+12(Spin(2n + 1))# ! ! 0
if s + b is even. Here n** is the largest odd integer satisfying n** < n - 1, *
*while
K[[n_2], n - 1] and C[[n_2], n - 1] are as in (5.7) and (5.8). If n = 2a + 1 is*
* odd, then xa
should be replaced by xa+ D in C[[n_2], n - 1]. If n = 2e+ 1, then xa should be*
* replaced
by xa + D in K[[n_2], n - 1].
The number of eta-towers in Es,2b+12(Spin(2n+1)) is 2+[log2(4(n-1)=3)]+ffiff(*
*n-1),1,
where ffi is the Kronecker delta and ff(m) denotes the number of 1's in the bin*
*ary
expansion of m.
Proof.With M = QK1(Spin(2n+1); Z=2)= im(_2) in 5.1b, we have, using 5.9, M=2 =
and `(xi) = xi+2and `(D) = 0. Thus coker(`|M=2)
and ker(`|M=2) = .
Let Sn = {i : [n_2] i n - 1} and, for e 0, let Sn(e) = {i 2 Sn : (i) *
*= e}.
By 5.3, M2 , where x0i= xi unless n = 2a + 1 and i = a, in which
case x0i= xi+ D. Let M2(e) = . Then ` induces automorphisms of
each M2(e) given by `(x02eu) = x02e(u+2)mod H, and so the result follows simila*
*rly to
40 BENDERSKY AND DAVIS
the proof of 5.6. If n = 2a + 1, then a is maximal in some Sn(e) if and only if*
* a is a
2-power.
One way to make the asserted count of the number of eta-towers is by comparis*
*on
with Sp(n - 1). The number of eta-towers is 2 plus the number of values of e wh*
*ich
occur as (i) for some i 2 [[n_2], n - 1], whereas it was shown in [11, 1.14] t*
*hat the
number of values of (i) in [[n-1_2] + 1, n - 1] is [log2(4(n - 1)=3)]. The two*
* intervals
are the same if n is even, while if n = 2a + 1, the interval considered here co*
*ntains a
as an additional element. This a will give an additional value of (-) if and o*
*nly if
it is a 2-power. ||
The above result can also be deduced from 11.3.
The following result will be useful for Spin(4a), in which _-1 = -1, but less*
* useful
for Spin(4a + 2).
Proposition 5.15. QK1(Spin(2n); Z=2) has basis {x1, .8.,.xn-2, D+, D-} with _2xi
2 (or s = 2 and b even), then there is a
short exact sequence
0 ! Z2 Z2 ! Es,2b+12(Spin(2n))# ! Es,2b+12(Spin(2n - 1))# ! 0,
where the two classes in the kernel are both D+ + D-, one in the ker-part and o*
*ne in
the coker-part in 5.1b.
Proof.One way of interpreting this result is to first note that the SES (4.8) r*
*emains
short exact after modding out by im(_2), and hence, by 5.1a, induces an exact s*
*e-
quence
! Es,2b+12(S2n-1)# ! Es,2b+12(Spin(2n))# ! Es,2b+12(Spin(2n - 1))#
-ffi! Es+1,2b+12n-1 #
2 (S ) ! .
PERIODIC HOMOTOPY GROUPS OF SO(N) 41
This exactness also follows from 2.2 and 2.3. We are, in effect, claiming that *
*ffi = 0
when n is even, so that this sequence is short exact.
The direct computation for Spin(2n) is extremely similar to the proof of 5.14,
performed for Spin(2n-1). The M=2-part, comprising the first paragraph, is chan*
*ged
only by replacing D by both D+ and D-. The M2-part, comprising the second
paragraph, has an extra D+ + D- in M2, and it appears in both ker(`) and coker(*
*`).
||
For Spin(2n) with n odd, _-1 6= -1 and so 5.1b does not apply. The generaliza*
*tion
is given by [9, 3.8], which states that there is a short exact sequence
0 ! coker(`b|Qs+b) ! Es,2b+12(Spin(2n))# ! ker(`b|Qs+b-1) ! 0,
(5.17)
where
ker((1 - (-1)m _-1)|M)
Qm = _____________________
im ((1 + (-1)m _-1)|M)
with M = QK1(Spin(2n))= im(_2), and `b = _3 - 3b. For this, we need more than
just mod-2 K-theory.
We use integral classes __xiwhich reduce mod 2 to the classes xiof Propositio*
*n 5.4,
and satisfy the same restriction formula as xi. By [8, 3.4], these classes mus*
*t be
defined by
`i j i j'
__x i-1X j 2i-1 2i-1
i= (-1) ,i-j j - j-1 . (5.18)
j=0
Proposition 5.19. If n is odd, QK1(Spin(2n)) has basis {__x1, . .,.__xn-2, D+, *
*D-} with
_-1(__xi) = -__xi, _-1(D ) = -D ,
8
n-2X < odd j = 2i
_2(__xi)= ffi,j_xj+ fij2n-1(D+ + D-) withffi,j
j=1 : even j 6= 2i
8
n-2X < = -1 j = n - 2
_2(D+) = fli,j_xj+ 2n-1D+ withfli,j
j=1 : even j 6= n - 2
_2(D+ - D-) = 2n-1(D+ - D-). (5.20)
Proof.The basis was derived in 4.1 and 4.9, and (5.20) is just (4.2). The mod 2
reduction of the coefficients is immediate from 5.15. That the D -part of _2(_*
*_xi)
involves only D+ + D- is in the proof of 4.9. That the coefficient of D+ + D- *
*in
42 BENDERSKY AND DAVIS
_2(__xi) is divisible by 2n-1 is a consequence of (4.8) and the fact that the c*
*oefficient
of D in _2(,i) in QK1(Spin(2n - 1)) is divisible by 2n. This äf ct" follows fro*
*m [8,
3.10], which says that in QK1(Spin(2n - 1)) we have
n-2X
,n-1 = (-2)nD + cj,j. (5.21)
j=1
The algorithm for _2,ibegins by expressing it as ,2i, and if 2i > n - 1, then r*
*elations
identical to Sj of Proposition 4.1 are used to express each ,j in terms of lowe*
*r ,'s
until it gets down to ,1, . .,.,n-1, and then (5.21) is used to eliminate ,n-1,*
* obtaining
a coefficient of D divisible by 2n.
That the coefficients of D+ and D- in _2(D+) are 2n-1 and 0, respectively, was
derived in (4.17). Finally, that the coefficient of __xn-2is -1 follows from (5*
*.11) and
the argument in the proof of 4.9. ||
Now we can obtain our final result enumerating eta-towers.
Proposition 5.22. If n is odd and s > 2, then the morphism
i#* s,2b+1 #
Es,2b+12(Spin(2n))# -! E2 (Spin(2n - 1))
satisfies 8
< if s + b even
ker(i#*) = :
0 if s + b odd
and 8
< if s + b even
coker(i#*) :
if s + b odd.
Thus, when n is odd, the number of eta-towers in Spin(2n) is 1 less than that*
* in
Spin(2n - 1), which was determined in 5.14.
Proof.We use (5.17) and begin by determining the groups Qm . We obtain that Qod
has generators __xifor 1 i n - 2, D+ + D-, and 2n-1D+ with relations 2__xii*
*f i is
odd, __xiif i is even, D+ + D-, and __xn-2+ 2n-1D+. The quotient Qod is a Z2-ve*
*ctor
space with basis {xi: i odd,1 i n - 4, __xn-2~ 2n-1D+}. Similarly, we find *
*that
Qevis a Z2-vector space with basis {1_2_2(__xi) : [n_2] i n - 2}. We use 5.*
*3 to think
of this _3-module as .
PERIODIC HOMOTOPY GROUPS OF SO(N) 43
Similarly to our previous cases, ` = _3 - 1 satisfies `(__x2eu) __x2e(u+2)m*
*od H, if u
is odd. Thus (5.17) becomes
0 ! ! Es,2b+12(Spin(2n))# ! K[[n_2], n - 2] ! 0
if s + b is odd, and
0 ! C[[n_2], n - 2] ! Es,2b+12(Spin(2n))# ! ! 0
if s + b is even. The morphism i#*of the proposition sends the K[-, -] and C[-,*
* -]
parts bijectively, and also x1 maps across. This, with 5.14, yields the claim.*
* Note
that n**becomes n - 4 here. ||
6.d3 on eta towers
Since j4 = 0 in homotopy, d3-differentials must annihilate all eta-towers, ex*
*cept for
a few elements at the bottom of the target tower. In this section, we determine*
* the
d3-differential on the eta towers.
The group ji(X) of eta-towers passing through Es,2(s+i)+12(X) (s > 2) was def*
*ined in
Definition 1.5. Note that d3 is a homomorphism from ji(X) to ji-2(X). As custom*
*ary
with Adams-type spectral sequence charts, we place Es,t2in position (x, y) = (t*
*-s, s),
so that (assuming convergence) ßi(X) has associated graded E*,*+i1. Then ji(X) *
*is a
tower of elements whose position satisfies x - y = 2i + 1.
It will be convenient to classify the eta-towers determined in Section 5 as ü*
* nstable"
or "stable" depending upon whether or not they are of the form _2(x)=2. Thus the
unstable classes in Spin(n) come from M2 if n 6 2 mod 4, and from Qevif n 2
mod 4. We will abbreviate ji(Spin(n)) as ji(n). We tabulate the elements found *
*in
Propositions 5.14, 5.16, and 5.22 in the following table.
44 BENDERSKY AND DAVIS
Table 6.1. This table describes all eta towers.
| j (4a - 2) |j (4a - 1) | j (4a) |j (4a + 1) |
_________|___i____________|_i____________|___i_________|_i________ |
i even, || x2a-3 || x2a-3 || x2a-3 || x2a-3 ||
__stable__||____________||____D_______||D+,_D+_-_D-__||____D_____||_
i even, C||[a - 1, 2a -C3][||a - 1, 2aC-[2]a||- 1, 2aC-[2]a||, 2a - 1] ||
_unstable_||____________||_____________||_D+_-_D-____||___________||
i odd, || x1 || x1 || x1 || x1 ||
__stable__||____________||____D_______||D+,_D+_-_D-__||____D_____||_
i odd, K||[a - 1, 2a - 3]K||[a - 1, 2aK-[2]a||- 1, 2aK-[2]a||, 2a - 1] ||
_unstable_||____________||_____________||_D+_-_D-____||___________||
Now we can state the main theorem of this section.
Theorem 6.2. For the eta towers as described in Table 6.1,
d3 : ji(4a + ffl) ! ji-2(4a + ffl), with - 2 ffl 1,
sends the following eta towers nontrivially to eta towers with the same name.
i even, stable:
x2a-3if i 0 mod 4;
D, D+, D+ - D- if i 2a mod 4;
i even, unstable:
all classes if i 0 mod 4;
i odd, stable:
x1 if i 1 mod 4;
D, D+, D+ - D- if i 2a + 1 mod 4;
i odd, unstable:
all classes if i 3 mod 4.
This theorem will be proved by comparing with known d3's in the BTSS of spher*
*es.
It is immediate from 2.2, 2.3, 2.5, and 2.10.i that there are exact sequences
j* s,t p* s,t 2n-1 ffi s+1,t
! Es,t2(Spin(2n - 1)) -! E2 (Spin(2n)) -! E2 (S ) -! E2 (Spin(2n - 1))*
* !
(6.3)
and
! Es,t+12(S2n) ! Es,t2(Spin(2n)) ! Es,t2(Spin(2n + 1)) ! Es+1,t+12(S2n) !
(6.4)
in which all morphisms respect differentials in the BTSS.
PERIODIC HOMOTOPY GROUPS OF SO(N) 45
The behavior of d3 in the BTSS of the odd spheres is stated in 6.5. In [12], *
*it was
shown that the BTSS for odd spheres agrees with the v1-periodic UNSS, which was
computed in [5] and also described in [7, p.488].
Proposition 6.5. The groups ji(S2n+1) equal Z2 Z2 with one tower beginning in
filtration 1 and called stable, and the other tower beginning in filtration 2 a*
*nd called
unstable. (The lowest class in each tower may have order greater than 2.) The s*
*table
towers map to one another under double suspension, while the unstable towers map
to 0 under double suspension (except perhaps on their lowest class). The differ*
*ential
d3 : ji(S2n+1) ! ji-2(S2n+1) is nonzero in the following cases:
_n__i___type____||i_mod_4
ev ev stable ||n + 2
ev ev unstable ||0
ev od stable ||n + 1
ev od unstable ||1
od ev stable ||n + 1
od ev unstable ||0
od od stable ||n + 2
od od unstable ||3
As for the even spheres, we have the following.
Proposition 6.6. In the EHP sequence
-P! Es,t 2n-1 E s,t+1 2n H s-1,t 4n-1 P s+1,t 2n-1
2 (S ) -! E2 (S ) -! E2 (S ) -! E2 (S )
(6.7)
of [4, 5.4] and [6, 7.1(ii)], the homomorphism P is 0 on eta-towers if n is eve*
*n, while
if n is odd, P sends the stable eta-towers of S4n-1to the unstable eta-towers o*
*f S2n-1,
and sends the unstable eta-towers of S4n-1to 0. The d3-differentials on the eta*
*-towers
of S2n agree with those of S4n-1(excluding the stable ones when n is odd) and S*
*2n-1
(excluding the unstable ones when n is odd) in (6.7).
Proof.We use the determination of v-11ß*(S2n) given in [26]. For a pair of eta *
*towers
in an odd sphere with d3(A) = B, the bottom few elements in the eta-tower B sur*
*vive
to periodic homotopy classes represented from the classical Adams spectral sequ*
*ence
viewpoint utilized in [26] by classes connected by diagonal lines near the top *
*or bottom
of vertical towers such as that pictured below. The ones at the top are stable *
*and the
ones at the bottom are unstable.
46 BENDERSKY AND DAVIS
r rr|
r r||
r|
r||r
rr|
r|
The exact sequence (6.7) is depicted (for S13 ! S14, which is representative *
*of
any odd value of n) on the left side of [26, p.233]. The boundary P from the st*
*able
classes on the large sphere to the unstable classes on the large sphere is appa*
*rent.
Similarly, the left diagram on [26, p.235] shows that the boundary morphism P i*
*s 0
on eta towers when n is even in (6.7). ||
Proof of Theorem 6.2.Case 1: x1.
The classes x1 are, of course, compatible under restriction. They pull back *
*to
Spin(7), and the bottom of the target eta-tower survives to give the Z2's in ß8*
*i(Spin)
and ß8i+1(Spin).6 One way to see the differential in Spin(7) is to use the 2-pr*
*imary
splitting Spin(7) ' G2 x S7 to see that the two eta towers in jod(Spin(7)) which
emanate from filtration 1 both have d3 : ji(Spin(7)) ! ji-2(Spin(7)) nonzero for
i 1 mod 4. The splitting cited here was proved in [35, 9.1], while the claims*
* about
d3 in G2 and S7 were proved, respectively, in [9, 4.8] and references cited jus*
*t before
6.5.
Case 2: D+ - D- in ji(4a).
Dualizing the exact sequence in the proof of 5.16, we obtain that the four fa*
*milies
of eta towers in Spin(4a) dual to D+ - D- map isomorphically to the eta towers *
*of
__________
6This statement seems slightly inconsistent with the result of [27], which st*
*ated
that these Z2's pull back to Spin(6).jThe(causeiof)the apparent discrepancy is *
*that
x1is in the image of jod(Spin(6)) -! odjod(Spin(7)), but it has a different na*
*me in
Spin(6). This seems to be related to the anomaly for the 1-line group of Spin(6)
discussed in Proposition 4.33. To see this, we let Mn = QK1(Spin(n))= im(_2) and
Q(M) = ker(1+_-1)= im(1-_-1) on M. Then the above morphism jod(i) is dual
to Q(M7) ! Q(M6). (Actually it is dual to coker(_3-1) applied to these modules,
but _3- 1 turns out to be 0 here.) We find that Q(M7) and
Q(M6) is the subspace of generated by x1
and 2(D+ - D-), and the morphism sends x1 7! x1 and D 7! D+ + D-. The
generator of Q(M6) is most naturally thought of as 2(D+ - D-), but the relations
make it equivalent to x1.
PERIODIC HOMOTOPY GROUPS OF SO(N) 47
S4a-1. The pattern of d3-differentials on these towers in Spin(4a) must be the *
*same
as in S4a-1, which was given in 6.5, with n of 6.5 replaced by 2a - 1.
