A FAITHFUL UNITARY REPRESENTATION OF THE
2-COMPACT GROUP DI(4)
KRZYSZTOF ZIEMIA'NSKI
Abstract.We construct a monomorphism from the 2-compact group DI(4)
into a 2-compact unitary group.
Let p be a fixed prime. A p-Compact group X is a triple (X, BX, "), where
o X is a p-complete space with finite Fp-homology,
o BX is a pointed p-complete space,
o " : BX ! X is a weak homotopy equivalence.
p-Compact groups has been defined by Dwyer and Wilkerson [6] and since they
possess many remarkable properties of compact Lie groups, there are considered
as homotopy-theoretic analogues of compact Lie groups. In particular, if G is a
compact Lie group such that ss0(G) is a finite p-group, then a triple (G^p, BG^*
*p, ")
is a p-compact group. It is an interesting question what properties of compact *
*Lie
groups generalize onto p-compact groups. Let us recall the theorem of Peter and
Weyl:
Theorem (Peter-Weyl). Every compact Lie group admits a monomorphism into a
unitary group U(n) for some integer n.
To formulate the analogue of this theorem we need some further definitions. If
X := (X, BX, "X ) and Y := (Y, BY, "Y ) are p-compact groups, then a homomor-
phism f : X ! Y is a pointed map Bf : BX ! BY . The homomorphism f is a
monomorphism iff the homotopy fibre of Bf has finite Fp-homology.
Conjecture. Every p-compact group X admits a monomorphism into a p-compact
unitary group U(n)^p.
This conjecture is known to be true in many cases. If X is a completion of
a compact Lie group it is obvious and Castellana [4] has proven it for simple p-
compact groups if p is odd. The main result of this paper is the following
Main Theorem. There is a monomorphism of 2-compact groups DI(4) ! U(246)^2.
The 2-compact group DI(4). DI(4) is the only known simple exotic (i.e. not
being a completion of a Lie group) 2-compact group. Its characteristic property*
* is
that the mod 2 cohomology algebra of its classifying space is an algebra of ran*
*k 4
mod 2 Dickson invariants. The space BDI(4) has been constructed by Dwyer and
Wilkerson [5] as a 2-completion of the homotopy colimit of the following diagram
(which is in fact a centralizer decomposition [7]):
GL4(F2)y GL3(F2)y GL2(F2)y
(0.1) B{ 1}4 ' B(T 3n { 1})^2' B(SU(2)3={ 1})^2' BSpin(7)^2.
1
2 KRZYSZTOF ZIEMIA'NSKI
The underlying category A is isomorphic to the opposite category of F2-vector
spaces of dimensions 1,. . . ,4 and monomorphisms. The groups Spin(7), SU(2)3={*
* 1},
T 3n { 1} and { 1}4 are centralizers of elementary abelian subgroups of Spin(7)
of orders 1,2,3,4 respectively and the maps between different spaces are induced
by inclusions. Automorphisms are more complicated and can be defined only after
completing classifying spaces.
The action of the Weyl group on the maximal torus. A Weyl group WDI(4)
of DI(4) is isomorphic to { 1} x GL 3(F2) and since Spin(7) and DI(4) have a
common maximal torus it contains WSpin(7)as a subgroup. Let a : WSpin(7)!
GL 3(Z) be a natural homomorphism and let V be a subgroup of WSpin(7)of index
2 such that the composition
i : V WSpin(7)a-!GL 3(Z) mod-2---!GL3(F2)
is a monomorphism (one can easily see that V is isomorphic to C2 o 3). By [6,
4.1] there is a section j of the mod 2 reduction GL 3(Z^2) ! GL 3(F2) such that
compositions
(0.2) V -! WSpin(7)a-!GL 3(Z) -! GL 3(Z^2)
and
(0.3) V -i! GL 3(F2) -j!GL 3(Z^2)
are equal. The original construction contains no explicit description of j; how*
*ever
it is sufficient to give a value of j on any element of GL (F2) of order 7. Ele*
*mentary
calculations show that j determined by
00 11 0 1
0 0 1 0 -3h + 1 -h
(0.4) j @@ 1 0 1AA = @-1 h - 1 -h A ,
0 1 0 0 h 3h - 1
where h is the only 2-adic integer satifying equation 4x2 - 3x + 1 = 0, satisfi*
*es the
required conditions. Now the homomorphism
(0.5) WDI(4)' { 1} x GL3(F2) 3 (e, M) ! e . j(M) 2 GL 3(Z^2)
provides an action of the Weyl group on that maximal torus of DI(4).
The decomposition diagram up to homotopy. The action of GL 4(F2) on
{ 1}4 is the obvious one. The action of GL 3(F2) on B(T 3n{ 1})^2is an extension
of an action j on B(T 3)^2' K((Z^2)3, 2). Finally, the following proposition p*
*ro-
vides an action of GL 2(F2) ' NWDI(4)(WSU(2)3={ 1}) NGL3(Z^2)(WSU(2)3={ 1})
on B(SU(2)3={ 1})^2:
Proposition 0.6. If H ' SU(2)n or H ' SU(2)n={ 1}, then the group of homo-
topy classes of self-equivalences of BH^2is isomorphic to NGLn(Z^2)(WH )=WH .
Proof.This is a direct corollary from [5, 5.5].
A FAITHFUL UNITARY REPRESENTATION OF THE 2-COMPACT GROUP DI(4) 3
Sketch of the proof of the main theorem. The main idea of the proof is to
start with a representation of a subgroup of DI(4) and then extend it successiv*
*ely
to larger and larger subgroups using homotopy decompositions. The first step is*
* to
find a suitable of the representation of the common 2-normalizer of the maximal
torus of both DI(4) and Spin(7) (4.6). Next we check that it extends to a homot*
*opy
compatible family of maps from the subgroup decomposition diagram of Spin(7)
(4.14) and then conclude (using the result of [20]) that it extends to a map fG*
* :
BSpin(7) ! BU(m)^2. A similar procedure is performed to produce an extension
of this map to BDI(4). The most difficult part is to show that the restriction *
*of
fG to B(SU(2)3={ 1})^2is GL 2(F2)-invariant; the problem is that there are many
non-homotopic maps from B(SU(2)3={ 1})^2to BU(m)^2which are homotopic
after restriction to the normalizer of a maximal torus. To overcome this problem
we classify maps B(SU(2)3={ 1})^2! BU(m)^2(Section 3) and introduce some
additional structure on the set of extensions of a given representation of a ma*
*ximal
torus normalizer (3.24). Technical results on obstruction theory applied here a*
*re
proven in Section 2.
Notation and terminology. Throughtout the whole paper D := DI(4), G :=
Spin(7), L := SU(2), H := L3={ 1}, p = 2 (unless stated otherwise) and A is a
ring of p-adic integers. The group T is a maximal torus of both G and H and its
2-completion is a maximal torus of D. Let WD , WG , WH be the Weyl groups of
D, G and H respectively. If R is a commutative ring and C is a small category,
then an R[C]-module M is a contravariant functor from C into the category of
R-modules and Hn(C; M) is an n-th higher limit of M. A trivial 1-dimensional
complex representation of any group is denoted by ` and a non-trivial irreducib*
*le
representation of an order 2 group is denoted by '.
