NOTES ON K(B4) AT Q = 4
NEIL P. STRICKLAND
In this document we describe the Morava K theory (with n = p = 2) of 4and its*
* subgroups in excruciating
detail. We use Chern classes and their transfers as generators, and describe t*
*he ring structure and all
transfer and restriction maps. Much of the calculation was done using Mathemat*
*ica. The Mathematica
program includes very extensive internal consistency checks, so I have quite a *
*high degree of confidence in
its correctness.
1.Group Theory
We think of {0; 1}2 as {0; 1; 2; 3} via (k; l) 7! k + 2l, or as the corners o*
*f the unit square in R2:
2 3
0 1
We take 4 to be the permutation group on this set.
= (12) = (03) = (01)(23)
= (012)
P = 22= {1; ; ; }
W = 2o 2= D4= P t P
G = 4= W t W t -1W
V = {1; ; ; } = W \ W C W ' C22
L = {1; } = V \ W
C = {1; ; ; } ' C4
Note that the wreath product W acts on the square as the dihedral group, equi*
*valently, it respects the
partition of the set {0; 1; 2; 3} of corners into two opposite pairs {{0; 3}; {*
*1; 2}}.
2.Character Tables
In this section we give the character tables of the groups mentioned above. L*
*ater, we will describe the
Morava K-theory of these groups in terms of Chern classes of these representati*
*ons. We are really only
interested in the 2-completion of the representation ring, so in the case of G *
*we can ignore the conjugacy
class of order 3 and discard one character.
2.1. P.
______________________
|____|1_ff__fi_fl_=_fffi_|
| 1 1| 1 1 1 |
| |1-1 1 -1 |
| |1 1 -1 -1 |
|__|_1_-1__-1_______1_|
ff2= fi2 = 1
1
2 NEIL P. STRICKLAND
2.2. V .
___________________
|_____|1__i______i_|
| 1 |1 1 1 1 |
| |1 1 -1 -1 |
| |1 -1 -1 1 |
|__|__1_-1____1_-1_|
i2 = 2 = 1
2.3. W.
_________________________
|_______|1_ffi_ffl_ffiffloe_|
| 1 |1 1 1 1 2 |
| ; |1 -1 -1 1 0 |
| | 1 1 1 1 -2 |
| ; | 1 -1 1 -1 0 |
|_;__|__1__1__-1__-1___0_|
ffl2= ffi2 = 1
ffloe = ffioe = oe
oe2 = 1 + ffl + ffi + fflffi
2oe = 1 - ffi + ffl + fflffi
oe is the obvious representation of D4 on R2, and ffi is its determinant. The*
* kernel of the linear character
ffiffl is P.
2.4. G.
__________________
|___1|__ffl_ae_fflae_|
| 11| 1 3 3 |
| 221| -1 1 -1 |
| 211| 1 -1 -1 |
|_4_1|_-1__-1___1_|
ffl2= 1
ae2= 2 + ffl + ae + fflae
ffl is the signature and ae is the reduced standard representation.
NOTES ON K(B4) AT Q = 4 *
* 3
3.Maps of Representation Rings
Restriction W to P:
ffi; ffl 7! fl
oe 7! ff + fi
Restriction W to V :
ffi 7! i
ffl 7! 1
oe 7! i +
Restriction W to C:
ffi 7! 1
ffl 7! 2
oe 7! + 3
Restriction G to W:
ae 7! oe + fflffi
Transfer P to W:
1 7! 1 + ffiffl
ff; fi 7! oe
fl 7! ffi + ffl
Transfer V to W:
1 7! 1 + ffl
i; 7! oe
i 7! ffi + fflffi
Transfer W to G:
1 7! 2 + ffl
ffl 7! 2ffl + 1
ffi 7! fflae
fflffi 7! ae
oe 7! ae + fflae
4.Morava K Theory
Let K be the extended version of K(2) with period 2, so we can choose a gener*
*ator of K*CP1 in degree
zero. This gives Chern classes in degree zero for complex bundles. We write K(B*
*G) = K0(BG).
