Bull. Kyushu Inst. Tech.
(Math. Natur. Sci.) No. 36, 1989, pp. 1-9
EQUIVARIANT SURFACES IN 3-MANIFOLDS WITH ABELIAN GROUP ACTIONS
Teruhiko SoMA
(Received November 10, 1988)
1. Introduction
Let G be a finite group acting on a closed, connected, orientable 3-manifold M as a group of orientation preserving diffeomorphisms. A closed orientable surface F embedded in M is said to be G-equivariant if, for each gEG, either gFnF=: ip or F=gF.
If M contains an incompressible 2-sphere S, that is, S does not bound any 3-ball in M, then by the Equivariant Sphere Theorem [12] M contains a G-equivariant, in- compressible 2-sphere. Here we consider the case where M contains no incompressible 2-spheres, i.e. M is irreducible. If M contains an incompressible torus, then by the
uniqueness of the torus decomposition (Jaco-Shalen [8], Johanson [9]) and by the minimal surface theory by (Freedman-Hass-Scott [2]) we have the following Equivariant Torus Theorem.
The Equivariant Torus Theorem ([8], [9], and [2]). Let M be a closed, connected, orientable 3-manifold which is irreducible and contains an incompressible torus. Let G be afinite group acting on M as a group of diffeomorphisms. Then either M is Seifert .fibered or contains a G-equivariant, incompressible torus.
Now we suppose that M contains no incompressible 2-spheres or tori but contains an incompressible surface. Such a surface is called simple and Haken. By Thurston's Uniformiztion Theorem (see [14]), every closed, simple Haken manifold is either Seifert fibered or hyperbolic.
In this paper we will deal with finite group actions on 3-manifolds with positive first Betti number. That is, we will consider the following problem.
Problem. Suppose M is a closed, connected, orientable 3-manifold with positive first Betti number, and G is a finite group acting on M as a group of orientation preserving diffeomorphisms. Under what conditions does M have a G-equivariant, incomppressible surface?
Here we note that Kojima [10] proved that, for every finite group G, there exists an
orientable, closed hyperbolic 3-manifold (with positive first Betti number) on which G
acts as a group of orientation preserving isometries.
2 Teruhiko SoMA
For a topological space X, we denote by 6i(X) the first Betti number of X.
Hempel [6] and Morita [15] gave examples of closed, orientable, hyperbolic 3- manifolds M containing no incompressible surfaces and having finite regular coverings p: IVI.M with 6,(M)ÅrO. Let G(=zi(M)/p.(ni(M))) be the group of covering transformations. If M contained a G-equivariant, incompressible surface F, then p(F) (or the boundary of a tubular neighborhood of p(F) if p(F) is 1-sided in M) would be an incompressible surface in M, a contradiction. Thus M is a hyperbolic 3-manifold with 6i(M)ÅrO but cosntains no G-equivariant, incompressible surfaces.
The action of G on M induces that on H.(M; Z) as a group of
isomorphisms. We say that ctEH.(M; Z) is almost G-invariant if, for each gEG, either
g (ct ) == ct or g (ct )= - ct.
Theorem 1. Let M be a closed, connected, orientable 3-manifold with 6i (M)ÅrO and let G be afinite group which acts on M as a group ofdilffeomorphisms. ijH2 (M; Z) has a non-trivial, almost G-invariant element ct, then M contains a G-equivariant, incompressible surface F such that [F]IO in H2(M; Z) (in general [F]tct).
Theorem 2. Let M be a closed, connected, orientable 3-manofold with 6i(M) ÅrO. Iet G be a.finite, abelian group such that each non-trivial element of G has order 2. ifG acts on M as a group oforientation preserving dilffeomorphisms, then M contains a G-equivariant, incompressible surface F with [F]tO in H2(M; Z).
If 6i(M) is odd, we have a more general result as follows.
Theorem 3. Let M be a closed, connected, orientable 3-manifold with 6i(M) odd, and let G be a finite abelian group acting on M as a group of orientation preserving di:ffeomorphisms. 7hen M contains a G-equivariant, incompressible surface F such that
[F]#O in H,(M; Z).
In the last section, we will give two examples which show that Theorem 3 fails when 6i(M) is even. More precisely, these show, for any positive even number 2k, there exist a closed, connected 3-manifold M with fii(M)==2k and a cyclic group G of order 2(2k+ 1 ) acting on M as a group of orientation preserving diffeomorphisms such that M contains no G-equivariant, incompressible surfaces. In Example 1, M is Seifert fibered, and in Example 2, M is hyperbolic.
