Extensions realising a faithful abstract kernel and their automorphisms. Paul Igodt and Wire Malfalt 1. We are interested in group extensions 1 --* N ~ E --~ F -+ I, ...
manuscripta
math.
84,
135
-
161
(1994)
manuscripta mathematica (~ Springer-Vcrlag 1994
Extensions realising a faithful abstract kernel and their automorphisms P a u l I g o d t a n d W i r e Malfalt 1
W e are interested in group extensions 1 --* N ~ E --~ F -+ I, for which the corresponding abstract kernel F ~ Out(N) is faithful. For these groups E, we develop commutative diagrams which are helpful to understand and to compute Aut(E, N ) (the group of all E-automorphisms mapping N into itself)and Out(E, N ) = Aut(E, N)/inn(E). Of course, ifN is characteristic
in E, A u t ( E , N ) =- Aut(E) and Out(E,N) -- Out(E). These conditions occur e.g. when studying almost crystallographic groups, which were in fact the initiating cases to us. Similar work has bcen done previously by Conner and Raymond ([3]) (several types where N = Z~k) and by Charlap ([1]) (for crystallographic groups). Although the approach in both works is rather different, we did an effort to obtain a description covering most aspects of both previously developed pictures. We include the results of an example computation for one family of isomorphism types of 3-dimensional almost crystallographic groups. For K(E, 1)-manifolds it is known that Out(E) has an important geometric meaning. In the closing section and for certain K(E, 1)-manifolds, we establish a gcometric interpretation of Out(E, N) and its subgroups from the determining diagrams.
1
Introduction
Our goal in this paper is primarily algebraic. Let N be a (not necessarily abe]Jan) group and consider a one to one abstract kernel ~b : F ,-o Out(N). Write p : A n t ( N ) -~ O u t ( N ) for the canonical projection. In section (3) we tResearch AssistantNationalFund For ScientificResearch (Belgium)
136
IG ODT-MA LFAIT
construct an action of the group AAr -- p-t(NOut(N)r on H$(F,N) which is the set of 2-cohomology classes compatible with tb. If 1 --* N ---* E ---* F ~ 1 is an extension realising !b and determining a eohomology class a E H~ (F, N), the stabiliser of a for this group action is of major importance to determine and to describe Aut(E, N) (the group of automorphlsms of E mapping N into itself) and Out(E, N) = Aut(E, N)/Inn(E)" In section (4) a description of Aut(E, N) and Out(E, N) is given in terms of different (we establish two approaches) 9 - d i a g r a m s (diagrams (3) and (4)). Our picture covers similar ideas developed earlier by Conner and Raymond ([3]) and Charlap ([1]) for situations with an abelian kernel N = .~k. We briefly indicate the particularity of the case of splitting extensions. A context for which this set-up is of explicit importance is that of the almost crystallographic (and almost Bieberbaeh) groups E. Now we assume N is a finitely generated, torsion-free, nilpotent normal subgroup of finite index in E which is maximal nilpotent in E. An extension 1 -* N -* E --* F -4 1, satisfying these conditions, is called essential. The corresponding abstract kernel is faithful and Aut(E, N) -- Aut(E) (Out(E, N) = Out(E)). Starting from an extension of this type, in section (5.1) we obtain some additional information which will simplify significantly the computation of the fundamental 9-diagrams. Up to isomorphism, there are only finitely many almost crystallographic groups E, containing a fixed group N as their maximal nilpotent subgroup. In section (5.2) we explain the explicit A u t ( E ) (Out(E)) computation for one family of isomorphism types of 3-dimensional almost crystallographic groups E. A complete list of the outer automorphism groups for all isomorphism types of almost crystallographic groups in dimension _ 4 is in preparation ([7]). If E is the fundamental group of an aspherical manifold M, O u t ( E ) is naturally isomorphic to the group of homotopy classes of homotopy self equivalences H0(e(M)) and a geometric meaning of it is of interest. In section (6) and for a specific situation, we indicate an interpretation of the subgroups of O u t ( E , N ) , occurring in our fundamental 9-diagrams. Here, we will briefly recall the concept of the Seifert Space Construction for a given set of data. Based on fundamental properties of this construction (existence, uniqueness and rigidity), an interpretation of Out(E, N) can be given in terms of fiber preserving homeomorphisms.
2
Preliminaries
If N is a group containing n, we will write /~(n) for the inner automorphism determined by n; i.e. g(n) maps an element z to n z n - t . # ( N ) is known as the inner automorphism group Inn(N), while O u t ( N ) -- Aut(N)/inn(N) is called the outer automorphism group N. We write p : A u t ( N ) ---* O u t ( N ) for the natural projection. Each group extension (or short exact sequence of groups) 1 --* N --* E -4 F --* 1 induces an abstract kernel r : F -~ O u t ( N ) in a well known way. We
IGODT-MALFAIT
] 37
say t h a t the extension is an algebraic realisation of r or that it is compatible with r It is well known that, in general, not every abstract kernel has an algebraic realisation. In some results we draw diagrams which should be regarded as c o m m u t a tive in the set-theoretic sense. This will be indicated by writing SET for the category of sets and set-mappings. Our c o m p u t a t i o n s will use certain aspects of group cohomology with non abelian coefficients. In such a context, usually one takes the structure of crossed modules as optimal coefficient structure (see e.g. [9]). Let us quickly review some basic facts a b o u t this. A crossed module is a 4-tuple (N, p, H, O) where N and H are groups and p : N -~ II and ff : H --* A u t ( N ) are group h o m o m o r p h i s m s satisfying compatibility conditions @op=#
and, for every z E H, ]*(x) o p = p o @ ( = ) .
