ContemporaryVMathematicsolume 00, 0000
Extensions of p-compact Groups
JESPER MICHAEL MOLLER
Abstract.The classification of short exact sequences of p-compact groups
and of rational isomorphisms of not necessarily connected p-compact grou*
*ps
is discussed.
1. Introduction
The concept of a p-compact group was introduced by Dwyer and Wilkerson
[6] as a homotopy theoretic version of a compact Lie group. In a subsequent
paper [7], they showed that the center of any p-compact group agrees with the
centralizer of the identity map. That result is the starting point of this note.
For given p-compact groups , X and Y , let Ext(X; Y ) denote the set of equi*
*v-
alence classes of short exact sequences
Y -! G -! X
of p-compact groups. Two such extensions of X by Y are declared equivalent if
there exists a homomorphism over X and under Y between them.
The discussion of Ext(X; Y ) proceeds along two parallel tracks. One track
is concerned with the case where Y is a (completely reducible [10 ]) connected
p-compact group while the other track deals with the case where Y = Z is
an abelian p-compact (toral) group. For fixed homotopy actions ae and i of
ss0(X) on Y and Z let Extae(X; Y ) Ext(X; Y ) and Exti(X; Z) Ext(X; Z)
denote the subsets of extensions realizing the actions ae and i, respectively. *
*As is
quickly seen, Exti(X; Z) is an abelian group and it turns out [Theorem 3.3] that
Extae(X; Y ) is an affine group with ExtZae(X; Z(Y )) as group of operators. He*
*re
Z denotes the conjugation action of the group of self-homotopy equivalences of
BY on the classifying space BZ(Y ) = map (BY; BY )B1 [7, Theorem 1.3] of the
center Z(Y ) of Y [12 , 7].
____________
1991 Mathematics Subject Classification. 55P35, 55S37.
Key words and phrases. Universal fibration, center, extension.
cO00000American0Mathematical0Societ*
*y0-0000/00 $1.00 + $.25 per page
1
2 J. M. MOLLER
The abelian group Exti(X; Z) enjoys nice bifunctorial properties. The affine
group Extae(X; Y ) is functorial in X by pull back but only restricted functori*
*al
in Y : Any equivariant rational isomorphism g :Y ! Y 0, where Y 0is a connected
p-compact group locally isomorphic to Y equipped with a homotopy action ae0
by ss0(X), induces a push forward map
g*: Extae(X; Y ) ! Extae0(X; Y 0)
which is affine [Lemma 3.7]. Given also a homomorphism h: X ! X0 , pull back
and push forward of extension classes provide obstructions to the existence of
homomorphisms
Y _____//_GO___//_X
g|| O h||
|fflffl fflfflO fflffl|
Y 0_____//G0____//X0
under g and over h. Indeed, such a homomorphism exists [Theorem 3.8] if and
only if g*(G) = h*(G0) in Ext(X; Y 0).
Fibrewise discrete approximations to fibered abelian p-compact groups are
briefly discussed in Section 4. Lemma 4.1-4.2 show that provided the identity
component X0 of X is simply connected, there is a group isomorphism
Exti(X; Z) ~=H2i(ss0(X); Z)
where Z is the discrete approximation to Z.
The above concepts are exploited in the final section for the classification*
* of
rational automorphisms of not necessarily connected p-compact groups. When
combined with [11 , Theorem 3.3] [10 , Theorem 3.5], the short exact sequence of
Theorem 5.2 could potentially lead to a fairly explicit classification of ratio*
*nal
automorphisms of any given p-compact group.
2. Universal fibrations
Thanks to the homotopy equivalence [7, Theorem 1.3] between the center
and the centralizer of the identity map of a p-compact group, the classification
of fibrations with p-compact group classifying spaces as fibres is surprisingly
manageable.
Let's first fix some notation. For any two p-compact groups, X and X0, put
Hom (X; X0) = [BX; *; BX0], the set of based homotopy classes of maps, and
Rep (X; X0) = [BX; BX0], the set of unbased homotopy classes of maps. A
homomorphism h 2 Hom (X; X0) is said to be a rational isomorphism if [11 ,
Definition 1] the map
H*(Bh0; Zp) Zp Qp: H*(BX00; Zp) Zp Qp ! H*(BX0; Zp) Zp Qp ;
induced by the restriction Bh0: BX0 ! BX00 of h to the identity components,
is an isomorphism. Let "Q(X; X0) Hom (X; X0) denote the subset of rational
isomorphisms.
EXTENSIONS OF p-COMPACT GROUPS 3
If X = X0, End(X) = Hom (X; X) is a monoid (under composition) containing
"Q(X) = "Q(X; X)as a submonoid and having Aut(X) as its group of invertible
elements. Out (X) denotes the invertible elements of the monoid Rep(X; X). If
X is connected or abelian, there is no difference between the based or unbased
case: End (X) = Rep(X; X) and Aut(X) = Out(X).
Turning to classifying fibrations, let Y be a p-compact group with center
[12 , 7] Z(Y ) and adjoint form P (Y ) = Y=Z(Y ). Then Z(Y ) is an abelian p-
compact toral group, P (Y ) is a centerfree p-compact group, and there exists a
fibration
BZ(Y ) -! BY -! BP (Y )
of classifying spaces. Using a Borel construction as in the proof of [6, Propos*
*ition
8.3] this fibration may be extended one step further to the right to give a fib*
*ration
(2.1) BY -! BP (Y ) Bk--!B2Z(Y )
which is universal for fibrations with fibre BY over simply connected base spac*
*es
[7, Remark 1.11].
Assume from now on that Y is connected and let g :Y ! Y 0be a rational
isomorphism into another connected p-compact group Y 0locally isomorphic [11 ,
Definition 3] to Y . Then g induces [11 , Corollary 3.2, Theorem 3.3] a fibre m*
*ap
BZ(Y ) _____//_BY_____//_BP (Y )
BZ(g)|| Bg || BP(g)||
fflffl| fflffl| fflffl|
BZ(Y 0)_____//BY 0____//BP (Y 0)
which also extends one step to the right.