Case 3: D in ji(4a + 1).
We use the exact sequence (6.3) with n = 2a + 1 and t = 2b + 1. For either
parity of s + b, the element D 2 Es,2b+12(Spin(4a + 1))# is obtained from Q = k*
*er(1 +
_-1)= im(1-_-1). In one parity, it is as an element of ker((_3-1)|Q), and in th*
*e other
as an element of coker((_3-1)|Q). In either case, ffi# (D) in Es,2b+12(S4a+1) i*
*s obtained
by pulling D back to D+ 2 QK1(Spin(4a + 2))= im(_2), applying 1 + _-1 to that,
obtaining D+-D-, and pulling that back to an element in P K1(S4a+1)= im(_2), wh*
*ich
will be in ker(1-_-1). This element can be chosen to be the generator of P K1(S*
*4a+1).
Thus ffi# (D) = g, and dually we obtain that ffi sends the dual class, that we *
*are also
calling D, to the stable class in E1,2b+12(S4a+1). Thus, for D 2 jb-s-1(Spin(4a*
* + 1)),
d3(D) is nonzero if and only if d3 is nonzero on the stable class of jb-s(S4a+1*
*), and
from 6.5 we see that this happens if b - s - 1 2a or 2a + 1 mod 4.
Case 4: D+ in ji(4a), and D in ji(4a - 1).
The morphisms ji(4a - 1) ! ji(4a) ! ji(4a + 1) are dual to
i*1 1 2 i*0 1 2
P K1(Spin(4a+1))=(2, _2) -! P K (Spin(4a))=(2, _ ) -! P K (Spin(4a-1))=(2, _ ).
These satisfy i*1(D) = D+ + D- and i*0(D ) = D. Thus d3 on the D+-towers in
Spin(4a) agrees with that on the D towers in Spin(4a + 1), and (since D+ + D-
D+-D- mod 2) d3on the D-towers in Spin(4a-1) agrees with that on the (D+-D-)-
towers in Spin(4a).
Case 5: All classes x2a-3.
We use the exact sequence (6.3) with n = 2a - 1 and t = 2b + 1. In QK1(Spin(4*
*a -
2))= im(_2, 1 - _-1), we have x2a-3~ 22a-2D+ ~ p*(22a-3g), using 5.19 and (2.4).
Since 22a-3g generates ker(1 + _-1)|QK1(S4a-3)= im(_2), we deduce that
p*
ker(`|Qod(S4a-3)) -! ker(`|Qod(Spin(4a - 2)))
sends 22a-3g to x2a-3. Then (5.17) says that dually p* : jev(Spin(4a-2)) ! jev(*
*S4a-3)
sends the class we call x2a-3to the unstable class. Now the fact that d3 is non*
*zero
on the unstable class in ji(S4a-3) if i 0 mod 4 implies the same for x2a-3in
48 BENDERSKY AND DAVIS
ji(Spin(4a-2)). That d3 behaves in the same way on x2a-3in Spin(4a-1), Spin(4a),
and Spin(4a + 1) follows by naturality.
Case 6: All elements in C[ff, fi] and K[ff, fi].
Using Proposition 6.6, the sequences (6.4) and (6.7) combine to
! jsti(S4a-3) juni+1(S8a-5) ! ji(Spin(4a -(2))6.8)
ffii,4a-2 st 4a-3 un 8a-5
-! ji(Spin(4a - 1)) ! ji-1(S ) ji (S ) !,
where st and un refer to stable and unstable classes, respectively, and
! ji(S4a-1) ji+1(S8a-1) ! ji(Spin(4a)) (6.9)
ffii,4a 4a-1 8a-1
- ! ji(Spin(4a + 1)) ! ji-1(S ) ji(S ) ! .
The morphisms OEi,4a-fflin the above exact sequences are closely related to n*
*atural
morphisms bCi,4a-ffland cKi,4a-ffldefined using Definitions 5.8 and 5.7.
Definition 6.10. Let C4a-2= C[a - 1, 2a - 3] and C4a = C[a - 1, 2a - 2], and
Cbi,2b: C2b! C2b+2the morphism obtained from ji(2b) ! ji(2b + 2) in Table 6.1.
Make a similar definition with all C's replaced by K's. Note that bCi,2bdoes *
*not
depend upon the value of i, but we will need to keep track of the value of i as*
* it relates
to ji(-). The exact sequences (6.8) and (6.9) can be interpreted as the followi*
*ng short
exact sequences, which preserve d3-differentials. Here dstand dun refer to D+ -*
* D-
in Table 6.1 in stable and unstable boxes. The value of i associated to the ele*
*ments
D and d agrees with that of the accompanying (co)ker(Cbor cK).
0 ! coker(cK2k+1,4a-2) ! jst2k(S4a-3) jun2k+1(S8a-5) ! ker(Cb2k,4a-2) !*
* 0
(6.11)
0 ! coker(Cb2k,4a-2) ! jst2k-1(S4a-3) jun2k(S8a-5) ! ker(cK2k-1,4a-2) !*
* 0
(6.12)
0 ! coker(cK2k+1,4a) ! j2k(S4a-1) j2k+1(S8a-1) ! ker(Cb2k,4a) ! 0
(6.13)
0 ! coker(Cb2k,4a) ! j2k-1(S4a-1) j2k(S8a-1) ! ker(cK2k-1,4a) ! 0
(6.14)
PERIODIC HOMOTOPY GROUPS OF SO(N) 49
The d3-differentials on the eta-towers in the spheres were tabulated in Theor*
*em 6.5.
The d3-differentials on the classes D in (6.11) and (6.12) were established in *
*Case 4,
and the d3-differentials on the classes dstand dunin (6.13) and (6.14) were est*
*ablished
in Case 2. In the short exact sequences (6.11)-(6.14), the classes D and d will*
* have
to match up with classes in spheres with agreeing d3. Then d3 on (co)ker(Cb or *
*cK)
must agree with that on the remaining classes in the spheres.
For example, for the class D in (6.11), d3(D) 6= 0 if 2k + 1 2a + 1 mod 4. *
*Also,
in jst2k(S4a-3), d3 6= 0 if 2k 2a mod 4, while in jun2k+1(S8a-5), d3 6= 0 if *
*2k + 1 3
mod 4. Although this may not always imply that the D-class maps to the stable c*
*lass
on S4a-3, (6.11) does imply that coker(cK2k+1,4a-2) ker(Cb2k,4a-2) does have *
*just one
nonzero element, and d3 is nonzero on it iff 2k + 1 3 mod 4. This yields the *
*first of
the four cases of Proposition 6.15.
As another example, in (6.13) the dstand dun have d3 6= 0 when 2k 2a and 0,
as do the two classes in j2k(S4a-1), while the two in j2k+1(S8a-1) have d3 6= 0*
* when
2k+1 4a+1 and 3; i.e., 2k 0, 2, all mod 4. This yields the third case of 6.*
*15. The
second and fourth cases follow similarly from (6.12) and (6.13), respectively, *
*yielding
the following result.
Proposition 6.15. For ffl = 0 or 2, let zffldenote an element on which d3 is no*
*nzero
if 2k ffl mod 4. Then
coker(cK2k+1,4a-2) ker(Cb2k,4a-2)
coker(Cb2k,4a-2) ker(cK2k-1,4a-2)
coker(cK2k+1,4a) ker(Cb2k,4a)
coker(Cb2k,4a) ker(cK2k-1,4a)
Suppose now that xj 2 C[ff, fi] ji(n). We will show in Proposition 6.16 tha*
*t xj
is in the appropriate C[ff, fi] groups for an interval of values of n. At the b*
*eginning
of that interval, it is in some coker(Cbi,2b), and at the end of the interval, *
*it is in
some ker(Cbi,2b0). We will see in Corollary 6.17 that at least one of these is*
* of the
first, second, or fourth type in Proposition 6.15, for which d3 is determined b*
*y the
proposition, and, indeed, is shown to be nonzero when i 0 mod 4, as claimed.
The same behavior (d3(xj) 6= 0 if i 0 mod 4) for all n in this interval follo*
*ws by
naturality. A similar argument will be performed for elements of K[ff, fi].
50 BENDERSKY AND DAVIS
The following proposition refers to the notation established in Definition 6.*
*10.
Proposition 6.16.
8
<2j + 2 if j is a 2-power
xj 2 K2b iffj + 2 b :
j + 2 (j)+1+ 1if j is not a 2-power.
8
C4j+6 =
Clearly xj is in C4j+4and not in C4j+6.
C4j-2 (j)+3+4=
C4j-2 (j)+3+6=
Clearly xj 62 C4j-2 (j)+3+4, while xj 2 C4j-2 (j)+3+6iff j = 2e+ A . 2e+1with 2*
*A 3,
which is true since j 6= 2e or 3 . 2e. The same sort of argument shows that xj *
*is in
C2bfor intermediate values of b, i.e. between 2j - 2 (j)+2+ 3 and 2j + 2. ||
From this, we can read off the following corollary.
Corollary 6.17. (1)xj 2 coker(cKi,2j+2) if j is even;
(2)xj 2 ker(cKi,2j+6) if j is odd;
(3)xj 2 coker(Cbi,2j+2) if j = 2e or 3 . 2e;
(4)xj 2 coker(Cbi,4j+4-2 (j)+3) if j 6= 2e or 3 . 2e.
Proof.Just use Proposition 6.16. For example, if j 6= 2e or 3 . 2e, then xj 2
C2(2j+3-2 (j)+2)but xj 62 C2(2j+2-2 (j)+2). ||
PERIODIC HOMOTOPY GROUPS OF SO(N) 51
Case 6 now follows from Corollary 6.17 and Proposition 6.15. Indeed, if xj is*
* as in
Corollary 6.17.1, then it is of the first type in Proposition 6.15 with i = 2k *
*+1, and so
d3(xj) 6= 0 iff i 3 mod 4. If xjis as in 6.17.2, then it is of the fourth typ*
*e in 6.15 with
i = 2k - 1, and again d3(xj) 6= 0 iff i 3 mod 4. Since these two cases compri*
*se all
values of j, we find that for all j there exists n such that xj 2 K[ff, fi] j*
*i(Spin(n))
has d3(xj) 6= 0 iff i 3 mod 4, and by naturality this holds for all values of*
* n for
which xj 2 K[ff, fi] ji(Spin(n)).
Similarly, if xj is as in 6.17.3, then it is of the second type in 6.15 with *
*i = 2k, and
so d3(xj) 6= 0 iff i 0 mod 4, while if xj is as in 6.17.4, then it is of the *
*fourth type in
6.15, and the same conclusion follows. Since these two types comprise all value*
*s of j,
we conclude by naturality that whenever xj 2 C[ff, fi] ji(Spin(n)), then d3(x*
*j) 6= 0
iff i 0 mod 4.
This completes the proof of Theorem 6.2. ||
7.Fine tuning
In this section, we determine the d3-differential on the 1-line and most of t*
*he
extensions (exotic multiplication by 2) in the BTSS of Spin(N). For the most pa*
*rt,
we are carrying out proofs of results stated in Section 3. This section also co*
*ntains
an important result, 7.2, regarding computing h1 on the 1-line.
We begin with the case N = 8a 1, where the results were stated in 1.4.
Proof of 1.4.2.Because both d3 : E2,8k+52! E5,8k+72and h1 : E4,8k+52! E5,8k+72a*
*re
bijective, it is equivalent to prove that for Spin(8a 1), h1 : E1,8k+32! E2,8*
*k+52is
nonzero on both summands. We use the isomorphism of E2 with ExtA of 5.1.a.
Letting X = Spin(8a 1) and N = (QK1(X)= im(_2))#, we obtain, similarly to [9,
3.6], a commutative diagram of exact sequences
#
0 ---! E1,8k+32(X)---! N - `--! N
?? ? ? ?
?y h1.??y j2??y j2??y
# 2,8k+5 `# (7.1)
N2 -`--! N2 ---! E2 (X) ---! N=2 - --! N=2
Here N2 = ker(2|N) and ` = _3 - 34k+1. The effect of h1. on elements of E12(X)
corresponding to elements of ker(`# |N) which are not divisible by 2 in N is cl*
*ear
52 BENDERSKY AND DAVIS
from the diagram. However, for other elements we need the following result, whi*
*ch
we prove after completing the proof of 1.4.2. Another approach to this result, *
*with
various extensions of the formula, is given in Theorems 11.5 and 11.18.
Proposition 7.2. In (7.1), if x 2 ker(æ2)\ker(`# ), then the corresponding elem*
*ent of
E12(X) satisfies h1.x = `# (x=2), which is well-defined as an element of coker(*
*`# |N2).
Note that this x=2 is not an element of N2; it is in N.
Now diagram chasing on (7.1) implies that h1. is injective on E1,8k+32(X)=2, *
*as
desired. Indeed, suppose h1 . x = 0. Then x corresponds to an element x 2 N
satisfying `# (x) = 0 and x = 2y. By Proposition 7.2, h1 . x = `# (y) considere*
*d as
an element of coker(`# |N2). Since this is assumed to be 0, we deduce `# (y) = *
*`# (z)
with 2z = 0. Then y - z 2 ker(`# ) and so it pulls back to an element y02 E1,8k*
*+32(X)
satisfying 2y0= x. Hence x = 0 2 E1,8k+32(X)=2. ||
Proof of Proposition 7.2.Using the exact sequence in A of [9, after 3.3],
p
0 ! U(M) -`! U(M) -! M ! 0,
(7.1) is, with M = QK1(X)= im(_2), S = QK1(S8k+3), and S0= QK1(S8k+5),
--`*-!Ext0 ffi 1 p* 1 `*
A (U(M),?S)- --! ExtA(M,?S) ---! ExtA(U(M),?S) ---!
? ? ?
h1?y h1?y h1?y
--`*-!Ext1 0 ffi0 2 0 p* 2 0 `*(7.3)
A(U(M), S )- --! ExtA (M, S )---! ExtA(U(M), S )---!
Here the vertical maps are Yoneda product with an element h1 2 ExtA(S, S0) de-
scribed in [9, 3.6]. Also, U : GInv ! A is left adjoint to the forgetful functo*
*r, where
GInv denotes the category of 2-profinite abelian groups with involution _-1, as*
* in
[9, x3]. The existence of U, its adjointness and exactness, and the fundamental*
* SES
*
* __
above all follow by Pontrjagin duality from the analogous results for the funct*
*or U in
[18, pp 145-6]. Here we use that GInv is Pontrjagin dual to the category of 2-t*
*orsion
abelian groups with involution, and our category A of stable 2-adic Adams modul*
*es
is dual to the category of stable 2-torsion Adams modules, as noted in [15, 10.*
*2]. The
__
properties proved for U restrict to properties on the 2-torsion subcategories w*
*hich
dualize to the properties of U that we need. For example, using results of [9, *
*x3], we
PERIODIC HOMOTOPY GROUPS OF SO(N) 53
have
Ext2A(U(M), S0) Ext2GInv(M, S0) Ext1Inv(Z(+)(2), M# ) M# =2.
If . .!.P1 ! P0 ! M ! 0 is a projective resolution in GInv , then . .!.
U(P1) ! U(P0) ! U(M) ! 0 is a projective resolution in A. This is true because *
*of
the left adjointness and exactness of U.
If . .!.P1 ! P0 ! M ! 0 is a projective resolution in A, then it is also a
projective resolution in GInv. This is true because free objects in A are also *
*free in
GInv , and a module is projective iff it is a direct summand of a free module.
Combining these, we obtain that if . .!.P1 ! P0 ! M ! 0 is a projective
resolution in A, then there is a SES of projective resolutions in A
?? ? ?
?y ??y ??y
p
0 ---! U(P1) --`-! U(P1) ---! P1 ---! 0
?? ? ?
?y ??y ??y
p
0 ---! U(P0) --`-! U(P0) ---! P0 ---! 0
?? ? ?
?y ??y ??y
p
0 ---! U(M) --`-! U(M) ---! M ---! 0
?? ? ?