If ~ E 11 and n E N , we write " n for @(~)(n). There is a trivial way to build a crossed module for each group N: take (N,/~, A u t ( N ) , Id). If not explicitly mentioned otherways, most of the time this is the crossed module we will work with. Given a group F , a 2-cocycle of F with coefficients in (N,p, H, r is a pair (~, c), where ~o : F ~ H and c : F x F ---* N are maps satisfying
{
=
for all z, y and z in F. It will always be assumed t h a t ~ and c are normalised i.e. ~(1) = 1 and c(z, 1) = c(1,~) = 1. We write Z2(F, N) for the set of all 2-cocycles of F with coefficients in ( N , p , II, 0). Clearly, each 2-cocycle (~o, c) determines a mozphism r : F ---* O u t ( N ) . If we concentrate on a fixed abstract kernel r we write Z,~(F,N) for the set of 2-cocycles inducing r Two 2-cocycles (~, c), (~o', c') are called cohomologous if there exists a normalised m a p )~ : F ---* N such t h a t for z, y E F :
{ ~'(~1 = p(X(~))~(=) ~'(:, V) = ;~(:) "~'(') X(V)" ~(:, V)" ;'(:V) -~ T h e set of cohomology classes of Z~(F,N) is called H2(F,N). Similarly, H,~(F, N) is the set of all classes inducing a fixed abstract kernel r It is well known that H2(F, N ) has an interpretation in terms of so-called H-crossed extensions [9]. Here, it is enough to recall t h a t a 2-cocycle (~,c) gives rise to an extension E = N x(~,c) F with group operation
V,~,~, e N, V~, Y E F:
(,~,~) .(~,o)(.~, Y) = ('~ "~'(~ m" c(z, Y), zY).
Conversely, if 1 ---* N ---* E --* F --* 1 is compatible with r by choosing a n o r m a n s e d section s : F ---* E, we obtain a 2-cocycle (~o, c) e Z ~ ( F , N).
IGODT-MALFAIT
138
There is also a concept of 1-cohomology with non abelian coefficients. Here, start with a fixed 2-cocycle (~,c) E Z 2 ( F , N ) . A 1-cocycle is a (normalised) map A : F ~ N satisfying
~(~) .~(') ~(y). c(~, y). ~(~y)-I = ~(~, y) for all x and y in F. The set of all 1-cocycles is denoted by Z 1 ( F , N ) . Two 1-cocycles A and ~? are called cohomologous if there exists ~nC)element n e N satisfying 77(z) = n . A ( z ) . ~ ( ' ) n -1, for all z E F. The set of all cohomology classes of 1-cocyeles is denoted by H~,,,c)(F, N), and is called the first cohomology of F with coefficients in the crossed module (N, p, II, ~2) with respect to the 2-cocycle (~, c). For a given r : F --~ Out(N), it is clear that Z ( N ) , the center of N, becomes an F-module in the classical sense. So, there is no problem to talk about H$ ( F, Z( N) ), the (classical) 1-cohomology-group.
3
A crucial
group
action
In this section, starting from a faithful abstract kernel, we define an action which will be of crucial importance to construct our fundamental automorphism diagrams. First, we introduce D e f i n i t i o n 3.1 If r : F --~ O u t ( N ) is an abstract kernel , we define the subgroup A4r of A u t ( N ) by
~
= v-~(Nou~(~)r
the inverse image in A u t ( N ) of the normaliser of r
in O u t ( N ) .
Let us establish a sufficient condition for an automorphism of N to belong to A4r L e m m a 3.2 If ~ E Aut(N) is an automorphism for which there ezisis a surjective mapping u' : F --* F and lifts ~, ~ : F -4 A u t ( N ) of r for which the following diagram commutes (in S E T )
F
-~ Aut(N)
F
~
~'1
~ ,(~)
hut(N)
then v E AAr Proof: Use p : A u t ( N ) --4 O u t ( N ) and verify that p(v) belongs to the normaliser in O u t ( N ) since u' is onto. ,,
of r
The following properties will be of practical importance for our work later on.
IGODT-MALFAIT
139
L e m m a 3.3 A s s u m e an injective abstract kernel ~b : F ~
Out(N).
1. There is a group h o m o m o r p h i s m * : A4r ---. A u t ( F ) defined by v* = r o U(P('-')) o r for each ~, E M~, 2. I / v E M e , then for each lif2 ~o : F ---* Aut(N) o r e , there ezists a unique lift ~ : F --+ A u t ( N ) (of r such that the following diagram c o m m u t e s
(i~ scT) F ~ ~" ~ F 3. I f ~' is another lift o r e ,
~
Aut(N) ~ u(~') Aut(N)
e.g. ~' = U(A)~o (A : F ~
N ) , then ~o~ =
~(v -1 o ,~ o v*)~o~. 4. I f v l , v 2 E A.~r
then for each lif-t ~o o r e , ~o(~o~,) = ( ~ ) ~ , .
Proof: 1. This follows directly from the injectivity of r and the definition of AA,). 2. and 3. The proof consists of rather elementary verifications which are left to the reader. Essentially, if v E AAr and ~o : F --* A u t ( N ) is a lif~ of r then one defines the lift ~o~ = ~(v -1) o ~o o v*. 4. As the following diagram commutes F
(~v~)vl A u t ( N )
~'~ ,L
~, ~(~)
F
~~ --~
Aut(N)
F
-~
Aut(N)
the uniqueness of the lifts implies that V~(~,,~,) = (V~,)vl. If we have an injeetive abstract kernel, we obtain an interesting group action, as we see now. P r o p o s i t i o n 3.4 A s s u m e an injective abstract kernel r : F ~ O u t ( N ) . There is a group action of A4r on H ~2 ( F , N ) , defined as follows:
.~,p • H,~(~,N)
--,
/-/,~(F,N)
(~, < ~, ~ >)
~
~ < ~, c >%1
where "c = ~ , o c o ( v " - I x v * - 1 ) .
