[ Pobierz całość w formacie PDF ]
the covering transformations of X contained a cyclic group of prime order, then it
would contradict the Lemma.
38 1. TRANSFORMATION GROUPS
1.15.5 Corollary ([?, Theorem 5.6], see also Theorem ?? ). If G = e is a compact
connected Lie group acting effectively on a closed aspherical manifold M, then G
is a toral group and acts injectively on M.
k
Proof. Let T be a maximal torus in G. If the restriction of the evaluation ho-
k
momorphism evx : À1(G, e) ’! À1(X, x) to the torus T is not an injection, then
"
k
there exists a circle subgroup C in T , for which the restriction of the evalua-
tion homomorphism is not injective. This contradicts the Lemma. Consequently,
k
the evaluation homomorphism evx : À1(T , e) ’! À1(M, x), which factors through
"
k
À1(G, e), is injective. But, evx : À1(T , e) ’! À1(G, e) (under the homomorphism
"
k
induced by inclusion) is injective if and only if G itself is T .
1.15.6 Remark. There is a subtlety in the argument above, namely, the validity of
Ø
the Künneth rule for Cech cohomology (with closed support). This is no problem if
F has the homotopy type of a CW-complex or if F is compact. Even if F is neither,
the rule is still valid because CP " (resp. BZ ) is a nice space, see [?, Chapter XVI,
p
§5].
The Vietoris mapping theorem states that under a closed mapping for which
Ø
inverse images of points are acyclic in Cech cohomology, then the mapping induces
1 1
an isomorphism in cohomology. Here we replace BS1 = Sx\CP " by Sx\CPn, where
x
1
n is very large. Then Sx\CPn is compact and Q-acyclic up to dimension n - 1.
These technical concerns all vanish if we assume that (S1, X) is a smooth action.
In fact, then F is a smooth Q-acyclic submanifold. A different proof of the lemma,
using Smith theory, can be found in [?, III,§10]. The lemma is also valid with
cohomology with integral coefficients and with the provision that (S1, X) has only
a finite number of non-isomorphic stability groups.
1.15.7 Lemma. If (S1, M) is a non-trivial action where M is a closed connected
aspherical manifold, then the action must be injective.
Proof. Assume that the kernel of evx is not trivial. Then there exists a finite
#
covering group S1 ’! S1 which acts non-trivially on M, and for which image of
evx is trivial. Then this action of S1 lifts to the universal cover M of M. Since M
#
is contractible, it is Q-acyclic. The group of covering transformations acts on M.
S1
By Lemma 1.15.3 , E = M must be Q-acyclic and by Lemma 1.15.1,
S1
½-1(F ) = E, and À1(M)\E = F = X . Since E is Q-acyclic, then
H"(À1(M)\E; Q) = H"(F ; Q) is the same as the group cohomology H"(À1(M); Q),
which is the same as H"(F ; Q) because M is aspherical.
If M is orientable, we have Hn(M; Q) = Q. But F = M, since the action
(S1, M) was assumed non-trivial, and Hn(F ; Q) = 0 because F is a proper closed
subset of M. This is a contradiction so the S1-action is injective.
If M is not orientable, we can lift the S1-action to the orientable double cover
M, because the elements of À1(M) which preserve the orientation of M are left
invariant by S1. Thus the S1-action lifts to orientable double cover M of M. Then
the S1-action on M is injective, and consequently it is injective on M.
1.15. COMPACT LIE GROUP ACTIONS 39
1.15.8 Proposition ([?, Corollary 6.2], see also Corollary ??). Let G be a finite
group acting effectively on a closed connected aspherical n-manifold M with fixed
point at x " M. Then the representation ¸ : G ’! Aut(À1(M, x)) induced by the
action of G on À1(M, x) is injective.
Proof. Suppose K is the kernel of ¸. If K = 1, let Zp be a cyclic subgroup
of K of prime order p. Then the action of Zp lifts to the universal cover-
p
ing M of M. Then MZ = " and is Zp-acyclic by Lemma 1.15.3. Further-
more, Zp commutes with the covering transformations À1(M, x) on M, by The-
orem 1.9.1. By Corollary 1.9.5??? , À1(M) acts freely as covering transforma-
p p
tions on MZ and covers C, the path component of MZ containing x. Therefore,
H"(À1(M, x); Zp) H"(M; Zp) H"(C; Zp). If M is orientable, this means that
= =
Hn(M; Zp) Hn(C; Zp) Zp. But C is a closed subset of M and this can only
= =
happen if C = M, which contradicts the effectiveness of G. Therefore K must be
trivial and ¸ is faithful.