?y ??y ??y
0 0 0
which yields the exact sequences of (7.3) in the usual way. In particular, the *
*following
derived diagram of SESs will be useful to us.
p* 0 `* 0
0 ---! Hom A(P1, S0)---! Hom A(U(P1), S )---! Hom A (U(P1), S )---! 0
?? ?? ??
d*?y d*?y d*?y
p* 0 `* 0
0 ---! Hom A(P2, S0)---! Hom A(U(P2), S )---! Hom A (U(P2), S )---! 0
x? x? x? (7.4)
h1?? h1?? h1??
p* `*
0 ---! Hom A(P1, S)---! Hom A (U(P1), S)---! Hom A(U(P1), S)---! 0
Let __x2 Ext1A(M, S) in (7.3) map to the given element x 2 ker(`*)\ker(h1), a*
*nd let
_z2 Hom * _
A(P1, S) and z 2 Hom A(U(P1), S) be representative cycles with p (z) = *
*z.
54 BENDERSKY AND DAVIS
Then, by the definition of ffi0, ffi0(y) = h1(__x) in (7.3) if and only if in (*
*7.4) there is
w 2 Hom A(U(P1), S0) such that `*(w) represents y and d*(w) = p*h1(_z). Proposi*
*tion
7.5 below will imply Proposition 7.2, since we have
p*h1_z= h1p*_z= h1z = d*(v(z=2)),
and so w = v(z=2) works and hence ffi0{`*(v(z=2))} = h1__x. ||
Proposition 7.5. If ! F2 -d2!F1 ! F0 ! M ! 0 is part of a free resolution in A,
there are natural isomorphisms
Hom A(F1, S) -v! Hom A(F1, S0)
satisfying d*2(v(z)) = 2h1z.
Proof.A free object of A is of the form N, where is a free object in A on*
* one
generator as in [9, 2.2], and N is a free K*-module. This is a free K*-module*
* on
elements ,k with k odd and positive, with _k,1 = ,k. There are isomorphisms for*
* all
K*-modules N and A-objects L
Hom K*(N, L) -'! Hom A( N, L)
defined by '(OE)(,k n) = _k(OE(n)). Note that '-1 is just given by restricti*
*ng to
,1 N. The morphism v is the composite
-1 j* 0 ' 0
Hom A( N, S) '-! Hom K*(N, S) -! Hom K*(N, S ) -! Hom A( N, S ),
where j : S ! S0 is the identity map of Z^2. Recall that S = QK1(S2m+1) and
S0= QK1(S2m+3), where m = 4k + 1.
Naturality for A-morphisms f : N1 ! N2follows since, for ` 2 Hom K*(N2, S*
*),
both vf* and f*v send '(`) to the element of Hom A( N1, S0) which sends x to
P m+1 P
ciki `(ni) if f(x) = ci,ki ni.
To define h1, we use the resolution of S which begins
0 S -ffl = C0 -d1 = C1
with ffl(,k) = km and d1(,k ,`) = ,k`- `m ,k. Since Ext1A(S, S0) = Z=2, its n*
*onzero
element h1 satisfies 2h1 = ffl0O d1, where ffl0 : ! S0 is the generator, sati*
*sfying
ffl0(,k) = km+1. Note that ffl0(d1(,k ,`)) = km+1`m (` - 1) is even.
PERIODIC HOMOTOPY GROUPS OF SO(N) 55
If ø 2 Hom A(F1, S) is a cocycle, then h1{ø} is the class of h1ø1 in the diag*
*ram
F1? -d2 F2?
? ?
. fi ?yfi0 ?yfi1
S -ffl C0 -d1 C1?
??
yh1
S0
Let F1 = R with R = ker(F0 ! M). Then ø(,k r) = km ø(r) and ø0(,k r) =
ø(r),k. By the definition of v and ffl0, we have
vø(,k r) = km+1ø(r) = ffl0ø0(,k r).
Thus
d*2(v(ø)) = ffl0ø0d2 = ffl0d1ø1 = 2h1ø1 = 2h1{ø},
as desired. ||
The following proof is easier.
Proof of 1.4.1.At first glance, this seems obvious from Diagram 1.3, using the *
*pic-
tured j-action and d3 from the 2-line. However, what we must rule out is that o*
*ne
of the Z2's in E2,8k+32(labeled 1 or D) supports a nonzero d3-differential into*
* the log-
classes. The class x1 is in the image from E1,8k+12(Spin(7)), where it does not*
* support
a differential, and hence d3(x1) = 0 in Spin(8a 1). The D-class in Spin(8a+1) m*
*aps
to 0 in Spin(8a + 3), while the log-classes inject. Thus there can be no differ*
*ential
from D to a log class in Spin(8a + 1).
The D-class and log-classes in Spin(8a - 1) inject into Spin(8a). Then D 7! 0*
* in
Spin(8a + 1) (see 6.1), while all but one of the log classes (x4a-3) map across*
* to log
classes in Spin(8a + 1). The only possible differential involving D in Spin(8a*
*) and
Spin(8a - 1) is to have d3(D) = x4a-3. However, D in E1,8k+12(Spin(8a)) is in *
*the
image from E1,8k+12(S8a-1) in (6.4), and d3 on this class in BTSS(S8a-1) is 0. *
*Hence
the same is true on D in BTSS(Spin(8a)) and BTSS(Spin(8a - 1)). ||
Now we settle the extension questions in the BTSS of Spin(8a 1).
56 BENDERSKY AND DAVIS
Proof of Proposition 1.4.3.The groups C1 in E1,8k+3
2 (Spin(n)) inject as n increas*
*es
and the classes x1 all correspond as n increases from 7. Thus it suffices to v*
*erify
the nontrivial extension in the BTSS of Spin(7). Localized at 2, Spin(7) ' G2x *
*S7.
By [9, 4.8], the BTSS of G2 has a nontrivial extension from filtration 1 to fil*
*tration
3 in dimension 8k + 2, and by [7, p.488], the same is true of S7. Analysis of *
*the
short exact sequence in QK1(-) for the fibration G2 ! Spin(7) ! S7 shows that t*
*he
C1-summand in E1,8k+32(Spin(7)) and the x1-summand in E3,8k+52(Spin(7)) are both
in the image from E2(G2), and so the extension in Spin(7) follows from that in *
*G2.
The extension from the Z=8 to the class D follows from an analysis of
Es,t2(Spin(8a - 1)) -i3!Es,t2(Spin(8a)) -i4!Es,t2(Spin(8a + 1)).
(7.6)
If s = 1, t = 8k + 3, then from 3.17, (7.6) is (ignoring C1-summands)
Z=8 -i3!Z=8 Z=8 -i4!Z=8,
with i3 injecting to the first summand, and i4 sending just the second summand
across. From Table 6.1 and the analysis surrounding it, if s = 3 and t = 8k + 5*
*, (7.6)
is, on stable classes and ignoring the x1-class,
Z2 -i3!Z2 Z2 -i4!Z2
with i3 mapping to the first summand, and i4 sending just the second summand
across. We wish to show that the first (resp. second) summand in E1,8k+32(Spin(*
*8a))
has a nontrivial extension into the first (resp. second) summand of E3,8k+52(Sp*
*in(8a)),
for then the extension in Spin(8a - 1) (resp. Spin(8a + 1)) follows by naturali*
*ty.
For the second summand, we use (6.3) with n = 4a. The summands of concern map
isomorphically to E2(S8a-1), and the extension there is known. (See, e.g., [7, *
*p.488].)
For the first summand, we use (6.4) with n = 4a. The summands of concern are in
the image of Es,t2(S8a-1) ! Es,t+12(S8a) ! Es,t2(Spin(8a)), and again the exten*
*sion in
E2(S8a-1) is known. ||
Now we prove the claims made earlier for Spin(8a).
Proof of Theorem 3.4.Most of the information about the 1-line groups was estab-
lished in 3.1, 3.3, and 3.17. That the initial summand in E1,8k-12(Spin(8a)) ha*
*s the
same order as that of E1,8k-12(Spin(8a - 1)) when (k) < 4a - 5 and (k - a) < *
*4a - 5
PERIODIC HOMOTOPY GROUPS OF SO(N) 57
follows from the fact that, in an exact sequence, e.g., (3.5), the order of a g*
*roup can
be no greater than the product of the orders of the groups on both sides of it.*
* If
(k) 4a - 5, the orders of the initial 1-line summands are known by 3.1 and 3*
*.3.
When (k - a) 4a - 5, na"ive consideration of (3.5) does not allow us to sett*
*le
whether the orders are equal or differ by 1. So we must resort to combinatorics*
* to
determine it. In section 8 we prove the following result, from which it follows*
* that the
1-line morphism is as claimed in this case. Note that k and a in the lemma corr*
*espond
to 2k and 2a in the above discussion.
Lemma 7.7. If (k - a) > 2a - 5, then R(2k - 1, 4a - 1) + 1 = R(2k - 1, 4a) =
((4a - 3)!). Here R(-, -) is as in 3.1 and 3.3.
For the d3-differentials on the 1-line, we use the exact sequence (6.3) with *
*n = 4a.
By 5.16, the eta towers of Spin(8a) are the direct sum of those of Spin(8a - 1)*
* and of
S8a-1. Since the morphisms commute with d3-differentials and extensions, the on*
*ly
thing that we have to worry about is that there could be a d3-differential from*
* a
class in E1,8k+12(Spin(8a)) which maps nontrivially to E1,8k+12(S8a-1) hitting *
*a class in
E4,8k+32(Spin(8a)) in the image from Spin(8a - 1).
In the notation of 6.1, the class in E1,8k+12which concerns us is ffi := D+ -*
* D-. It
maps nontrivially to Spin(8a + 1), but goes to 0 in Spin(8a + 2). Its image und*
*er d3
must map to 0 in E4,8k+32(Spin(8a + 2)). There is one nonzero class which does *
*so,
x4a-32 K[2a - 1, 4a - 2]. If d3(ffi) = x4a-3in BTSS(Spin(8a)), then it must be *
*the
case that h21(ffi) = ffi +x4a-3in E2(Spin(8a)). This is true because the bases *
*of the eta-
towers have been chosen to match up under d3; i.e., d3 : E3,8k+52! E6,8k+72(Spi*
*n(8a))
satisfies d3(ffi) = 0 and d3(x4a-3) = x4a-3. We use h21rather than just h1 in o*
*rder to
get fully into the region of the eta-towers.
Let __xiin QK1(Sp(n)) and QK1(Spin(2n)) be defined as in 5.18. Using 4.1, we
obtain, similarly to 5.9, the following useful result.
Proposition 7.8. There are bases {__x1, . .,.__xn} and {__x1, . .,.__xn-2, D+, *
*D-} of QK1(Sp(n))
and QK1(Spin(2n)), respectively, such that
oUnder the inclusion map Sp(n - 1) ! Sp(n), __xi7! __xiif i < n,
while __xn7! 0;
58 BENDERSKY AND DAVIS
oThere is an A-morphism, QK1(Sp(n)) -ffi!QK1(Spin(2n)) such
P __
that OE(__xi) = __xifor i n-2, OE(__xn-1) = 2n-1(D+ +D-)+ ffixi
P __
with ffieven, and OE(__xn) = fin-1OE(__xn-1) + fiixiwith fiieven.
Dualizing, we obtain a morphism
^ffi
P K1(Spin(2n)) -! P K1(Sp(n))
of K*K-comodules whose mod-2 reduction factors through P K1(Sp(n - 2)) Z2.
Thus there is a morphism
E2(Spin(2n); Z2) ! E2(Sp(n - 2); Z2)
which, when followed into E2(Sp(n); Z2), is the mod 2 reduction of ^OE*. Reduc*
*tion
mod 2 induces a morphism
j
E2(Spin(2n)) -! E2(Spin(2n); Z2),
which sends eta-towers injectively, since they are of order 2.
In our case, n = 4a, the composite E2(Spin(8a)) ! E2(Sp(4a - 2); Z2) on the
eta-towers is K[2a - 1, 4a - 2] ! K[2a, 4a - 2], in the notation of 5.6. In par*
*ticular,
x4a-3is mapped nontrivially. However, the 1-line class ffi defined above maps *
*to 0
in E2(Sp(4a - 2); Z2). Since this morphism respects h1-action, we deduce that h*
*21ffi
cannot equal ffi + x4a-3. ||
Now we prove the results claimed for Spin(8a + 3). The argument works verbatim
for Spin(8a + 5).
Proof of 3.8.Diagram 3.7 is a consequence of 5.14, 6.2, [8, 1.5], and, for the *
*G-groups,
[9, 3.1] and the fact that the kernel and cokernel of a morphism of finite abel*
*ian groups
have the same order. The extension in dimension 8k - 2 is deduced from (6.4) as
follows.
By 3.17, E1,8k-12(Spin(8a + 2)) -i*!E1,8k-12(Spin(8a + 3)) has kernel given b*
*y the
element of order 2 in the C2-summand. The element which hits this class in (6.*
*4)
supports a d3-differential in the BTSS of S8a+2. This can be seen by noting tha*
*t the
element pulls back to S8a+1and the differential there follows from 6.5 (stable *
*class
with n 0 mod 4 and i 3 mod 4). This implies that in the homotopy exact
sequence corresponding to (6.4) the image of the element of ker(i*) is the elem*
*ent of
PERIODIC HOMOTOPY GROUPS OF SO(N) 59
E3,8k+1 4,8k+1 8a+2
2 (Spin(8a + 3)) which maps in (6.4) to the element of E2 (S ) hit by*
* the
d3-differential. By 6.1, this element of E3,8k+12(Spin(8a + 3)) is D.
It remains to determine d3 on E1,8k22+1. This is done similarly to the way it*
* was
done for Spin(8a 1), using the action of h1, but here it is more delicate, be*
*cause
some of the elements in the target of h1 support d3-differentials and others do*
* not. If
g is a generator of a summand of E1,8k22+1, then d3(g) 6= 0 iff h1g equals an e*
*lement
which supports a nonzero differential. Thus this last remaining part of Theorem*
* 3.8
will follow from the following result. ||
Proposition 7.9. For Spin(8a + 3), in h1 : E1,8k-12! E2,8k+12, D is a summand of
h1(g2) iff (k)+4 n, and is a summand of h1(g1) iff n < (k)+4. In h1 : E1,8k*
*+32!
E2,8k+52, h1(g1) contains nonzero summands other than D, while h1(g2) does not.
Proof.We will work with the dual h#1of h1. With M = QK1(Spin(8a + 3))= im(_2),
the dual of (7.1) is the following commutative diagram of exact sequences, in w*
*hich
` = _3 - 32`+1.
# j `
0 --- E1,4`+32--- M --- M
x? x x x
?? h#1??? i??? i???
# (7.10)
M=2 --`- M=2 --- E2,4`+52--- M2 --- M2
Dual to Proposition 7.2 (or using 11.5), we have the following interpretation*
* of h#1.
Suppose x 2 M=2 satisfies `(x) = 0 2 M=2. Represent x by ~x2 M. Then `(~x) = 2y
# #
for some y 2 M. If ^x2 E2,4`+52maps to x, then h1(^x) = æ(y). One easily verifi*
*es
#
that this is well-defined. In (7.10), the elements x4a-1and D of E2,4`+52come f*
*rom
M=2, while the log-classes, represented by the big o in Diagram 3.7, come from *
*M2.
We consider first the case where 4` + 3 = 8k - 1. Since only the class D in
# #
E2,8k+12supports a nonzero d3, we need E1,8k-12=(h1(D)), and this is obtained by
#
adjoining to the four relations of [8, 3.18] which yield E1,8k-12the additional*
* relation
(_3 - 34k-1)(D)=2. Since the relation [8, (3.21)] is (_3 - 34k-1)(D), it means*
* that
this fourth relation of [8, 3.18] is divided by 2. Using 8.1 for the first, [8,*
* 3.18] for
#
the second and third, and 8.11 for the fourth, the relations which yield E1,8k-*
*12are,
with n = 4a + 1, A12n,1, A22n,1 - 2n+1D, A32n,1 - 2nD, and u2n,1 + 2 D, with
= (k) + 4, Aiintegers, and u an odd integer by 8.11. If n, then one summ*
*and
60 BENDERSKY AND DAVIS
of the group presented is Z=2 and the other is obtained by subtracting multipl*
*es of
the fourth relation from the others to remove D, and observing the smallest exp*
*onent
of 2; if the fourth relation is divided by 2, the Z=2 -summand has order divide*
*d by
2, while the other is unchanged. If > n, then one summand is Z=2n generated by
2-n times the last relation, and the other summand is Z=2n obtained from 2-n ti*
*mes
the third relation; if the fourth relation is divided by 2, then the first of t*
*hese Z=2n's
becomes Z=2n-1, while the second is unchanged. Thus h#1hits the element of order
2 in the C2- (resp. C1-)summand if n (resp. > n); dually h1 is nonzero on*
* the
stated summand.