Proof: Assume (V',~) is a 2-cocycle in Z~,(F, N). We verify the clam in three steps: S t e p 1" (!o,c) E Z ~ ( F , N )
~ (tou-2,Uc) E Z ~ ( F , N ) .
140
IG O D T - M A L F A I T
Take x, y, z in F . F i r s t , there is ,[~(~,y)]
=
uo,[c(u*-z(~),u*-1(y))]ou
-~
=
~o [~(~'-l(~))~(~'-l)(y)~(~*-~(~y))-~]
=
~,,_, (~)~,,_~ (y)~,~_, ( : ~ ) - ~
o ~-~
a n d , s e c o n d l y we verify,
~c(=, y) .~(~y, z) = ~[~(~*-~(~), ~'-~(y)). ~(~*-~(~y), ~*-~(z))] = ~[~(~'-'(=))~(~*-l(y), ~.-~(~)). c(~*-~(~), ~*-~(y~))] = ~-,(-)[~(y, ~)] .~ :(~, y~) Step ~. (~, :) ~ (r ~') ~ (~,~-~ , ~ ) ~ (~,'_~,~d). Let A : F --* N be t h e cochain such t h a t ~# = ]z()Q~.
T h e n ~_~, =
,(~ o ~ o ~'-~)~_, (~.3) and, for each ,, y e F ~'(~, y)= ~(d(~ "-~(~), ~*-~ (y))) = ~[A(~.-~(~))~(~'-'(=))(~(~*-~(y)))~(~*-~(~),~.-~(y))~(~'-l(~y))-~] = (,., o,~ o ~,*-~)(~).~%-~ (=)(~,o ;,o ~,'-' )(y).~' ~(~, y). (~, o,,, o ~,*- ~) (~y) -~ . S t e p 3: We have a g r o u p action. Take vl a n d v2 in M e .
Then
v2o,,,~ < (p~ c >
4
Automorphisms ful abstract
V2OVl
=
< ~(v;-,o~,;-~),
=
u ' ( v ' < ~,c >).
of extensions
c >
realising
a faith-
kernel
F r o m now on, we fix a n a b s t r a c t kernel r : F --* O u t ( N ) a n d a g r o u p e x t e n s i o n 1 --* N ---* E ~ F "--* 1 realising r It is our a i m to s t u d y A u t ( E , N ) , the g r o u p of those a u t o m o r p h i s m s of E which c a r r y N into N , a n d m o r e p a r t i c u l a r l y 0 u t ( E , N ) = A u t ( E , Y ) / i n n ( E ) . A n a u t o m o r p h i s m o" in A u t ( E , N ) restricts to an a u t o m o r p h i s m of N , a n d c o n s e q u e n t l y induces a n a u t o m o r p h i s m o f F . Let us write A : A u t ( E , N ) A u t ( N ) for the r e s t r i c t i o n to g a n d B : A u t ( E , N ) --4 A u t ( F ) for the corresp o n d l n g m o r p h i s m . So, each ~r gives rise to a c o m m u t a t i v e d i a g r a m o f g r o u p s and group homomorphisms : 1 1
---* N ---*
-4
E
IA(~)
~~
N
E
-*
--*
F
---*
1
~B(~) --*
F
--~
1
IG O D T - M A L F A I T
4.1
A fundamental
Aut(E,N)
141
diagram
We start with the following, rather elementary observations. L e m n a a 4.1 A s s u m e an abstrac~ kernel r sion 1 -~ N
~
: F ---* O u t ( N ) and an eztenE --+ F --~ 1 compatible with r F i z an au~omorphism
9 A u t ( E , N). 1. I f s : F ---* E is a section inducing a lift ~o : F ---* A u t ( N ) , then there ezists a lift ~ , o r e , making the following diagram c o m m u t a t i v e (in S ~ T )
F
~
Ant(N)
F
~
Aut(N)
2. Im(A) C A,4,k. 3. I r e is injective, then f o r each section s inducing a lift ~ o r e , and A(r = B(cr). Consequently, Ker(A) C Ker(B).
Io~.(r
= ~r
Proof: 1. The section s~ : F -* E, defined by r o sr = s o B(cr), will determine a lift ~o~ having the desired property. 2. This follows at once from l e m m a 3.2. 3. As r is injective we can use the notations of l e m m a 3.3 such that clearly A(~r)* = B(o') and !oA(r = ~r D e f i n i t i o n 4.2 A s s u m e r F r O u t ( N ) injective and suppose that H $ ( F , N ) is non-empty. For each element ~ = < ~ , c >E ~ ( F , N ) , we d e n e the s~bgroup A4r
of A4r as A4@,= ---- StabA4,a = {u e A4@ II v < ~o,c >~-< ~ , c >}.
P r o p o s i t i o n 4.3 Let r : F ~
O u t ( N ) be an injec~ive abstract kernel. A s s u m e 1 ---* N --* E --* F --* 1 an eztension realising r and determing a cohomology
class ~ = < ~, c > e H~(F, N). Then, Ira(A) = ~ V , o . Proof." Take E = N x(~,c) F and consider ~ E A u t ( E , N). Choose as in l e m m a 4.1, the section s~-~ = o" o s o B(~) -I, which induces the liR ~ - I .
the m a p ~ :
Construct
F -~ N defined by or(l, ~) = ( ~ ( z ) , B(cr)(x)) (~ E F).