Lemma 2.1. Any rational automorphism g of Y extends to a fibre self map
BY _____//_BP (Y_)Bk_//B2Z(Y )
Bg|| BP(g)|| |B2Z(g)|
fflffl| fflffl| fflffl|
BY 0_____//BP (Y 0)Bk_//B2Z(Y 0)
of the universal fibration ( 2.1).
Proof. The claim is that B2Z(g)OBk and Bk OBP (g) are homotopic. Since
looping provides a bijection : [BP (Y ); B2Z(Y 0)] ! [P (Y ); BZ(Y,0)]this fol-
lows from the extension one step to the left of the fibre map (BZ(g); Bg; BP (g*
*))
shown above. __|_|
The fibration which is universal for fibrations with fibre BY over arbitrary
base spaces has the form
BkhOut(Y )2
(2.2) BY -! BP (Y )hOut(Y )------! B Z(Y )hOut(Y )
4 J. M. MOLLER
where Out(Y ) = ss0aut(BY; *) = ss0aut(BY ) is the group of homotopy classes of
homotopy self-equivalences of BY and the homotopy orbit space BP (Y )hOut(Y )
(B2Z(Y )hOut(Y))denotes the classifying space of the group-like topological
monoid aut(BY; *) (aut(BY )) of based (free) homotopy self-equivalences of BY .
The monodromy action associated to the homotopy orbit space B2Z(Y )hOut(Y )
is induced from the conjugation action of Out(Y ) on BZ(Y ) ' map (BY; BY )B1,
i.e. from the action Z :Out (Y ) ! Out(Z(Y ))of [11 , Corollary 3.2].
Suppose now that the locally isomorphic p-compact groups, Y and Y 0, are
equipped with homotopy actions, ae: ss ! Out(Ya)nd ae0:ss0! Out(Y 0), by dis-
crete groups, ss and ss0.
Pulling back the universal fibration (2.2) along the maps Bae: Bss ! BOut (Y*
*,)
Bae0:Bss0! BOut (Y 0)produces fibrations
Bkhaess 2
BY _________//BP (Y )haess_____//B Z(Y )haess
| O O
Bg || OO OO
fflffl| fflffl Bkhae0ss0 fflffl
BY 0 _______//_BP (Y 0)hae0ss0_//_B2Z(Y 0)hae0ss0
that are universal for fibrations with fibre BY and with monodromy action
restricting to ae, ae0. The projection map Bkhaess, Bkhae0ss0is a map over Bss,
Bss0 since the universal projection map BkhOut(Y )is a map over BOut (Y ).
Thus the first obstruction to extending Bg to a fibre map Bkhaess-!Bkhae0ss0
is that g be O-equivariant, i.e. g . ae(fl) = ae0(O(fl)) . g holds in "Q(Y; Y 0*
*)for all
fl 2 ss, for some group homomorphism O: ss ! ss0. Provided the mapping space
map (BP (Y ); BP (Y 0))BP(g) is contractible, as is this case if Y and Y 0are c*
*om-
pletely reducible [10 ] p-compact groups, this is in fact the only obstruction *
*to
extending.
Lemma 2.2. Suppose that g 2 "Q(Y; Y 0)is a O-equivariant rational isomor-
phism between the locally isomorphic completely reducible p-compact groups Y
and Y 0.
(i)There exists, up to vertical homotopy, exactly one extension
BP (g)hO: BP (Y )haess! BP (Y 0)hae0ss0of BP (g): BP (Y ) ! BP (Y 0)to a
map over BO.
(ii)There exists, up to vertical homotopy, exactly one extension
B2Z(g)hO: B2Z(Y )haess! B2Z(Y 0)hae0ss0of B2Z(g): B2Z(Y ) ! B2Z(Y 0)
to a map over BO such that
Bkhaess
BP (Y )haess______//B2Z(Y )haess
BP(g)hO|| B2Z(g)hO||
fflffl| fflffl|
BP (Y 0)hae0ss0Bk_//_B2Z(Y 0)hae0ss0
hae0ss0
commutes up to vertical homotopy.
EXTENSIONS OF p-COMPACT GROUPS 5
The proof is based on the fibred mapping space construction occuring e.g. in
[1, 2]:
Let p: U ! A and q :V ! B be fibrations over connected and pointed base
spaces. Suppose that g :p-1(*) ! q-1(*)is a map between the fibres and
h: (A; *) ! (B; *)a map between the base spaces such that the pair (g; h) re-
spects the monodromy action in the sense that g . i = ss1(h)(i) . g holds in
[p-1(*); q-1(*)] for all i 2 ss1(A; *). The question of whether (g; h) comes fr*
*om
a fibre map can be turned into a section problem.
Define the set
a
fibmap(U; V )gh= map (p-1(a); q-1(h(a)))ga
a2A
where ga 2 [p-1(a); q-1(h(a))] is the homotopy class making
g -1
p-1(*)______//_q (*)
i|| h(i)||
|fflffl fflffl|
p-1(a)__ga_//q-1(h(a))
homotopy commutative for any path i from the base point * to a 2 A. Equipped
with topology as in [1] we obtain a fibration
map (p-1(*); q-1(*))g -!fibmap(U; V )gh-! A
whose based section space is [1, Proposition 2.1] the space of maps of U into V
under g and over h. Of course, fibmap(U; V )gh= fibmap(U; h*V )g1where h*V is
the pull back of V along h and 1 denotes the identity map of A.