If M is not orientable, the action of Zp lifts with fixed point to the orientable [ Pobierz całość w formacie PDF ]
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the covering transformations of X contained a cyclic group of prime order, then it
would contradict the Lemma.
38 1. TRANSFORMATION GROUPS
1.15.5 Corollary ([?, Theorem 5.6], see also Theorem ?? ). If G = e is a compact
connected Lie group acting effectively on a closed aspherical manifold M, then G
is a toral group and acts injectively on M.
k
Proof. Let T be a maximal torus in G. If the restriction of the evaluation ho-
k
momorphism evx : À1(G, e) ’! À1(X, x) to the torus T is not an injection, then
"
k
there exists a circle subgroup C in T , for which the restriction of the evalua-
tion homomorphism is not injective. This contradicts the Lemma. Consequently,
k
the evaluation homomorphism evx : À1(T , e) ’! À1(M, x), which factors through
"
k
À1(G, e), is injective. But, evx : À1(T , e) ’! À1(G, e) (under the homomorphism
"
k
induced by inclusion) is injective if and only if G itself is T .
1.15.6 Remark. There is a subtlety in the argument above, namely, the validity of
Ø
the Künneth rule for Cech cohomology (with closed support). This is no problem if
F has the homotopy type of a CW-complex or if F is compact. Even if F is neither,
the rule is still valid because CP " (resp. BZ ) is a nice space, see [?, Chapter XVI,
p
§5].
The Vietoris mapping theorem states that under a closed mapping for which
Ø
inverse images of points are acyclic in Cech cohomology, then the mapping induces
1 1
an isomorphism in cohomology. Here we replace BS1 = Sx\CP " by Sx\CPn, where
x
1
n is very large. Then Sx\CPn is compact and Q-acyclic up to dimension n - 1.
These technical concerns all vanish if we assume that (S1, X) is a smooth action.
In fact, then F is a smooth Q-acyclic submanifold. A different proof of the lemma,
using Smith theory, can be found in [?, III,§10]. The lemma is also valid with
cohomology with integral coefficients and with the provision that (S1, X) has only
a finite number of non-isomorphic stability groups.
1.15.7 Lemma. If (S1, M) is a non-trivial action where M is a closed connected
aspherical manifold, then the action must be injective.
Proof. Assume that the kernel of evx is not trivial. Then there exists a finite
#
covering group S1 ’! S1 which acts non-trivially on M, and for which image of
evx is trivial. Then this action of S1 lifts to the universal cover M of M. Since M
#
is contractible, it is Q-acyclic. The group of covering transformations acts on M.
S1
By Lemma 1.15.3 , E = M must be Q-acyclic and by Lemma 1.15.1,
S1
½-1(F ) = E, and À1(M)\E = F = X . Since E is Q-acyclic, then
H"(À1(M)\E; Q) = H"(F ; Q) is the same as the group cohomology H"(À1(M); Q),
which is the same as H"(F ; Q) because M is aspherical.
If M is orientable, we have Hn(M; Q) = Q. But F = M, since the action
(S1, M) was assumed non-trivial, and Hn(F ; Q) = 0 because F is a proper closed
subset of M. This is a contradiction so the S1-action is injective.
If M is not orientable, we can lift the S1-action to the orientable double cover
M, because the elements of À1(M) which preserve the orientation of M are left
invariant by S1. Thus the S1-action lifts to orientable double cover M of M. Then
the S1-action on M is injective, and consequently it is injective on M.
1.15. COMPACT LIE GROUP ACTIONS 39
1.15.8 Proposition ([?, Corollary 6.2], see also Corollary ??). Let G be a finite
group acting effectively on a closed connected aspherical n-manifold M with fixed
point at x " M. Then the representation ¸ : G ’! Aut(À1(M, x)) induced by the
action of G on À1(M, x) is injective.
Proof. Suppose K is the kernel of ¸. If K = 1, let Zp be a cyclic subgroup
of K of prime order p. Then the action of Zp lifts to the universal cover-
p
ing M of M. Then MZ = " and is Zp-acyclic by Lemma 1.15.3. Further-
more, Zp commutes with the covering transformations À1(M, x) on M, by The-
orem 1.9.1. By Corollary 1.9.5??? , À1(M) acts freely as covering transforma-
p p
tions on MZ and covers C, the path component of MZ containing x. Therefore,
H"(À1(M, x); Zp) H"(M; Zp) H"(C; Zp). If M is orientable, this means that
= =
Hn(M; Zp) Hn(C; Zp) Zp. But C is a closed subset of M and this can only
= =
happen if C = M, which contradicts the effectiveness of G. Therefore K must be
trivial and ¸ is faithful.
If M is not orientable, the action of Zp lifts with fixed point to the orientable [ Pobierz całość w formacie PDF ]