For every (n,x) E E, (n,z)= (n, 1)"(~,c)(1,~)= (1,x)"(~,c)(i~ Therefore we find that
r
1)-(~,r ~(1, z)
= =
(A(r 1)"C~,,) (~(z), B(r (aC~)Cn).~(=),B(~)(~))
l) 9
42
IG OD T - M A L F A I T
but also t h a t
~(1, ~) .(~,,r ~(~(')-'n, 1)
=
(~(x),B(~)(~)).(~,:)
(A(c~)(~'(~)-~n), 1)
=
(~(~) .~,(s(r174 (A(~r)(~(~)-~n)), B(~r)(x)).
So ~= must satisfy V~ E F :
I z ( ~ ( x ) -1) o A((r) -- ~o(B(cr)(a:)) o A(cr) o ~o(x) -1
or, because B(c~) C A u t ( F ) ,
v~ e F :
~ ( ~ ( B ( ~ ) - ~ ( ~ ) ) -~) = ~(~) o A(~) o ~(B(~)-~(~)) -~ o A(~) -~.
T h e reader can easily verify, using l e m m a 4.1, t h a t this is equivalent to ~ . - , = ~ ( ~ o B ( ~ ) -~) o ~.
(1)
Now, take x, y E F and find that
~[(1, ~) .(~,o) (1, y)]
=
~(:(~, y), ~y)) (r[(c(x, y), 1) -(~,:) (1, ~y)]
(A(cQ(c(x, y)). ~(xy), B(o')(~y)) which must be equal to
~(1, ~) .(~,o) ~(1, y) = (~(~), B(~)(~)).(~,o)(~(y), ~(~)(y)) = (~(~).~(s(~)(,))(~(y)). c(B(~)(~), B(~)(y)),B(~)(~.V)) So we have t h a t
A(o')(c(~, y)) = ~ ( ~ ) .~,(B(~)(,)) (~a(y)). c(B(~)(~), B(cr)(y)). ~c,(zy) -1. (2) R e m a r k t h a t we have not used the injectivity of the a b s t r a c t kernel to obtain (1) and (2). If r is one to one however, ~.4(~)-1 = ~ - 1 and A(tr)* = B(o') (see l e m m a 4.1), or (2) translates into
AC~)c(x, y) = ~ ( B ( u ) - I (x)) .~(~) (~(BC~)-l(y)) 9c(x, y). ~ ( B ( ~ ) - l ( x y ) ) -1 We conclude that A(a) < ~, c > = < ~, c > or A(r
E 2vLr
Conversely, take v E A4r and consider E as N x(~,,~) F. Since ~< ~ , c > = < ~ , c >, there is a cochain A~ : F --~ N such t h a t 8~ :
is a group isomorphism.
N x(~%_,,~c) F
--*
N x(~,,~)F
(n,~)
~
(n. ~ ( ~ ) , ~ )
$GODT-MALFAIT
~43
Moreover, we can introduce another group isomorphism ~, as follows
~ Let n, m E N ,
:
x,yEF.
N
x(~,~) F
--+
N
(n,~)
~
(~(n), ~* (~))
x(~_~,~)
F
Then
~(n, ~).(~ _,,~o) ~(m, y) = (~(n), ~*(~))"(~-, ,~o) (~(m), ~*(y)) = (~(~) .~v-, (~'c,)) (~(m)) .~ c(~" (~), ~* (y)), ~" C~y)) = (~(n) 9 ~(~(')(m)) 9~(c(~, y ) ) , , ' ( ~ y ) ) = ~[(n, ~) .(~,0) (.~, y)] or ~ is a homomorphism which is clearly bijective. Consequently, by taking o'~ = 8~ o ~ we obtain an automorphism of E such that A(o-~) = ~,. ,, D e f i n i t i o n 4.4 Let us write A u t ~ for Ker(A) n Ker(B), the subgroup of A u t ( E ) consisting of those automorphisms which induce the identity both on the I r ( g ) and on the right (F). Aut~
can be understood in a nice equivalent way. This is the purpose of
P r o p o s i t i o n 4.5 Take r : F --* O u t ( N ) and a group eztension 1 --* N --* E ~ F -~ 1 realising r
i. A u t ~
iso.~o~phie to Z$(F, Z(N)).
2. If r is one to one, then Ker(A) n I n n ( E ) -- Aut~
n I n n ( E ) and is
isomorphic to B~(F, Z(N)). 3. Consequently, i r e is injective, then Aut~
n I n n ( E ) ~ H$(F, Z ( N ) ) .