1. Homotopy representations of compact Lie groups
Let p be a fixed prime and let S be a compact Lie group. By a homotopy
representation of S we mean a map from BS into the p-completion of the classi-
fying space of the unitary group BU(m)^p. In this section we recall the method
of constructing homotopy representations using a subgroup decomposition due to
Jackowski, McClure and Oliver [10]. This approach is presented with more details
for example in [12].
Subgroup decomposition. We say that a group is p-toral iff it is an extension
of a torus by a finite p-group. A p-toral subgroup P S is stubborn iff NS(P )*
*=P
is finite and has no non-trivial normal p-subgroups. Let Rp(S) be a category of
S-orbits having the form S=P for p-stubborn P and S-maps. By [10] the map
(1.1) hocolimS=P2Rp(S)ES xS S=P -! BS
induced by projections induces isomorphism on mod p homology.
Dwyer-Zabrodsky theorem. A group is p-discrete toral if it is an extension
of a p-discrete torus (Z=p1 )r be a finite p-group. Every p-toral group P has a
unique (up to conjugacy) dense p-discrete toral subgroup P 1 called a p-discrete
approximation of P . The following theorem provides an algebraic description of
homotopy representations of p-toral groups:
Theorem 1.2 ([11, Thm. 1.1]). Let P be a p-toral group and let H be a compact
connected Lie group. Then:
4 KRZYSZTOF ZIEMIA'NSKI
(1) The maps
Hom (P 1, H)= Inn(H) =: Rep(P 1, H) -B![BP 1, BH^p] - [BP, BH^p]
are bijections.
(2) For any ' : P 1 ! H the pairing BCH ('(P 1)) x BP 1 ! BH induces a
homotopy equivalence
BCH ('(P 1))^p-! map(BP 1, BH^p)B' .
Complex representations of p-discrete toral groups. By 1.2 for any p-toral
group P we have [BP, BU(n)^p] ~= Rep(P 1, U(n)). A group P 1 is not finite
and therefore the classical representation theory cannot be applied to calculate
Rep(P 1, U(n)). However, p-discrete groups are countable locally finite groups *
*and
therefore their representations behave similarly to the finite case. Namely ev*
*ery
complex representation of P 1 admits a unique unitary structure ([20, 1.9]) so
Rep(P 1, U(n)) ~=Rep (P 1, GLn(C)), and every complex representation is semi-
simple and its decomposition into irreducible factors is unique up to permutati*
*on
of summands ([17], [20, 1.4]). Moreover, for every ' : P 1 ! U(n) the centraliz*
*er
of '(P 1) is a product of unitary groups with one factor for every isomorphism
class of irreducible subrepresentations of the rank equal to a multiplicity of *
*a given
representation ([20, 1.11]).
Rp(S)-invariant representations. Let NS be a p-normalizer of a maximal torus
TS of S. A representation ' : N1S ! U(n) is Rp(S)-invariant iff B'^pdetermines a
homotopy compatible family of maps {fP : BP ! BU(n)^p}S=P2Rp(S)(or, equiva-
lently the compatible family of representations {'P 2 Rep(P 1, U(n))}). Obvious*
*ly
every homotopy representation of S determines an Rp(S)-invariant representation
of N1S but a map
(1.3) [BS, BU(n)^p] -! limS=P2Rp(S)Rep(P 1, U(n)) ~=Repinv(N1S, U(n))
(where Repinvstands for the set of Rp(S)-invariant representations) is in gener*
*al
neither surjection nor injection.
Obstruction theory. Fix an Rp(S)-invariant representation ' of N1S. An ex-
istence of a homotopy representation of S being an extension of ' (resp. and its
uniqueness) is controlled by obstructions lying in groups Hi+1(Rp(S); 'i) for *
*i > 0
(resp. Hi(Rp(S); 'i)), where
(1.4) 'i(S=P ) := ssi(map (BP, BU(n)^p)B'^p|BP^p).
By Theorem 1.2
(1.5) 'i(S=P ) = ssi(BCU(n)('(P 1))) = ssi-1(CU(n)('(P 1)))
Since CU(n)('(P 1)) is a product of unitary groups, then '1(S=P ) = '3(S=P ) *
*= 0.
Let IR( ) be a set of isomorphism classes of irreducible representations of a g*
*roup
, and let IR( , !) IR( ) be a subset of isomorphism classes of irreducible
subrepresentations of !. There is a functorial isomorphism [20, 2.3]
1
(1.6) '2(S=P ) ' A[IR(P 1, resNSP1')].
As a consequence we obtain the following criterion of extensibility:
A FAITHFUL UNITARY REPRESENTATION OF THE 2-COMPACT GROUP DI(4) 5
Corollary 1.7. Let ' : N1S ! U(n) be an Rp(S)-invariant representation. If
H3(Rp(S); A[IR((-)1 , ')]) = 0 and Hi(Rp(S); M) = 0 for i > 4 and any A[Rp(S)]-
module M, then B'^2extends to a map BS ! BU(n)^p .
2. Obstruction theory
Let C be a small category and let F : C ! Sp be a functor into the category
of spaces. Fix a space Z and a homotopy compatible family of maps f := {fC :
X(C) ! Z}. Assume that for every C 2 C a space map(X(C), Z)fC is 1-connected
and let fi(C) := ssi(map (X(C), Z)fC). A theorem provided by [18] states that
if Hi+1(C; i) = 0 for all i > 1, then there is a map "f: hocolimCF ! Z which
extends f. If additionally Hi(C; i) = 0, then f"is determined uniquely up to
homotopy. In this section we consider the case when
o Hi+1(C; fi) = 0 for all i
o Hi(C; fi) = 0 for all i 6= 2.
We will prove that the set of homotopy classes of extensions Ef of f to hocolim*
*CF
has a structure of a free and transitive H2(C; f2)-set and this structure is f*
*unctorial
in some sense.
Definitions and notation. Let N(C) denotes the nerve of C and let N(C)ibe its
i-th skeleton. Define a cochain complex
Y f
Cji= i(oe(0))
oe2N(C)j
and let for u 2 Cji
j+1X
ffiji(u)(oe) = F (oe(0 ! 1))*u(d0(oe)) + (-1)ku(dk(oe)) 2 Cj+1i.
k=1
By [14, Lemma 2], H*(C*i, ffi*i) = H*(C; fi). For each i let Zji, Bji, Hjideno*
*te the
cocycles, the coboundaries and the cohomology of the cochain complex C*i(note
that Hji= Hj(C; fi)). Denote for short X := hocolimCF and Xi:= hocolim(i)CF .
