Description of the vector $G$-bundles over $G$-spaces with quasi-free proper action of discrete group $G$

0.14 MB·English

by  Alexander S. Mishchenko

Description of the vector G-bundles over G-spaces
9 with quasi-free proper action of discrete group G
0
0
2 Mishchenko , A.S.∗ Morales Mel´endez, Quitzeh

n
a January 21, 2009
J

1
2

Abstract
]
T WegiveadescriptionofthevectorG-bundlesoverG-spaceswithqusi-
A free properaction of discrete group G in terms of theclassifying space.
.
h
t 1 The setting of the problem
a
m
[ This problem naturally arises from the Conner-Floyd’s description ([2]) of the

bordismswiththeactionofagroupGusingtheso-calledfix-pointconstruction.
1
Thisconstructionreducestheproblemofdescribingthebordismstotwosimpler
v
8 problems: a)descriptionofthefixed-pointset(or,moregenerally,thestationary
0 pointset),whichhappenstobeasubmanifoldattachedwiththestructureofits
3 normal bundle and the action of the same group G, however, this action could
3
have stationary points of lower rank; b) description of the bordisms of lower
.
1 rank with an action of the group G. We assume that the group G is discrete.
0 Lets ξ be an G-equivariant vector bundle with base M.
9
0
ξ
:
v (1)

Xi My
r
a Lets H <G be a normal finite subgroup. Assume that the action of the group
G over the base M reduces to the factor group G =G/H:
0
G×M −→ M
k (2)


G0y×M −→ M
suppose,additionally,thatthe actionG ×M−→M isfreeandthereisnomore
0
fixed points of the action of the group H in the total space of the bundle ξ.
∗Partly supported by the grant of RFFI No.08-01-00034-a, NSh-1562.2008.1, Program
2.1.1/5031
1
So, we have the following commutative diagram
G×ξ −→ ξ
(3)
 
 
G0y×M −→ My
Definition 1 As in [6, p. 210], we shall say that the described action of the
group G is quasi-free over the base with normal stationary subgroup H.
Reducing the action to the subgroup H, we obtain the simpler diagram:
H ×ξ −→ ξ
(4)
 
 
My = My
Following[4],letρ :H−→U(V )betheseriesofalltheirreducible(unitary)
k k
representationof the finite groupH. Then the H-bundle ξ can be presented as
the finite direct sum:
ξ ≈ ξ V , (5)
M(cid:16) kO k(cid:17)
k
where the action of the group H over the bundles ξ is trivial, V denotes the
k k
trivial bundle with fiber V and with fiberwise action of the group H, defined
k
using the linear representation ρ .
k
Lemma 1 The group G acts on every term of the sum (5) separately.
Proof. Consider now the action of the group G over the total space of the
bundle ξ. Fix a pointx∈M. The actionofthe elementg ∈G is fiberwise,and
maps the fiber ξ to the fiber ξ :
x gx
Φ(x,g):ξ −→ξ .
x gx
Also, for a par of elements g ,g ∈G we have:
1 2
Φ(x,g g )=Φ(g x,g )◦Φ(x,g ), (6)
1 2 2 1 2
Φ(x,g g ):ξ Φ−(x→,g2) ξ Φ(g−2→x,g1) ξ
1 2 x g2x g1g2x
In particular, if g =h∈H <G, then g x=hx=x. So,
2 2
Φ(x,h) Φ(x,g)
Φ(x,gh):ξ −→ ξ −→ ξ
x x gx
2
Analogously, if g =h∈H <G, then g gx=hgx=gx. So
1 1
Φ(x,g) Φ(gx,h)
Φ(x,hg):ξ −→ ξ −→ ξ
x gx gx
According to [4] the operator Φ(x,h) does not depends on the point x∈M,
Φ(x,h)=Ψ(h): ξ V −→ ξ V ,
M(cid:16) k,xO k(cid:17) M(cid:16) k,xO k(cid:17)
k k
here,sincetheactionofthegroupH isgivenovereveryspaceV usingpairwise
k
different irreducible representations ρ , we have
k
Ψ(h)= Id ρ (h) .
M(cid:16) O k (cid:17)
k
In this way, we obtain the following relation:
Φ(x,gh)=Φ(x,g)◦Ψ(h)=Φ(x,ghg−1g)=Ψ(ghg−1)◦Φ(x,g). (7)
Lets write the operatorΦ(x,g) using matricesto decomposethe space ξ as
x
the direct sum
ξ = ξ V :
x M(cid:16) k,xO k(cid:17)
k
Φ(x,g) ··· Φ(x,g) ···
1,1 k,1
 .. .. .. 
. . .
Φ(x,g)=  (8)
 Φ(x,g)1,k ··· Φ(x,g)k,k ··· 
 