4.1. P.
a = c1(ff) a4= 0
b = c1(fi) b4= 0
c = c1(fl) = a + b + a2b2 c4= 0
K(BP) = K[a; b]=(a4; b4)
4.2. V .
z = c1(i) z4 = 0
x = c1() x4= 0
K(BV ) = K[z; x]=(z4; x4)
4 NEIL P. STRICKLAND
4.3. W.
d = c1(ffi) d4= 0
e = c1(ffl) e4= 0
u = c1(ffiffl) = d + e + d2e2 u4= 0
res(d) = res(e) = c res(u) = 0
t = c1(oe) res(t) = a + b
s = c2(oe) res(s) = ab
bk = trWP(bk) resWP(bk) = ak+ bk
b0= trWP(1) = u3
v = b1
xk = trWV(xk)
x0= e3
K(BW) is generated by v; s and u subject to
uv = 0
u4 = 0
v4 = s2u3
s4 = s2u + su3
s3v = 0
s2v2 = s3u3
sv3 = s2v
A basis is as follows:
v3 v2 v 1 u u2 u3
sv2 sv s su su2 su3
s2v s2 s2u s2u2 s2u3
s3 s3u s3u2 s3u3
Other interesting elements are
e = v + s2+ u2s
d = u + v + s2+ u2s + s2u3
b0 = u3
b1 = v
b2 = v2+ su3
b3 = v3+ sv
x0 = e3= v3+ s2u2+ s3u3
x1 = u + s2+ sv2+ s3u + s2u3
x2 = s2u + u2+ s3u2+ su3+ s2v
x3 = s3+ su + u3+ s3u3
t = v + u + su2
NOTES ON K(B4) AT Q = 4 *
* 5
4.4. G.
ck = ck(ae)
w0= trGP(1)
w1= trGP(a)
K(BG) is generated by c2; w1 and w0 subject to
c72 = c2w0
w51 = w30= w1w0= 0
c32w0 = w20
c22w1 = c2w31
c22w0 = w41
c2w41 = w20
c2w20 = 0
A basis is as follows:
1 c2 c22c32c42c52c62
w1 c2w1
w21c2w21
w31c2w21
w41
w0 c2w0
w20
The ideal of transfers from P is generated by w0 and w1, and is spanned by thos*
*e basis elements which are
visibly divisible by w0 or w1.
Other interesting elements are
c1(ae)= c52+ w1
c3(ae)= c32+ c62+ c2w41+ w0
c1(ffl)=c21+ c51+ w1
The image of K(BG) in K(BW) is spanned by the following elements:
1 s + s2u + u2su su2+ v
s2 s2u + su3 s2u2 s2u3
s3 s3u s3u2 s3u3
v2 v3 sv sv2
s2v
5. Maps of Morava K-Theory
Restriction G to W:
w17! v + su2+ s3u
w07! s3+ su + s3u3
c17! su2+ v
c27! s + u2+ su3
c37! su
Restriction W to P:
u 7! 0
v 7! a + b
6 NEIL P. STRICKLAND
s 7! ab
t 7! a + b
d; e 7! c = a + b + a2b2
bk 7! ak+ bk
Restriction W to V :
u 7! z + x + z2x2
v 7! z3x + z2x2+ zx3
s 7! zx
t 7! z + x
d 7! z + x + z2x2
e 7! 0
bk 7! zk(z + x)3
Restriction W to C:
K(BW) = K[r]=r16
u 7! r4
v 7! r13
s 7! r2+ r5+ r11
t 7! r4+ r10
d 7! 0
e 7! r4
bk 7! rk+12
Transfer P to W:
1 7! u3
akbl7! skbl-k (k l)
Transfer V to W:
1 7! e3
xkzl7! skxl-k (k l)
Transfer W to G:
1 7! 1 + c62+ w31
u 7! c22+ c52+ c2w21+ w41
u27! c42+ c2w31+ c2w0
u37! w0
v 7! w1
This determines everything because K(BW) is generated by {1; u; u2} as a module*
* over K(BG).