Let g : Xi! Z be any map extending f (i.e. such that g|F(C)= fC ) and let ski j
denote i-skeleton of j. For any oe 2 N(C)j let
(2.1) Adoe(g) : ski j ! map(F (oe(0)), Z)foe(0)
be an adjoint map to a composition
ski jx F (oe(0)) ~={oe} x ski jx F (oe(0)) -! Xn -g!Z.
Obviously any map g : Xn ! Z (resp. g : X ! Z) is determined uniquely by a
collection of maps Adoe(g), where oe 2 N(C)i, 0 i n (resp. i 0) which sat*
*isfies
suitable compatibility conditions. Now define an obstruction cochain on(g) 2 Cn*
*+1n
by
(2.2) on(g)(oe) = Adoe(g)*[@ n+1] 2 ssn+1 map(F (oe(0)), Z)foe(0).
(Note that we do not need to care about basepoints since map (F (oe(0)), Z)foe(*
*0)is
supposed to be 1-connected).
6 KRZYSZTOF ZIEMIA'NSKI
For a 1-connected space Y , a map a : Dn ! Y and ! 2 ssn+1(Y ) let a + ! :
Dn ! Y be a map such that a|Sn = (a + !)|Sn and
(2.3) Sn+1 ~=Dn [Sn Dn a[(a+!)------!Y
represents ! (obviously g + u is defined only up to mod Sn homotopy). For any
u 2 Cnnand any g : Xn ! Z which extends f let g + u : Xn ! Z be a map such
that g + u|Xn = g|Xn and Adoe(g + u) = Adoe(g) + u(oe) for each oe 2 N(C)n. Not*
*ice
that the set of homotopy classes mod Xn-1 of extensions of g|Xn-1 to Xn is a fr*
*ee
and transitive Cnn-set.
Properties. Here follow well-known properties of concepts introduced above:
Proposition 2.4. Fix g : Xn ! Z such that g|F(C)= fC for each C 2 C. Then
(a)g extends to Xn+1 if and only if on(g) = 0.
(b)on(g) 2 Zn+1n.
(c)on(g + u) = on(g) + ffinn(u) for each u 2 Cnn
(d)g|Xn-1 extends to Xn+1 if and only if on(g) 2 Bn+1n
(e)Fix u 2 Cnn. Then g is homotopic to g + u mod Xn-2 if and only if u 2 Bnn.
Proposition 2.5. Let g, g0: X ! Z be any extensions of f. Then
(a)g|X1 ~ g0|X1.
(b)If g|X2 ~ g0|X2, then g ~ g0.
(c)If u 2 C22, then g|X2 + u extends to X if and only if u 2 Z22.
Proof.First statement follows from 1-connectivity of the mapping spaces. Let
i > 2 and assume g|Xi-1 ~ g0|Xi-1. We can replace g0 by a homotopic map and
assume that g0|Xi-1 = g|Xi-1. Let u 2 Ciisuch that g0 = g + u. Moreover,
0 = oi(g0|Xi) = oi(g|Xi)+ffiii(u) = ffiii(u). Thus u 2 Ziiand since Hii= 0 also*
* u 2 Bii.
By 2.4.(e) we have g|Xi ' g0|Xi and an induction on i starting from 2 implies (*
*b).
To prove (c) note that the condition u 2 Znnis necessary to extensibility of g|*
*X2 + u
to X3 (by 2.4.(a) and 2.4.(c)) and each map from X3 extends to X (by 2.4.(b) and
2.4.(d)).
Corollary 2.6. The action of H22on Ef is transitive and free.
Proof.Transitivity follows form 2.5. Since for each C 2 C the mapping spaces
map(F (C), Z)fC are 1-connected, then any two homotopic maps g ~ g0: X2 ! Z
are homotopic modulo X0. Now the second part follows from 2.4.(e).
Functoriality of H22-action.
Definition 2.7. The category of diagrams on a category A, denoted by Diag A,
is the category whose objects are pairs (CA , A), where CA is a small category *
*and
A : CA ! A is a functor. A morphism ' from A : CA ! A to B : CB ! A is a
pair (T', t'), where T' : CA ! CBis a functor and t' : A ! B O T' is a natural
transformation.
Remark. Every morphism ' : A ! B in the category Diag Sp induces a map
'* : hocolimCAA ! hocolimCBB.
A FAITHFUL UNITARY REPRESENTATION OF THE 2-COMPACT GROUP DI(4) 7
Obviously, F is an object in the category Diag Sp. Let ' : F ! F be an
automorphism in the category Diag Sp such that for each C 2 C maps fC and
('*f)C := fT'(C)O t'(C) are homotopic. As before Ef (resp. E'*f) denotes a set
of homotopy classes of extensions of f (resp. '*f). Obviously there is a natural
bijection Ef ~=E'*f. The morphism ' induces a bijection '* : Ef ! Ef by
(2.8) Ef 3 [g] 7! [X '*--!X g-!Z] 2 E'*f ~=Ef
and automorphisms of chain complexes C*kby
(2.9) ('*(u))(oe) = t*'(u(T'*(oe))), for u 2 Clk, oe 2 N(C)l
which obviously induces automorphisms '* : H22! H22.
Now we are ready to prove the main result of this section:
Theorem 2.10. The diagram
H22x Ef _____Efw+
| ||
| * * | *
|' x' |'
| |
|u + |u
H22x Ef _____Efw
commutes.
Proof.Let g : X ! Z be a map extending f, and let g0 : X ! Z be a map
homotopic to '*g such that g0|X1 = g|X1.
Note that for any simplex oe we have
Adoe('*g) = t*'(oe(0)) O AdT'oe(g)
Then for any u 2 Z22and any oe 2 N(C)2
Adoe('*(g + u)) = t*'u(oe(0)) O AdT'oe(g + u)
= t*'u(oe(0)) O [AdT'oe(g) + u(T'(g))]
= t*'u(oe(0)) O AdT'oe(g) + t*'(u(T'(g))) = Adoe('*g) + '*(u)(oe)
Then '*(g + u)|X2 = '*(g)|X2 + '*u. Now the conclusion follows by 2.5.(b).
3.Homotopy representations of SU(2)n and SU(2)n={ 1}
2-Stubborn subgroups of Ln. By [13] there are two (up to conjugacy) 2-stubborn
subgroups of L, namely
o` 2ssit ' ` ' AE
(3.1) NL := e0 e-02ssit, -01 10
which is in fact the normalizer of the maximal torus TL of L, and
o ` ' ` 'AE
(3.2) Q := mi:= i0 -0i , mj := -01 10
which is a group of order 8 isomorphic to the quaternion group. Morphisms in J *
*:=
R2(L) are given by Aut(L=Q) ' Out(Q) ' 3, Aut(L=N) = 1, Mor(L=N, L=Q) =
; and Mor(L=Q, L=N) ' 3= 2. By [10, 1.6] there are equivalences of categories
(3.3) J n= R2(L)n 3 (L=P1, . .,.L=Pn) 7! Ln=(P1 x . .P.n) 2 R2(Ln)
8 KRZYSZTOF ZIEMIA'NSKI
(3.4)
R2(Ln) 3 Ln=(P1 x . .P.n) 7! (Ln={ 1})=(P1 x . .P.n={ 1}) 2 R2(Ln={ 1}).