 ... ... ... 
If k 6=l then Φ(x,g) =0, i.e. the matrix Φ(x,g) its diagonal,
k,l
Φ(x,g)= Φ(x,g) : ξ V −→ ξ V ,
M k,k M(cid:16) k,xO k(cid:17) M(cid:16) k,gxO k(cid:17)
k k k
Φ(x,g) : ξ V −→ ξ V ,
k,k (cid:16) k,xO k(cid:17) (cid:16) k,gxO k(cid:17)
as it was required to prove.
2 Description of the particular case ξ = ξ V
0
N
Here we will consider the particular case of a G-vector bundle ξ =ξ ⊗V with
0
base M.
ξ


My
3
where the action of the group G is quasi-free over the base with finite normal
stationary subgroup H <G.
We will assumethatthe groupH acts triviallyoverthe bundle ξ . By V we
0
denote the trivialbundle with fiber V andwith fiberwise actionof the groupH
given by an irreducible linear representation ρ.
Definition 2 A canonical model for the fiber in a G-bundle ξ = ξ V with
0
N
fiber F ⊗V is the product G ×(F ⊗V) with an action of the group G
0
φ
G×(G ×(F ⊗V)) −→ G ×(F ⊗V)
0 0
 
 µ 
G×yG0 −→ Gy0
where µ denotes the natural left action of G on its quotient G , and
0
φ([g],g ):[g]×(F ⊗V)→[g g]×(F ⊗V)
1 1
is given by the formula
φ([g],g )=Id⊗ρ(u(g g)u−1(g)). (9)
1 1
where
u:G−→H
is a homomorphism of right H-modules by multiplication, i.e.
u(gh)=u(g)h, u(1)=1, g ∈G,h∈H.
Lemma 2 The definition (9) of the action of G is well-defined.
Proof. It is enoughto prove that that a) the formula (9) defines an action,
i.e.
φ([g],g g )=φ([g g],g )◦φ([g],g ),
2 1 1 2 1
and b) that the formula (9) does not depends on the chosen representative
gh∈[g]:
Id⊗ρ(u(g g)u−1(g))=Id⊗ρ(u(g gh)u−1(gh))
1 1
for every g ∈G and h∈H.
In fact,
φ([g],g g )=Id⊗ρ(u(g g g)u−1(g))=
2 1 2 1
Id⊗ρ(u(g g g)u(g g)u−1(g g)u−1(g))=
2 1 1 1
=Id⊗ρ(u(g g g)u(g g))◦Id⊗ρ(u−1(g g)u−1(g))=
2 1 1 1
=φ([g g],g )◦φ([g],g ),
1 2 1
what proves a), and, recalling the equation u(gh)=u(g)h for every g ∈G and
h∈H, it is clear that
u(g gh)u−1(gh)=u(g g)hh−1u−1(g)=u(g g)u−1(g),
1 1 1
which is a sufficient condition for b) to be true.
4
As it is well known, for the actions we are studying, we can alwaysconsider
over the base M an atlas of equivariant charts {O },
α
M = O ,
[ α
α
[g]O =O , ∀[g]∈G .
α α 0
Iftheatlasisfineenough,theneverychartcanbepresentedasadisjointunion
of its subcharts:
O = [g]U ≈U ×G ,
α G α α 0
[g]∈G0
i.e. [g]U ∩[g′]U = ∅ if [g]6= [g′], and when α 6= β, if U ∩[g ]U 6= ∅, then
α α α αβ β
the element g is the only one for which that intersection is non-empty, i.e. if
αβ
[g]6=[g ], then U ∩[g]U =∅, i.e.
αβ α β
O ∩O ≈(U ∩[g ]U )×G ,
α β α αβ β 0
foreveryα,β. We use these facts andnotations to formulatethe next theorem.
Theorem 1 The bundle ξ = ξ V is locally homeomorphic to the cartesian
0
N
product of some chart U by the canonical model. More precisely, for a fine
α
enough atlas, there exist G-equivariant trivializations
ψ :O ×(F ⊗V)→ξ| (10)
α α Oα
where
O ×(F ⊗V)≈U ×(G ×(F ⊗V))
α α 0
and the diagram
g
ξ| −→ ξ|
Oα Oα
ψ ψ (11)
x α x α
 Id×φ(g) 
Uα×(G0× (F ⊗V)) −→ Uα×(G0× (F ⊗V))
is commutative where g ∈ G, Id : U → U , and φ(g) denotes the canonical
α α
action.
Proof. Usinganatlasasintheremarksatthebeginningofthetheorem,we
shall construct the trivialization (10) starting from an arbitrary trivialization
ψ :U ×(F ⊗V)→ξ|
α α Uα
in such a way, that the diagram
g
ξ| −→ ξ|
Uα [g]Uα
ψ ψ
x α x α
 