Proof of Lemma 2.2. Composition with the maps Bkhaessand Bkhae0ss0in-
duces, since Bk O BP (g) ' B2Z(g) O Bk by Lemma 2.1, fibre maps
map (BP (Y ); BP (Y 0))BP(g)__//fibmap(BP (Y )haess; BP (Y 0)hae0ss0)BP(g)BO/*
*/_Bss
||
Bk_|| || ||
fflffl| fflffl| ||||
map(BP (Y ); B2Z(YO0))B(kP(g))//_fibmap(BPO(Y )haess; B2Z(Y 0)hae0ss0)B(kP(g))*
*BO//_Bss
| OO| ||
___Bk'| | ||
| | ||
| ||2
map (B2Z(Y ); B2Z(Y 0))B2Z(g)//_fibmap(B2Z(Y )haess; B2Z(Y 0)hae0ss0)BBZ(g)O//*
*_Bss
___
of fibred mapping spaces. The map Bk is easily seen to be a homotopy equiva-
lence and the fibre map (BP (Y ); BP (Y 0))BP(g) is contractible [7, Theorem 1.*
*3]
[10 , Theorem 3.11] since Y and Y 0and with them their adjoint forms P (Y )
and P (Y 0) are completely reducible. Thus there exists up to vertical homotopy
exactly one section of the upper fibration inducing a corresponding section of
the lower fibration. __|_|
6 J. M. MOLLER
Note that the fibre map (BP (g)hss; B2Z(g)hss) of point (ii) of Lemma 2.2 is
an extension of the fibre map (BP (g); B2Z(g)) of Lemma 2.1 and thus restricts
to the map Bg :BY ! BY 0 on the fibres.
The above constructions pertaining to the connected p-compact groups can
also be carried out for abelian p-discrete or p-compact toral groups [6, Defini*
*tion
6.3, Definition 6.5].
Let Z be an abelian p-discrete toral group and Z its closure [6, Definition
6.6]. The group Aut(Z ) of abelian group automorphisms of Z acts by based
homeomorphisms on B2Z so we may apply the Borel construction to the path
fibration P B2Z -! B2Z to obtain the fibration
(2.3) B2Z -! (P B2Z )hAut(Z)-oe0!(B2Z )hAut(Z)
which is universal for fibrations with BZ as fibre. Note that both the total
space and the base space are spaces over and under BAut (Z ) and that the
projection map oe0is a map over and under BAut (Z ). Since p-completion induces
isomorphisms End (Z ) -! End(Z) and Aut(Z ) -! Out(Z) [12 , Proposition 3.2],
fibrewise completion of (2.3) results in the fibration
(2.4) BZ -! BOut (Z) oe0-!(B2Z)hOut(Z)
which is universal for fibrations with BZ as fibre. The projection map oe0 is a
map of spaces over and under BOut (Z).
The abelian group structure on Z induces on B2Z the structure of an
abelian topological group. Let r :B2Z x B2Z ! B2Z be the addition map and
: B2Z ! B2Z the inversion map such that r O o = r and r O ( x 1) O = 0
where o is the switch and the diagonal map. The p-completions of r and ,
r: B2Z x B2Z ! B2Z and :B2Z ! B2Z , promote B2Z to an abelian group-
like space. Moreover, since Aut(Z ) acts on B2Z through group isomorphisms,
r and extend to maps over and under BAut (Z )
(2.5) r : *(B2ZhAut(Z)x B2ZhAut(Z)) ! B2ZhAut(Z)
(2.6) : B2ZhAut(Z)! B2ZhAut(Z)
where is the diagonal on BAut (Z ). The fibrewise p-completion of these maps
are maps over and under BOut (Z)
(2.7) r: *(B2ZhOut(Z)x B2ZhOut(Z)) ! B2ZhOut(Z)
(2.8) :B2Z ! B2Z
extending the structure maps r and on B2Z.
Suppose now that Z0is another p-discrete toral group and that Z and Z0sup-
port group actions i :ss ! Aut(Z,)i0:ss0! Aut(Z0). Any O-equivariant abelian
group homomorphism _:Z ! Z0 extends to a topological group homomorphism
EXTENSIONS OF p-COMPACT GROUPS 7
B2_:B2Z ! B2Z0 and thus to a map
(2.9) B2_hO:B2Zhiss! B2Z0hi0ss0
over and under BO such that
r0 O *(B2_hOx B2_hO) = B2_hOO r; 0 O B2_hO = B2_hOO
where r0 and 0 are the structure maps for B2Z0.
Let Z0 denote the ablian p-compact toral group which is the closure of Z0.
Fibrewise p-completion of B2_hO is a map
(2.10) B2jhO: B2Zhiss! B2Z0hi0ss0
over and under BO such that
(2.11) r0O *(B2jhO x B2jhO) = B2jhO O r; 0O B2jhO = B2jhO O
where r0and 0 are the structure maps on B2Z0.
3. Short exact sequences
This section contains information about fibrations of p-compact group classi-
fying spaces.
Let X and Y be p-compact groups with classifying spaces BY and BX and
let cdFp(-)denote mod p cohomological dimension [6, Definition 6.13].
Lemma 3.1. Let BY ! BG ! BX be a fibration sequence. Then G is a
p-compact group and cdFp(G)= cdFp(X)+ cdFp(Y ).
Proof. As the base space as well as the fibre are p-complete spaces, the Fib*
*re
lemma [4, II.5.1-5.2] implies that also the total space BG is p-complete.
Let Y0 denote the identity component of Y . By pulling back the fibration
G ! X ! BY to the universal covering space BY0 we obtain a fibration denoted
G ! X|BY0 ! BY0. Extending this fibration one step to the left gives the
fibration Y0 ! G ! X|BY0 with connected fibre. The action of the fundamental
group of any component of the base on Hi(Y0; Fp); i 0, is nilpotent because
it factors through the finite p-group ss0(Y ) (acting on Hi(Y ; Fp)). Hence [6,
Lemma 6.16] the corresponding Serre spectral sequence is concentrated in a
rectangle of dimensions cdFp(X) by cdFp(Y )and the group in the upper right
corner is nontrivial. The fact that G is Fp-finite and the formula for its mod p
cohomological dimension now follows as in the proof of [6, Proposition 6.14].
This shows [6, Lemma 2.1, Definition 2.2] that G is a p-compact group. __|_|
It is a consequence of Lemma 3.1 that the composition of two epimorphisms
[6, 3.2] is an epimorphism.
A fibration BY ! BG ! BX of based maps where X and Y (and hence also
G) are p-compact groups is called [6, 3.2] a short exact sequence of p-compact
groups and is often referred to simply as the short exact sequence Y -! G -! X.