The case where 4` + 3 = 8k + 3 is handled similarly. In this case, all the el*
*ements
#
in E2,8k+52except D support a nonzero d3. Thus we wish to mod out E1,8k+32by the
image under h#1of all elements except D. This is accomplished by dividing the f*
*irst
three of the four relations in [8, 3.18] by 2. The relations have the same form*
* as the
four of the previous paragraph, except now = 3. Since < n, the Z=8 summand
will be unchanged if the first three relations are divided by 2, but the other *
*summand
will be divided by 2. Thus h#1hits the element of order 2 in C1; dually h1 is n*
*onzero
on the C1-summand, as claimed. ||
Now we prove the results stated earlier for Spin(4a + 2).
Proof of Theorem 3.11.The eta towers and d3 between them were established in 6.1
and 6.2. When s = 2, (5.17) must be modified according to [9, 3.8]; the Qs+b-1in
(5.17) must be replaced by coker(1 - (-1)b_-1).
For E2,2b+12(Spin(4a + 2)), we compare with the short exact sequences at the *
*end
of the proof of 5.22. For either parity of b, the left part of the SES is the s*
*ame as it
is when s > 2, which is the case described there. This accounts for the class l*
*abeled
1 in (8k + 1, 2), while in (8k - 3, 2) this class is not depicted because it su*
*pports a
d3-differential. It also accounts for the big o's in (8k - 1, 2) and (8k + 3, *
*2); these
represent the group C[[n_2], n - 2] with n = 2a + 1.
The quotient part of the SESs must have the same order as the groups E1,2b+12*
*(Spin(4a+
2)), because one is the cokernel and the other the kernel of the same endomorph*
*ism
of a finite abelian group, namely `b on coker(1 - (-1)b_-1). If b is odd, this *
*is just
represented in our chart by the groups labeled G, the group structure of which *
*we
PERIODIC HOMOTOPY GROUPS OF SO(N) 61
do not attempt to determine. The cyclicity of this group when b is even require*
*s the
following calculation.
Let b = 2c. Using 5.19, we obtain a description of coker(1 - _-1)|QK1(Spin(4a*
* +
2))= im(_2) as
,
with `(xi) xi+2mod i + 2>, and `(D+) = (32a- 32c)D+, using 4.1 for
_3(D+). If 3+ (a-c) 2a, then ker(`) Z=24+ (a-c)generated by 22a-3- (a-c)D+ -
x2a-3, while if 3 + (a - c) > 2a, then ker(`) Z=22a+1generated by D+.
The extension from C1 in 8k + 2 follows by naturality from Spin(4a + 1). One *
*way
to establish the d3 from the C0in (8k + 3, 2) to h31gC0is to use that
E22(Spin(4a + 2)) ! E22(S4a+1)
sends the C0and h31gC0surjectively, and the d3 is present in S4a+1by [7, p.488]*
*. The
extension from C0into (8k - 1, 4) is trivial since the morphism
E2,8k+22(S4a+2) ! E2,8k+12(Spin(4a + 2))
of (6.4) sends one summand injectively onto the multiples of 2 in the C0-summan*
*d,
and the extension on this summand in the BTSS of S4a+2is trivial, by comparison
with the computation of v-11ß*(S2n) in [26].
That d3 = 0 on E1,8k+12follows for the class labeled 1 by naturality from Spi*
*n(4a+1),
and for the group labeled C by pushing into Sp(2a - 1) Z2, similarly to the p*
*roof
of 3.4. As in that proof, d3 6= 0 iff h1 6= 0. We must use h1 because the morph*
*ism is
only algebraic. The C-group maps to 0, but the log classes which form the putat*
*ive
target under h1 map bijectively. The groups are both K[a, 2a - 1] in the notati*
*on of
5.7.
To determine d3 on E1,8k+32, we first observe that
E4,8k+52(Spin(4a + 2)) ! E4,8k+52(Spin(4a + 3))
is injective. This can be seen in 6.1, where we have i even and the a in that t*
*able
corresponds to our a+1 here. Note that C[a-1, 2a-3] ! C[a-1, 2a-2] is injective.
Thus the generators of E1,8k+32(Spin(4a + 2)) support nonzero d3 iff their imag*
*e in
Spin(4a + 3) does. By 3.17, the Z=8 maps by .2, so its image does not support
a nonzero differential. The condition stated in the theorem that R(4k + 1, 4a *
*+ 2)
62 BENDERSKY AND DAVIS
equals R(4k + 1, 4a + 3) exactly says that the C1-summand in E1,8k+3
2 (Spin(4a + 2*
*))
maps onto that of E1,8k+32(Spin(4a + 3)). Since it was shown in 3.8 and 1.4 tha*
*t C1
in E1,8k+32(Spin(4a + 3)) supports a nonzero d3, the claim follows. ||
Now we prove the claims made earlier for Spin(8a + 4).
Proof of Theorem 3.14.The claims about the 1-line groups and homomorphisms fol-
low as in the proof of 3.4.
In the first of the four cases of 3.14, d3 is nonzero on the first summand of
E1,8k+32(Spin(8a+3)), and its possible targets map injectively to E4,8k+52(Spin*
*(8a+4))
by Table 6.1, implying d3 6= 0 on the first summand of E1,8k+32(Spin(8a + 4)). *
*That
d3 = 0 on the third summand holds since this class is the image of a class in
Spin(8a + 3) on which d3 = 0.
To see that d3 = 0 on the second summand, we must show that it does not hit
one of the classes in the image from E4,8k+52(Spin(8a + 3)). To do this, we sho*
*w that
h1 times this class does not equal an element of E2,8k+52(Spin(8a + 4)) support*
*ing
a nonzero d3. This is done by dualizing and using Diagram (7.10). We must show
that the order of this summand in E1,8k+32(Spin(8a + 4))# is not decreased when
the relations for the elements of E2,8k+52(Spin(8a + 4))# supporting nonzero d3*
*'s are
divided by 2. The argument leading to (4.32) shows that the relation for Z=2 (k*
*-a)+5in
E1,8k+32(Spin(8a+4))# involves _2 and _3-34k+1applied to D+ and D+ -D-. These
are not the relations that will be divided by 2, since D+ -D- comes from E2(S8a*
*+3)#,
while D+ 2 E2,8k+52(Spin(8a+4)) does not support a nonzero d3, inasmuch as it c*
*omes
from D in Diagram 3.7.
In the second of the four cases of 3.14, d3 is nonzero on the first summand b*
*ecause
it maps onto an element of E1,8k+32(S8a+3) on which d3 6= 0. The nonzero d3 on *
*the
second summand is a consequence of its being the image of the first summand of
E1,8k+32(Spin(8a + 3)), on which d3 is nonzero into classes mapping injectively*
* under
i*. That d3 = 0 on the third summand is true because it is the image of a class*
* (the
second summand of E1,8k+32(Spin(8a + 3))) on which d3 = 0.
In the third of the four cases of 3.14, d3 is zero on the first summand and n*
*onzero on
the third by naturality from Spin(8a + 3). The nonzero part requires the observ*
*ation
PERIODIC HOMOTOPY GROUPS OF SO(N) 63
that E4,8k+1 4,8k+1
2 (Spin(8a + 3)) ! E2 (Spin(8a + 4)) is injective by 6.1. Naturali*
*ty
from Spin(8a + 4) ! S8a+3implies d3 nonzero on the second summand.
Finally, in the fourth case, naturality from Spin(8a + 3) implies d3 is zero *
*on the
second summand of E1,8k-12(Spin(8a + 4)) and nonzero on the third, while natura*
*lity
from Spin(8a + 4) ! S8a+3implies it nonzero on the first. ||
8. Combinatorics
In this section we present some combinatorial arguments used earlier in the p*
*aper.
We begin with the proof of Lemma 4.22. For the first part, we have the follow*
*ing
sharper result.
P kijj m
Proposition 8.1. For any nonnegative integers m and j, k(-1) k k is divisib*
*le
by j!.
Note that the numbers whose minimal 2-exponent define eSp(m, n) are like the
sum in 8.1 with j > 2n and without the terms having k even. These omitted terms
will be divisible by 2m , and we consider m to be large enough that these terms*
* will
not affect the divisibility. (e.g. m > n.) Proposition 8.1 is sharper than w*
*hat is
required for 4.22 since (j!) = j - ff(j), where ff(j) denotes the number of 1'*
*s in the
binary expansion of j, and j - ff(j) n if j > 2n.
Proof of Proposition 8.1.The proof is by induction on m and j. The result is tr*
*ivially
true if j = 1 or m = 0. We have
X ijj X ij-1j X ijj X ij-1j
(-1)k k km+1 = j (-1)k k-1 km = j (-1)k k km -j (-1)k k km .
By the induction hypothesis, both terms are divisible by j!. ||
Next we prove the part of 4.22 which states (P1(m, n)) n. The second double
sum in P1(m, n) is the same (with n here corresponding to n + 1 there) as the s*
*um
in [8, (3.20)], which was shown to be divisible by 2n+1 in [8, 3.18].7 Thus thi*
*s second
__________
7The statement in [8, 3.18] was divisibility by 2n, but the argument implied
divisibility by 2n+1.
64 BENDERSKY AND DAVIS
double sum in P1(m, n), with its factor of 2, is divisible by 2n+1. The first *
*double
sum in P1(m, n) can be evaluated as
_ !
X X i 2n j X X i 2n-1 j i 2n-1 j
km n-1-k-2t= km n-1-k-2t+ n-2-k-2t
k t 0 k t 0
X X i 2n-1 j
= km n-1-k-t. (8.2)
k t 0
This is divisible by 2n by the divisibility of [8, (3.19)] proved in [8, 3.18].
The desired divisibility result for P3(m, n) follows from Lemma 8.19, complet*
*ing
the proof of 4.22.
The following lemma was used in the proof of Theorem 3.3.
Lemma 8.3. Let m be a fixed odd positive integer, and define
æ ` X i j ' oe
At = min jkkm : j t ,
æ ` k
X ii tj i tjj ' oe
Bt = min j-k - j+k km : [t=2] j < t .
k
Then A2n= A2n+1 B2n+1 B2n.
In fact, we conjecture that the four expressions are equal, but all we need i*
*s the
weaker result stated in 8.3.
Proof.The equality of A2n+1and A2nwas established in [10, 1.4], using a topolog*
*ical
argument. That A2n+1 B2n+1was established in [8, 3.6], using another topologic*
*al
argument. ` '
P i tj i tj m
Let f(t, j) = k j-k- j+k k . The following facts are elementary:
f(t + 1, j)= f(t, j) + f(t, j - 1) (8.4)
f(2n, n)= 0 (8.5)
Choose minimal j n + 1 such that B2n= (f(2n, j)). Using (8.5) in case j = n *
*+ 1,
we have (f(2n, j)) < (f(2n, j - 1)). Thus, using (8.4) in the middle equality*
*, we
have
B2n+1 (f(2n + 1, j)) = (f(2n, j)) = B2n.
||
The following result immediately implies the _3 part of Proposition 5.4.
PERIODIC HOMOTOPY GROUPS OF SO(N) 65
Proposition 8.6. If bases {,i: i 1} and {xi: i 1} of a vector space over *
*Z2
are related by
[i=2]Xij
xi= ij,i-2j, (8.7)
j=0
then the endomorphism _3 defined by _3(,i) = ,3isatisfies
X iij
_3(xi) = jxi+2j. (8.8)
j 0
Proof.Substituting (8.7) into (8.8) shows that it suffices to prove the followi*
*ng equiv-
alences mod 2, for positive integers i and m, and 0 ffl 2:
8i j
X i iji i+2j j < mi if ffl = 0
j j-i+3m+ffl :
j 0 if ffl = 1, 2.
This is immediate from the following integral analogue, which we will prove.
X iiji i+2j j X 3k-ffli2m+2kjiij
j j-i+3m+ffl= 2 2m-k+fflm+k (8.9)
j k
Note that sums involving binomial coefficients are, unless specified to the con*
*trary,
taken over all values of the summation variable for which the terms are nonzero*
*. The
RHS of (8.9) has a possibly odd term only if ffl = 0, the term with k = 0.
We prove (8.9) by showing that both sides satisfy the same recurrence relation
(3i - 3m - ffl)(3i - 3m - ffl - 1)(3i - 3m - ffl - 2)f(i) (8.10)
= (49i2- (87 + 22ffl)i - 66im + (87 + 54ffl)m + 81m2+ 44 + 29ffl + 9ffl2)if(*
*i - 1)
-(17i + 15m - 28 + 5ffl)i(i - 1)f(i - 2) - 5i(i - 1)(i - 2)f(i - 3)
for 3i - 3m - 2 - ffl > 0, with initial values
8
>>>1 if ffl = 0 and i = m
>>>
>><0 if ffl = 0 and i < m
f(i) = >4(m + 1)(2m + 1) if ffl = 1 and i = m + 1
>>>
>>>4(m + 1) if ffl = 2 and i = m + 1
>:
0 if ffl 2 {1, 2} and i <.m + 1
The initial values are easily verified. The equation (8.9) was discovered by *
*comput-
ing the LHS of (8.9) for many values of i and m and observing the pattern of it*
*erated
differences. To prove (8.9), we used the software associated to the book [38] t*
*o find
66 BENDERSKY AND DAVIS
the recurrence relation satisfied by both sides of (8.9), and observing that th*
*ey are
the same recurrence relation. This software is a batch of Maple programs which *
*can
be downloaded from www.math.temple.edu/~zeilberg. If the downloaded program
zeil is run using as input the formula being summed on either side of (8.9), it*
* will
say that the recursion relation (8.10) is satisfied by the sum. Although the au*
*thors
have not done so, this relation is simple enough that one could probably verify*
* it by
hand. The recurrence relation has been verified for several values of i and m,*
* but
the strongest verification of this relation is that this same relation was foun*
*d for the
disparate sums in the two sides of (8.9), which had been empirically observed t*
*o be
equal by computing the value of the LHS in more than 100 cases, and using this *
*to
determine the RHS. ||
The following result was used in the proof of 3.2 and 7.9.
Proposition 8.11. For any positive integer n,
X X i 2n+1 j
n-1-k-3t
kkodd>0t 0
is odd.
The proof requires several subsidiary results.
Lemma 8.12. For n 0, the coefficient of xn in (1 + x)2n+3=(1 + x3) is odd.
Proof.The proof is by induction on n. The validity for n = 0 or 1 is elementar*
*y.
Assume the result is true for n - 1. Working mod 2, the desired coefficient is
X i2n+3j X ii2n+1j i2n+1 jj
n-3i n-3i + n-3i-2
i 0 i 0
X i2(n-1)+3j
= n-1-j
j6 0 (3)
j 0
X i2(n-1)+3jX i2(n-1)+3j
= n-1-j - n-1-3i .
j 0 i 0
The first sum on the last line equals 2n, while the second sum is odd by the in*
*duction
hypothesis. ||
PERIODIC HOMOTOPY GROUPS OF SO(N) 67
Corollary 8.13. Let
8
<1 if i > 0, i 6 0 (3) X i j
g(i) = : and h(n) = g(n - 2j) nj.
0 otherwise j 0
Then h(n) is odd for n 1.
Proof.We work mod 2. For ffl = 0 or 1, let Gffl(n) g(2n + ffl). Then
X
(x1-ffl+ x2)=(1 + x3) = Gffl(i)xi.
i 0
Hence, using Lemma 8.12 at the last step, we have
X i2n+fflj
h(2n + ffl)= Gffl(n - j) j
j 0
coef(xn, (x1-ffl+ x2)(1 + x)2n+ffl=(1 + x3))
coef(xn, x1-ffl(1 + x)2n+1+2ffl=(1 + x3))
= coef(xn+ffl-1, (1 + x)2(n+ffl-1)+3=(1 + x3))
1.
||
Proposition 8.14. Suppose f(n, k) 2 Z2 is defined for n 0 and k 2 Z by
8
<1 k < 0, k 6 0 (3)
f(0, k) = : f(n, k) = f(n-1, k-1)+f(n-1, k+1).
0 otherwise,
Then
X
f(n, k) = 1 forn 2.
k>0
k odd
Proof.We begin by using Corollary 8.13 to deduce that f(n, 0) = 1 for n 1.
This is done by noting that f(0, -i) = g(i) of the corollary, and that f(n, 0) =
P inj
f(0, -n + 2i) i , so that f(n, 0) = h(n) of the corollary.