/Aut~
Proof: Again let E -- N • and consider o" E A u t ~ 9 If we take, again as in l e m m a 4.1, the section s~-~, we obtain the lift ~ - ~ , but for r E A u t ~ , ~ - ~ = ~. Therefore (1) implies that the m a p ~,, : F --* N, defined by r z) = ( ~ ( z ) , z), takes its values in Z ( N ) , the center of N. Consequently, translating (2) for o" E A u t ~ we obtain
~(~y) =
~(~).~(-)
(~(y))
which means that ~a E Z$(F, Z ( N ) ) . We claim that
~: Aut~
-~ Z$(E, Z(N)) : ~ ~
~
is an group isomorphism. Here, we leave it to the reader to show t h a t ~ is injective. To see t h a t it is also onto, take 7 E Z$(F, Z ( N ) ) , construct the m a p ~ : E--* E : ( n , x ) ~ ( n . 7 ( ; ~ ) , $ ) ,
IGO DT-MALFAIT
144
and verify that o- is an automorphism of E belonging to Aut~ . Evidently, it follows that ~ = 7. Let us now investigate what happens to ~ if o" belongs to Inn(E). Assume cr -- , ( n o , x o ) 9 Aut~ n Inn(E). Then, A(o') = # ( n o ) o ~(mo) -- 1, and hence, projecting to Out(N), we find that r = 1. As 15 is one to one, zo = 1 and no 9 Z ( N ) . It follows that, for 9 9 F, ~,(~) = no "~(~) no ~, which is equivalent to saying that (= belongs to B ~ ( F , Z ( N ) ) . If conversely, 7 : F -* Z ( N ) is given by 7(~) = no .~(~) n o 1, for some no 9 Z ( N ) , then or: E - , E defined by a(n, z) = ( n . 7(x), x) will belong to Aut~ n Inn(E). 9 We are now ready to summarise these results in the following T h e o r e m 4.6 I f r : F r O u t ( N ) is an abstract kernel which is one ~o one and 1 --+ N ---* E ---* F ---* 1 is an eztension compatible wi~h r determining a cohomoIogy class a E H $ ( F , N ) , then there is a commutative diagram of groups and group h o m o m o r p h i s m s such that both the rows and the columns are ezact sequences: 1
1
1
1
-~
B~(~,Z(N))
-~
1
---*
Z~(F,Z(N))
---* A n t ( E , / )
1
-~
H$(F,Z(N))
--*
1
Inn(E)
Out(E,N)
A -+
E/ZE H ~ ( F , N ) . Then, I. I m ( B ) = .A~.~,~ (the image of ]~4r
under , ) .
2 1 ~ Aut~ of groups.
-~ Ker(B) A K e ~ ( ' ) n m ~ , o
s
n Inn(E) -~ Ker(B) n Inn(E) A j - ~ ( Z ( F ) ) / Z ( N )
1 -. Aut~
-~ 1 is a short e~act sequence
-* 1 is
a short e~act sequence of groups. Proof: 1. Return to (4.3) and notice that, for each o" E A u t ( E , g ) , A(~)* = B(~r). 2. Again, since A(cr) E A4r and A(cr)* = B(~r), it is clear that A(Ker(B)) C A4r n Ker(*). Conversely, if v belongs to A4r and v* = 1, then, using proposition 4.3, v can be found as the image under A of an automorphism o'v. Now, 1 = v* = A ( ~ ) * = B ( ~ ) implies that v E A(Ker(B)). 3. Just realise that if, for (n0, x0) E f = N x(~,,,) F, , ( n 0 , z0) E Ker(B), then ~o E Z ( F ) or Ker(B) M Inn(E) ~ j - ' ( Z ( F ) ) / Z ( E ), Moreover, as r is one to one, Ker(A) ['1Ker(B) N I n n ( E ) = Ker(A) M I n n ( E ) and isomorphic to C~(N)/z(E) = Z(N)/z(E). 9 Combining these properties, we conclude with Theorem
4.8 Let r : F r
O u t ( N ) be an injective abstrac~ kernel and 1 --*
N ---, E i_~ F --* 1 an eztension reaIising r and determining a cohomology class
146
IGO DT-MAL FAIT
a = < ~o,c > 6 H $ ( F , N ) . ezac~ rows and columns:
Then there are two commulalive diagrams buil~ wiih 1
1
1 --*
g t = Ker(B) N I n n ( E )
1 -*
K~ = Ker(B)
-*
1
Inn(E)
~ I n n ( F ) --* 1
---, Aut(E, N) ~
1 --* K . = K e r ( B ) / K e r ( B ) A Inn(E) --* Out(E, N) -*
1
1
M,~,~ ---+ 1 Qt
(4)
---* 1
1
and 1
1
1
1
--+
B~(F,Z(N))
--.
Kt
A
1
--*
Z$(F,Z(N))
-*
K2
A_~
1
--*
H$(F,Z(N))
--*
K8
--*
1
j_t(Z(F))/Z(N)
Ker(*)N.AAr Q2
1
--.
1
(5)
---* 1 --*
1
1
where
Q2 = (Ker(*) N A A r with (Ker(*) n A~fr
) N Inn(E)) ~ (Ker(*) N M r
= (Ker(*) N Adr
)
C 0 u t ( N ) , and
Q~ ~-~ ( C O u t ( N ) t ~ ( F ) ) / Z ( F ) .
Proof: We leave it to the reader to obtain these diagrams. For Q2, the reader can proceed as in (4.6). Notice that
Z4r
N Ker(*) _C Z4,p N Ker(*) = p-1(Cout(N)r
This result shows that, to compute A u t ( E , N) (Out(E, N)) based on diagrams (4) and (5), somehow, we have to go one level deeper - compared to the (3)-situation - in terms of 9-diagrams. Let us close here with remarking another 9-diagram in which we learn more about those somewhat mysterious lower right corners of diagrams (3), (4) and (5), which we called Q, Q1 and Q2 resp.. The verification is left to the reader.
IGO D T - M A L F A I T
4 . 9 Suppose a situation where the diagrams (3), (4) and (5) are
Remark
valid. Then r
is a commutative diagram built with ezact rows and columns: 1
1
--~ j - I ( Z ( F ) ) / Z ( N )
1
---*
1
~
1 --,
E/Z(N )
~ ~
l Ker(*) n A4r Q~ 1 4.3
147
Splitting
1 -,
Inn(R)
A4~,~
--,
Q
--*
--*
1
A4~,~
---,
1
Q~
-*
1
1
1
1
(6)
1
extensions
Let us have a closer look at the s i t u a t i o n of a split extension. 4 . 1 0 Let r : F ~ O u t ( N ) be one to one. Assume 1 --* N -* E --* F --* 1 is a splitting ez~ension realising ~b and determining a cohomology class a E H~ (F, N). Then, Proposition
Mr
= {v 6 M e [] ~v-~ can be induced by a splitting section Sv-~ : F -* E}.