In this paper we work in the smooth category and suppose that all 3-manifolds and surfaces are oriented. And we' refer to Hempel [6] for fundamental definitions and notatlons.
2. Proof of Theorem 1
Let M be a closed, connected 3-manifold and let G be a finite group acting on M as
a group of diffeomorphisms. One can assume that M has a G-invariant
metric. Suppose ctEH2(M; Z) is a non-trivial, almost G-invariant element We denote
Equivariant Surfaces in 3-Manifolds 3
by F the set of all closed (possiblly non-connected) surfaces F which are immersed in M and [F]=ct in H2(M; Z). Since F has the pull back metric, one can define the area of F, Area (F). By Lawson [11, Corollary 3.3], there exists FoEF such that
Area (F,)= inf{Area (F )l FEF},
and such that the image of Fo in M is an embedded surface (possibly disconnected) in M. That is, there exists a finite collection of mutually disjoint, connected surfaces C,,..., Ck embedded in M such that Image (Fo)==Ciu•••uCk, [Ci]#O for all i and k
[F]= 2 ni[Ci]
i=1
for some positive integers ni. We set So =CiU•••UCk•
Now we suppose that So is not G-equivariant, so there exists an element gEG such that gSo#So and gSoASo #sZS. By the argument similar to that of [2, Lemma 1.3], one can assume that So meets gSo transversely. We will orient So and gSo so that [Fo]==
-[gFo]. Note that this orientation on gSo may not be that defined by g.[So], where g.: H2(M; Z).H2(M; Z). Let N(So) and N(gSo) be tubular neighborhoods of S, and gSo respectively. To each component of N(So)-So (resp. N(gSo)-gSo), we
associate a sign which is determined by M and So (resp. gSo), that is, if 7 is an oriented arc which goes from the negative side of So (resp. gSo) through the positive side of So
(resp. gSo ), then the local intersection number ofy and So (resp. gSo) at the intersection
point is +1. Let Xi,...,X. be the components ofM-SougSo. We may assume that Xi has an intersection with the negative side of So or gSo. Let x be a point in Xi. Let 7 be an orientated curve in M starting from x and satisfying the following (iF (iv). (For a set Y, we denote by #(Y) the number of elements of Y.)
(i) yA (SoAgSo ) -= sZS•
(ii) 7 meets So and gSo transversely.
(iii) For each point y of 7ASo (resp. 7AgSo), the local intersection number of 7 and So (resp. gSo) at y is +1.
(iv) Either 7 has the maximal #(7A(SoUgSo)) among all oriented curves starting from x and satisfying (i)-(iii), or #(7nÅë(SoUgSo))== oo.
Let 6. (2 S. sÅq# (yA (SougSo ))+2) be the compo nents of 7-7A (SoUgSo ).
Assertion 1. if 6,, 6,cXi for some i (ISiÅq=m), then s=t.
Proof If there exist j,, 6, (s 7E t) contained in Xi, then for pE6, and qE6,, there exist two arcs cti, ct2 such that Octi ==Oct2 ={p, q}, cticy and ct2cXi. We orient a close loop l
=ctiU2 so that the orientation oflconsiststs with that of 7 in ct,. By (iii), the local
intersection number of l and Fo (resp. gFo) at yEIASo (resp. yEIAgSo) is niÅrO if yECi (resp. yEgC,). So we have[l]'[FoUgFo]ÅrO. This contradicts that [FougFo]=[Fo]
+[gFo]=O. U
4 Teruhiko SoMA
By assertion 1, we have #(7A(SoUgSo))Åqm. By the maximality (iv), we have the following.
Assertion 2. 7here exists a component Xo ofM-SoAgSo such that each side ofSo or gSo meeting non-trivially Xo is positive. O
-
Let X, be the closure of Xo in M. We set P==XoUSo and 9=XoAgSo, so OXo
.- pue•
First we suppose that Area(P)).Area(9). We may assume that Ci,..., Ci are the components of So meeting P. By Assertion 2, for each i, 1 S.iÅq,.. I, all sides of Ci meeting Xo are positive. Therefore a closed surface X=(Ciu•••uCi-P)u9 is orientable, and one can orient E so that [E]=[Ci]+•••+[Ci]. Let FicM be a piecewise smooth, CO-immersion defined by
lk
Fi= 2(ni-1)Ci+.2]+ Z njCj.
i-- 1 j=l+ 1
Then we have [F,] = [F,]. Since Area (P ) ). Area (9 ), Area (F,)$ Area (F, ). By rounding the corners of Fi, we can construct F2EF which is homotopic to Fi in M and Area(F2)ÅqArea(Fi). This contradicts the minimality of Area(F,).