For n 2, we have
X X X
f(n, k)= f(n - 2, k - 2) + f(n - 2, k + 2)
k>0 k>0 k>0
k odd k odd k odd
= f(n - 2, -1) + f(n - 2, 1) = f(n - 1, 0) = 1.
68 BENDERSKY AND DAVIS
This looks like an induction proof, but it isn't. At the second step, we are us*
*ing that
all except the initial terms appear twice and hence cancel. At the first and th*
*ird steps
we are using the recursive formula for f, and at the last step we use the resul*
*t of the
first paragraph. ||
P i 2*
*n+1 j
Proof of Proposition 8.11.We continue to work mod 2. Define ~f(n, k) = t 0n-1*
*-k-3t.
We will show that ~fsatisfies the same formulas that define f of Proposition 8.*
*14, and
hence the desired result follows from the conclusion of 8.14.
P i 1 j
We first note that f~(0, k) = t 0 -1-k-3t, and this is 1 iff -1 - k 0 and
-1 - k 0, 1 (3), which is true iff f(0, k) = 1. Finally, we have
ii j i jj
~f(n, k)= X 2n-1 + 2n-1
n-1-k-3t n-3-k-3t
t 0
X ii 2(n-1)+1 j i 2(n-1)+1 jj
= (n-1)-1-(k-1)-3t+ (n-1)-1-(k+1)-3t
t 0
= f~(n - 1, k - 1) + ~f(n - 1, k + 1).
||
Proof of Lemma 7.7.We are assuming that k and a are fixed integers satisfying *
*(k-
a) > 2a - 5. There is also the implicit assumption that k > 2a, as discussed af*
*ter
P iijj2k-1
3.3. Let g(j) = i(-1) ii . We will prove, in notation of 3.3,
(P1(2k - 1, 2a)) ((4a - 3)!) (8.15)
(P2(2k - 1, 2a))= ((4a - 3)!) - 1 (8.16)
(P3(2k - 1, 2a)) (4a - 2)!) (8.17)
(g(j)) ((4a - 2)!) forj 4a - 1.(8.18)
The lemma follows from these results and the definitions.
Proposition 8.1 immediately implies (8.18). The inequality (8.17) is implied*
* by
Lemma 8.19. The proofs of (8.15) and (8.16) will occupy the remainder of this s*
*ection
(after the proof of 8.19). ||
Lemma 8.19. If n is positive, then
X i2nj
(-1)ii2n n-i = (-1)n(2n)!=2, (8.20)
PERIODIC HOMOTOPY GROUPS OF SO(N) 69
while if d is positive and even, then P (-1)ii2n+di2nj
n-i is divisible by (2n)!=2.
Proof.Both parts of the lemma utilize the following lemma, which is a standard
combinatorial result (e.g. [21, pp. 243-245]). The coefficients in these polyno*
*mials
are known as Eulerian numbers.
Lemma 8.21. There are polynomials
Xn
pn(x) = ai,nxi
i=1
satisfying
(1)p1x = x;
(2)pn(x) = x(1 - x)p0n-1(x) + nxpn-1(x), where p0denotes the de-
rivative;
P n i n+1
(3) i 1i x = pn(x)=(1 - x) ;
(4)ai,n= iai,n-1+ (n - i + 1)ai-1,n-1;
(5)ai,n= an+1-i,n;
(6)pn(1) = n!.
Proof.If the polynomials pn are defined by (1) and (2), then (4) is immediate a*
*nd
(6) is easily proved by induction on n. The symmetry property (5) is easily obt*
*ained
P n i P n-1 i 0
from (4), while (3) is proved by induction on n using that i x = x( i x ) *
*. ||
To prove (8.20), we note that, by 8.21.3, the left hand side of (8.20) is the*
* coefficient
of xn in
p2n(x) n X i
(-1)n(1 - x)2n__________= (-1) p2n(x) x .
(1 - x)2n+1 i 0
This coefficient equals
(-1)n(a1,2n+ . .+.an,2n) = (-1)n1_2p2n(1) = (-1)n1_2(2n)!,
using parts 5 and 6 of 8.21.
The second part of 8.19 is proved by induction on d and n, with the case d = 0
being (8.20) and n = 1 being trivial. Let C(n, d) denote the coefficient of xn*
* in
70 BENDERSKY AND DAVIS
p2n+d(x)=(1 - x)d+1. By 8.21.3, we must prove (C(n, d)) ((2n)!) - 1. We will
prove that, for d 2 and n > 1,
C(n, d) = n2C(n, d - 2) + 2n(2n - 1)C(n - 1, d),
(8.22)
from which our desired conclusion follows by induction. To prove (8.22), we cal*
*culate
X id+ij
C(n, d) = an-i,2n+dd
i 0
X i
= (n - i)2an-i,2n+d-2+ (2(n2- i2) + 2dn - (d + 1)(2i + 1))an-i-1,2n*
*+d-2
i 0
ji j
+(n + d + i + 1)2an-i-2,2n+d-2d+ii
X i id+ij
= an-i,2n+d-2(n - i)2 i + (2(n2- (i - 1)2) + 2dn
i 0
i j i jj
-(d + 1)(2i - 1)) d+i-1d+ (n + d + i - 1)2 d+i-2d
X i id-2+ij id+i-1jj
= an-i,2n+d-2n2 d-2 + 2n(2n - 1) d
i 0
= n2C(n, d - 2) + 2n(2n - 1)C(n - 1, d).
At the first step, we made two applications of 8.21.4; other steps were just al*
*gebraic
manipulation. The equality of coefficients of an-i,2n+d-2in the next-to-last st*
*ep can
be verified by considering separately terms involving n2, n1, and n0. ||
Next we prove (8.15). Referring to 1.1, we have P1(m, n) = S1(m, n) - S2(m, n*
*),
where
X n-1-`Xi2n-1j
S1(m, n) = `m i
odd ` i=0
X X i 2n j
S2(m, n) = 2 `m n-2-`-4t. (8.23)
odd ` t 0
We show that both S1 and S2 are sufficiently divisible. First note that S1 is t*
*he same
as the sum in [8, (3.19)] with n replaced by n - 1. Using the alternate expres*
*sion
below the middle of [8, p.54], the required divisibility for S1(2k - 1, n) foll*
*ows from
Lemma 8.24. The divisibility of S2(2k - 1, n) is included in Lemma 8.31. These
lemmas then will imply (8.15).
PERIODIC HOMOTOPY GROUPS OF SO(N) 71
Lemma 8.24. If n is even, k > n, j 2, and (n) > (k), then
_ ii j i jj !
X ik-1j
n+1-jj- n-1-jj-2 8i i fj(i) 4j - 3 - ff(n - 2),
i j-1
where
j-2X i j i ji
fj(i) := (-1)t 2j-1t(2j - 2t - 1) j-t2.
t=0
The condition (n) > (k) here is much less restrictive than the condition (*
*k -
n_
2) > n - 5 of (8.15).
The proof of 8.24 requires the following three lemmas.
Lemma 8.25. The expression fj(i) of 8.24 equals
X Yj i`je`
(2j - 1)! . 2-(j-1) 2 .
|_e|=i-j+1`=2
P _ *
* _
Here |_e| = `e`, and the sum is taken over all e= (e2, . .,.ej) with the pres*
*cribed |e|
and ei 0.
Proof.The proof is very similar to that of [8, 4.23]. We show that, for i j -*
* 1, the
system
i j i j ij
a0 22+ a1 32+ . .+.aj-2 j2= 0
i j2 i j2 ij2
a0 22 + a1 32 + . .+.aj-2 j2 = 0
..
.
i jj-2 i jj-2 ijj-2
a0 22 + a1 32 + . .+.aj-2 j2 = 0
i ji i ji iji X Yj ije`
a0 22 + a1 32 + . .+.aj-2 j2= (2j - 1)!21-j `2
|_e|=i-j+1`=2
i j
has solution as-2= (-1)j+s 2j-1j-s(2s - 1) for 2 s j. The last equation is *
*then
the content of the lemma. That this solution (or any multiple of it) is the sol*
*ution
of all but the last equation was proved in [8, 4.23], but it seems convenient t*
*o prove
them all together.
72 BENDERSKY AND DAVIS
The Vandermonde method easily implies that
0 1
x1 . . . xn
BB x2 . . . x2 C
B 1 . n CC nY Y X e e
detBB .. CC= xi (xj- xi) x11. .x.nn.
B@ n-1 n-1C i=1 1 i mj,
then __mprecedes __m0. This order is not unique, but it does not matter how the*
* tuples
P
with equal mjare ordered. Form a matrix A with these tuples in this order lab*
*eling
the rows and columns. Let the entries in the __mcolumn be the entries of the va*
*rious
__
g__m0's in the expansion of fm . The matrix is lower triangular since if g__m0h*
*as nonzero
__ __ __ P P
coefficient in fm , then either m 0= m or m0j< mj. By the claim proved in t*
*he
Q
previous paragraph, the __mrow is divisible by mj! and its diagonal entry equ*
*als
Q
mj!.
_*
*_0
The columns of A-1 give the unique way of writing each g__min terms of the fm*
* 's.
Q
Let B be obtained from A by dividing the g__mrow by mj!. Then B is an integral
triangular matrix with 1's on its diagonal. Hence so is B-1. But A-1 is obtai*
*ned
Q
from B-1 by dividing the __mcolumn by mj!. The proposition follows. ||
The other lemma needed in the proof of 8.24 is the following result about exp*
*onents
of 2.
Lemma 8.27. For e 1 and j 2,
8
>>>0 if j 2 mod 4
>>
_ j ! >>> (a) + 1if e is even and |j - 4a| 1
X ikje <
2 = > (a) + 1 if e = 1 and j = 4a
k=2 >>>>
>>> (a) + 2if e is odd and j = 4a 1
: (a) + 2 if e > 1 is odd and j = 4a.
i j
Proof.If e = 1, the sum equals j+13, from which the result follows easily.
Let e > 1. The proof is by induction on j. By consideration of the next term
added, it is easy to see that validity for j = 4a - 1 implies validity for j = *
*4a, 4a + 1,
and 4a + 2. We will prove, for t 2,
_ 2t-1_ !e! 8
X 2t+1b + k 0 follows from the case 4a - 1 = 2t+1b - 1 *
*of the
lemma and (8.28).
74 BENDERSKY AND DAVIS
It remains to prove (8.28). We prove it by induction on t, and assume it prov*
*en
for t - 1. We work mod 2t+1. Combining the summands for k = 2` and 2` + 1, the
desired sum equals
2t-1-1X 2t-1-1X
(2tb+`)e((2t+1b+2`-1)e+(2t+1b+2`+1)e) `e((2`-1)e+(2`+1)e).
`=0 `=0
One can verify that the summands for ` = 2t-2- s and ` = 2t-2+ s are congruent.
By this symmetry, the desired sum becomes
2t-2-1X
2 `e((2` - 1)e+ (2` + 1)e).
`=0
This sum, without the factor of 2, is, by induction, congruent mod 2tto 2t-2if*
* e is
even and to 2t-1if e is odd. Doubling this yields our claim. ||
Now we can prove Lemma 8.24.
P Qj i`je` in+1-jj in-1-jj
Proof of Lemma 8.24.Let gd(j) = |_e|=d`=22 . Since j - j-2 =
i j i j i j
n_n-1-j k-1 i+1_k
j j-1 and i = k i+1, our desired inequality is implied by the statement
that, for all i j - 1,
i j i j
(n_jn-1-jj-1) + 3i + (i + 1) - (k) + ik+1 (8.29)
+j - ff(2j - 1) + (gi-j+1(j))4j - 3 - ff(n - 2).
We have also usedi8.25.j Using the hypothesis that (n) > (k) and well-known i*
* j
formulas for n-1-jj-1and ff(2j - 1), and removing the nonnegative quantity *
*ik+1,
(8.29) will be implied by
1 - (j) + ff(j - 1) + ff(n - 2j) - ff(n - 1 - j) + 3i
+ (i + 1) - ff(j - 1) - 1 + (gi-j+1(j))3j - 3 - ff(n - 2).
i j
Next we note that since ff(n - 2j) + ff(n - 2) - ff(n - 1 - j) = 2n-2-2jn-2j *
* 0, this
inequality will be implied by
(gi-j+1(j)) (j) - (i + 1) - 3(i - j + 1).(8.30)
This inequality is true (0 0) if i = j - 1. If i > j - 1, then it is certain*
*ly true
unless (j) > 3. Let d = i - j + 1 > 0 and j = 4a.iByjLemma 8.26, gd(j) equals
_1_ P j ke
d!times an integral polynomial in Se(j) := k=2 2 for various e > 0. By Lemma
PERIODIC HOMOTOPY GROUPS OF SO(N) 75
8.27, (Se(j)) (a) + 1. Thus (gd(j)) (a) + 1 - (d!). Hence (8.30) follo*
*ws
from the observation that for d > 0
3d (d!) + 1.
||
Finally we prove (8.16). Reverting to the notation of 1.1, we will show that *
*if n is
even and (m+1-n) > n-4, then (P2(m, n)) = ((2n-3)!)-1 = 2n-5-ff(n-2).
The hypothesis implies that (m + 1) = (n) n - 3 (for n > 4). Now P2(m, n)
is the sum of two terms. The first has the same 2-divisibility as 1_2S2(m, n) o*
*f (8.23),
while, by 8.11, the exponent of 2 in the second is 2n - 4 - (n). This latter *
*is
2n - 5 - ff(n - 2), with equality if and only if n is a 2-power. Thus (8.16) *
*follows
from Lemma 8.31, which will complete the proof of 7.7.
Lemma 8.31. If n is even and (m + 1 - n) > n - 4, then
8
<> 2n - 4 - ff(n - 2)if n is a 2-power
(S2(m, n)) :
= 2n - 4 - ff(n - 2)if n is not a 2-power.
Proof.We use the expression of S2(m, n) given (with minor notational modificati*
*ons)
in [8, (4.20)]. With fj(i) as in 8.24, the lemma will follow from the statement*
* that
_ i j !
X i(m-1)=2j
n-j-1j 8i i fj(i) 4j - 3 - ff(n - 2)
i j-1
with equality iff n is not a 2-power and j = (n - 2 (n))=2. With d = i - j + 1 *
*and
gd(j) as in the proof of 8.24, this will follow from, for n even, 2 j < n=2, *
*and d 0,
i j i j
n-j-1j+ 2jm-1-2+2d+ 3d - ff(2j - 1) + (gd(j)) + ff(n - 2)
8
>>>= 1n = 2e, j = 2e-2, and d = 0
>><
> 1 n = 2e, j, d otherwise
>>>= 0ff(n) > 1, j = (n - 2 (n))=2, and d = 0 (8.32)
>>:
> 0 ff(n) > 1, j, d otherwise.
Here we haveiusedj8.25 to relate fj(i) and gd(j).
We use ab = ff(b) + ff(a - b) - ff(a), (a!) = a - ff(a), ff(a - 1) = ff(a)*
* - 1 +
(a), ff(2a) = ff(a), and ff(2e- k) = e - ff(k - 1) without comment. For our fi*
*rst
simplification of the LHS of (8.32), we note that m - 1 = n - 2 + with high*
*ly
76 BENDERSKY AND DAVIS
2-divisible. Then i m-1 j i n-2 j
2j-2+2d = 2j-2+2d unless 2j + 2d > n, iniwhichjcase d > 0,
and the inequalities of (8.32) are easily established, for 2jm-1-2+2dwill be *
*roughly
( ). Thus the LHS of (8.32) becomes
i j
n-j-1j+ ff(j - 1 + d) + ff(n - 2j - 2d) + 3d - ff(j) - (j) + (gd(j)).
(8.33)
If n = 2e, then (8.33) equals e - 1 + 3d - (j) + (gd(j)). Since (gd(j)) *
*0 and
g0(j) = 1, and j < 2e-1, the first two cases of (8.32) follow.
Next we consider the case where ff(n) > 1 and d = 0. In this case, (8.33) bec*
*omes
ff(n - 1 - 2j) - ff(n - 1 - j) + ff(j) - 1 + ff(n - 2j)
= ff(n - 1 - 2j) - ff(2n - 1 - 2j) + ff(2j) + ff(n - 2j)
= (n - 1 - 2j, 2j, n - 2j),
where the last expression denotes the exponent of 2 in a multinomial coefficien*
*t. This
exponent is 0 and is 0 iff the binary expansions of n - 1 - 2j, 2j, and n - 2*
*j are
disjoint. One readily verifies that this is the case iff n = 2e+ A2e+1with A > *
*0 and
2j = A2e+1.