NOTES ON K(B4) AT Q = 4 *
* 7
6.Generalised Character Table
Let F be the obvious spectrum with ss*F = WF4[u1 ], where |u| = 2 and u3= v2,*
* and let R be obtained
from WF4 by adjoining a full set of roots for [4](x). Write = (Z=21 )2, so * *
*= Z22, and choose an
identification of the points of order 4 in with the roots of the [4]-series.
If {a; b; c} are the nonzero roots of [2](x) in R then
a + b + c = ab + bc + ca = 0
c = a +F b = -a - b ab = c2
There is an element o 2 Z2 with o = 2 (mod 8) such that
abc = a3= b3= c3= -o
Conjugacy classes of maps * -!W biject with isomorphism classes of *-sets X, *
*equipped with an
equivariant partition into two blocks of order_two_(the action is allowed to ex*
*change the two blocks of the
partition, but not to mix them up). We write X for the set of blocks, consider*
*ed as a *-set. We will
typically draw X as a square, with the blocks as diagonals. If the action of * *
*is transitive, then it is forced
to be Euclidean.
Suppose a 2 . Then a* = (a)* ' *= ann(a) is a *-set of size equal to the orde*
*r of a. We also write n
for the trivial *-set of order n.
These isomorphism classes can be classified as follows:
__ r r
(1)X ' 4, X ' 2_(one_class, denoted by r r). r r
(2)X ' a*t a*, X_' 2 (three classes, denoted by rrr@). *
* r r
(3)X ' 2 t a*, X_' 2 (three classes (one for each pointrarof exact order 2),*
* denoted by r@r).
(4)X ' 2 x a*, X_' a* (three classes, denoted by by r|r|).rr
(5)X ' a*t b*, X_' 2, a 6= b (three classes, denoted by r@r).rr_
(6)X ' a*x b*,_X ' (a +F b)* (three classes, denoted by by r|r|_@).
(7)Xr'ra*,_X ' [2](a)* where a is a point of exact order 4, determined up to*
* sign (six classes, denoted
r|r|_). In this case we write _a= [-1](a).
Generalised character theory assigns to each X as above and each y 2 K(W) an *
*element O(X; y) as follows:
_______________________________________
| | |
| | u v s t d e |
|_____|________________________________|
| rr | |
| rr | 0 0 0 0 0 0 |
|_____|________________________________|
| rrr| |
| rr@| 0 2a a2 2a 0 0 |
|_____|________________________________|
| rr | |
| rr@| 0 a 0 a a a |
|_____|________________________________|
| rr | |
| rr|||a 0 0 a a 0 |
|_____|________________________________|
| rr | |
| rr@| 0 a + b ab a + b a +F ba +F b |
|_____|________________________________ |
| rr_| |
| rr||_@|a +F0b ab a + b a +F b 0 |
|_____|________________________________|
| rr_| _ _ |
| rr||_|[2](a)0 aa a + a 0 [2](a) |
|_____|________________________________ |
8 NEIL P. STRICKLAND
7. Formal Formulae
*
* P k
Here we work with the usual formal group law over F0= WF4, whose logarithm is*
* kx4 =2k.
