~lventione$
mathematicae
Invent, math. 87, 457-476 (1987)
9 Springer-Verlag 1987
0(2) actions on the 5-sphere
Ronald Fintushel 1,, and Ronald J. Stern 2,**
I Tulane University, Department of Mathematics, New Orleans, LA 70118, USA
2 University of Utah, Department of Mathematics, Salt Lake City, UT 84112, USA
1. Introduction
A pseudofree Sl-action on a sphere S 2 k - 1 is a smooth Sl-action which is free
except for finitely many exceptional orbits (whose isotropy types Z,,, .... ~ , .
have pairwise relatively prime orders). Linear examples of these actions are
obtained by viewing S 2k- 1 as the unit sphere in ~k and defining the Sl-action
t . ( z 1. . . . . z~) = (t ~ z 1. . . . . t~kzk)
where a I ..... a k are positive pairwise relatively prime integers with product
---a 1.... , ak>__2. This action has one exceptional orbit for each ai->2. Since all
linear pseudofree Sa-actions on S 2k- a are obtained in this fashion, we see that
such a linear action on S 2g- t can have no more than k exceptional orbits.
For k = 2 Seifert [S] showed that each pseudoffee St-action on S 3 is linear
and hence has at most two exceptional orbits. In contrast to this, Montgomery
and Yang [ M Y ] showed that given any positive pairwise relatively prime
collection of integers an, ..., a,, there is a pseudofree Sa-action on a homotopy
7-sphere whose exceptional orbits have exactly those orders. Petrie [P] has
proven similar results in all higher odd dimensions. This leaves what we like to
call:
The Montgomery-Yang problem
M u s t a pseudofree S l - a c t i o n on S s have <=3 e x c e p t i o n a l orbits?
In this paper we use techniques of gauge theory to give an affirmative answer
to this question under the added hypothesis that the pseudofree St-action
extends to an action of 0(2).
* Partially supported by National Science Foundation Grant DMS8501789
** Partially supported by National Science Foundation Grant DMS8402214
458
R. Fintushel and R.J. Stern
The orbit space X=SS/S 1 of a pseudofree Sl-action is a pseudofree orbifold
(see [FS]); i.e. a 4-manifold with isolated singularities whose neighborhoods
are cones on lens spaces corresponding to the exceptional orbits of the S xaction. It is easy to check that X is a simply connected rational homology
manifold whose intersection form on H z ( X ; Q)~(l~ has matrix (l/s) where
= a t, . . . , a , is the product of the orders of the exceptional orbits. An exceptional orbit in S s with isotropy type Z a has an equivariant tubular neighborhood which may be identified with ~ • ~ • S 1 with Sl-action
t.(z, w, u)=(tr z, tSw, tau)
where r and s are relatively prime to a. The "slice type" (a;r,s) is then
associated to the corresponding singular point in X. If D(X) denotes the
complement of cone neighborhoods of the singular points of X, then the
restriction of the orbit map Ss--*X over D(X) is a principal Sl-bundle projection whose Euler class is a generator of HZ(D(X); Z)~Z. This points out that
the slice types are in fact determined by the orbit space X. The quotient by S 1
of the above tubular neighborhood of the 7/a-exceptional orbit is the cone
cL(a;r,s)cX. It is well-known (see [ M Y ] ) that pseudofree Sl-actions on S 5
are classified by their corresponding orbifolds.
In [FS] we associated to X=Ss/S x (or equivalently to the SX-action) an
integral invariant
where there are n exceptional orbits with slice types (ai; ri, si) and ~=a~, ..., a,,
and we showed that R(X)<0. The odd integer R(X) is the expected dimension
of the moduli space of self-dual SO(3)-connections on a singular bundle over X
(computed using a branched cover) whose existence ( R ( X ) > 0 ) would give a
contradiction. (See [FSw
Our technique in this paper is to study the antiself-duality equations on the same bundle over X. Since this bundle has a
positive Pontryagin number one would expect to find no solutions to these
equations on X. Indeed, if X were actually a manifold, the expected dimension
of the moduli space of solutions to the anti-self-duality equations would be
-2pt-6<0.
However, for the pseudofree orbifold X the expected dimension
turns out to be 2 n - 9 - R ( X ) where n is the number of singular points of X.
Since R(X) must be negative 2n-9-R(X)>O if n > 4 . We are able to parlay
this into a contradiction if n > 4 and the corresponding St-action on S 5 extends
to an O(2)-action.