If d = 1, then (8.33) equals
i j
n-j-1j+ 3 - (j) + ff(n - 2j - 2) + (g1(j)),
which could be 0 only if j 0 mod 4, in which case (g1(j)) (j) - 1, by t*
*he
argument at the end of the proof of 8.24. So (8.33) is positive in this case.
The case d = 2 is handled similarly. This time (8.33) equals
i j
n-j-1j+ 7 - (j) - (j + 1) + ff(n - 2j - 4) + (g2(j)).
This could possibly be 0 only if j or j + 1 is highly 2-divisible, in which c*
*ase
(g2(j)) max( (j), (j + 1)) - 2.
Finally we consider the case d 3. This case is different because (8.33) has*
* a term
- (j + 2) which could be very negative without compensation from (g3(j)), beca*
*use
of the way 8.27 comes out when j 2 mod 4. For any d, (8.33) equals
i j
n-j-1j+ 4d - 1 + ff(n - 2j - 2d) - (j . .(.j + d - 1)) + (gd(j)).
(8.34)
PERIODIC HOMOTOPY GROUPS OF SO(N) 77
This is positive unless, perhaps, (j + ffi) is very large for some ffi satisfy*
*ing 1 < ffi
d - 1. If so, let j + ffi = B2k+1+ 2k, ffl = d - ffi, and n = 2(j + ffi) + D. *
* We have
0 < ffl < d and D 0. Now (8.34) is
ff(B)-ff(ffi-1)+ff(D+2ffi-1)+3d-ff(j+2ffi+D-1)+ff(D-2ffl)+ff(d-1).
We drop the nonnegative term ff(d - 1), replace j + ffi by B2k+1+ 2k in the fif*
*th term,
and add 0 in the guise of
ff(D+ffi-1)-ff(2D+2ffi-1-2ffl)-(2ffl-1)+ ((2D+2ffi-2ffl) . .(.2D+2ffi-2)),
and replace this last (-) by ffl, which it certainly exceeds. We obtain now th*
*at (8.34)
is
i j
3d - ff(ffi - 1) - (2ffl - 1) + 2D+2ffi-1-2fflD+2ffi-1+ ff(B) + ffl
- ff(B2k+1+ 2k + D + ffi - 1) + ff(D + ffi -(1).8.35)
Now we write D + ffi - 1 = C2k+1+ E2k + F with E = 0 or 1 and 0 F < 2k. The
sum of the last two terms of (8.35) is ff(C) + E - ff(B + C) - 1. Thus (8.34)*
* is
i j i j
3d - ff(ffi - 1) - ffl + 2D+2ffi-1-2fflD+2ffi-1+ E + B+CB .
The only negative terms are much smaller than 3d, completing the proof that (8.*
*34)
is positive. This completes the proof of 8.31. ||
9.Comparison with J-homology approach
In the late 1980's, the second author and Mahowald attempted to compute the
groups v-11ß*(SO(n); 2) by using charts for v-11ß*(S2m+1; 2) derived from J-hom*
*ology,
and exact sequences of fibrations. In [27], this approach was applied to obtain*
* mod 2
v1-periodic homotopy groups8 of SO(n) for n = 5, 7, and 9, and in [26], it was *
*used
to compute v-11ß*(Sp(2); 2) and v-11ß*(Sp(3); 2). In this section, we use our *
*BTSS
results here to draw some conclusions about this J-homology approach to v1-peri*
*odic
homotopy groups.
The J-homology approach is simpler for SO(2n+1) than for SO(2n+2). The latter
has more interacting towers than does the former. It seems very difficult, at *
*best,
__________
8Mod 2 does not mean (integral) 2-primary periodic homotopy groups. Mod 2
does not contain the important information about large 2-torsion summands.
78 BENDERSKY AND DAVIS
to see from the J-homology approach that v4k-2(SO(2n + 1)) consists of exactly *
*two
cyclic summands plus a certain number of Z=2's associated to multiplications by*
* the
Hopf map j. (Here we have initiated the abbreviation v*(-) for v-11ß*(-), which*
* we
will utilize throughout this section.) On the other hand, this is readily appar*
*ent from
the BTSS charts 1.3 and 3.7. The small third summand in v4k+2(SO(4a)), described
explicitly in Theorems 3.3, 3.4, and 3.14, results, in the J-homology approach,*
* from
some complicated interaction of the towers, but seems virtually impossible to d*
*educe
from that perspective. So we restrict our comparisons here to SO(2n + 1).
The J-homology approach builds a chart for v*(SO(2n+1)) from those of v*(SO(2*
*n-
1)) and v*(V2n+1,2) using the exact sequence associated to the fibration
SO(2n - 1) ! SO(2n + 1) ! V2n+1,2. (9.1)
A chart for V2n+1,2can be obtained from the fibration
S2n-1! V2n+1,2! S2n,
using charts of v*(S2n-1) and v*(S2n), such as those of [26]. We obtain as a ch*
*art for
v*(V2n+1,2) a sum over all integers k of Diagram 9.2. Our filtration convention*
* is to
use a filtration-preserving isomorphism
v*(S2n+1) v-11J*( 2n+1P 2n).
This puts many elements in the chart for v*(S2n+1) in filtration one less than *
*their
Adams filtration; e.g. j has filtration 0. The differentials between adjacent t*
*owers
indicated in the diagram might not be quite accurate when they are near their m*
*ax-
imum value. The indicated formula is for the differential in S4n-1. The tower*
*s in
V2n+1,2are slightly taller than those of S4n-1; we make no assertion about the *
*differ-
ential in cases when it is 0 in S4n-1. The big dots establish the coordinates f*
*or the
two parts of the diagram.
PERIODIC HOMOTOPY GROUPS OF SO(N) 79
Diagram 9.2. A summand of v*(V2n+1,2)
| |
|______________|
| |r|
| | |
| r | |
| | |
| r |r| |
| | | |
| |r r | | |towers
| | | | |
| |rr | | |areZ=22n
| | | | |
||w | | | |
|| | d (|4|k|+2-2n)
||r | | | |
|______________| |||||| ||||||
| r| || | | | |
| | || | |r|r|
| r r | || | | |
| | | d||(4|k-2rn)|r |
| rw|rr | || | | |
| | || | r |r |
| r r r | || | |
| | | ||r |rr |
| r |r | || |
| | ||r r |
| r | || |
|______________| ||r |
|______________|
w2 (2n + 8k, 4k) w 2 (8k - 2, 4k + 1 - 2n)
The difficult part in computing v*(SO(2n + 1)) is the determination of the boun*
*dary
morphism
v*+1(V2n+1,2) @*-!v*(SO(2n - 1))
and the extensions in forming v*(SO(2n + 1)) from coker(@*) and ker(@*-1).
In a 1988 e-mail to the second author, Mahowald wrote "In SO(n), there are two
phenomena going on at the same time. The first deals with the `stable' stuff in*
* the
sphere, and this just makes up the metastable homotopy of the stunted projective
space like the Barratt-Mahowald theorem says. The unstable `S4n-1' which goes w*
*ith
each S2n is busy making up the stable homotopy of SO. It does so in a way very
similar to Sp."
The "stable stuff" to which he refers is essentially the way that the left pa*
*rts of
Diagram 9.2 build up and go out, which is indeed very similar to the way in whi*
*ch
v-11J*(P22ba+1) is built from v-11J*(P22ii-1) for a < i b. The ü nstable" par*
*t is the way
in which the right parts of Diagram 9.2 interact.
The "stable stuff" mainly builds the regular second summands of E1,4k-12(Spin*
*(2n+
1)) (Theorem 3.1) together with the occasional d3-differential on them and the *
*occa-
sional extension on them into E32(Spin(2n + 1)), as described in 1.4 and 3.8. I*
*t also
80 BENDERSKY AND DAVIS
involves the eta towers which begin in filtration 1 in Diagrams 1.3 and 3.7. Th*
*rough-
out this section, we talk about the BTSS of Spin(2n + 1) and the (J*-approach) *
*chart
for SO(2n + 1), keeping in mind that v*(Spin(m)) v*(SO(m)).
When the differentials and extensions are taken into account, the morphisms of
these second summands of v4k-2(SO(2n + 1)) are
2n + 1 = 8a - 1 8a + 1 8a + 3 8a + 5 8a + 7
n (k) + 3 Z=24a-1-.4!Z=24a,! Z=24a+2-.4!Z=24a+3-.4!Z=24a+3
(9.3)
(k) + 3 < n Z=2e -.2!Z=2e -! Z=2e -.2!Z=2e -.4!Z=2e,
(9.4)
where e = (k)+4. These groups and homomorphisms agree exactly with v-11ßs4k-1(*
*P24n+1n+1).
This is consistent with, but probably not implied by, the Barratt-Mahowald theo*
*rem
to which Mahowald alluded in his 1988 e-mail. The Barratt-Mahowald theorem ([3])
states that, if q < 2(m - 1) and m 13, then
ßq(SO(m)) ßq(SO(2m)) ßq+1(V2m,m),
i.e. that the homotopy sequence of the fibration
V2m,m! SO(m) ! SO(2m)
splits in this range.
Because the Barratt-Mahowald theorem only makes a statement about homotopy
groups in a limited range of dimensions, while v1-periodic homotopy groups depe*
*nd
on all homotopy groups, one cannot really use it to draw a conclusion about
v-11ß*+1(V2m,m) ! v-11ß*(SO(m)).
Moreover, the relationship with v-11ßs*+1(Pm2m-1) is via the stable splitting m*
*ap of
j 2m-1
James ([31]), V2m,m-! QPm , which induces a homomorphism
j* -1 s 2m-1 -1 2m-1
v-11ß*(V2m,m) -! v1 ß*(Pm ) v1 J*(Pm ).
Our observation is that, with m = 2n+1, for n 6 and all integers k, v-11ßs4k-*
*1(P24n+1n+1)
is isomorphic to the regular summand of v-11ß4k-2(SO(2n+1)), and both are mapped
to from v-11ß4k-1(V4n+2,2n+1). Note that, by Proposition 11.4, it is apparentl*
*y not
PERIODIC HOMOTOPY GROUPS OF SO(N) 81
true that v-11ßs4k-2(P24n+1n+1) appears as a direct summand in v-11ß*(SO(2n + 1*
*)); i.e.
the splitting is valid in certain periodic homotopy groups but not others.
Building charts for v*(SO(2n + 1)) inductively using the fibrations (9.1) and*
* the
charts 9.2 is a complicated business. The pattern of differentials and extensi*
*ons
involving the interacting towers from the right side of Diagram 9.2 is particul*
*arly q
*
* q
inscrutable. Another delicate matter is the pattern by which the short eta-towe*
*rs (q )
in Diagram 9.2 cancel out. We will use our BTSS work to show the way in which t*
*he
regular (second) summands of v4k-2(SO(2n+1)) (the ones described in the precedi*
*ng
paragraphs) are built, and the pattern of differentials among the eta-towers. *
*Two
complicating factors are that the charts for v*(SO(2n + 1)) are particularly cr*
*owded
when n is small, and an anomaly for SO(9) noted in [8, 4.21].
The cases Spin(3) = S3, Spin(5) = Sp(2), and Spin(7) '2 S7 x G2 have been dea*
*lt
with thoroughly in [26] and [9]. A comparison of the J-homology approach and BT*
*SS
approach was useful for Spin(7) in [9]. We begin with Spin(9).
The BTSS of Spin(9) is essentially given in Diagram 1.3. The big o there repr*
*esents
Z2 Z2 (e.g. by 5.14). The anomaly occurs in the 1-line, which is given in [8, *
*4.21]
to be 8
>>0 s = 0
>>>
>>>Q(1 + (-1)m m ) s = 1
>>> 0 1
>>> m
>>>G@ A Q(2 m ) s = 2, m odd
>>>
>< 0 1
Exts,2m+1A(M= im(_2)) > B mC
>>>GB@ CA Q(2 m ) s + m odd, s 3
>>>
>>> 2
>>> 0 1
>>> m 0
>>> BB CC
>>>Q@ 2 0 m A s + m even, s 2
: 0 2 -
The Pontryagin dual is given by
8
>>>0 s = 0
>>> 0 1
>>> 1 + (-1)m
>>> B T C
>>>G B@ CA s = 1
>>> T
>>> m
>>> 0 1
>>> B 2 C
>>>Q( T T) G B T C s = 2, m odd
< m @ A
Exts,2m+1A(M= im(_2))# > Tm
>>> 0 1
>>> 2
>>> T T BBT CC
>>>Q( m 2) G @ As + m odd, s 3
>>> T
>>> 0 1 m
>>> T
>>> B 2 0 C
>>>Q B T 0 2 C s + m even,s 2.
>>: @ m A
0 Tm - T
A basis-free form is given as follows, where `m = _3 - 3m . We have
Exts,2m+1A(M= im(_2))#
8
>>>M= im(1 + (-1)m , _2, `m ) s = 1
>><
ker(`m |M=21 ) \ ker(_2|M=21 ) M= im(2, _2,s`m=)2, m odd
>>>ker(` |M=2) \ ker(_2|M=2) M= im(2, _2, `s )+ m odd, s 3
>>: m`m-_2 _2+`m m
H(M=2 -! M=2 M=2 -! M=2) s + m even, s 2
Here H(-) refers to the homology of the short sequence.
PERIODIC HOMOTOPY GROUPS OF SO(N) 103
The description of part of Ext2,2m+1 2 i mj
A (M= im(_ )) as G when m is odd is par-
ticularly useful. Our previous description of this was as the Pontryagin dual o*
*f the
kernel of `m on M= im(_2), and this was felt to be somewhat intractable. It see*
*ms
quite possible that exploitation of this result might allow complete calculatio*
*n of
E2,2m+12(Spin(n)), which has been the only gap in our complete knowledge. It se*
*ems
even more likely that we could use this result to obtain complete information a*
*bout
the group structure of E2,2m+12(SU(n)), both at p = 2 and at odd primes, where *
*only
the orders of the groups were determined in [7] and [23].
The fact that this new description of the 2-line reduces mod 2 to the eta tow*
*ers,
which are known, improves upon our previous understanding that h1 acts surjecti*
*vely
on the 2-line ([5, 5.4]), which had been used to give a lower bound on the numb*
*er
of summands of the 2-line group. Now we can say that the number of summands
of the relevant part of the 2-line group equals thei(known)jnumber of eta tower*
*s. A
relatively straightforward Maple row reduction of m yields the following res*
*ult.
Proposition 11.4. Let m be odd, and e = e(m, n) = (|E1,2m+12(Spin(2n + 1))|),
which is given in [8, 1.5] in terms of sums of binomial coefficients, and which*
* is given
explicitly for n 12 in 3.22. Then
8
>>>Z=23 Z=2e-3 n = 3
>>> e-1
>>>Z=2 Z=2 n = 4
>>> 2 e-3
> Z=23 Z=2e-3 n = 6, m 1 mod 4
>>> 5 e-5
>>>Z=2 Z=2 n = 6, m 3 mod 4
>>> 6 e-7
>>>Z=2 Z=2 Z=2 n = 7, m 1 mod 4
: Z=2 Z=27 Z=2e-8 n = 8, m 3 mod 4
The next result expresses the action of h1 in terms of the above descriptions*
* of
ExtA and Ext#A. This includes a new proof of Lemma 7.2 and its implementation in
7.9, as well as generalizations. The extension of this result to modules which *
*do not
necessarily satisfy _-1 = -1 is given in 11.18.
Note that in all cases in which a submatrix 2 occurs in one of the matrices o*
*f 11.3,
the group/summand depends only on M=2 and its Adams operations, in which case
m m+1. The following description identifies m and m+1 mod 2.
104 BENDERSKY AND DAVIS
Theorem 11.5. Under the identifications of Theorem 11.3, h1 : Exts,2m+1! Exts+1*
*,2m+3
# s,2m+1#
and h#1: Exts+1,2m+3! Ext satisfy
oif s 3 or s = 2 and m even, then h1 = 1 and h#1= 1;
oif s = 1 and m is even, then h1 is inclusion into the second
summand, and h#1is the dual projection;
oif s = 2 and m is odd, then h1 = æ2 1 and h#1= '2 1, where
æ2 is reduction mod 2 and '2 is inclusion into elements of order
2;
oif s = 1 and m is odd, then
0 1
m+1 0
h1 : Q(0 m ) ! Q B@2 0 m+1 CA (11.6)
0 2 -
satisfies
_ m
h1(0, w , w m ) = (w , w m , -3 w ) (11.7)
_ 1 _ 1
= (w , 0, _2w m+1) + (0, w m , -_2w m ).