If A4r = A,fr Q2 ~ ~--" C O U t ( N ) r
and Q ~- N O U t ( y ) r
"
Proof: As 1 ---* N ---* E ---* F --* 1 splits, we can fix a s p l i t t i n g section s : F --* E i n d u c i n g a 2-cocycle (~o, 1). Let v E A4r such t h a t ~ v - , is i n d u c e d by a h o m o m o r p h i s m s v - , : F --* E . T h e r e is a m a p Av : F ~ N such t h a t W9
s~-,(z)=A~Cz).s(4
,
a n d c o n s e q u e n t l y ~ v - , (z) = / ~ ( A ~ ( z ) ) o ~o(z). As b o t h sections are h o m o m o r phisms, we easily o b t a i n t h a t , for each z , y 9 F , Av(xy) = A ~ ( = ) - ~ ( = ) A v ( y ) . T h e r e f o r e v < ~p, 1 > = < p, 1 > or z~ 9 A4r Conversely, t a k e v 9 A4r As ~ < ~, 1 > = < ~, 1 > , there exists a cochain A~: F - * N such t h a t ~ - ~ ( z ) = / ~ ( A ~ ( z ) ) o ~ ( z ) a n d A~(zy) = A~(z).~'(=)Av(y), for ~, y 9 F . Take sv-~ = Av 9 s : F --* N . T h e r e a d e r can verify t h a t sv-~ is a s p l i t t i n g section a n d induces ~ - , . For Q a n d Q2, in the case M e = M e , a , the r e a d e r can easily see t h a t the e m b e d d i n g s a n n o u n c e d in reap. (4.6) a n d (4.6) t u r n into i s o m o r p h i s m s . 9 4 . 1 1 In the "abelian" case (i.e. if N is abelian), obviously A4~,a = M e if the cohomology class a corresponds ~o a split-eztension. In a situation where N is non abelian, the above proposition implies that this equality (again for a splitting ez~ension) is not automatic. As the following ezample shows, ~here are indeed, si~ua$ions like this for which A4~ r A4r in other words, ~he condition ezplicited in the proposition is non-trivial. Remark
48
[G O D T - M A L F A I T
Example
4 . 1 2 Take N:
< a , b , c l [ [ b , a ] = c 2, [ c , a ] = 1, [ c , b ] = 1 >
and F ~ z~Y2 x g 2 , given as { 1 , a , f l , afl}. F ---, A u t ( N ) given by b C
b- c ~
Consider the h o m o m o r p h i s m T :
and
b
C
C
b ~
C -1
It is not h a r d to see t h a t there are no inner a u t o m o r p h i s m s in the image of ~; consequently, ~ induces a faithful abstract kernel r : F ---* O u t ( N ) . Let 1 ---* N ~ E ~ N:~ F --* F --* 1 be the semi-direct product determined by ~. As presentation of E one can take e.g. E:
1}.
It is well known that ~ , (1 < k < c + 1), is characteristic in N ([12]). Given 1 --~ N ~ E --* F ---* 1 and F is a characterictic subgroup of N, clearly we can factor out r to obtain 1 ~ N / r --. E l f --, F -4 1. For an essential extension, it is well known that I' can be chosen such that the resulting extension (of quotients) again is essential and with N I p of lower nilpotency class. In fact, as seen in [5] at least two choices are available: 9 taking I" = ~ will reduce the nilpotency class substantially and brings us to an abelian kernel situation, while
IGODT-MALFAIT
150
9 taking r = ~
reduces the nilpotency class exactly one level.
Since r is assumed characteristic in N, each automorphism v E A u t ( N ) induces automorphisms C(v) and D(~) of P and N / p resp. , such that the diagram below is commutative: 1
---+
r
--*
N
-4
NIp
-4
1
N
--,
NIp
---,
1
c(.) 1
-4
r
--.
To understand and to compute the fundamental diagram (3), the two rather hard points are the groups A4r and Q. Let us state now two results which will be helpful to overcome some difficulties. L e m m a 5.4 Let 1 ---* N ---* E -4 F --~ 1 be an eztension, realising r : F O u t ( N ) and d e t e r m i n i n g class a E H $ ( F , N ) . A s s u m e r is characteristic in g and write e r : F -4 Out(N/r ) for the abstract i e r n e t induced by 1 -4 N / p - 4 E f t -4 F -4 1 and a r E HSr (F, N / p ) f o r the corresponding cohomology class. Consider D : A u t ( N ) -4 A u t ( N / r ) as introduced above. T h e n D(A4r
C_ A4r
r.
O n the problem of computing Q (in diagram (3)), the following brings a useful reduction.
L e m m a 5.5 Zet 1 -4 N -4 E -4 F ---* 1 be an essential e z t e n s i o n compatible with r : F ~ O u t ( N ) and determining class a = < ~,c > e H $ ( F , N ) . Take r = N~ / - ~ N ) and D : Aut(N) -4 A u t ( N / r ) as introduced before. T h e n Q (see diagram (3)) fits in a shor~ ezac~ sequence
1 ---* (Ker(D) n Agr
--* Q --* D ( A d r
---* 1.