We shall see that the existence of such O(2)-actions is related to certain
cobordism type properties of 2-bridge knots and links in S 3. The last section of
this paper studies these properties, one of which is concerned with the question
of when a disjoint union of 2-bridge knots can bound a punctured KI.IP2 in a
punctured S 4.
We close this introduction with the description of some pseudofree S 1actions on S s which extend to O(2)-actions. Of course each linear action
0(2) actions on the 5-sphere
459
extends, but many nonlinear actions extend as well. For example, consider the
algebraic variety V={z"o+Z'~+2zzz3=O } in II~4, then Vc~Sv_~S 5 [ H M , p. 106].
1
i
(To see this change coordinates by Yz = ~ 2 ( z 2 + z3), Y3 = ~ ( z z - z 3 ) ' )
Vz
Vz
action
The
St -
t'(z0, Zt, 22, Z3) = ( t m z o , t n z l , t ran- t Z2 ' t z 3 )
on II;3 restricts to Vc~ST~-S s to give a pseudofree S~-action with one exceptional orbit, and it clearly extends to an action of 0(2) using complex conjugation in II~4. The quotient by S ~ of a tubular neighborhood of the exceptional orbit is a cone on the lens space L ( m n - l ; m , n ) . Since a linear action
with one exceptional orbit always has corresponding neighborhood a cone on
some L(p; 1, 1), this construction gives many examples of nonlinear actions
which extend to actions of 0(2).
Actually, every pseudofree St-action on S s which is known to us also
extends to an action of 0(2). The nonlinear examples which we are aware of
(other than those described in the last paragraph) have orbit spaces which are
constructed from Seifert fibered 3-manifolds which bound rational balls. (See
[FS].) We now describe why the actions extend to actions of 0(2) in these
examples. A Seifert fibered homology lens space is the 2-fold branched cover of
a "Montesinos link" in S 3. Quite often the techniques which can be used to
show that a given Seifert fibered homology lens space bounds a rational ball
will also show that the involution given by the branched cover extends over
the rational ball. See e.g. [CH]. This process will then construct an involution
on Ss/S t which arises from the extension of the action of S t on S s to an action
of 0(2). In forthcoming work we shall use the techniques of this paper to study
these Montesinos links.
2. Circle actions and involutions
Consider a pseudofree
relatively prime) orders
that this action extends
is an involution z of S 5
St-action on S 5 with exceptional orbits of (pairwise
a t .... , a, and let ~ = a l , . . . , a, be the product. Suppose
to an action of 0(2), in other words suppose that there
such that
zt(z)=t-lz(z)
for every t e S 1 and z~S s. It is clear that 9 induces an involution, which we shall
again call z, on X = S S / S 1. Let D X = X -
0 cL(ai; ri, si) be the nonsingular part
i=1
of X. The next proposition describes Y = X/~.
(2.1) Proposition. (i) The orbit space Y is a 4-manifold with the homotopy type
of S ~.
(ii) The orbit map ~: X ~ Y is a 2-fold branched covering with branch set X \
a projective plane if ~ is odd or a pinched torus if ~ is even.
460
R. Fintushel and R.J. Stern
(iii) The inclusion r c X ~ Y
is an embedding with singularities; for each
singular point in X there is one singularity of ~zX~ ~ Y which is a cone on a 2bridge knot or link.
Proof. To see that Y is a manifold, consider an Sl-orbit in S 5 that is invariant
under z. (Note that each exceptional orbit is invariant under z.) If we identify
such an orbit with S t c l E , then the equation z t = t - i z shows that the action of
z is equivalent to complex conjugation on the orbit. Hence each z-invariant S 1orbit has a pair of z-fixed points. A slice of the O(2)-action at such a fixed
point may be identified with ~,4, and the isotropy group of the O(2)-action at a
z-fixed point is the dihedral group D2a where 7Za is the isotropy group of the
S~-orbit. (Identify Z~ = { 1}.) The orbit space Y at the image of our z-fixed point
may be locally identified with the quotient of the slice F, 4 by a linear action of
the isotropy group D2.. The quotient of such a linear action is easily seen to
be R 4. In fact, for the Z a c D 2 a , ~ 4 / Z a ~ c L ( a ; r , s ) (where (a;r,s) is the slice
type of the S~-orbit), and cL(a; r, s ) / z ~ - c S 3 = ~ 4. The orbit map
L(a; r , s ) ~ L(a; r,s)/z~-S 3
is just the standard branched double covering of a 2-bridge knot if a is odd or
a 2-bridge link if a is even [Sch]. (To see this first note that one can conjugate
the slice representation of D2a o n R z~ until it is in standard form, i.e. Za ~ D 2 a
is generated by the rotation generating the action with slice representation
(a;r,s) and z is given by complex conjugation on C2---R 4. Write S a = S 1
• D 2 u D 2 • S1; the action of z on each solid torus has fixed set equal to a pair
of arcs and has orbit space a 3-ball. The same is true of each solid torus after
taking the quotient by Z,. Hence the orbit map is as claimed.) Notice that our
argument also implies that the fixed point set X ~ of z is a surface, nonsingular
if ct is odd and with one singularity (of type a cone on two circles) if ~ is even.