Uα×(F ⊗V) −→ [g]Uα×(F ⊗V)
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commutes for every g ∈ [g], where the left and upper arrows are given and we
have to construct the down and right arrows.
Fromsuchaconstruction,theequivariancewillfollowautomaticallyandthe
proofofthetheoremreducestoshowthattheconstructeddownarrowcoincides
with that on (11).
Evidently, for a given trivialization ψ : U ×(F ⊗V) → ξ| , there are
α α Uα
several ways to define a trivialization ψ : [g]U × (F ⊗V) → ξ| , since
α α [g]Uα
there are several elements g ∈G sending ξ| to ξ| .
Uα [g]Uα
Thus, consider a set-theoretic cross-section
′
p :G −→G,
0
to the projection p in the exact sequence of groups
p
1−→H−→G−→G ,
0
p◦p′ =Id:G −p→′ G−p→G .
0 0
Put
′ ′
g =p ◦p:G−→G.
Without loss of generality, we can take g′(1)=1.
In this case
g′(g)=gu−1(g),
where
u:G−→H
is a homomorphism of right H-modules by multiplication, i.e.
u(gh)=u(g)h, g ∈G,h∈H.
In particular, this means that
′ ′
g (gh)=g (g), h∈H.
Lets
ψ˜ :U ×F−→ξ |
α α 0 Uα
be some trivialization. We define the trivialization ψ in (10) by the rule: if
α
[g]x ∈[g]U , i.e. x ∈U , then, the map
α α α α
ψ ([g]x ):[g]x ×(F ⊗V)−→ξ ⊗V
α α α [g]xα
is given by the formula
ψ ([g]x ) =Φ(x ,g′(g))◦ ψ˜ (x )⊗Id =
α α α (cid:16) α α (cid:17)
(12)
=Φ(x ,gu−1(g))◦ ψ˜ (x )⊗Id .
α (cid:16) α α (cid:17)
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where, from the first equality, it is clear that the definition does not depend on
the representative g ∈[g].
In particular, for [g]=1, we recover the initial trivialization
ψ (x )=ψ˜ (x )⊗Id
α α α α
since Φ(x,g′(1))=Φ(x,1)=1.
Using this trivialization the action of the group G can be carried to the
cartesian product O ×(F ⊗V):