8 J. M. MOLLER
Let Ext(X; Y ) denote the set of equivalence classes of all such p-compact group
short exact sequences. Here, Y -! G -! X and Y -! H -! X are considered
equivalent if there exists a homomorphism of the form
Y _____//G_____//X
|| | ||
|| | ||
|| fflffl|||
Y _____//H_____//X
between them.
Associated to the short exact sequence Y -! G -! X is a homotopy action
ae: ss0(X) ! Out(Y.)Observe that this monodromy action is an invariant of the
equivalence class so that it makes sense to let Extae(X; Y ) denote the subset *
*of
Ext(X; Y ) represented by all short exact sequences realizing the action ae.
Assume from now on that Y is a connected p-compact group.
Let [BX; B2Z(Y )hOut(Y])Baedenote the set of vertical homotopy classes of li*
*fts
B2Z(Y4)hOut(Y4)
i i
i ii |
i i |
i i ii fflffl|
BX _Bss0//_Bss0(X)_Bae_//_BOut(Y )
of the map Bae O Bss0.
Similarly, if Z is an abelian p-compact group with discrete approximation
Z -! Z and i :ss0(X) ! Aut(Z ) = Out(Z)an action, let [BX; B2ZhAut(Z)]Bi
and [BX; B2ZhOut(Z)]Bi denote the sets of vertical homotopy classes of lifts of
Bi O Bss0.
Define Exti(X; Z) to be the set of equivalence classes (with respect to fibre
homotopy equivalences under BZ and over BX) of fibrations BZ -! BG -! BX
with monodromy action i.
Lemma 3.2. Let Y be a connected and X any p-compact group. Then there
are bijections
[BX; B2Z(Y )hOut(Y])Bae-!Extae(X; Y )
[BX; B2ZhAut(Z)]Bi -! Exti(X; Z)
[BX; B2ZhOut(Z)]Bi -! Exti(X; Z)
defined by pulling back the universal fibrations ( 2.2), ( 2.3), and ( 2.4), re*
*spec-
tively.
Proof. The base space of the fibration of based mapping spaces
map *(BX; B2Z(Y )hOut(Y))-! map*(BX; BOut(Y ))
EXTENSIONS OF p-COMPACT GROUPS 9
is homotopically discrete. Therefore, the total space is homotopically equivale*
*nt
to the disjoint union over all homomorphisms ae: ss0(X) ! Out(Yo)f the spaces
of based lifts of Bss0 O Bae.
By classification theory, pull back of the universal bundle (2.2) provides a
bijection
ss0(map *(BX; B2Z(Y )hOut(Y))-! Ext(X; Y )
which by the above remarks restricts to a bijection between the based and verti*
*cal
homotopy classes of lifts of Bss0 O Bae and Extae(X; Y ). However, since the fi*
*bre
B2Z(Y ) is simply connected, the clause that the maps be based is superfluous.
Similar arguments apply in the remaining two cases. __|_|
For the following, assume that Y -! G -! X and Y -! H -! X are two
short exact sequences realizing the same homotopy action ae: ss0(X) ! Out(Y.)
Choose [Lemma 3.2] based lifts (also denoted) G; H :BX ! B2Z(Y )hOut(Y )of
Bae O Bss0 classifying the two fibrations. Define
B2Z(Y ) -! B(H; G) -! BX
to be the fibration whose fibre over any point b 2 BX is the space of vertical
(i.e. having constant projection in BOut (Y )) paths in B2Z(Y )hOut(Y )from G(b)
to H(b). This fibration represents an element in ExtZae(X; Z(Y )).
Theorem 3.3. Let G 2 Extae(X; Y ) where Y is a connected and X an arbi-
trary p-compact group. Then the map
(-; G): Extae(X; Y ) ! ExtZae(X; Z(Y ))
is a bijection.
Proof. Pulling back to BX the two fibrations shown as downward pointing
arrows in the diagram
B2Z(Y8)hOut(Y8) B2Z(Y )hOut(Z(YO))O
Gqqqqq | oe ||
qqq | 0||
qqq |fflffl fflffl||
BX ___B(aes//s0)_BOut(Y_)BZ___//BOut(Z(Y ))
provides two sectioned fibrations, B2Z(Y )haess0X-!BX and B2Z(Y )hZaess0X-!
BX, with fibre B2Z(Y ). These two spaces over and under BX are equivalent
in the sense that there exists up to homotopy over and under BX exactly one
extension of the identity map of B2Z(Y ) to a map
uG :B2Z(Y )haess0X! B2Z(Y )hZaess0X
over and under BX. This follows from the fact that the fibre of the based fibred
mapping space
21
map *(B2Z(Y ); B2Z(Y ))B21 -!fibmap*(B2Z(Y )haess0X; B2Z(Y )hZaess0X)BB1 -!BX
10 J. M. MOLLER
is contractible. Composition with uG induces a bijection
~=
(uG )*: Extae(X; Y ) -! ExtZae(X; Z(Y ))
identical to the map (-; G). __|_|
It is a consequence of Theorem 3.3 that Extae(X; Y ) is an affine group [3, *
*x9,
no 1].
To see this, note that the structure maps (2.7) and (2.8) make Exti(X; Z)
into an abelian group. The neutral element of this group is represented by the
short exact sequence Z -! Zo iX -! X classified by the map oe0 O Bi O Bss0.
The sum of two short exact sequences Z -! A -! X and Z -! B -! X, with
classifying maps A; B :BX ! B2ZhOut(Z) , is the short exact sequence classified
by the lift r O (A x B) O of Bi O Bss0. The inverse -A is classified by O A.
Note that the projection B(Zo iX) -! BX admits a based section, i.e. that
the short exact sequence
(3.1) Z _____//Zo iX_____//Xoo_
is a split short exact sequence, and that there exist homomorphisms
Z x Z _____//*(A x B) _____//X
(3.2) r || || ||||
fflffl| |fflffl ||
Z _________//A + B_______//X
Z _____//_A____//_X
(3.3) || || ||||
fflffl| fflffl| ||
Z ____//_-A____//_X
of short exact sequences. These properties are characterizing.