Representations of discrete approximations. There are 5 isomorphism classes
of irreducible representations of Q, namely the trivial one `, three non-trivia*
*l 1-
dimensional representations
oi:= resQQ=', oj := resQQ=', ok := oi oj
and one 2-dimensional representation ~. The group AutJ(N=Q) ' 3 permutes oi,
oj, ok and fixes ` and ~.
Denote K := N1L. All irreducible representations of K can be found as subrep-
resentations of representations induced from TL1. For any k 2 A define
` _l_ ' ` '
(3.5) %k : TL1 3 exp(2ssi2s)0exp(02ssi-l_7! exp 2ssi_kl_s2 U(1).
2s) 2
Obviously IR(TL1) = {%k}k2A. A representation indKT1L%k which splits to a sum
` o if k = 0; otherwise it is irreducible. Furthermore, indKT1L%k ' indKT1L%l*
* iff
k = l. Therefore
IR(K) = {`, o} [ {indKT1L%k}k2(Z^2)*={ 1}
Here follows a table of restriction of representations of K to Q:
8
>>>` for ! ' `
>>>
` oi for ! ' indKT1L%4k, k 6= 0
>>> K
>>:oj okfor ! ' indT1L%4k+2
~ for ! ' indKT1L%2k+1,
Next, we provide a useful criterion of J -invariance of representations of K.
Define
IR+ (K) := {o} [ {indKT1L%k}k24Z^2\{0}
IR- (K) := {indKT1L%k}k22+4Z^2
IR0(K) := {`} [ {indKT1L%k}k21+2Z^2.
Proposition 3.7. A representation ' of K is J -invariant iff a total multiplici*
*ty
of irreducible factors of ' in IR+(K) is equal to the total multiplicity of fac*
*tors in
IR-(K).
Let n := {1, 2, . .,.n} and let ' be a representation of Kn. For each k 2 n t*
*here
is a unique presentation
M
(3.8) ' ' ! ~_!.
!2IR(N1 )n\{k}
Proposition 3.9. The representation ' is J n-invariant if and only if for each k
and each ! 2 IR(N1 )n\{k}the representation _! is J -invariant.
Proof.If ' is J n-invariant, then it obviously satisfies the conditions above. *
*Since
each morphism of J nis a composition of morphisms with all coordinates but one
being an identity, the inverse follows.
A FAITHFUL UNITARY REPRESENTATION OF THE 2-COMPACT GROUP DI(4) 9
In particular irreducible J n-invariant representations of Kn are not necessa*
*rily
tensor products of J -irreducible representations of N1 .
Obstruction modules. Fix a J n-invariant representation ' : Kn ! U(m). As
mentioned before (Section 1), obstructions to existence (uniqueness) of an exte*
*n-
sion of B'^2to a map from BLn lie in groups Hi+1(J n; 'i) (resp. Hi(J n; 'i)).
Furthermore, 2(Ln=P ) ' Z^2[IR(P 1, ')] and '1= '3= 0. The following lemma
states that in fact obstruction modules for Ln and Ln={ 1} are isomorphic:
Proposition 3.10. If _ is an J n-invariant representation of Kn={ 1}, then there
Kn={n1}_
is a natural equivalence of J n-modules _i~= resKi .
Proof.By 1.2 we have
_i((Ln={ 1})=(P={ 1})) = ssi(BCU(m)(_(P 1={ 1})))
n={ 1} 1 resKn={n1}_ n
= ssi(BCU(m)(resKKn _(P ))) = i K (L =P ).
Spectral sequence. The groups H*(J n; '*) are calculated in two steps: first we
calculate cohomology of J nwith coefficients being atomic functors (i.e. conce*
*n-
trated on a single object). Next, we use a spectral sequence provided by [9]. D*
*efine
a function
(3.11) ht : Ob(J n) 3 Ln=(QB x Kn\B) 7! |B| 2 Z
By [9, 1.3] there is a spectral sequence of a cohomological type with the first*
* term
M '
(3.12) Es,t1:= s+t(Aut J n(Ln=P ); 2(L=P ))
ht(L=P)=s
which strongly converges to H*(Rp(L); '2). By 1.6 we have
M n
(3.13) Es,t1:= s+t(A[IR(QB x Kn\B); resKQBxKn\B']).
ht(L=P)=s
If M is an atomic A[J n]-module concentrated on Ln=P , then H*(J n; M) =
*(Aut J n(Ln=P ); M(Ln=P )), where group * were defined by [10]. Let V be a
free rank 3 A-module with 3 acting by permutations on its basis.
Proposition 3.14. Let B n and for every b 2 B let Mb be a A[ 3]-module
isomorphic either to V or to A. Then
O
l( B3; ~ Mb) = 0
b2B
for each l 6= |B|. If Mb ' V for every b 2 B, then and the homomorphism
O O~
A ' A B ~= 1( 3; V ) -! |B|( B3; Mb)
b2B b2B
N
is an isomorphism; otherwise |B|( B3; ~ a2BMb) = 0.
Proof.By [10, 6.2.ii] i( 3; V ) ' A if i = 1 and it vanishes otherwise and *(*
* 3; A) =
0. The isomorphism follows from [10, 6.1].
10 KRZYSZTOF ZIEMIA'NSKI
Description of groups E*,*1. Let Z'(n) be a subset of (IR(K)[{*})n containing
all elements (!1, . .,.!n) such that there is a sequence (j1, . .,.jn) 2 IR(K)n*
* which
satisfies the following conditions:
(a)j1~ . .~.jn ',
(b)If !l2 IR(K), then jl= !l,
(c)If !l= *, then jl2 IR+(K).
Note that (c) is equivalent to the condition that resKQjlcontains a subrepresen*
*tation
isomorphic to oi (or oj or ok). For each B n let
(3.15) ZB'(n):= {(!1, . .,.!n) 2 Z%(n) : (!l= *) , (l 2 B)}
[
(3.16) Zr'(n):= ZB%(n)
|B|=r
Proposition 3.17.
(
n A[ZB'(n)] for r = |B|
r( B3; A[IR(QB x Kn\B); resKQBxKn\B']) ~=
0 for r 6= |B|
Proof.Since resKnQBxKn\B' is B3-invariant, then we have a presentation
_ ! l((~a),( b))
n M iO~ j i O~ j
resKQBxKn\B' ~= ~a ~ b
(~a)2(IR(Q) 3)Ba2B b2n\B
( b)2IR(K)n\B
where IR(Q) 3 = {`, ~, oi oj ok}. Then we obtain
n M iO~ j
A[IR(QB x Kn\B; resKQBxKn\B')] = a2B V
(~a)2(IR(Q) 3)B ~a=(oi oj ok)
( b)2IR(K)n\B
l((~a),( b))>0
Finally, by 3.14
n M B
|B|( B3; A[IR(QB x Kn\B); resKQBxKn\B']) = A = A[Z% (n)]
(~a)=oi oj ok
( b)2IR(K)n\B
l((~a),( b))>0
and r( B3; A[IR(QB x Kn\B); ') = 0 if r 6= |B|.