α
Φ (g):O ×(F ⊗V)−→O ×(F ⊗V).
α α α
Lets x ∈U , g ∈G, then
α α
Φ ([g]x ,g ):[g]x ×(F ⊗V)−→[g g]x ×(F ⊗V)
α α 1 α 1 α
is given by the formula
Φ ([g]x ,g )=(ψ ([g g]x ))−1Φ([g]x ,g )ψ ([g]x ).
α α 1 α 1 α α 1 α α
Applying (12), we obtain
−1
Φ ([g]x ,g )= Φ(x ,g gu−1(g g))◦ ψ˜ (x )⊗Id ◦
α α 1 (cid:16) α 1 1 (cid:16) α α (cid:17)(cid:17)
◦Φ([g]x ,g )◦Φ(x ,gu−1(g))◦ ψ˜ (x )⊗Id =
α 1 α (cid:16) α α (cid:17)
−1
= ψ˜ (x )⊗Id ◦
(cid:16) α α (cid:17)
◦Φ(x ,g gu−1(g g))−1◦Φ([g]x ,g )◦Φ(x ,gu−1(g))◦
α 1 1 α 1 α
◦ ψ˜ (x )⊗Id =
(cid:16) α α (cid:17)
−1
= ψ˜ (x )⊗Id ◦
(cid:16) α α (cid:17)
◦Φ(x ,u−1(g g))−1◦Φ(x ,g g)−1◦Φ([g]x ,g )◦
α 1 α 1 α 1
◦Φ(x ,g)◦Φ(x ,u−1(g))◦
α α
◦ ψ˜ (x )⊗Id =
(cid:16) α α (cid:17)
−1
= ψ˜ (x )⊗Id ◦
(cid:16) α α (cid:17)
◦Φ(x ,u−1(g g))−1◦Φ(x ,u−1(g))◦
α 1 α
◦ ψ˜ (x )⊗Id ;
(cid:16) α α (cid:17)
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−1
Φ ([g]x ,g )= ψ˜ (x )⊗Id ◦
α α 1 (cid:16) α α (cid:17)
◦(Id⊗ρ(u(g g)))◦ Id⊗ρ(u−1(g)) ◦
1
(cid:0) (cid:1)
◦ ψ˜ (x )⊗Id =
(cid:16) α α (cid:17)
−1
= ψ˜ (x )⊗Id ◦
(cid:16) α α (cid:17)
◦ Id⊗ ρ(u(g g)u−1(g)) ◦
1
(cid:0) (cid:0) (cid:1)(cid:1)
◦ ψ˜ (x )⊗Id =
(cid:16) α α (cid:17)
=Id⊗ρ(u(g g)u−1(g)).
1
The operator
Φ ([g]x ,g )=Id⊗ρ(u(g g)u−1(g))=φ(g ,[g]).
α α 1 1 1
does not depend on the point x ∈U . So, the theorem is proved.
α α
ByAut (G ×(F ⊗V))wedenotethegroupofequivariantautomorphisms
G 0
of the space G ×(F ⊗V) as a vector G-bundle with base G , fiber F ⊗V and
0 0
canonical action of the group G.
Corollary 1 The transition functions on the intersection
O ∩O ≈(U ∩[g ]U )×G ,
α β α αβ β 0
i.e. the homomorphisms Ψ on the diagram
αβ
(U ∩[g ]U )×(G ×(F ⊗V)) −Ψ→αβ (U ∩[g ]U )×(G ×(F ⊗V))
α αβ β 0 α αβ β 0
 