= (w , w m+1, 0) + (0, 2 . 3m w, -3m w )
Here w is a rational vector such that w and w m are inte-
gral. The basis-free h#1in this case sends (x1, x2) to (_2(x1) +
`m (x2))=2.
The different expressions in (11.7) can each be useful in different situation*
*s. In
the description of Q(-) given after Definition 11.1, we have, if s = 1 and m is*
* odd,
h1 : Q(N1) ! Q(N2) is given by
h1(wN1) = (0, 1_2w , 1_2w m )N2 = (w, 0, 3m w)N2,
(11.8)
where N1 and N2 are the matrices in (11.6).
Before proceeding with the proof of Theorems 11.3 and 11.5, we illustrate them
with M = P K1(F4=G2), which was studied in [9]. We have
_ ! _ !
27 0 37- 3m 0
= 120 211 and m = 5 . 37 311- 3m .
PERIODIC HOMOTOPY GROUPS OF SO(N) 105
Let m = 2k + 1 be odd, and define = (k - 3) + 3 and 0 = (k - 5) + 3. Then
N1 = (0 m ) is, up to unit multiples,
_ !
0 0 27 0 2 0
0 ,
0 0 23 211 1 2
which pivots to
_ 0 0 !
0 0 2min(7,3+ )211+ 0 2 +
M0= 3 11 0 ,
0 0 2 2 1 2
where min0(a, b) = min(a, b) if a 6= b, while min0(a, a) > a.
If k is even, then = 0= 3, and so Q(0 m ) = Z=26 with generator 1=26 ti*
*mes
the first row of M0, while if k 3 mod 4, Q(0 m ) = Z=27, generated by 1=27
times the first row of M0. The case k 1 mod 4 is more delicate, and will be o*
*mitted
from this illustration.
We have 0 1
27 0 37- 3m+1 0 0 0
0 1 BB120 211 5 . 37 311- 3m+1 0 0 CC
m+1 0 BB 7 m+1 C
2 0 0 0 3 - 3 0 C
N2 = B@2 0 m+1CA= BB0 2 0 0 5 . 37 311- 3m+1CC.
0 2 - BB 7 CC
@ 0 0 2 0 -2 0 A
0 0 0 2 -120 -211
Since m + 1 is even, (37 - 3m+1) = (311- 3m+1) = 1. Let Ri denote the ith row
of N2. Then R1 is in the span of R3 and R5, while 2R2 is in the span of R3, R4,
R5, and R6. Because of the units in positions (2, 3) and (4, 5), we can deduce *
*that
Q(N2) Z2 Z2 generated by g1 = 1_2R3 and g2 = 1_2R6.
In the fourth part of Theorem 11.5, let w be the vector whose components are *
*the
numbers by which the rows of N1 must be multiplied to give the generator of Q(N*
*1).
Thus 8
<(5 . 37=26 (3m - 37)=26)if k even
w = :
(5 . 37=27 (3m - 37)=27)if k 3 mod 4,
and, mod 2,
8
<(0 0 1 0 0 1)if k even
g := (0 w w m ) :
(0 0 1 0 0 0)if k 3 mod 4.
106 BENDERSKY AND DAVIS
From (11.8), we obtain, with the first equivalence mod 1,
8
<(0 0 1_0 0 1_)N2 = g1+ g2if k even
h1g : 21 2
(0 0 _20 0 0)N2 = g1 if k 3 mod 4.
This provides an alternate argument for some d3-differentials in the BTSS of F4*
*=G2
established by another method in the first paragraph of the proof of [9, 4.15].
We begin now to work toward the proofs of 11.3 and 11.5. We will construct a *
*free
A-object and a small resolution of an ASR object M (not assuming _-1 = -1) to
which applying Hom A(-, K1(S2m+1)) yields the following.
Lemma 11.9. Assume M is an ASR A-object, and B any basis of M. Let k and
m denote the matrices of _k and _3 - 3m , respectively, with respect to B. Th*
*en
Exts,2m+1A(M= im(_2)) is the homology of a sequence of free Z^2-modules
C0 -d1!C1 -d2!C2 -d3!C3 ! . .,.
where the transposes of the matrices of ds are given by
( -1 - (-1)m 2 m )
for s = 1, and for s 2 by
0 -1 s+m 2 1
+ (-1) m 0
BB 0 - -1 + (-1)s+m 0 C
B@ 0 0 - -1 + (-1)s+m - m2 CCA
0 0 0 -1 + (-1)s+m
with the last row deleted if s = 2.
Note that, if rank(M) = n, then rank(C0) = n, rank(C1) = 3n, and rank(Cs) = 4n
for s 2.
Proof of Theorem 11.3.Since M is ASR, d1 is injective and hence Ext0 = 0. Also
since M is ASR and E2(S2n+1; Q) = 0, the rational homology of the sequence of 1*
*1.9
is 0, and hence the homology at Cs is given by dividing elements in im(ds) as m*
*uch as
possible and using im(ds) as the relations. If N is the matrix of ds, then the *
*columns
of N are im(ds), and so Q(NT ) measures the homology as just described. Since
_-1 = -1, the matrices -1 + (-1)s+m will be 0 or 2. The desired homology
is obtained by substituting these into the matrices displayed in Lemma 11.9 and
PERIODIC HOMOTOPY GROUPS OF SO(N) 107
applying Q. For example, if s = 2 and m is odd, the homology is
0 1
-2 m 0 _ !
m
Q B@0 0 0 m CA Q(2 m ) Q .
0 0 0 -
The desired result here follows from our remark about Q = G for matrices whose
rank equals their number of columns. Other cases follow similarly.
We obtain Exts,2m+1A(M= im(_2))# as the homology of the sequence
d#1 # d#2 # d#3
C#0- C1 - C2 - . .,.
where C#s= Hom (Cs, Q=Z) and the matrix of d#sis the matrix listed in 11.9. Sin*
*ce the
cohomology of the sequence of Hom (Cs, Q) is acyclic, the cohomology exact sequ*
*ence
induced by
0 ! Z ! Q ! Q=Z ! 0
implies that Exts,2m+1A(M= im(_2))# is the homology at C*s-1of the sequence
d*1 * d*2 * d*3
C*0- C1 - C2 - . .,.
where C*s= Hom (Cs, Z) and the matrix of d*sis that listed in 11.9. Thus
0 1
1 + (-1)m
Ext1(-)# = coker(d*1) = G B@ CA,
m
while for s 2, Exts(-)# is given by dividing elements in im(d*s) as much as p*
*ossible
and using im(d*s) as relations. Thus it is given by applying Q to the transpose*
*s of the
matrices displayed in 11.9. This is as claimed in 11.3, once we replace Q by G *
*for
matrices whose rank equals their number of columns.
Finally we give the proof of the basis-free interpretation of Ext(-)#. The c*
*ase
s = 1 and the second summand when s + m is odd are immediate since G(N) is the
cokernel of the transformation with matrix NT . _ !
m
Next note that Q( Tm T) is Pontryagin dual to G , which is coker(M
`Tm _T
M -! M). Hence it is
`m+_2 # # # 2 #
ker(M# -! M M ) = ker(`m |M ) \ ker(_ |M ).
108 BENDERSKY AND DAVIS
Since M is a free 2-primary module, M# may be replaced by M=21 . Similarly,
Q(2 `Tm _T) = ker(2|M=21 ) \ ker(`|M=21 ) \ ker(_2|M=21 )
= ker(`|M=2) \ ker(_2|M=2)
since ker(2|M=21 ) = M=2.
Finally, we need
0 1
T 2 0
`m-_2 _2+`m
Q B@ T 0 2 CA H(M=2 -! M=2 M=2 -! M=2).
0 T - T
To see this, first note that
0 1 0 1
T 2 0 T 2 0
Q B@ T 0 2 CA = Q B@ T 0 2CA
0 2 T -2 T T T - T T 0 0
_ !
T 2 0
= Q T 0 2
_2+`m
ker(M=2 M=2 -! M=2)
with (vi, wj) 2 M=2 M=2 corresponding to the row
1_ 2
2((_ ei, 2ei, 0) + (`m ej, 0, 2ej)).
Then note that the homomorphism
0 1 0 1
_T 2 0 T 2 0
Q B@ T 0 2 CA! Q B@ T 0 2 CA
0 2 T -2 T 0 T - T
has kernel spanned by all elements (0 `m ej -_2ej), which corresponds to (`m *
*vj, -_2vj)
under the above correspondence, establishing the claim. ||
In order to prove 11.5 and 11.9, we need to describe the free A-resolution. *
*We
begin with the following description of the free objects.
Theorem 11.10. Define an object in the category A of stable 2-adic Adams modu*
*les
by letting S = {(i, j) : i 0, j 2 {0, 1}} and = (Z^2)S with
_-1(f)(i, j)= f(i, 1 - j)
8
< f(i - 1, j)i 1
(_3 - 1)(f)(i, j)=:
0 i = 0.
PERIODIC HOMOTOPY GROUPS OF SO(N) 109
Then is free on one generator.
Proof.[18, p.145] says that forgetting other odd operations on 2-torsion stable*
* Adams
modules (not 2-adic) is a categorical isomorphism. Indeed, it states that a 2-t*
*orsion
object of A(2)corresponds to an object in the category A3(2)of 2-torsion abelia*
*n groups
___
with a locally nilpotent operator _3 = _3 - 1 and a commuting involution _-1.
__
For a 2-torsion object G 2 Inv, there is a universal A-object U (G) correspondi*
*ng
to the object of A3(2)which is G G . .w.ith _-1 acting componentwise and
___
_3(g1, g2, . .).= (g2, g3, . .).. Here Inv and GInv are as in the proof of 7.2.*
* (See also
[9] and [18].)
__
The functor U is right adjoint to the forgetful functor, and so it sends inje*
*ctives to
injectives. Recall that GInv (resp. A) is Pontryagin dual to the torsion subcat*
*egory
of Inv (resp. A(2)). Since Q=Z Q=Z with _-1 reversing the factors is injectiv*
*e in
__
Inv, applying U to it yields an injective in A(2). Applying Pontryagin duality *
*yields
the desired projective objects in GInv and A. ||
The generator of can be taken to be the element f0 defined by f0(0, 0) = 1 *
*and
ffi
f0(i, j) = 0 otherwise. A morphism -!N in A is determined by OE(f0), which wi*
*ll
be used implicitly.
We will prove the following result later in this section.
Theorem 11.11. Suppose M is an ASR 2-adic Adams module. Let B = {v1, . .,.vn}
be any basis of M over Z^2. Let F be a free A-module with basis {g1, . .,.gn}. *
*Thus
F is the sum of n copies of the object described in Theorem 11.10.
There is a free A-resolution
0 M= im(_2) -fflR0 -@1 R1 -@2 R2 -@3 . .,.
(11.12)
where 8
>>F F F s = 1
>:
F F F F s 2,
ffl(gi) = vi, and the matrix of @1 is given by
___
( -1 - oe1 2 3 - oe3),
110 BENDERSKY AND DAVIS
while, for s 2, that of @s is
0 -1 s 2 ___ 1
+ (-1) oe1 3- oe3 0
BB 0 - -1 + (-1)soe 0 ___-3oe C
B@ 1 -1 s 2 3 CC
0 0 - + (-1) oe1 - A
0 0 0 -1 + (-1)soe1
with the last row deleted if s = 2, where
ooe1 : ! satisfies oe1(f)(i, j) = f(i, 1 - j);
ooe3 : ! satisfies oe3(f)(i, j) = f(i - 1, j) if i > 0, while
oe3(f)(0, j) = 0;
ooej : F ! F does oej on each summand;
___
o k (resp. 3) : F ! F has matrix with respect to {gj} the
same as that of _k (resp. _3 - 1) on M with respect to {vj}.
Proof of Lemma 11.9.The complex (Cs, ds) is obtained as (Hom A(Rs, K1S2m+1), @**
*s).
We need merely to observe that the duals of oe1 and oe3 are (-1)m and 3m - 1,
respectively. To see this, first note that Hom A( , K1S2m+1) is cyclic on gener*
*ator fl
satisfying
X
fl(f) = (-1)mj(3m - 1)if(i, j).
i,j
Then
oe*3: Hom A( , K1S2m+1) ! Hom A( , K1S2m+1)
satisfies
oe*3(fl)(f)=fl(oe3f)
X
= (-1)mj(3m - 1)i(oe3f)(i, j)
i,j
X
= (-1)mj(3m - 1)if(i - 1, j)
i,j
= (3m - 1)fl(f),
and similarly for oe1. ||
PERIODIC HOMOTOPY GROUPS OF SO(N) 111
Proof of Theorem 11.5.We focus on the most difficult case, s = 1 and m odd. The
Yoneda product with h1 is defined using the diagram
M= im(_2) --ffl- F -@1-- F F F --@2- F F F F
?? ??
fi1?y fi2?y
0 @01
K1S2m+1 --ffl- ---
??
?yh1
K1S2m+3
P j m i
where is as in Theorems 11.10 and 11.11, ffl0(f) = (-1) (3 - 1) f(i, j), a*
*nd
@01= (-1 - oe1 3m - 1 - oe3). Note that there is no _2-summand in the resoluti*
*on of
K1S2m+1 since it is not a resolution of K1S2m+1= im(_2).
Similarly to [9, x3], h1 2 Ext1A(K1S2m+1, K1S2m+3) is the sole nonzero elemen*
*t and
h1 is defined by
h1 = 1_2ffl00O @01,
P m+1 i
where ffl00: ! K1S2m+3 satisfies ffl00(f) = (3 - 1) f(i, j). Thus
X
h1(f, g)= - (3m+1 - 1)i(f(i, 0) + f(i, 1)) (11.13)
Xi
+ 1_2(3m+1 - 1)i((3m - 1)g(i, j) - g(i - 1, j))
i,j
X X
= - (3m+1 - 1)i(f(i, 0) + f(i, 1)) - (3m+1 - 1)i3m g(i, j).
i i,j
The map ø1 is a lifting over ffl0of a map ø : F F F ! K1S2m+1, and ø sati*
*sfies
2bø = OE O @1 for some OE : F ! K1S2m+1 and some b 1. This latter is due to t*
*he
characterization of cocycles that we have been using throughout, that some mult*
*iple
of them equals a coboundary. This defines a vector
v = (OE(g1), . .,.OE(gn)) 2 Z^2n,
where {g1, . .,.gn} is the basis of F used in Theorem 11.11. Next we note that
_ 2 3 m
2bø = (0, v , v( - 3 )).
Here we are using the description of @1 given in Theorem 11.11 and the fact that
_-1 = -1 in both M and K1S2m+1. The oe3 in the third component becomes 3m - 1
___
in K1S2m+1, which is subtracted from the matrix 3.
112 BENDERSKY AND DAVIS
The lifting ø1 can be chosen to satisfy the same formula
_ 2 3 m
2bø1 = (0, v , v( - 3 )).
The difference between this formula and the one for ø resides in the Adams oper*
*ations
in the target. Note that v 2 (and also v( 3- 3m )) is a 1-by-n matrix (ff1, . .*
*,.ffn)
with entries in Z^2. Its meaning as a morphism n ! is the usual matrix of a
linear transformation, while as a morphism from n ! K1S2m+1, v( 3- 3m ) sends
P m i j -1*
* 3
(f1, . .,.fn) to i,j,kffk(3 - 1) (-1) fk(i, j), reflecting the action of _ *
* and _ - 1
on K1S2m+1.
Using the formula for @2 in 11.11, we find that
ø1@2 : F F F F !
is
o0 on the first summand,
o2-bv 2(1 + oe1) : ! on the second summand,
o2-bv( 3- 3m )(1 + oe1) on the third summand, and
___
o2-bv( 2( 3 - oe3) - ( 3- 3m ) 2) on the fourth.
The formula on the fourth summand simplifies to 2-bv 2(3m - 1 - oe3).
Now ø2 can be chosen as
o0 on the first summand,
o-2-bv 2 into the first summand on the second summand,
o-2-bv( 3-3m ) into the first summand on the third summand,
and
o2-bv 2 into the second summand on the fourth summand.