Proposition 3.18. If t 6= 0 or s 62 {0, 1, . .,.n}, then Es,t1= 0. In particula*
*r, the
spectral sequence E*,**degenerates to an exact sequence:
d0,011,0d1,01 dn-2,01n-1,0 dn-1,01n,0
(3.19) 0 -! E0,01--!E1 --! . .-.---! E1 ----! E1 -! 0.
Proof.Immediate from 3.17.
For each ! 2 IR(K) [ {*} let
(
|!| = 0 for ! 2 IR(K).
1 for ! = *
A FAITHFUL UNITARY REPRESENTATION OF THE 2-COMPACT GROUP DI(4) 11
L n
Then di,01= j=1ffiij, where
ffiij(!1, ..., !n) =
8
><(-1)|!1|+...+|!j-1|(!1, ..., !j-1, *, !j+1,!...,j!n)2 IR+(K)
|!1|+...+|!j-1|+1(! , ..., ! , *, !! ,2...,I!R)-(K)
>:(-1) 1 j-1 j+1j 0 n
0 !j 2 IR (K) [ {*}
Proposition 3.20. Then Hk(J n; '2) = 0 for k n.
Proof.Since Ek,01= 0 for k > n it suffices to prove that Hk(J n; '2) = 0. The
set Zn'(n) is either empty or it contains only (*, . .,.*). But if Zn'(n) 6= ;*
*, then
there is ! 2 IR- (N1 ) such that (!, *, . .,.*) 2 Zn-1'(n). Hence (*, . .,.*) =
dn-1,01(!, *, . .,.*).
Corollary 3.21. Let ' be an J n-invariant representation of Kn (resp. of Kn={ 1*
*}).
If n 3, then ' extends to a homotopy representation of SU(2)n (resp. SU(2)n={*
* 1}).
If n 2, then the extension is unique.
Proof.It is straightforward from 3.20, 1.7 and 3.10.
The case n = 3. As proven above, each J 3-invariant representation ' of K3
extends to a map (BSU(2)3)^2! BU(m)^2, although it possibly exists more than
one extension. Let E' denote the set of extensions and let
(3.22) Z2'(3) := {(*, . .,.*, !, *, . .,.*) 2 Z2'(3) : ! 2 IR(K)+ [ IR(K)- }
Z2'(3)0 := Z2'(3) \ Z2'(3)
Let be a relation on Z2'(3) given by
(oe1, *, *) (*, oe2, *) , (oe1, oe2, *) 2 Z1'(3)
(oe1, *, *) (*, *, oe3) , (oe1, *, oe3) 2 Z1'(3)
(*, oe2, *) (*, *, oe3) , (*, oe2, oe3) 2 Z1'(3)
Let ~ be an equivalence relation spanned by and let ~A[Z2'(3) = ~] be a kern*
*el
of the augmentation A[Z2'(3) = ~]! A.
Proposition 3.23. H2(J 3; A[Z*'(3)]) ' ~A[Z2%(3) = ~].
Proof.Let j : A[Z2'(3) ] ! A[Z2'(3) ] be an automorphism defined as follows:
j(*, . .,.*, !i, *, . .,.*) = (-1)i-1sgn(!i)(*, . .,.*, !i, *, . .,.*)
where sgn(!i) = 1 for !i2 IR+(K) and sgn(!i) = -1 for !i2 IR-(K). If b b0,
then we have
j(b - b0) = (-1)i-1sgn(!i)(*, . .,.*, !i, *, . .,.*)+
0
+ (-1)isgn(!i0)(*, . .,.*, !i0, *, . .,.*) =
= d1,01(*, . .,.*, !i, *, . .,.*, !i0, *, . .,.*) 2 *
*imd1,01
and hence imd1,01= j(**b b0) = j(****b~b0). The argument similar to t*
*he
one used in proof of 3.20 shows that ~A[Z2'(3)0] imd1,01. Moreover,
d2,01(j((*, . .,.*, !i, *, . .,.*)) = (*, . .,.*) 2 Z3'(3).
12 KRZYSZTOF ZIEMIA'NSKI
P 2,0 P
It implies that j( bi) 2 kerd1 if and only if bi2 ~A[Z2'(3) ].
Here follows the main theorem of this section:
Theorem 3.24. (a) E' is a free and transitive ~A[Z2'(3) = ~]-set.
(b)Fix a permutation oe 2 3 and assume that ' ' oe*'. If oe acts trivially on
Z2'(3) = ~, then it acts trivially on E'.
Proof.Part (a) follows immediately from 3.23, 2.10 and 3.10. Let F : J ! Sp be a
decomposition diagram of L and let (Toe, toe) be an element of AutDiagSp(F 3) w*
*hich
corresponds to oe. By 2.10 and the assumption the map oe* : E' ! E' preserves a
structure of ~A[Z2'(3) = ~]-set. But the only such an automorphism of E' having
a finite order is an identity.
4. A homotopy representation of Spin(7)
In the present section we construct a representation ' of N1G and then extend*
* it
to a map BG^2! BU(m)^2. We use a representation having a huge dimension. The
main reason for this particular choice is that it is easy to prove R2(G)-invari*
*ance
of '.
Bases of 2-discrete tori. Let n be a positive integer and let l = |n_2|. Denote
` '
(4.1) e(t) := - cos2ssitsin2ssitsin2ssitcos2ssit2 SO(2)
and let i : Rl! TSpin(n)be a lift of a map
(4.2) Rl3 (t1, . .,.tl) 7! diag(e(t1), . .,.e(tl)) 2 TSO(n)
Let S ' (Z=21 )r be a p-discrete torus. A basis of S is an epimorphism fi :
Z[1_2]r ! S. Any representation ! : S ! U(1) has a lift of ! along the expotent*
*ial
map Z=21 3 x 7! e2ssix2 U(1) (which is of course unique). Now the composition
"!O fi 2 Hom (Z[1_2]r, Z=21 ) ' (Q^2)r determines a sequence of p-adic rationals
(k1, . .,.kr) which will be called a root of ! in the basis fi. Let %k1,...,kr*
*be a
representation corresponding to root (k1. .,.kr).
We use the following three bases of T :
(4.3) fi : Z[1_2]3 3 (x1,7x2,!x3)i(x1, x2, x3),
(4.4) fi0: Z[1_2]3 3 (x1, x2,7x3)! i(x2 + x3, x1 + x3, x1 + x2),
(4.5) fi00: Z[1_2]3 3 (x1,7x2,!x3)i(x1+x2_2, x1-x2_2, q-1x3),
where q is an odd 2-adic integer such that q2 - q + 2 = 0 (note that -q-1_2= q-1
and q 3 mod 8). The bases fi, fi0 and fi00are natural bases of maximal tori *
*of
respectively SO(7), G and H.