 Id 
(Uα∩[gαβy]Uβ)×G0 −→ (Uα∩[gαβy]Uβ)×G0
(13)
are equivariant with respect to the canonical action of the group G over the
product of the base by the canonical model, i.e.
Ψ (x)◦φ(g ,[g])=φ(g ,[g])◦Ψ (x)
αβ 1 1 αβ
for every x∈U ∩[g ]U , g ∈G, [g]∈G , In other words,
α αβ β 1 0
Ψ (x)∈Aut (G ×(F ⊗V)).
αβ G 0
NowwegiveamoreaccuratedescriptionofthegroupAut (G ×(F ⊗V)).
G 0
By definition, an element of the group Aut (G ×(F ⊗V)) is an equivariant
G 0
mapping Aa, such that the pair (Aa,a) defines a commutative diagram
Aa
(G ×(F ⊗V)) −→ G ×(F ⊗V)
0 0
 
 a 
Gy0 −→ Gy0,
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whichcommuteswiththecanonicalaction,i.e. themapa∈Aut (G )satisfies
G 0
the condition
a∈Aut (G )≈G , a[g]=[ga], [g]∈G ,
G 0 0 0
and the mapping Aa =(Aa[g])[g]∈G0,
Aa[g]:[g]×(F ⊗V)→[ga]×(F ⊗V)
satisfies a commutation condition with respect to the action of the group G:
Aa[g]
[g]×(F ⊗V) −→ [ga]×(F ⊗V)
φ(g ,[g]) φ(g ,[ga]) ,
 1  1
[g1g]×y(F ⊗V) A−a[→g1g] [g1ga]×y(F ⊗V)
φ(g ,[ga])◦Aa[g]=Aa[g g]◦φ(g ,[g]) (14)
1 1 1
i.e.
(Id⊗ρ(u(g ga)u−1(ga)))Aa[g]=Aa[g g](Id⊗ρ(u(g g)u−1(g))) (15)
1 1 1
where [g]∈G , g ∈G.
0 1
Lemma 3 One has an exact sequence of groups
1→GL(F)−→Aut (G ×(F ⊗V))−→G →1. (16)
G 0 0
Proof. To define a projection
pr :Aut (G ×(F ⊗V))−→G
G 0 0
we send the fiberwise map
Aa :G ×(F ⊗V)−→G ×(F ⊗V)
0 0
to its restriction over the base a : G → G , i.e. a ∈ Aut (G ) ≈ G . So, this
0 0 G 0 0
is a well-defined homomorphism.
Weneedtoshowthatpr isanepimorphismandthatitskernelisisomorphic
to GL(F). Lets calculate the kernel.
For [a]=[1] we have
(Id⊗ρ(u(g g)u−1(g)))A1[g]=A1[g g](Id⊗ρ(u(g g)u−1(g))) (17)
1 1 1
In the case g =h∈H, we obtain
1
(Id⊗ρ(u(hg)u−1(g)))A1[g]=A1[g](Id⊗ρ(u(hg)u−1(g)))
9
Since the representation ρ is irreducible, by Schur’s lemma, we have
A1[g]=B1[g]⊗Id.
On the other side, assuming in (17) that g =1, we have
(Id⊗ρ(u(g)))A1[1]=A1[g](Id⊗ρ(u(g))),
i.e.
(Id⊗ρ(u(g)))(B1[1]⊗Id)=(B1[g]⊗Id)(Id⊗ρ(u(g))),
or
(B1[g]⊗Id)=(B1[1]⊗Id).
So, the kernel kerpr is isomorphic to the group GL(F).
Inthe genericcase,i.e. [a]6=1,we cancompute the operatorAa[g]interms
of its value at the identity Aa[1] from the formula (15): assuming g = 1, we
obtain (changing g by g):
1
(Id⊗ρ(u(ga)u−1(a)))Aa[1]=Aa[g](Id⊗ρ(u(g))), (18)
i.e.
Aa[g]=(Id⊗ρ(u(ga)u−1(a)))Aa[1](Id⊗ρ(u−1(g))), (19)
Therefore, the operator is completely defined by its value
Aa[1]:[1]×(F ⊗V)→[a]×(F ⊗V)
at the identity g =1.
Now we describe the operatorAa[1] in terms of the representationρ and its
properties.
We have a commutation rule with respect to the action of the subgroup H:
Aa[1]
[1]×(F ⊗V) −→ [a]×(F ⊗V)
φ(h,[1]) φ(h,[a]) ,
 
 Aa[1] 
[1]×(yF ⊗V) −→ [a]×(yF ⊗V)
Equivalently
Aa[1]◦φ(h,[1])=φ(h,[a])◦Aa[1],
i.e.
10