Proof: First notice that Inn(N) 4 ( A u t ( L ) • 7-/(W)) ----7 t ! (L x W ) is a t r a n s f o r m a t i o n group of the p r o d u c t space L • W; i.e. is (z, w) E L x W, then (x,g,n)(x,w) = ( g ( z ) . A(h(w)),h(w)). One considers L as s u b g r o u p of 7~](L x W ) via the embedding t : L --* X Y ( L x W ) : z ~-+ (x,lz(x), i). Then, it is easily seen that L acts on L x W as (left) translations in the L-factor. D e f i n i t i o n 6.1 Assume L and W as above. A quadruple (N, E, p, W ) is called a L - s e t o f d a t a f o r a S e i f e r t C o n s t r u c t i o n if and only if
1. N is a lattice in L and ( N , L ) has UAEP 2. E contains N as normal subgroup with quotient E / N = F 3. p : F ---+? t ( W ) is a properly discontinuous action such that, for every e~tension i -* L -~ E -* F -* 1 (compatible with r
F -* O u t ( L ) ) , t h e n
exists a homomorphism ~ making the following diagram commutative 1--*
L
--+
E
It
---*
I~
F
--*1
lCxp
1 -~ A4(W,L*)>~Inn(L) --* X J ( L • W )
---* O u t ( L ) •
---+ 1
and such that ~ is unique up to conjugation by elements of A4(W, L*). If clear from the context, we speak of a set of data and avoid to m e n t i o n L explicitly. Example
6.2
1. I f L is nilpotent containing a lattice N, for every extension 1 --* N ---* E ---+ F --* 1 and properly discontinuous action o f f on W (via p : F --+ 7-/(W)), ( N , E , p , W ) is a set of d a t a ([8, w 2. If L is solvable with connected center and N is a lattice of L such t h a t ( N , L ) has UAEP, for every extension 1 --+ N --* E --4 F --* 1 and properly discontinuous action of F on W (via p), (N, E, p, W ) is a set of d a t a ([11, Th.3]).
For a given set of d a t a (N, E,p, W), a S e i f e r t C o n s t r u c t i o n is a homom o r p h i s m @ : E ~ 7( 1 (L x W) making the following d i a g r a m c o m m u t a t i v e : 1
---*
N
i
-~
Z~(W,L*)~Inn(L)
---+ -.
E XI(L x W)
---+ -.
F Out(L) x X(W)
---* 1 -.
i
IG ODT-MALFAIT
158
(notice that r : F -4 Out(L) is well determined by the extension 1 --~ N -4 E ---* F ---* 1 and the UAEP of (N, L)). Now, let us agree on the following terminology: we will say that L has the Unique Lattice Isomorphism Extension Property (ULIEP) if and only if every isomorphism N --* N I between lattices in L extends uniquely to an automorphism of L. E.g. it is well known that a simply connected, connected nilpotent Lie group L has ULIEP (see [13]). More general, a simply connected, connected solvable Lie-group of type (K) also has ULIEP (see [6]). This Seifert Construction concept is most frequently referred while having 3 important properties in mind. As they are of primary importance, also in the perspective of this section, we remind them briefly. E x i s t e n c e : For each set of data (N, E,p, W), there exists a Seifert Construction (based on the UAEP o / ( N , L)). U n l q u e n e s s - l " Let (N,E,p, W) be a set of data and fix an embedding e : N ~-~ L of N in L. Then, a Seifert construction 9 with respect to this embedding is unique up to conjugation by elements of AA(W, L*). U n l q u e n e s s - 2 : Let ~1 and ~2 be Seifert Constructions for a set of data (N,E,p,W), corresponding resp. to embeddings r : N ~ L. If L has ULIEP, there exists a unique g E Aut(L) such that r -- g o el and ~ . = ~(A,g) o ~ 1 for a A e ~ ( w , L*). R i g i d i t y : Assume L has ULIEP and let (N, E, p, W) and (N', E', p', W) be two L-sets of data. Assume 8 : E ~ E ' an isomorphism inducing ~ : N ---+ N ' and 0 : F --+ F ' such that there exists h E ~ ( W ) for which Iz(h) o p -- p' o 8. Then, for each couple of Seifert Constructions ~ and 9 ~ for these sets of data (resp.), there exists h E 7~I (L • W) such that o e =
o
If k~ is a Seifert Construction for a given L-set of data (N, E, p, W), E acts properly discontinuously on L x W. It is well known that, if e.g. E is torsionfree, q is injective. E acts with compact quotient if and only if F acts with compact quotient on W. So, if W and L are contractible, F \ W is compact and E is torsion-free, the given set of data will determine a K(E, 1)-manifold M. Many families of aspherical manifolds arising in this way, and their topological and geometrical properties, have been studied in the literature of the past 15 years. Now, assume we are in this case and write 7~! (M) for the group of homeomorphisms of M whose likings to the universal cover L • W belong to 7~! (L • W). Such liftings automatically belong to the normaliser of I'II(M) -- E in ~ ! ( L • W). In fact, 7~!(M) can also be seen as the group of homeomorphisms of M arising by conjugation in N ! =- N~t(L• Write e(M) for the space of self-homotopy equivalences of M. It is known that Wo(e(M)) is isomorphic to O u t ( I I l ( M ) ) = Out(E). Let us write 9 for the
159
IGODT-MALFAIT
natural homomorphism + : N ! -- N u q L •
) --* 7~] ( M ) ~-~ e ( M ) -+ Out(E).
If we consider E as a group of covering transformations of L • W, conjugating E with a homeomorphism f E N ! will induce an automorphism jr. of E which represents + ( f ) E Out(E). It was already shown in ([8, w that, if the F-action on W is rigid, and L has ULIEP, the image of # is precisely Out(E, N). We now come back to the fundamental 9-diagrams developed earlier in view of obtaining information on Out(E, N). In particular, we give a geometrical interpretation for the subgroups of Out(E, N) occurring in diagrams (3) and (4). P r o p o s i t i o n 6.3 Assume (N, E, p, W ) a L-set of data, determining a K(E,1)manifold M via a Seifert Construction 9 . I f the abstract kernel ~b : F --* Out(N), compatible with 1 --+ N --* E --* F ~ 1, is faithful, then 9 (AA(W, L*) n N f) = A u t ~
N Inn(E) -~ H$(F, Z ( N ) ) .