Mod 2; [2](a); [2](b):
a4= 0
a +F b = a + b + a2b2
Mod 8; [2](a); [2](b):
a4+ 2a = 0
a +F b = a + b - 2ab(a + b) - 3a2b2
Mod 8; [4]=[2](a):
a12- 2a9+ 4a6+ 2 = 0
_a= [-1](a) = 3a - 3a4+ 2a7- a10
b = a_a= 3a2- 3a5+ 2a8- a11
[2](a) = 2a + a4+ 4a10= b2+ 3b5
<2>(a) = 2 + a3+ 4a9
a + _a= b2
b6- 2b3+ 2 = 0
8. A Little Justification
By the last part of [1], the spectral sequence of the extension P -!W -fflffi*
*!C2collapses. Moreover, K(BW)
is spanned by the transfers of elements akblwith k < l, together with a free mo*
*dule over K(BC2) = K[u]=u4
on elements which restrict to akbk. We may take these elements to be {1; s; s2;*
* s3}. Note that
tr(akbl) = tr(res(sk)bl-k) = skbl-k
We thus have a basis for K(BW) as follows:
{skul| 0 k; l < 4} t {skbl| k 0; l > 0; k + l < 4}
If k l then
bkbl= tr((ak+ bk)bl) = skbl-k+ bk+l
b0bl= 0
b21= b2+ su3
b1b2 = b3+ sb1
b1b3 = sb2
b22= s2u3
b2b3 = s2b1
b23= s3u3
ubk = tr(res(u)bk) = 0
v = b1
b2= v2+ su3
b3= v3+ sv
Using the above, one can check that the following is also a basis for K(BW), *
*as claimed earlier:
{skvl| k; l 0 ; k + l < 4} t {skul| k 0 ; l > 0 ; k; l < 4}
NOTES ON K(B4) AT Q = 4 *
* 9
It is well-known that trC21(1) = [2](x)=x = x3, where x is the usual generato*
*r of K(BC2). A naturality
argument gives
b0= u3
Using skbl= 0 for k + l > 3, we find
s3v = 0
s2v2 = s3v3
sv3 = s2v
P
For any bundle V we write c(V; x) = kck(V )xk. I claim that c( 2oe; x4) = c*
*(oe; x)4. To see this, analyse
the Chern classes of 2V using the splitting principle and the fact that [2](x)*
* = x4.
Using 2oe = 1 - ffi + ffl + fflffi we get
c(1 - ffi + ffl + fflffi; x4) = (x2+ tx + s)4
Moving c(-ffi; x4) to the right hand side and putting y = x4 we get
y(y + e)(y + d + e + d2e2) = (y + d)(y2+ t4y + s4)
This means that d is a root of the left hand side, so
d2e + de2+ d3e3= 0
Multiplying by d2e gives d3e3= 0, and thus de(e + d) = 0.
de(d + e) = 0 = d3e3
d + e = u(1 + e3) = u(1 + d3)
u = (d + e)(1 + e3) = (d + e)(1 + d3)
s4= e(e + d) = eu
t4= d2e2
Some delicate calculations in generalised character theory give
e = v + s2+ u2s
t = v + u + su2
x1= u + s2+ sv2+ s3u + s2u3
We can use the above to get the complete ring structure of K(BW) just by bash*
*ing the relations in a
simple-minded manner.
We have double-coset formulae as follows:
resWVtrWP(x) = trVVr\PesPV(\Px)
resWPtrWV(x) = trPVr\PesVV(\Px)
In both cases, the right hand side involves only transfers between Abelian grou*
*ps. These are easy to calculate,
because the corresponding restriction maps are epi. Using this and the naturali*
*ty of Chern classes, we can
calculate the restriction map K(BW) -!K(BV ). Combining this with our knowlege *
*of x1, we can compute
the transfer map K(BV ) -!K(BW).
It is well known that res:K(BG) -! K(BW) is mono. To identify the image, we *
*restrict further to
K(BV ); the image is fixed under the action of by conjugation (note that V is *
*normal in G). Calculation
shows that the set of elements of K(BW) with this property has dimension 17, wh*
*ich we know by generalised
character theory to be the rank of K(BG). This shows that the set in question i*
*s precisely the image of
K(BG) -!K(BW).
10 NEIL P. STRICKLAND
Naturality of Chern classes gives
resGWc2(ae) = c2(oe + fflffi) = c2(oe) + c1(oe)c1(fflffi) =*
* s + tu
We also have the double-coset formula
resGWtrGW(x) = x + trWVc*resWV(x)
This reduces everything to monomial-bashing.
References
1.Michael J. Hopkins, Nicolas J. Kuhn, and Douglas C. Ravenel, Generalised grou*
*p characters and complex oriented cohomology
theories, To appear.