Next we suppose that Area (P)ÅqArea (e). Since g: M.M is isometric, we again have a contradiction by the argument as above. Therefore So is G-equivariant, hence so are Ci's.
If Ci is incompressible in M, then the proof is complete. We suppose that Ci is compressible in M. Let N(Ci) be a G-equivariant tubular neighborhood of Ci. We set R=M-int(GN(C,)). The action of G on M induces that on R naturally. Since OR is compressible in R, by the Equivariant Dehn's Lemma [13] there existrs a G- equivariant, compressing disk D for OR. We do G-equivariant surgery on 0R along GD, and Jet x(OR) be the resultant surface. Since [x(OR)]==2[Ci] in H2(M; Z), there exists a component ei of x(OR) such that [e,]#O. Since e, is a G-equivariant and genus(ei)Åqgenus (Ci), by repeating the process finite (Åqgenus (Ci)) more times, one can obtain a desired G-equivariant surface. This completes the proof of Theorem
1. D
3. Proofs of Theorems 2 and 3
Let GL(n, C) be the general linear group of degree n over C, and let AEGL(n, C). Let e(A)be the set of eigenvalues of A. For ctGe(A), we denote by pt(A, ct)the multiplicity of ct as a characteristic root We define pt(A, ct)=O if ctÅëe(A). Let [C"]G be the set of points of C' fixed by the action of subgroup G of GL(n, C). Then [C"]G is a C-subspace of C". The following lemma is an elementary exercise (for example, see [16, Chapter III]).
Lemma 1. Suppose that AEGL(n, C) is periodic, i.e. AM==I. for some mEN, where
I. is the unit matrix. Let G be a.finite cyclic group generated by A. 7hen we have
Equivariant Surfaces in 3-Manifolds 5
(i) dim[C"]G ="(A, 1), and
(ii) If e (A ) =- {ct}, then A=: ctI.. D
Let X be a locally compact, Hausdorff space, and let G be a finite cyclic group acting on X. Let g be a generator of G, and let g': H'(M; C).H'(M; C) be the induced homomorphism. By Borel [1, Chapter III, Corollary 2.3], H'(X/G;
C)=[H'(X: C)]G, where X/G is the orbit space. By Lemma 1 (i), dim[H'(X: C)]G
=pt(g", 1). So we have the following lemma.
Lemma 2. 6i(X/G)="(g",1). D
We denote by o(G) the order ofa finite group G.
Proof of 7heorem 2. Let G be a finite abelian group such that each nonLtrivial element has order 2. Then one can set
G=Åqg,: gi=1ÅrÅ~•••Å~Åqg,: g,2=1År.
We proceed by the induction on o(G). Since (gi )2 is the identity map, e(gi)c{1, -1}, where gf• : H'(M; C).H'(M; C).
First we suppose that e(gi )Dl for some i. The orbit space M/G, is a closed, connected 3-manifold and, by Lemma 2, 6i(M/Gi)ÅrO, where Gi==Åqgi: g?• =1År. Let p:
M.M/Gi be the quotient map, and let G(i)=G/Gi. The action of G on M induces that of G(i) on M/Gi. It is not hard to show that M/Gi has a differentiable structure so that G(i) acts on M/Gi as a group of diffeomorphisms. By our induction, M/Gi contains a G(i)-equivariant surface F with [F];O in H2(M/Gi; Z). Therefore p-'(F) is a G- equivariant surface in M with [p"(F)]IO in H2 (M; Z). As in the proof of Theorem 1, one can obtained a desired G-equivariant surface by G-equivariant Dehn surgery [13].
Next we suppose that e(g,"• )={-1} for all i. By Lemma 1 (ii), (gl ),(x)== -x for all xEH' (M; C). Since Hi (M; Z) is a free abelian, one can regard it as a subgroup of Hi(M; C). Therefore, for all xEH'(M; Z), we have (gi )z(x)=-x, where (gi )z:
H'(M;Z).H'(M;Z). Let [M]EH,(M; Z)be the fundamental class of M. Since (gi),[M]=[M], we have the following commutative diagram.