Lemma 3.4. Suppose that Z -! C -! X is a short exact sequence representing
an element of Exti(X; Z).
(i)If there exists a splitting
Z _____//C____//Xoo_
then C = 0 in Exti(X; Z).
(ii)If there exists a short exact sequence homomorphism
Z x Z _____//*(A x B)_____//X
r || || ||||
fflffl| fflffl| ||
Z ___________//C________//X
then C = A + B in Exti(X; Z).
EXTENSIONS OF p-COMPACT GROUPS 11
(iii)If there exists a short exact sequence homomorphism
Z _____//A____//X
|| || ||||
fflffl|fflffl| ||
Z _____//C____//X
then C = -A in Exti(X; Z).
Proof. (i) The based fibred mapping space
map *(BZ; BZ)B1 -! fibmap*(B(Zo iX); BC)B1B1-!BX
admits a section because its fibre is contractible.
(ii) Precomposition with the short exact sequence homomorphism (3.2) deter-
mines a fibre map
map (BZ; BZ)B1 _________//fibmap(B(A + B); BC)B1B1___//_BX
__r'| | ||||
| | ||
fflffl| fflffl| ||
map (BZ x BZ; BZ)r _____//fibmap(B*(A x B); BC)rB1____//_BX
__
which is a fibre homotopy equivalence since r is a homotopy equivalence. As
the lower fibration admits a section, so does the upper one.
(iii) Similar to (ii). __|_|
In case Z = Z(Y ) is the center of the connected p-compact group Y , the dif-
ference map from Theorem 3.3 and the additive structure in ExtZae(X; Z(Y ))
are nicely related.
Lemma 3.5. Let G; H; K 2 Extae(X; Y ). Then
(K; G) = (K; H) + (H; G)
in ExtZae(X; Z(Y )).
Proof. Since composition of paths defines a map
*(B(H; G) x B(K; H)) -! B(K; G)
over BX and under the H-space structure on B2Z, this formula follows from
Lemma 3.4. __|_|
The formula of Lemma 3.5 implies that
(G; G) = Z(Y )o ZaeX; -(H; G) = (G; H)
for all G; H 2 Extae(X; Y ). More formally
Corollary 3.6. Extae(X; Y ) is an affine group with the abelian group
ExtZae(X; Z(Y )) as its group of operators.
12 J. M. MOLLER
Let's now look at functorial properties of the Ext-affine groups.
Let Y 0-! G0 -! X0 be another short exact sequence of p-compact groups
with associated homotopy action ae0:ss0(X0) ! Out(Y.0)Any p-compact group
homomorphism h: X ! X0 induces a map
(3.4) h*: Extae0(X0; Y 0) ! Extae0ss0(h)(X; Y 0)
defined by pull back. Note that h*(G0) is indeed a p-compact group by Lemma 3.1
and that h extends to a morphism
Y 0____//h*(G0)____//_X
|| | |
|| | h|
|| fflffl| fflffl|
Y 0______//G0______//X0
of short exact sequences.
As to functorial properties in the second variable, assume now that Y 0is
connected, completely reducible, locally isomorphic to Y , and that g :Y ! Y 0
is a rational isomorphism which is O-equivariant for some group homomorphism
O: ss0(X) ! ss0(X0). Let
(3.5) g*: Extae(X; Y ) ! Extae0O(X; Y 0)
be the map induced by composing classifying maps with the essentially
uniquely determined map B2Z(g)hO: B2Z(Y )haess0(X)! B2Z(Y 0)hae0Oss0(X)from
Lemma 2.2. Note that the rational isomorphism g extends to a short exact
sequence homomorphism
Y ______//_G______//X
g|| || ||||
fflffl| fflffl| ||
Y 0_____//g*(G)____//X
where the middle arrow is induced from BP (g)hO.
There are similar functorial properties in the abelian case. Let *
* Z
and Z0 be abelian p-compact toral groups equipped with homotopy ac-
tions i :ss0(X) ! Out(Z), i0:ss0(X0) ! Out(Z0). Pull back along the map
Bh: BX ! BX0 induces a map
h*: Exti0(X0; Z0) ! Exti0ss0(h)(X; Z0)
which clearly is a group homomorphism. Also, if j :Z ! Z0 is a O-equivariant
homomorphism, composition with the map B2jhO: B2Zhiss0(X)! B2Z0hi0Oss0(X)
over and under Bss0(X) from (2.10) induces
j*: Exti(X; Z) ! Exti0O(X; Z0)
which is a group homomorphism by the identities (2.11).
EXTENSIONS OF p-COMPACT GROUPS 13
The O-equivariant rational isomorphism g :Y ! Y 0induces [11 , Corollary 3.2]
a O-equivariant rational isomorphism Z(g): Z(Y ) ! Z(Y 0).
Lemma 3.7. Let h: X ! X0 be a homomorphism and g :Y ! Y 0a O-
equivariant rational isomorphism.
(i)The pull back ( 3.4) along h is an affine map with
h*: ExtZae0(X0; Z(Y 0)) ! ExtZae0ss0(h)(X; Z(Y 0))
as its corresponding operator group homomorphism.
(ii)The push forward ( 3.5) along g is an affine map with
Z(g)*: ExtZae(X; Z(Y )) ! ExtZae0O(X; Z(Y 0))
as its corresponding operator group homomorphism.
Proof. (i) It is immediate that (h*G0; h*H0) = h*(G0; H0) for all G0; H0 2
Extae0(X0; Y 0).
(ii) In the diagram
B2Z(Y )haess0X__uG___//B2Z(Y )Zaess0X
B2Z(g)hX|| |B2Z(g)hX|
fflffl| fflffl|
B2(Y 0)hae0Oss0Xug*(//G)_B2Z(Y 0)hZae0Oss0X
the left vertical map is the one defined in Lemma 2.2 and the right vertical
map is, despite the notational coincidence, the one defined in formula (2.10).