Proof: Fix A E A~(W, L * ) n N 1. In 7-/1(L x W) we observe that, for each n E N,
~(~) o A =
(n, . ( n ) , l )
o (A,I,1) = ( ( . ( ~ )
o
A).n, . ( . ) , 1 ) = ( ~ . A , . ( ~ ) , I ) =
~ 09(.).
So, conjugating with A in 7/] (L • W) clearly induces an automorphism A, of E restricting to the identity on N. As r : F -+ Out(N) is one to one, A. also induces the identity on F and consequently, @(A) E A u t ~ N Inn(E)" Conversely, let r E Aut~ 9 As r induces the identity on both N and F, ~o~r is a Seifert Construction for the same data (N, E, p, W). Using uniqueness1, there exists A e A4( W, L* ) such that xI' o ~ = Iz( A ) o g2 or ~. = cr. 9 P r o p o s i t i o n 6.4 Assume (N, E, p, W ) a L - s e t of data, determining a K(E,1)manifold M via a Seifert Construction ~ with respect ~oa certain embedding e : N r L. Suppose that L has ULIEP and thai the abstract kernel r : F --* Out(N) is faithful. LeE a ~ H$(F, N ) be the cohomoIogy class de$ermined by 1 -~ N -~ Z -~ F -~ 1. W~i~e Aut(L)0 for the subg~,oup of A u t ( L ) containing all auZomorphisms of L which ave eztensions of automorphisms in Ker(*) n A4r a. Then ~(AA(W, L*) >~Aut(L)o n N f) = K e r ( B ) / K e r ( B ) n Inn(E)" proof: Using the UAEP, an element of Aut(L)0 can be written as A(~), for c r e Ker(B). For this r q ocr is a Seifert Construction for (N, E,p, W) with respect to the embedding N A(~o~ L. Use uniqueness-2 to conclude that ~I'o~r -#(A=,A(o-)) o if2 for a +~= e A4(W, L*). So, + maps (A~,A(cr)) onto the class of r in Ker(B)/Ker(B) n Inn(E)"
160
IGODT-MALFAIT
Now, remark that, for A E A,/(W, L'), (A, A(cr),l)= (A.A;1,1,1)(Aa, A(cr),l), and consequently, using (6.3) @((A, A(~r), 1) e Ker(B)/Ker(S) N Inn(E)" E x a m p l e 6.5 We give two situations where the conditions mentioned above are satisfied. Take W the 1-point-space and F finite. Then 1. if L is simply connected, connected nilpotent containing a lattice N the resulting manifolds are known as the inf~a-nilmanifolds. 2. if L is simply connected, connected solvable of type (R) ([6]) containing a lattice N, for the resulting infra-solvmanifolds we can apply both propositions.
References [1] Charlap, L. S. Bieberbach Groups and Flat Manifolds. Springer-Verlag, New York Inc., 1986
Universitext.
[2] Conner, P. E. and Raymond, F. Manifolds with few periodic homeomorphisms. Lect. Notes in Math. 299. , Proceedings of the Second Conference on Compact Transformation Groups, Springer-Verlag, 1-75, (1971) [3] Conner, P. E. and Raymond, F. Deforming Homotopy Equivalences to Homeomorphisms in Aspherical Manifolds. Bull. A.M.S., 83 (t), pp. 3685, (1977) [4] Dekimpe, K., Igodt, P., Kim, S., and Lee, K. B. Affine siruelures for closed 3-dimensional manifolds with NIL-geometry. (1993). to appear in Quart. :J. Math. Oxford [5] Dekimpe, K., Igodt, P., and Maifait, W. On the Fitting subgroup of almost crystallographic groups. Tijdschrift van her Belgisch Wiskundig Genootschap, B 1, pp. 35-47, (1993) [6] GorbaceviS, V . V . Lattices in solvable Lie groups and deformations of homogeneous spaces. Math. USST Sbornik, 20 (2), pp. 249-266, (1973) [7] Igodt, P. and Malfait, W. The outer aulomorphism groups for almost Bieberbach groups of dimension < 4. (1994). Preprint. [8] Kamishima, Y., Lee, K. B., and Raymond, F. The Seifert construction and its applications $o infra-nilmanifolds. Quarterly J. of Math. (Oxford),
34, pp 433-452, (19s3) [9] Lavendhomme, R. and Roisin, J. R. Cohomologie non abdlienne de s~ructures algdbriques. Journal of Algebra, 67, pp. 385-414, (1980)
IGODT-MALFAIT
161
[10] Lee, K. B. There are only finitely many infra-nilmanifolds under each nilmanifold. Quart. J. Math. Oxford, (2) 39, pp. 61-66, (1988) [11] Lee, K. B. and Raymond, F. Seifert Manifolds Modelled on Principal Bundles. Transf. Groups (Osaka, 1987); Lect. Notes in Math., 1375 pp. 207-215, (1989) [12] Passman, D.S. The Algebraic Structure of Group Rings. Pure and Applied Math. John Wiley & Sons, Inc. New York, 1977 [13] Raghunathan, M. S. Discrete Subgroups of Lie Groups. Springer-Verlag, 1972 [14] Segal, D. Polycyclic Groups. Cambridge University Press, 1983
Katholieke Unlversiteit Leuven Campus Kortrijk Unlversltalre Campus B-8500 K O R T I ~ I J K (Belgium) E-mail: P a u l . I g o d t ~ k u l a k . a c . b e
(Received June 28, 1993; in revised form March 21,
1994)