A representation of N1G. Throughout the present section we use the basis fi.
Define 0 1
1 O
(4.6) ' := indNGT1@ (` w*%1,0,0)A.
w2WD=(WD)%1,0,0
Obviously ' is not a restriction of a continuous representation of NG since coo*
*rdi-
nates of its roots are not integers (in any basis).
A FAITHFUL UNITARY REPRESENTATION OF THE 2-COMPACT GROUP DI(4) 13
Proposition 4.7. The representation ' is faithful. Its character is WD -invaria*
*nt
on T 1 and vanishes on N1G \ T 1.
Proof.It is an immediate consequence of the definition.
Proposition 4.8. The isotropy group (WD )%1,0,0 WD has order 8. An WD -orbit
of %1,0,0contains exactly the representations %k1,k2,k3, where
(k1, k2, k3) 2 {( 1, 0, 0), (0, 1, 0), (0, 0, 1),
( q_2, 1_2, 1_2), ( 1_2, q_2, 1_2), ( 1_2, 1_2, q_2),
( q-1_2, q-1_2, 0), ( q-1_2, 0, q-1_2), (0, q-1_2, q*
*-1_2)}.
Proof.By acting with a matrix (0.4) and WG we can easily produce the represen-
tations listed above. On the other hand, there are eight obvious elements in WG
which fix %1,0,0. Therefore the orbit contains exactly 42 elements.
Corollary 4.9. dim ' = 246.
Throughout the rest of this section we prove that ' is R2(G)-invariant.
Stubborn subgroups of orthogonal groups. Let
` ' ` '
(4.10) A := 10 -01 , B := 01 10 .
For i < n define the matrices Ani, Bni2 O(2n) by
Ani:= I2i A I2n-i-1, Bni:= I2i B I2n-i-1.
Let
2n := <-I2n, An0, . .,.Ann-1, Bn0, . .,.Bnn-1> O(2n)
~ 2n:= <{X I2n-1}X2SO(2), An0, . .,.Ann-1, Bn0, . .,.Bnn-1> O(2n)
Let Tprod(k) be a set of all product of wreath products 2no Cr12o . .o.Crm2(wh*
*ere
n 6= 1 and (n, r1) 6= (0, 1)) and ~ 2no Cr12o . .o.Crm2(where n > 0). By [13,
Thm. 8] every 2-stubborn subgroup of O(k) is conjugate to an element of Tprod(k*
*).
Moreover, every 2-stubborn subgroup of SO(k) is conjugate to P \ SO(k), where
P 2 Tprod(k) [13, Prop. 11] and every 2-stubborn subgroup of Spin(k) has the fo*
*rm
ss-1(Q), where Q is stubborn in SO(k).
Proposition 4.11. Fix P 2 Tprod(k). Let Q := P \ SO(k) and let Q0 Q be
a unity component. Assume that Q0 := g-1Qg NSO(k). Then for each x =
i(t1, . .,.tl) 2 (Q \ Q0) \ TSO(k)one of the following conditions holds:
(1) ti 1_2mod 1 for some i
(2) ti tj 1_4mod 1_2for some i 6= j
Proof.By [13, Prop. 9] and [13, Thm. 12] we can assume that g permutes irre-
ducible factors of P . Consider the following cases:
o P = 1 = O(1) or P = ~ 2= O(2) _ obvious.
o P = 2n and n > 1. Then Q0\ T = and
(Q0\ Q00) \ T = {i(1_4, . .,.1_4), i(1_2, . .,.1_2), i(3_4, . .,.3_*
*4)}.
o P = ~ 2n. Then Q0\ T = Q00.
14 KRZYSZTOF ZIEMIA'NSKI
o P = P 0o Ct2where P 0 O(2n) and n > 1. Then Q0\ TSO(k)= (P 0\
TSO(2n))2t. Then x = (y1, . .,.y2t) where yj 2 P 0\ TSO(2n)for all j and
since x 62 P0 we have yl2 (P 0\P00)\TSO(2n)for some l. Then the conclusi*
*on
follows by induction. t-1
o P = 1 o Ct2, t > 1. Then Q0\ TSO(k)= ({ 1}2 n C2) \ SO(2)) and
either ti 0 mod 1_2for all i, or ti 1_4mod 1_2for all i.
o P = P10x . .x.Pl0x { 1}s and Pi06= { 1}, where Pi02 SO(k)i. Then
x = (y1, . .,.ys, z) 2 (P10\ TSO(k1)) x . .x.(P10\ TSO(ks)) x { 1}s \ TSO(s)
If there is i such that yi62 (Pi0)0, then the conclusion follows from in*
*duction
(since ki> 1). Otherwise z is not unit and has the form diag(e(z1), e(z2*
*), . .).
where zj 2 {0, 1_2}. Hence some zj equals 1_2.
Proposition 4.12. Let S = SO(k) or S = Spin(k), P 2 Tprod(k), Q := P \SO(k)
and g 2 SO(k). Assume that g-1P g NSO(k). Then there exists w 2 WSO(k)such
that for each t 2 P0 holds g-1tg = w(t).
Proof.If S = SO(k) it is immediate since g permutes irreducible factors of P . *
*The
case S = Spin(7) follows from the former since the projection ss : Spin(k) ! SO*
*(k)
induces an equivalence R2(Spin(7)) ~=R2(SO(7)).
R2(G)-invariance of '.
Proposition 4.13. Let x = i(t1, t2, t3) 2 G. If any of numbers ti or ti tj (wh*
*ere
i 6= j) equals 1_2mod 1, then O'(x) = 0.
Proof.Notice that for each w 2 WD ' contains ` w*%1,0,0as a tensor factor. If
t1 1_2mod 1, then O%1,0,0(x) = -1. Thus
O'(x) = O` %1,0,0(x)O_(x) = (1 - 1)O_(x) = 0,
By replacing succesively %1,0,0by %0,1,0, %0,0,1, %q-1_2, q-1_2,0, %q-1_2,0, q-*
*1_2, %0,q-1_2, q-1_2
and using an analogous argument we obtain the conclusion in the other cases (no*
*tice
that q 3 mod 8).
Proposition 4.14. Let P 2 Tprod(O(7)), Q := P \ SO(7) and Q" := ss-1(Q).
Assume that ss(g)-1Qss(g) NG for some g 2 G. Then for each x 2 "Q1 we have
O'(x) = O'(g-1xg). In particular, ' is R2(G)-invariant.
Proof.If ss(x) 62 Q0, then by 4.11 O'(x) = O'(g-1xg) = 0. If ss(x) 2 Q0, then t*
*he
conclusion follows by 4.12.
Proposition 4.15. The representation ' extends to a map fG : BG ! BU(246)^2.
Proof.As a consequence of [19] every R2(G)-invariant representation of N1G has
an extension.
5.A homotopy representation of H
Let fH : BH^2! BU(m)^2be the composition of fG with an inclusion BH^2!