Hi(M; Z)lk'f'Z Hi(M; Z)
in[M] in[M]
H,(M;Z) M")" H,(M;Z)
Hence, for all cteH2(M; Z), (gi).(ct)=-ct. By Theorem 1, M contains a desired G- equivariant surface. This completes the proof. O
Proof of 7hoerem 3. Let G and M be a group and a 3-manifold satisfying the
assumptions of Theorem 3. We prove the theorem by the induction on o(G). If m
6 Teruhiko SoMA
and n are relatively prime integers, then Åqt: tM"=1År is isomorphic to Åqu: uM=1ÅrÅ~Åqv: v"
=1År. Therefore one can set
G= Åqtl: ti =1År Å~ ''' Å~ Åq t.: t,2 = 1År Å~ Åqul: uiki = 1År Å~ •••
Å~ Åqu,: u,2 kq =: 1År Å~ ÅqVl: VT' == 1År Å~ ''' Å~ ÅqVr: VrM' = 1År,
where kiÅr1 for 1$iÅq..,q and mj is odd for ISjÅq=r.
First we suppose that rÅrO. Since the characteristic polynomial for vr: Hi (M; C) .Hi(M; C) has integer coeMcients, the number of the linear factors without real coeMcient is even. That is, 2 pt(vr, ct) is even, where S2=e(vf)A(C-R). Note that ae n e(vf) c9U{1, - 1} and det (vf ) == (- 1 )"("f- '). Since 1 = det ((vf )Mi )= (- 1 )mip("l•- ')
and since m, is odd, pt(vf, -1) must be even. Since 6i(M)=2 pt(vf, ct)+pt(vf, 1) orEn
+pt(vf, -1) is odd, so is pt(vf, 1). By Lemma 2, M/Vi is a closed, connected 3- manifold with 6i(M/Vi) odd, where Vi=Åqvi: vTi=1År. Since the action of G on M induces that of G/Vi on M/Vi and o(G/Vi)Åqo(G), one can obtain a desired G- equivariant surface in M by the argument similar to that in Theorem 2.
Next we suppose that qÅrO. Since kiÅr1, Ui=Åqui: uik'=1År has a non-trivial
subgroup W, = Åqu? : (ub2ki - i = 1År of index 2. Since 1 == det (det (uf ) )2 = det ((u? )" )=
(- 1)pa ((u?)"•'- i), pt ( (ub", - 1 ) is even, so pt((u? )", 1 ) : 6, (M/ Wi ) is odd. Hence one can obtain a desired G-equivaiant surface as above.
If q=r= O, then each non-trivial element of G has order 2. Hence Theorem 2 completes the proof of Theorem 3. D
4. Examples
For a kont K in S3, E(K) =S3-int N(K, S3) is called the exterior of K in S3, where N(K, S3) is a tubular neighborhood of K. We refer to Rolfsen [17] for fundamental noations and definitions on the kont theory.
Example 1. For a given integcr kÅrO, let K be the torus knot of type (2, 2k +1). Then E(K) is diffeomorphic to the mapping torus FÅ~I/h, where F is a compact,
connected surface with genus(F)=k and OF is connected, and h: F-F is a diffeomorphism of period 2(2k+1). Hence there exists a diffeomorphismf p-`(F)-F
xS', where p: M-ÅrS3 is a 2(2k+1)-fold cyclic covering of S3 branched over K. Therefore we have 6,(p-'(F))=2k+1. Since K is a torus knot, M is a Seifert fibered space. We set V=p-'(N(K, S3)). Note that Vis a solid torus with ameridian pt such that, for each pES',f(pt) meets Fx{p} transversely in one point. From the Mayer-Vietoris sequence for p-'(E(K)) and Vwith real coeMcients, we have 6i(M)
=2k. Let r: N.S3 be a 2-fold cyclic branched covering of S3 branched over K, and let
q: M--ÅÄN be a (2k+1)-fold cyclic branched covering of N branched over
r-' (K). Hence p=:roq. Let T(resp. S) be the group of the covering transformations for
Equivariant Surfaces in 3-Manifolds 7
p (resp. r). Then Tis isomorphic to Åqt: t2(2k")==1År and S isomorphic to Åqs: s2=
1År,