The singularity occurs because for a i even L(ai; ri, sl) is the 2-fold branched
cover of a 2-bridge link, not knot.
Since z has fixed points in X and X is simply connected, so is Y [B; II.6.3].
N o w let D Y = n ( D X ) and let D X ~ denote the fixed point set of z on DX. We
have
2z(DY ) = )~(OX) + z(D X ~)
i.e.
2(2 + b 2 ( Y ) - n ) = 3 - n + x(D X ~)
(2.2)
1 + 2b2(Y ) - n = x ( D X ~)
Suppose first that ct is odd; so each DX~r~L(ai; r~,s~) is a circle and D X ~ is a
surface with n boundary components. Then the fixed set X ~ is the corresponding closed surface. By a standard result of Smith theory [B; 111.4.1]
rkHo(Y,~z(X~); Z z ) + ~ r k n , ( x ' ; Z 2 ) < ~ r k H i ( S ; ~2)
i~o
i.e.
rkHi(z~; Z2)_<-3.
i>-o
i~o
0(2) actions on the 5-sphere
461
This implies that X ~ is a connected surface. Thus
z(DX~)={22-2g-n,
g - n,
X~ orientable,
X ~ nonorientable,
g>O
g > 1.
+2ba(Y)=S2-2g,,
~2 - g,
x " orientable,
X" nonorientable,
g>0
g ~ 1.
-
So from (2.2):
1
Hence X ~ is nonorientable of genus l, X ~ R I P 2, and b2(Y)=0.
If ct is even then DX ~ is a surface with n + 1 boundary components, and X ~
is the singular surface obtained by identifying two points in the closed surface
corresponding to DX ~. An argument completely analogous to the one just
given shows that in this case X ~ is a pinched torus, DX ~ is a sphere with n + 1
holes, and again b2(Y)=0. Hence Y is a homotopy 4-sphere. []
Our next goal is to lift the involution r on X to a branched cover. The next
proposition is a first step.
Proposition. Each class in H2(DX; 71)~71 can be represented by an embedded surface S which is z-invariant, i.e. z(S)=S.
(2.3)
Proof. The action of z , on H2(DX; ~ ) ~
is multiplication by - 1 since
H2(DX; ~)~=H2(DY; F,0=0 by (2.1 (i)). Hence the action of ~, on Hz(DX; 71)
is also multiplication by - 1 . N o w we wish to show that z, is not the identity
on Hz(DX, DX~;712). For
if z , = i d ,
then ( i d + z , ) x = x + x = O
for
x~Hz(DX, DX~; 712). So if, using the terminology of Smith Theory, we let o = 1
+z, then a , ~ - 0 on H2(DX, DX~; 712). But, letting DY~=n(DX'), from Smith
Theory (e.g. [B; III.3.5]) we have:
H2(DX, DXr;
712) -~*
~
H~(DX, DX~; 7Z2)
H2(DY, DYe; 712) ind.
= ' H2(DY, DY~;712).
So if a , =0, then also 7t, =0. But we also have
0-------~ Hz(DX; 7 1 2 ) -
0
' H2(DX, DX~; 7~2)
' HI(DX~;
, H2(DY, DY~;712)
, HI(DY~;TZ2)-------~O
712)-
' 0
Now ~ is surjective and HI(DX~; Z2)---(~)Za; thus n, cannot be 0. Hence z, is
not the identity on H2(DX, DX~; Z2).
Consider now the sequence with Z-coefficients
0--* He(DX; Z) J*, H2(DX, DX~; I)
o , HI(DX~ ,z)___, 0
462
R. Fintushel and R.J. Stern
For any xeH2(DX, DX~;Z), O(x-z,x)=O since z, is the identity on
HI(DX~; Z). Hence there is a unique yeH2(DX; Z) satisfying j , y = x - z , x .