BG^2. The main result of this section is that fH is homotopy GL 2(F2)-invariant,
where the action of GL 2(F2) comes from the centralizer decomposition diagram of
DI(4).
A FAITHFUL UNITARY REPRESENTATION OF THE 2-COMPACT GROUP DI(4) 15
GL 2(F2)-action on BH^2. Since H and G have a common maximal torus, then
WH WG . It appears that the group NWD (WH )=WH acts on BH^2. Since
NWD (WH )=WH ' GL 2(F2), this determines the action of GL 2(F2) on BH^2.
The group GL 2(F2) is generated by an order 2 element which stabilizes the
inclusion BH^2! BG^2and by some element of order 3, say b. The invariance of
fH under the action of s is clear. The main effort will be to show that fH O b *
*' fH .
Elementary calculations show that b is represented (in the basis fi00) by the m*
*atrix
0 1
0 0 1
(5.1) @ 1 0 0A.
0 1 0
1
Denote ! := resNGN1H'
Proposition 5.2. The morphism b stabilizes the representation !. In particular,
both maps fH and b O fH restrict to the same R2(H)-invariant representation !.
Proof.Strightforward from 4.7.
Extensions of !. By 3.24 the set E! of extensions of ! to a map from B(L3={ 1})*
*^2
is a free and transitive H2(J 3; !2)-set. By 3.24 E! ' Z~^2[Z2!(3) = ~], whe*
*re
Z2!(3) is the set defined in 3.22. For simplicity denote Z := Z2!(3), Z := Z2*
*!(3) .
In order to calculate the set Z and prove that b acts trivially on it we need t*
*o de-
scribe the set of irreducible subrepresentations of !. Let Rt(!) (Q^2)3 be th*
*e set
of roots of ! (in dual basis fi00). Obviously WD acts of Rt(!).
Proposition 5.3. The set Rt(!) contains all combinations of roots having the fo*
*rm
w*(1_2, 1_2, 0) for w 2 WD .
Proof.Since the sequence (1, 0, 0) in dual basis fi has the form (1_2, 1_2, 0) *
*in basis
fi00, then the conclusion follows from the definition of ' (4.6) and properties*
* of the
tensor product.
Roots having the form w*(1_2, 1_2, 0) will be called basic roots. The set of *
*all basic
roots will be denoted by BRt(!). Put c = 1_2, d = q-1_4. Notice that c d *
*1_2
mod 2.
Proposition 5.4. We have
BRt(!) = {( c, c, 0), ( c, 0, c), (0, c, c),
( 2d, 0, 0), (0, 2d, 0), (0, 0, 2d),
( (c + d), d, d), ( d, (c + d), d), ( d, d, (c + d))}
Proof.It is the list (4.8) converted into the basis fi00.
Directly from the definition (4.6) we obtain
Corollary 5.5.
8 9
< X =
Rt(!) ~=: l(r1,r2,r3)(r1, r2, r3) : 0 l(r1,r2,r3). 16
(r1,r2,r3)2BRt(!) ;
Proposition 5.6. If even integers lc and ld satisfy inequalities |lc| 16 . 8,*
* |ld|
16 . 14 and |lc- ld| 16 . 14, then (lcc + ldd, 0, 0) 2 Rt(!).
16 KRZYSZTOF ZIEMIA'NSKI
Proof.Introduce a relation 1~on BRt(!) by
(r1, r2, r3) 1~(r01, r02, r03) , r1 = r01^ r2 = -r02^ r3 = -r03.
Let SRt(!) Rt(!) be the set all combinations
X
l(r1,r2,r3)(r1, r2, r3)
(r1,r2,r3)2BRt(!)
such that the coefficients l(r1,r2,r3)are constant on equivalence classes of th*
*e relation
1~. Then SRt(!) contains exactly sequences
(mc(2c) + md(2d) + mc+d(2c + 2d), 0, 0),
where the integers mc, md and mc+d satisfy inequalities -2 . 16 mc 2 . 16,
-5 . 16 md 5 . 16, mc+d = -2 . 16 mc+d 2 . 16.
Proposition 5.7. If odd integers lc, ld satisfy inequalities |lc| 8.16, |ld| *
* 14.16
and |lc- ld| 14 . 16, then (lcc + ldd, c + d, c + d) 2 Rt(!).
Proof.By 5.6 we have ((lc - 1)c + (ld - 1)d, 0, 0) 2 Rt(!) and it may be written
without using basic roots (c + d, d, d) and (0, c, c) (since lc 8 . 16 - 2 an*
*d ld
14 . 16 - 2). Then
((lc-1)c+(ld-1)d, 0, 0)+(c+d, d, d)+(0, c, c) = (lcc+ldd, c+d, c+d) 2 Rt(!).
Elementary arguments lead to the following corollaries
Corollary 5.8. If (l1cc + l1dd, l2cc + l2dd, l3cc + l3dd) 2 Rt(!) is a root wit*
*h (2-adic)
integer coordinates, then for i = 1, 2, 3 we have lic lidmod 2. Moreover, num*
*bers
licand lidsatisfy inequalities analogous to the ones in the formulation of 5.6.
Corollary 5.9.
Rt(!) [ (Z^2)3 = {(l1cc + l1dd, l2cc + l2dd, l3cc + l3dd) : lic, lid2 Z,
lic lid mod 2 ^ |lic| 8 . 16 ^ |lic| 14 . 16 ^ |lic- lid| *
*14 . 16}
Proposition 5.10.
Z ~={(lcc + ldd, *, *), (*, lcc + ldd, *), (*, *, lcc + ldd) : lc, ld 2 Z
^ lc ld mod 2 ^ |lc| 8 . 16 ^ |ld| 14 . 16 ^ |lc- ld| 14 .*
* 16}
Proof.Recall that Z is a set of symbols (j1, *, *), (*, j2, *), (*, *, j3) su*
*ch that
j1~ j2~ j3 !, and ji2 IR+(K) [ IR-(K) for each i. Now the conclusion follows
from 5.9 (we identify a representation indKT1L%k with a 2-adic integer k and o *
*with
0).
Proposition 5.11. There are two equivalence classes of relation ~ on the set Z *
* ,
namely
Z0 = {(lcc + ldd, *, *), (*, lcc + ldd, *), (*, *, lcc + ldd) :
lc, ld 2 2Z ^ |lc| 8 . 16 ^ |ld| 14 . 16 ^ |lc- ld| 14 *
*. 16}
and
Z1 = {(lcc + ldd, *, *), (*, lcc + ldd, *), (*, *, lcc + ldd) :
lc, ld 2 1 + 2Z ^ |lc| 8 . 16 ^ |ld| 14 . 16 ^ |lc- ld| 14 *
*. 16}.