Here we have used that oeicommutes with scalar multiplication. Following by (11*
*.13)
yields that the element h1{ø} 2 Ext2,2m+3A(M= im(_2)) is represented by the map
F F F F ! K1S2m+3 defined by
o0 on the first summand,
o2-bv 2 on the second summand,
o2-bv( 3- 3m ) on the third summand, and
o-3m 2-bv 2 on the fourth summand.
PERIODIC HOMOTOPY GROUPS OF SO(N) 113
This yields the first equality of 11.7, with w = 2-bv, while the second follows*
* from
-3m = 1_2(_3 - 3m+1 - (_3 - 3m )).
Finally, we prove the basis-free form of h#1. Let C = {Cs} and C* = {C*s} =
{Hom (Cs, Z^2)} be the complexes used in the proof of 11.3 involving `m , and l*
*et eC
and eC*be the analogous complexes involving `m+1 instead of `m . We will use the
Universal Coefficient Theorem and the formula (11.7) for
H1(C) -h1!H2(Ce) (11.14)
to deduce the desired basis-free formula for
h*1 *
H1(Ce*) -! H0(C ), (11.15)
which becomes h#1: H2(Ce)# ! H1(C)# under the isomorphism used in the proof of
11.3. Note that the shift of indices is opposite to that of the usual UCT, sinc*
*e the
boundary morphisms in the chain complex C whose homology is being considered
increase the grading.
We consider the commutative diagram
Ext(H1C, Z^2)--u-! H0(C*)
x? x?
(h1)*?? h*1?? (11.16)
Ext(H2eC, Z^2)-eu--!H1(Ce*)
where u and euare the homomorphisms of the UCT, which are isomorphisms here
because C is assumed to be rationally acyclic. The (h1)* in the diagram is dual*
* to
(11.14). Recall that the UCT homomorphism u is induced by applying Hom (-, Z^2)
to C0 -@!B1, noting that
Ext(H1C, Z^2) Hom (B1, Z^2)= Hom (Z1, Z^2).
Here and elsewhere Bidenotes the boundaries and Zithe cycles in Ci.
`m+1-_2 _2+`m+1
The basis-free version of H1(Ce*) is H(M=2 -! M=2 M=2 -! M=2). Let
(xj, yk) be a pair of basis vectors representing a cycle. If a sum of basis ve*
*ctors is
required, either a change of basis or an obvious adaptation of the argument wil*
*l yield
the result. It is often the case that only one of xj and yk is needed to repre*
*sent a
class. The argument in such a case is slightly easier.
114 BENDERSKY AND DAVIS
Let V denote a free Z^2-module of rank n. We consider
0 1
0 0 0
B T C
Ce1= V V V -e@2!eC2= V V V V, (@e2) = BB 2 0 CC
@ Tm+1 0 2 A
0 Tm+1 - T
In the proof of 11.3, it is shown that (xj, yk) corresponds in H1(Ce*) to 1_2(r*
*n+j+r2n+k)
in Q(@e2), where rt denotes the tth row of the matrix. Let eB2= im(@e2) denote *
*the
column space of (@e2), and let c` denote the `th column. Under the isomorphism *
*euof
(11.16), 1_2(rn+j+ r2n+k) corresponds to the morphism eB2! Z^2sending
8
>><1 if ` = n + j or 2n + k
c`7! >0 if n + 1 ` 3n, ` 6= n + j, ` 6= 2n + k
>:1_f f
2((@2)n+j,`+ (@2)2n+k,`))if 1 ` n.
The latter element is an integer since _2xj+ `m+1(yk) 0 mod 2.
The formula for h1 already derived induces h1 : B1 ! eB2sending
(0, v , v m ) 7! (0, v , 0, 1_2v m+1) + (0, 0, v m , -1_2v m ).
These image vectors were previously viewed as columns of (f@2). We deduce that
(h1)*eu-1(xj, yk) : B1 ! Z^2
sends (0, e` , e` m ) to 1_2( )`,j+ 1_2( m )`,k, and so u(h1)*eu-1(xj, yk) send*
*s e` to the
`th component of 1_2(_2xj+`m yk). This is what is meant by 1_2(_2xj+`m yk) 2 H0*
*(C*).
||
Proof of Theorem 11.11.We will prove that (11.12) is acyclic when M = QK1(S2n+1*
*).
It follows that (11.12) is acyclic when M is ASR by induction on the rank of M.
To see this, first note that there is a short exact sequence of 2-adic Adams *
*modules
0 ! QK1S2n+1! M ! M0! 0
with M0 ASR and rank(M0) 0. Note that (f001, f002, f003) (f1, f2, f3) in R1= im*
*(@2). We compute
8
>><-f001(i, 0) + (3n - 1)f003(i, 0) - f003(ij-=1,0*
*0), i > 0
@1(f001, f002, f003)(i, j) = >-f001(i, 0) j = 1
>: 00 n 00 n 00
-f1(0, 0) + 2 f2(0, 0) + (3 - 1)f3(0,i0)= j = 0.
If @1(f001, f002, f003) = 0, then the (j = 1)-part implies f001(i, 0) = 0 for*
* all i. Now we
obtain
f003(0, 0) = (3n - 1)f003(1, 0) = (3n - 1)2f003(2, 0) = . ...
Since 3n - 1 is even, this implies that f003(0, 0) is infinitely 2-divisible, a*
*nd hence is
0, and hence so are all f003(i, 0). Finally, since 2nf002(0, 0) + (3n - 1)f003(*
*0, 0) = 0, we
deduce f002(0, 0) = 0. Thus (f001, f002, f003) = 0, as desired. ||
Now we consider the generalization of the above work to the situation when _-1
is any involution, no longer assumed to equal -1. Lemma 11.9 and Theorem 11.11
were already done in this generality. The analogue of Theorem 11.3 is given bel*
*ow. It
follows immediately from 11.9 and the UCT argument used in the paragraph contai*
*n-
ing (11.14). The simplifications which were made in 11.3 do not apply in the ge*
*neral
case, nor does the äb sis-free" version, which relied on the simplifications.
Theorem 11.17. Let M be as in Lemma 11.9. Then Exts,2m+1A(M= im(_2)) is ob-
tained by applying the functor Q to the matrices displayed in 11.9, while the P*
*on-
tryagin dual of these Ext groups are obtained by applying Q to the transposes o*
*f the
matrices displayed in 11.9.
Finally, we give the generalization of Theorem 11.5. We restrict our attentio*
*n to
h1 and h#1between the 1-line and 2-line.
Theorem 11.18. Let M be ASR, N1 = ( -1 - (-1)m 2 m ) and
0 1
-1 + (-1)m+1 2 m+1 0
N2 = B@ 0 - -1 + (-1)m+1 0 m+1CA
0 0 - -1 + (-1)m+1 - 2.
Then
h1 : Ext1,2m+1A(M= im(_2)) ! Ext2,2m+3A(M= im(_2))
PERIODIC HOMOTOPY GROUPS OF SO(N) 117
is the homomorphism h1 : Q(N1) ! Q(N2) defined by
h1(wN1) = ((-1)m+1w 0 3m w)N2.
Here w is a 1-by-n matrix of rational numbers such that wN1 is integral. The Po*
*n-
tyragin dual
h#1 1,2m+1 2 #
Ext2,2m+3A(M= im(_2))# -! ExtA (M= im(_ ))
is the homomorphism h#1: Q(NT2) ! Q(NT1) defined by
h#1(qNT2) = ((-1)m+1q0- 3m q1)NT1,
where q = (q1, . .,.q4n) is a 1-by-4n matrix of rationals such that qNT2is inte*
*gral,
q0 = (q1, . .,.q3n), and q1 = (q2n+1, . .,.q4n, 0, . .,.0). Note that q0 and q1*
* are 1-by-
3n matrices; there are n 0's at the end of q1.
The reader can perform the simple verification that the formulas for h1 and h*
*#1are
well-defined; i.e., that integrality of wN1 implies that of ((-1)m+1w 0 3m w)*
*N2,
and that integrality of qNT2implies that of ((-1)m+1q0 - 3m q1)NT1. The identi*
*ty
that makes this work appears later in the proof.
118 BENDERSKY AND DAVIS
Proof.We use the diagram at the beginning of the proof of Theorem 11.5, and will
follow along that proof. We have now
X
ffl0(f)= (-1)mj(3m - 1)if(i, j),
@01= ((-1)m - oe1 3m - 1 - oe3),
X
ffl00(f)= (-1)(m+1)j(3m+1 - 1)if(i, j),
X
h1(f, g)= (-1)m (-1)(m+1)j(3m+1 - 1)if(i, j)
i,j
X
-3m (-1)(m+1)j(3m+1 - 1)ig(i, j),
i,j
2bø = (v( -1 - (-1)m ), v 2, v( 3- 3m )),
2bø1 = (v( -1 - (-1)m ), v 2, v( 3- 3m )),
ø1@2 = 2-bv( -1 - (-1)m )( -1 + oe1) on first summand,
2-bv 2(-(-1)m + oe1) on second,
___ m -1 m
2-bv(( 3 - oe3)(-(-1) + oe1) + ( - oe1)(3 - 1 - oe3)) on third,
2-bv 2(3m - 1 - oe3) on fourth,
ø2 = -2-bv( -1 - (-1)m ) into first summand on first summand,
-2-bv 2 into first on second,
___
-2-bv( 3 - oe3) into first on third,
+2-bv( -1 - oe1) into second on third,
2-bv 2 into second on fourth,
h1{ø} = 2-bv(-1)m+1( -1 - (-1)m ) on first summand,
2-bv(-1)m+1 2 on second,
2-bv((-1)m+1( 3- 3m+1) - 3m ( -1 + (-1)m )) on third,
-3m 2-bv 2 on fourth.
With w = 2-bv, this yields the claim for h1.
To prove the claim for h#1, we use (11.16). The element qNT22 H1(Ce*) corresp*
*onds
under euto the morphism from the column space eB2of NT2into Z^2which sends the
jth column to the jth component of qNT2. The morphism H1(C) -h1!H2(Ce) was
just seen to be given by h1(wN1) = ((-1)m+1w 0 3m w)N2. This corresponds to
PERIODIC HOMOTOPY GROUPS OF SO(N) 119
a morphism B1 ! eB2of boundaries in the chain complexes, or equivalently from t*
*he
column space of NT1to that of NT2sending the jth column of NT1to
(-1)m+1(jth column ofNT2) + 3m ((2n + j)th column ofNT2).
Composing, we obtain that (h1)*eu-1(qNT2) is the morphism B1 ! Z^2sending the
jth column of NT1to
(-1)m+1(qNT2)j+ 3m (qNT2)2n+j= ((-1)m+1q0NT1- 3m q1NT1)j,
(11.19)
where q0 and q1 are as in the statement of the result being proved, and subscri*
*pts
on a vector denote the indicated component of the vector. To verify (11.19), we
note that (q1, . .,.qn) is multiplied by (-1)m+1(( -1)T + (-1)m+1) on both side*
*s,
(qn+1, . .,.q2n) is multiplied by (-1)m+1( 2)T on both sides, (q3n+1, . .,.q4n)*
* is multi-
plied by -3m ( 2)T on both sides, while (q2n+1, . .,.q3n) is multiplied by (-1)*
*m+1 Tm+1+
3m (-( -1)T + (-1)m+1) on the left side and by (-1)m+1 Tm- 3m (( -1)T - (-1)m )
on the right side, and these are easily verified to be equal.
Hence h*1(qNT2) = u(h1)*eu-1(qNT2) is the element ((-1)m+1q0 - 3m q1)NT1, as
claimed. ||
References
[1]J.F. Adams, Lectures on generalized cohomology, Springer-Verlag Lecture
Notes in Math 99 (1969) 1-138.
[2]D.W. Anderson, Universal Coefficient Theorems for K-theory, unpublished.
[3]M. G. Barratt and M. Mahowald, The metastable homotopy of O(n), Bull
Amer Math Soc 70 (1964) 758-760.
[4]M. Bendersky, Derived functors of the primitives for BP*( S2n+1), Trans
Amer Math Soc 276 (1983) 599-619.
[5]________, The v1-periodic unstable Novikov spectral sequence, Topology 31
(1992) 47-64.
[6]M. Bendersky, E.B. Curtis, and D.C. Ravenel, EHP sequences in BP theory,
Topology 21 (1982) 373-391.
[7]M. Bendersky and D. M. Davis, 2-primary v1-periodic homotopy groups of
SU(n), Amer Jour Math 114 (1991) 529-544.
[8]________, The 1-line of the K-theory Bousfield-Kan spectral sequence for
Spin(2n + 1), Contemp Math AMS, 279 (2001) 37-56.
[9]________, A stable approach to an unstable homotopy spectral sequence, sub-
mitted.
[10]________, The unstable Novikov spectral sequence for Sp(n), and the power s*
*e-
ries sinh-1(x), Proc Adams Symposium, London Math Society Lecture Notes
176 (1992) 73-86.
120 BENDERSKY AND DAVIS
[11]M. Bendersky, D. M. Davis, and M. Mahowald, v1-periodic homotopy groups
of Sp(n), Pac Jour Math 170 (1995) 319-378.
[12]M. Bendersky and R. D. Thompson, The Bousfield-Kan spectral sequence for
periodic homology theories, Amer Jour Math 122 (2000) 599-635.
[13]________, Some properties of the K-completion, Contemp Math Amer Math
Soc 293 (2002) 35-44.
[14]R. Bott, Lectures on K(X), Benjamin (1969).
[15]A. K. Bousfield, The K-theory localization and v1-periodic homotopy groups *
*of
H-spaces, Topology 38 (1999) 1239-1264.
[16]________, On the homotopy theory of K-local spectra at an odd prime, Amer
Jour Math 109 (1985) 895-932.
[17]________, Nice homology coalgebras, Trans Amer Math Soc 148 (1970) 473-
489.
[18]________, A classification of K-local spectra, Jour Pure Appl Alg 66 (1990)
121-163.
[19]________, Cosimplicial resolutions and homotopy spectral sequences in model
categories, preprint.
[20]A.K. Bousfield, D.M. Kan Homotopy limits, Completions and Localizations,
Lecture Notes in Mathematics 304, Springer-Verlag.
[21]L. Comtet, Advanced Combinatorics, Reidel (1974).
[22]D. M. Davis, From representation theory to homotopy groups, to appear in
Mem Amer Math Soc.
[23]________, v1-periodic homotopy groups of SU(n) at odd primes, Proc London
Math Soc 43 (1991) 529-541.
[24]________, Computing v1-periodic homotopy groups of spheres and some com-
pact Lie groups, Handbook of Algebraic Topology, Elsevier (1995) 995-1048.
[25]________, The K-completion of E6, submitted.
[26]D. M. Davis and M. Mahowald, v1-periodic homotopy of Sp(2), Sp(3), and
S2n, Springer-Verlag Lecture Notes in Mathematics 1418 (1990) 219-237.
[27]________, The SO(n)-of-origin, Forum Math 1 (1989) 239-250.
[28]________, Some remarks on v1-periodic homotopy groups, Lecture Notes, Lon-
don Math Society 176 (1992) 55-72.
[29]L. Hodgkin, On the K-theory of Lie groups, Topology 6 (1967) 1-36.
[30]D. Husemoller, Fibre bundles, Springer-Verlag (1966).
[31]I. M. James, Spaces associated with Stiefel manifolds, Proc London Math Soc
9 (1959) 115-140.
[32]M. Mahowald, The image of J in the EHP sequence, Annals of Math 116
(1982) 65-112.
[33]M. Mahowald and R. D. Thompson, A commentary on "The image of J in the
EHP sequence", Springer-Verlag Lecture Notes in Mathematics 1286 (1987)
298-304.
[34]J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Annals of
Math 81 (1965) 211-265.
[35]M. Mimura, The homotopy groups of Lie groups of low rank, Jour Math Kyoto
Univ 6 (1967) 131-176.
[36]J. C. Moore and L. Smith, Hopf algebras and multiplicative fibrations II, A*
*mer
Jour Math 90 (1960) 1113-1150.
PERIODIC HOMOTOPY GROUPS OF SO(N) 121
[37]C. M. Naylor, Cohomology operations in the K-theory of the classical groups,
Port Math 38 (1979) 145-153.
[38]M. Petkovsek, H.S.Wilf, and D. Zeilberger, A=B, A.K. Peters (1996).
[39]D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres,
Academic Press (1986).
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