Define 7(x)=y; so y: H2(DX, DX~;Z)~H2(DX;Z) is a homomorphism defined by j,7(x)=x-~,x. If zeH2(DX; 7~), then j, T j , ( z ) = j , z - T , j , z = j , ( z z,z)=j,(2z). So ~j,(z)=2z. Thus Im(~j,)=2ZcZ=H2(DX; Z).
Since T, is not the identity on H2(DX, DX~;Z)=H2(DX, DX';Z)|
there is an x~H2(DX, DX~; Z) with x - z , x 4 : 0 in Hz(DX, DX~; Z2). Hence
7xr
Z). So 7 x = 2 r + 1, say, in Z, thus 1 =V(x-rj,(1)) and so 7 is
surjective.
Given y~Hz(DX; Z), let xoEHz(DX, DX'; Z) be such that Vxo=Y. Since
Oxo can be represented by a finite disjoint union of embedded circles, x o can
be represented by an embedded surface C whose boundary lies on DX ~. It can
be arranged that C intersects 9 C only in transverse double points. The closed
surface C - z C represents 7Xo=y and any double point can be removed
equivariantly by a local cut and paste, increasing the genus, but not changing
the homology class of C c~zC. Thus we obtain an equivariant embedded
surface S c D X with [ S ] = y . []
As we have pointed out earlier, the Sa-action over DX is free; so it is
classified by an Euler class eeH2(DX;Z)~-Z. It is easily seen that e is a
generator of H2(DX; Z), for example see [MY]. Also Hz(DX, ODX;Z)_~7~
and the map
j*: H2(DX, c3DX; Z) ~ H2(DX ; Z)
may be identified with multiplication by ~. Let f be the unique class in
H2(DX, ODX;Z) such that j*f=ae.
Then
eZ=l-(euf)[DX, c3OX]= I-
[FS; w2]. (For later reference note that f is a generator of H 2(DX, ODX; Z).)
As in [FS] we can find a regular c~-fold branched cyclic cover
2: M-* X
where M is a smooth closed 4-manifold and 2 is branched over the n singular
points of X and a z-equivariant surface F embedded in DX, where the
homology class [F]~H2(DX; Z) is the Lefschetz dual of f~H2(DX, ~OX; Z).
It is easily checked that H~(M; Z ) = 0 .
Proposition. The involution ~ on X is covered by an involution of M.
Furthermore, this involution combines with the action of Z, to give an action of
the dihedral group G=D2, on M.
(2.4)
Proof We first lift z to an involution, still called 3, on 2-t(DX-F). A lift
exists if and only if
z , 2 , nl(2-1(DX-F))~2, ~x(2-1(DX-F)).
Furthermore, such a lifted map will be an involution since (DX-F)~4:r (See
[B; 1.9.2].) Since IF] generates H2(OX; Z), and since [ F ] 2 = ~ it is easy to see
that H t ( D X - F ; Z)_---Z~. Because 2 is an a-fold cyclic branched cover
2,nx(2-1(DX-F))=ker(~I(DX-F)--*HI(DX-F; Z ) ~
Z,);
0(2) actions on the 5-sphere
463
i.e. 2,~zx(2-X(DX-F)) is the c o m m u t a t o r s u b g r o u p of ~ I ( D X - F ) , which is
invariant u n d e r any a u t o m o r p h i s m of g , ( D X - F ) a n d therefore is invariant
u n d e r z , . So r lifts to an involution z of 2 - ' ( D X - F ) ,
a n d the lifted ~ has a
nontrivial fixed point set by virtue of the choice of lifting. (Again see [B; 1.9.2].)
N o w H t ( D X - F ; 7g)~-Z, is generated by the b o u n d a r y circle of a n o r m a l
disk to F in DX. We saw in the proof of (2.3) t h a t Fc~DX ~ is a n o n e m p t y
u n i o n of circles. Let xoeFc~DX ~ a n d consider a z-invariant n o r m a l disk D2o to
F at x o. Since D X ~ is a surface, DZoC~DX~ is an arc. So ~ acts as a reflection on
D2o , a n d on the generator [0D2o] of H t ( D X - F ; 7g), t [ ~ D 2 o ] = - [ 8 D 2 o ] ; i.e. t
=-1
o n H t ( D X - F ; Z). The deck transformations of D X - F can be identified with H t ( D X - F ; Z ) ; it follows that if we write Z~ multiplicatively we
have zt t = t / - t t for all t/~Z~. Thus z a n d 7 / g e n e r a t e a n action of the dihedral
group D2~ on 2 - ~( D X - F).