A FAITHFUL UNITARY REPRESENTATION OF THE 2-COMPACT GROUP DI(4) 17
Proof.For even lc, ld, by 5.6 (lcc + ldd, 0, 0) 2 Rt(!) (and therefore (0, lcc +
ldd, 0), (0, 0, lcc + ldd) 2 Rt(!)). Moreover, obviously (0, 0, 0) 2 Rt(!). Then
(lcc + ldd, *, *) ~ (*, 0, *) ~ (0, *, *) ~ (*, lcc + ldd, *) ' (*, *, lcc*
* + ldd)
Then all elements of Z0 are in the same equivalence class as (0, *, *). Similar*
*ly, if
lc and ld are odd, then
(lcc + ldd, c + d, c + d), (c + d, lcc + ldd, c + d),
(c + d, c + d, lcc + ldd), (c + d, c + d, c + d) 2 Rt(!*
*).
Therefore all elements of Z1 are in the same equivalence class as (c + d, *, *).
Proposition 5.12. The action of b on E! is trivial.
Proof.The action of b on the set Z is trivial since b has order 3 and Z has o*
*nly
2 elements. The conclusion follows by 3.24.(b)
Proposition 5.13. The map fG is A-invariant, i.e. it extends to a homotopy
compatible family of maps from the centralizer decomposition functor of BD into
BU(m)^2.
Proof.By 5.12 the map fH = fG |BH^2is GL2(F2)-invariant. The map fG |B(T3o{ 1})*
*^2
is, by Dwyer-Zabrodsky1theorem, the completion of the classifying map of the re*
*pre-
sentation resN1(T3o{ 1})1', which is GL 3(F2)-invariant (since the action of GL*
* 3(F2)
1
is the restriction of the action of the Weyl group). Since resN1{ 1}4' is a su*
*m of
regular representations, then the map fG |B{ 1}4 is GL 4(F2)-invariant.
6.Proof of the main theorem
Let F : A ! Sp be the centralizer decomposition diagram of BD and let Ai,
i = 1, . .,.4 be objects of A. We have proven that the map
fG : F (A1) ' BG^2! BU(m)^2
is A-invariant. Now we have to check that the obstructions to the existence of *
*an
extension BDI(4) ! BU(m)^2in groups Hi+1(A; i) vanish, where
fi(Ar) := ssi(map (F (Ar), BU(m)^2)fG|F(Ar)).
Oliver [14] provided a powerful tool for calculating this kind of cohomology
groups:
Theorem 6.1. ([14, Thm. 1, Prp. 6]) Let A := Ap(X) be the centralizer decompo-
sition category of a p-compact group X. Let Aibe the set of objects of A which *
*have
the form B(Z=p)i ! BX. Then for any Z^p[A]-module F there is a isomorphism
H*(A; F ) ' C*St(F ), where
Y
CiSt(F ) ~= Hom AutA(A)(StA , F (A))
A2Ai+1
However, this theorem applies only to the case when the coefficient functor h*
*as
abelian values (obviously i(Ar) is abelian for i > 1). For r = 3, 4 the spaces
F (Ar) are 2-completed classifying spaces of 2-toral groups. Therefore, by 1.2 *
*the
spaces map (F (Ar), BU(m)^2)fG|F(Ar)are classifying spaces of products of unita*
*ry
groups. Hence f1(Fr2) = 0. For r = 1, 2 this argument fails, since the spaces *
*F (A1)
and F (A2) are not classifying spaces of 2-toral groups.
18 KRZYSZTOF ZIEMIA'NSKI
Proposition 6.2. The fundamental group of map (BH^2, BU(m)^2)fH is abelian.
Proof.By the Dwyer-Zabrodsky Theorem (1.2) we have
map (BH^2, BU(m)^2)fH ~=map (hocolimH=P2J 3(EH xH =P )^2, BU(m)^2)fH
= holimH=P2J 3map((EH xH H=P )^2, BU(m)^p)fH|BP^2
' (holimH=P2J 3BCU(m)(!(P 1))^2)^2
The second term of the Bousfield spectral sequence [3, XI,7.1], which converges*
* to
the homotopy groups of the homotopy inverse limit above has the second term
Ep,q2= Hp(J 3; ssp+q(map (EH xH H=(-), BU(n)^2)fH|B(-)^2)
= Hp(J 3; ssp+q(BCU(m)('((-)1 )^2)) ) q(A2)
The spaces BCU(m)('((-)1 )^2are products of 2-completed classifying spaces of
unitary groups (cf. Section 1), where the number of multiplies1is the number of
isomorphism classes of irreducible subrepresentations of resNGP1' and the rank *
*of
every one is the multiplicity of the corresponding irreducible representation. *
*Then
the groups ssp+q(BCU(m)('((-)1 )^2) are abelian. Moreover, for p + q = 1, 3 the*
*y1
vanish and for p+q = 2, 4 they are isomorphic to !2, where as before ! = resNG*
*N1H',
since all irreducible subrepresentations of ' appear with multiplicity at least*
* 16.
By 3.20 we have H3(J 3; !2) = 0. Thus Ep,12= 0 for p 6= 1. Therefore f1(A2) i*
*s a
subquotient of the abelian group E1,12and hence is abelian.
The following proposition is a straightforward consequence of 6.1:
Proposition 6.3. For any Z^2[A]-module M there is an isomorphism
H*-1(A; M) ~=H*(Hom GL*(F2)(StGL*(F2), M(A*))),
where St is the Steinberg module of the group .
In particular, Hi(A; M) = 0 for i > 3.
Proposition 6.4. For each i 1 holds Hi+1(A; fi) = 0.
Proof.If i 3, then the conclusion is obvious, so it is sufficient to consider*
* cases
i = 1, 2 only. Note that the full subcategory of A with objects A1 and A2 is
isomorphic to J . Since fi(Ar) = 0 for r = 3, 4, then we have H2(A; f1) ~=
H2(J ; resAJ f1) = 0. If i = 2, then by 6.3 H3(A; f2) is a quotient of the gro*
*up
Hom GL4(F2)(StGL4(F2), f2(A4)).
A Z^2-module f2(A4) = ss2(map (B{ 1}4, BU(m)^2)resN11 ) is free and has di-
{ 1}4'
mension not larger than 24 (there are 24 isomorphism classes of irreducible rep-
resentations of { 1}4). The Steinberg module StGL4(F2)is a second homology
group of a geometrical realization of a poset of all non-trivial proper subspac*
*es
of F42. By an Euler characteristic argument it has dimension 64 and is an ir-
reducible Z^2[GL 4(F2)]-module (cf. [16]). Hence there is no non-zero homomor-
phism StGL4(F2)! f2(A4). Hence Hom GL4(F2)(StGL4(F2), f2(A4)) = 0 and thus
H3(A; f2) = 0.
As a consequence we obtain the main theorem of this paper:
A FAITHFUL UNITARY REPRESENTATION OF THE 2-COMPACT GROUP DI(4) 19
Theorem 6.5. The map fG : BG^2! BU(m)^2extends to a faithful complex
representation of the 2-compact group DI(4).
Proof.By 6.4 the map fG extends to BDI(4). By 4.7 and [8, 3.2] the extension is
a classifying map of a monomorphism of 2-compact groups.
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