However, all maps in this diagram are maps over and under BX and as such
maps are essentially unique, cfr. the proof of Theorem 3.3, B2Z(g)hX O uG and
ug*G O B2Z(g)hX are homotopic over and under BX. Hence Z(g)*(-; G) =
(g*(-); g*G). __|_|
For the final result of this section, suppose that the rational isomorphism
g :Y ! Y 0is ss0(h)-equivariant such that push forward along g and pull back
along h
*
Ext ae(X; Y ) g*-!Extae0ss0(h)(X; Y 0) h- Ext ae0(X0; Y 0)
have the same target.
Theorem 3.8. Assume that Y and Y 0are locally isomorphic connected, com-
pletely reducible p-compact groups. Then there exists an extension homomor-
phism of the form
Y _____//_G____//_X
g|| || h||
|fflffl fflffl| fflffl|
Y 0_____//G0____//X0
if and only if g*(G) = h*(G0) in Extae0ss0(h)(X; Y 0).
14 J. M. MOLLER
Proof. Precomposition with the map BG -! B(g*G) under Bg and over
BX and postcomposition with the map B(h*(G0)) -! BG0under BY 0and over
Bh induce a fibre map
map(BY 0; BY 0)B1____//fibmap(B(g*G); B(h*(G0))B1B1_//BX
___Bg| | ||||
| | ||
fflffl| fflffl| ||
map (BY; BY 0)Bg_________//_fibmap(BG; BG0)BgBh_____//BX
of fibred mapping spaces. Since Y and Y 0are completely reducible, this is a
fibre homotopy equivalence [11 , Theorem 3.11]. Hence one of the two fibrations
admits a section if and only if the other one does. __|_|
A Lie group version of the material contained in this section can be found in
Notbohm [13 ].
4. Approximations
In this section obstruction theory is used to equate Ext-affine groups in ce*
*rtain
advantageous situations.
As in the previous sections, X is any p-compact group, Y is a connected
p-compact group, Z is an abelian p-compact (toral) group with discrete ap-
proximation Z, and ae: ss0(X) ! Out(Ya)nd i :ss0(X) ! Aut(Z ) = Out(Z)are
homotopy actions.
Lemma 4.1. Suppose that the identity component X0 of X is simply con-
nected. Then the component homomorphism ss0: X ! ss0(X)induces bijections
ss*0:Extae(ss0(X); Y ) ! Extae(X; Y )
ss*0:Exti(ss0(X); Z) ! Extae(X; Z)
of equivalence classes of extensions.
Proof. Since the composite map BX0 -! BX -! Bss0(X) is nonessential,
the homotopy orbit space B2Z(Y )hX0 ' B2Z(Y ) x BX0 and the homotopy
fixed point space B2Z(Y )hX0 ' map (BX0; B2Z(Y )) ' B2Z(Y ) because BX0 is
3-connected by Browder [5] [12 , Corollary 5.6]. Hence [6, Lemma 10.5, Remark
10.8]
B2Z(Y )hX ' (B2Z(Y )hX0)haess0(X)' B2Z(Y )haess0(X)
and these homotopy equivalences induce bijections
Extae(X; Y ) = ss0(B2Z(Y )haess0X) = ss0(B2Z(Y )haess0(X)) = Extae(ss0(X); Y*
* )
of Ext-sets. This proves the lemma for extensions of X by Y ; extensions of X
by Z are handled similarly. __|_|
EXTENSIONS OF p-COMPACT GROUPS 15
The structure maps (2.5) and (2.6) on B2ZhAut(Z)make Exti(X; Z) into an
abelian group and the map
e*: Exti(X; Z) ! Exti(X; Z);
induced by fibrewise completion e: B2ZhAut(Z)! B2ZhOut(Z), into an abelian
group homomorphism.
The next lemma shows that extensions of X by Z have unique fibrewise dis-
crete approximations if the identity component of X is semisimple.
A connected p-compact group is said to be semisimple if its fundamental group
or, equivalently [12 , Theorem 5.3], its center is finite.
Lemma 4.2. The above group homomorphism e* is surjective and also injec-
tive provided the identity component X0 of X is semisimple.
Proof. The sets Exti(X; Z) and Exti(X; Z) correspond to vertical homo-
topy classes of the lifts indicated by dashed arrows in the diagram
Bss0(X)Ooo___B2Zhiss0(X)ooe__B2Zhiss0(X)O88
p p g g g g33
Bss0|| p p g g g g
| pgpgg g
BX
where the two spaces to the right are total spaces for the pull backs of the
classifying fibrations (2.3) and (2.4) along Bi :Bss0(X) ! BAut (Z ) = BOut (Z).
The obstruction to lifting a map BX -! B2Zhiss0(X)to B2Z (Y )hiss0(X)
lives in H3(BX; V ) as the fibre of ehss0(X)is B2V for some rational vector
space V [7, Proposition 3.2]. Since ss3(BX) = ss2(X) = 0 [5], [12 , Corol-
lary 5.6], there exists a 4-connected map BX -! B to some 2-stage Postnikov
tower B. Hence H3(BX; V ) ~= H3(B; V ) ~= H0(ss0(X); H3(ss1(X); 2; V )) ~=
H0(ss0(X); Hom (H3(ss1(X); 2); V )) = 0 as H3(ss1(X); 2) = 0 [14 , Theorem
V.7.8]. This shows that Exti(X; Z) maps onto Exti(X; Z).
The obstruction to lifting a vertical homotopy to B2Zhiss0(X)lives *
*in
H2(BX; V ) ~= H2(B; V ) ~= H0(ss0(X); Hom (ss1(X); V )) which vanishes if the
fundamental group ss1(X) is finite. This shows that the map in the lemma is
injective provided X0 is semisimple. __|_|
If X and Z are p-compact tori of rank 1, the map e* of Lemma 4.2 is not
injective.
There exists a version of Lemma 4.2 allowing the fibres to be arbitrary, not
just abelian, p-compact toral groups.
We now know that in case the identity component X0 is semisimple and
Extae(X; Y ) 6= ;, there are bijections
Extae(X; Y ) (-;G)-----!~ExtZae(X; Z(Y )) - ExtZae(X; Z(Y ))
= ~=
16 J. M. MOLLER
where the right hand group is isomorphic to the cohomology group
H2Zae(BX; Z(Y )). If X0 is even simply connected, there are bijections
Extae(X;OYO) ExtZae(ss0(X); Z(Y ))
~=ss*0|| |~=|
| (-;G) fflffl|
Extae(ss0(X); Y_)~=__//_ExtZae(ss0(X); Z(Y ))
where the upper right corner group is isomorphic to the cohomology group
H2Zae(ss0(X); Z(Y )). Note, however, that the bijection (-; G), depending on
the choice of the extension G, is noncanonical.
Now follows an alternative description of the Theorem 3.3 difference (H; G)
between two short exact sequences Y -! G -! X and Y -! H -! X from
Extae(X; Y ).
Proposition 4.3.There exists a homotopy equivalence
: B(H; G) ! fibmap(BG; BH)B1B1
over BX.
Proof. Let Bk = BkhOut(Y ):BP (Y )hOut(Y )! B2Z(Y )hOut(Yd)enote the
projection map of and : W ! map (I; BP (Y )hOut(Y))a connection [14 , p. 29]
for the universal fibration (2.2); i.e. assigns to any element of
W = {(x; u) 2 BP (Y )hOut(Y )x map(I; B2Z(Y )hOut(Y))| Bk(x) = u(0)}
a path (x; u) in BP (Y )hOut(Y )starting at (x; u)(0) = x and lying over
Bk((x; u)) = u.
The fibres over any b 2 BX of BG -! BX, BH -! BX are the fibres
Bk-1(G(b)), Bk-1(H(b)) and the fibre of B(G; H) -! BX is the space of
vertical paths u in B2Z(Y )hOut(Y )from G(b) to H(b). Define
: B(H; G) ! fibmap(BG; BH)B1B1
as the map over BX taking u to the map (-; u)(1): Bk-1(G(b)) ! Bk-1(H(b)).
The restriction of to the fibre over the basepoint (where the classifying maps
G and H have the same value) is the monodromy B2Z(Y ) -! map(BY; BY )B1
for the universal fibration (2.1), hence a homotopy equivalence. __|_|
Thus also
fibmap(BG; -)B1B1:Extae(X; Y ) ! ExtZae(X; Z(Y ))
is a bijection.
EXTENSIONS OF p-COMPACT GROUPS 17
The evaluation map B: BY x map(BY; BY )B1 ! BY is a left action
: Y x Z(Y ) ! Y of Z(Y ) on Y . Using the alternative description of Proposi-
tion 4.3 of the difference (H; G) it is immediate that extends to a morphism
Y x Z(Y )_____//*(G x (H; G)) _____//X
|| || ||||
fflffl| fflffl| ||
Y _______________//_H__________//_X
of extensions. This property characterizes (H; G) as an operator on
Extae(X; Y ).
Corollary 4.4. Let Z(Y ) -! -! X be a short exact sequence representing
an element 2 ExtZae(X; Z(Y )). Then G + = H in Extae(X; Y ) if and only
if the action extends
Y x Z(Y )_____//*(G x ) ____//_X
|| || ||||
fflffl| fflffl| ||
Y _____________//H________//_X
to a morphism over X.
Proof. The fibrewise adjoint of such a fibre map is an equivalence between
and (H; G) = fibmap(BG; BH)B1B1. __|_|
Corollary 4.4 concludes this section.
5. Rational automorphisms of non-connected p-compact groups
The purpose of this section is to investigate the monoid of rational automor-
phisms of not necessarily connected p-compact groups.
Let Y -! G -! ss be a short exact sequence of p-compact groups, where Y
is connected and ss is a finite p-group, representing an element G 2 Extae(ss; *
*Y ),
ae: ss ! Out(Yb)eing the monodromy.
The pull back diagram
mapEnd(ss)(BG; BG)_____//map(BG; BG)
| |
| |
fflffl| |fflffl
End (ss)__________//map(BG; Bss)
where the bottom map takes O: ss ! ssto BG -! Bss -BO-!Bss, serves as defini-
tion of the space in the upper left corner. Thus map End(ss)(BG; BG) consists of
self-maps of BG over maps BO: Bss ! Bss induced from endomorphisms of the
group ss.
18 J. M. MOLLER
Recall that End (G) = [BG; *; BG] denotes the monoid of based homotopy
classes of based self-maps of BG.
Lemma 5.1. ss0map End(ss)(BG; BG) ~=End(G).
Proof. The above construction of the space of self-maps of BG over induced
maps also has a based version, map End(ss)(BG; *; BG), of based self maps of BG
over induced maps of Bss. In the diagram
mapEnd(ss)(BG; BG)- mapEnd(ss)(BG; *; BG) -! map(BG; *; BG)
the right hand arrow is a homotopy equivalence since End(ss) -! map(BG; *; Bss)
is a homotopy equivalence and the left hand arrow is a ss0-bijection since BY is
simply connected. __|_|
The submonoid "Q(G) End (G) = ss0map End(ss)(BG; BG) of rational au-
tomorphisms of G thus consists of all vertical homotopy classes of fibre maps
of the form (f; BO) where f :BG ! BG restricts to a rational automorphism
f|BY :BY ! BY on the fibre BY . Consider the monoid homomorphism
: "Q(G)! End(ss) x "Q(Y )
that to the fibre self-map (f; BO) associates the pair consisting of O 2 End (s*
*s)
and the restriction f|BY 2 "Q(Y ).
Let End Q(ss; Y ) End(ss) x "Q(Y )denote the submonoid consisting of
pairs (O; g) where g is O-equivariant. EndQ(ss; Y ) contains the submonoid
End Q(ss; Y )G of elements (O; g) for which O*(G) = g*(G) in ExtaeO(ss; Y ).
Theorem 5.2. Suppose that the connected component Y of G is completely
reducible. Then there exists a short exact sequence
0 -! H1Zae(ss; Z(Y )) -! "Q(G)-! End Q(ss; Y )G -! 1
of monoids. For any pair (O; g) 2 EndQ (ss; Y )G , there is a bijection between*
* the
inverse image -1(O; g) and the cohomology group H1ZaeO(ss; Z(Y )).
Proof. The image of equals "Q(ss; YG)by Theorem 3.8. The kernel of
is the group of vertical homotopy classes of maps over Bss with restriction to
BY homotopic to the identity. As a set, ker() is in bijection with the vertical
homtopy classes of sections of the fibration
(5.1) map(BY; BY )B1 -! fibmap(BG; BG)B1B1-!Bss
of mapping spaces. If f; g :BG ! BG are maps over Bss representing elements
of ker(), let sf; sg denote the corresponding sections of fibration (5.1). Note
that sfg = f_O sg where f_denotes the self-map over Bss of fibmap(BG; BG)B1B1
given by composition with f. By Lemma 4.2, fibration (5.1) has a fibrewise
discrete approximation
(5.2) BZ (Y ) -! BZ (Y ) -! Bss
EXTENSIONS OF p-COMPACT GROUPS 19
and obstruction theory implies that there is a bijection, induced by fibrewise
completion, between vertical homotopy classes of sections and self-maps over
Bss of the two fibrations. Thus we may view sf as a section and f_as a self-map
over Bss also of fibration (5.2). Associate to f the primary difference [14 , *
*p.
299] ffi1(sf; s1) 2 H1Zae(ss; Z(Y )) between the section corresponding to f and*
* the
section corresponding to the identity map. Then
ffi1(sfg;=s1)ffi1(sfg; sf) + ffi1(sf; s1)
= ffi1(f_O sg; f_O s1) + ffi1(sf; s1)
= (f_)*ffi1(sg; s1) + ffi1(sf; s1)
= ffi1(sg; s1) + ffi1(sf; s1)
because f_is homotopic to the identity map on the fibre BZ (Y ). This computa-
tion shows that the bijection ker() -! H1Zae(ss; Z(Y )): f -! ffi1(sf; S1) is a*
* group
homomorphism.
For an arbitrary pair (O; g) 2 End Q(ss; Y )G , the inverse image -1(O; g)
is in bijection with the vertical homotopy classes of sections of th*
*e fi-
bration fibmap(BG; BG)BgBO -! Bss, or, equivalently (see the proof of
Theorem 3.8), the vertical homotopy classes of sections of the fibrat*
*ion
fibmap(B(g*G); B(O*G))B1B1-!Bss. Taking primary differences, as above, with
respect to some fixed section provides a (noncanonical) bijection -1(O; g) -!
H1ZaeO(ss; Z(Y )). __|_|
More explicitly, the elements of -1(O; g) are represented by maps of the form
BG -! B(g*G) -! B(O*G) -! BG
where the outer maps are fixed as the canonical ones and the middle arrow varies
over all fibre homotopy equivalences over Bss and under the homotopy class of
the identity map of BY .
The space of self-maps of BG over BO and homotopic to Bg on the fibre BY is
homotopy equivalent to the section space of fibmap(B(g*G); B(O*G))B1B1-!BX,
i.e. to the homotopy fixed point space
map (BY; BY )hssB1' BZ(Y )hss
which is a disjoint union of classifying spaces of p-compact toral groups.
Let Aut(ss; Y ) and Aut(ss; Y )G denote the subgroups of invertible elements*
* of
End Q(ss; Y ) and EndQ (ss; Y )G , respectively. The monoid short exact sequenc*
*e of
Theorem 5.2 restricts to a short exact sequence
0 -! H1Zae(ss; Z(Y )) -! Aut(G) -! Aut(ss; Y )G -! 1
of groups. The equivalence class of this group extension is unknown but it is
perhaps worth noting that a somewhat similar group extension is determined by
a differential in a Lyndon-Hochschield-Serre spectral sequence [9].
20 J. M. MOLLER
Corollary 5.3. Suppose that Y is connected, completely reducible, and cen-
terfree p-compact group. Then Ext ae(ss; Y ) = {Y -! G -! ss}, "Q(G) =
End Q(ss; Y ) and Aut(G) = Aut(ss; Y ).
Proof. Apply Lemma 3.2 and Theorem 5.2. __|_|
Finally, a couple of examples to illustrate the use of Theorem 5.2.
Example 5.4. (1) The 2-compact group SO (2n + 1)^2; n 2, is centerfree.
Hence Extae(ss; SO(2n + 1)^2) contains [Corollary 5.3] exactly one element G wi*
*th
"Q(G) = EndQ (ss; SO(2n+1)^2) and Aut(G) = Aut(ss; SO(2n+1)^2) for any given
homotopy action ae: ss ! Out(SO (2n + 1)^2).
(2) The center of the 2-compact group SO(2n)^2; n > 4, is cyclic of order 2 so *
*the
affine group Extae(ss; SO(2n)^2) of equivalence classes of short exact sequence*
*s of
2-compact groups
SO (2n)^2-! G -! ss
realizing a fixed homotopy action ae: ss ! Out(SO (2n)^2)has H2(ss; Z=2) as
group of operators. Assume that the homotopy action ae is injective. Since
"Q(SO (2n)^2)= Aut(SO (2n)^2) is abelian [8], it follows that EndQ(ss; SO(2n)^2*
*) =
Aut (ss; SO(2n)^2) = Aut(SO (2n)^2) consists of all automorphisms. For any
(equivariant) automorphism g of SO(2n)^2, Z(g) is the identity map, so
[Lemma 3.7] g* fixes G if and only if it fixes all elements of Extae(ss; SO(2n)*
*^2).
The short exact sequence of Theorem 5.2 implies that "Q(G) = Aut(G).
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EXTENSIONS OF p-COMPACT GROUPS 21
Matematisk Institut, Universitetsparken 5, DK-2100 Kobenhavn O
E-mail address: moller@math.ku.dk