Next consider a b o u n d a r y c o m p o n e n t L(ai; ri; s~) of DX. The preimage
2-tL(a~; r~; s~)= ~ S a. Some of the copies of S a are preserved by t a n d others
~t/at
are interchanged by t. T h a t is t ( S 3 ) = q ( S 3) for some tleZ ~ (still multiplicative)
for each c o m p o n e n t S 3. Even if t/=l=1, q t defines an involution of S 3 a n d so is
equivalent to a linear involution. Thus t extends over 2 - t (c L(a~; r~; s~))= ~ . B 4
covering the involution t o n cL(ai; r~; s~).
Finally, since F is a t - i n v a r i a n t surface, we may assume t h a t t sends n o r m a l
disks of F to n o r m a l disks of F. Thus the same is true for p u n c t u r e d n o r m a l
disks of 2 - t (F) in i - t ( D X - F). Hence t extends over F. [ ]
W h e n the pseudofree S~-action of S s is pulled back using 2: M - ~ X , it
becomes a principal St=SO(2) bundle P over M. Let E be the associated
SO(3)-vector b u n d l e E = P x 1t 3. T h e n E is a 7~-equivariant SO(3)-vector
SO(2)
bundle
over
M
with
a
Z,-invariant
SO(2)-reduction.
We
have
pt(E)
=(2* e)2[m] =ee 2 = 1.
It will be i m p o r t a n t for us to count the Z , - i n v a r i a n t reductions of E to
SO(2). W e shall do this in the general situation of [-FS] where we assume only
that X=QS/S t is the orbit space of a pseudofree S~-action o n some 5-manifold
with exceptional orbits of orders a t .... , a,. Then if or=at .... ,a,, there is still a
Z , b r a n c h e d cover M which is a s m o o t h manifold, and pulling the St-action
back over M we obtain a principal S~-bundle a n d we can form the SO(3)vector bundle E as above. As before, Q ~ X when restricted over D X = X
-0
cL(ai, b~) has
an
Euler
i=1
class
e,
Now
8 D X = 0 L(a~,b~). Let
j=l
ij: L(ai, b~)--*DX be the inclusion.
Proposition. Let X=QS/S t be a pseudofree orbifold with e-fold branched
cyclic cover M and SO(3)-vector bundle E over M as above. Suppose that
Ht(DX; Z 2 ) = 0 and i*(Tor I-I2(DX; 7,))=0. Then, up to orientation, the number
of 7g~-invariant reductions of E is just tt(e).[Ha (D X ; Z)I where
(2.5)
t t ( e ) = 8 9 { f eFrH2(DX; ~ . ) l f 2 = e 2, f = e ( m o d 2) and i ' f = +_i'e, j = 1..... n}.
(Here we take an arbitrary splitting H2(DX ; Z)= Fr H2(DX ; Z ) ~ Tor HZ(DX ; Z)
into free and torsion parts.)
464
R. Fintushel and R.J. Stern
Proof. This is just Proposition 4.2 of [FS]. However, the number #(e) was
incorrectly calculated in [FS] so we give the correction here. The mistake
arose from [FS, Prop. 4.1] where condition (iii) should read: i*(d)= +_if(e) for
each j. The proofs of [FS; 4.1 and 4.2] then go through to prove our proposition once we correct the proof of [FS, 4.1]. This is done as follows. In that
proof we had a 7~-equivalence of L e9 e with L~ (9 e which gave an equivalence
of the representations of Za~ on the 1/3 fibers of Le(~e and L~G e over each
point in M sitting over the cone point of X of order aj. These representations
take the form (kJo) O and (~'J~0 where ( is rotation of 112 by e 2'ti/aJ and 0 is
the trivial 1-dimensional real representation. From this one concludes that ~j=
___ki (not ~ j = k i as was erroneously stated in [FS, 4.1]). (The point here is that
when complexified (RJ400 has eigenvalues e 2nikdas, e -2nik/aJ, and 1.) Thus
corrected, the proofs of [FS, 4.1, 4.2] give our proposition. []
(2.6) Remark. Once the changes pointed out above in the definition of /l(e)
and the statement of Proposition 4.1 are made in [FS], the rest of that paper
goes through without change. In particular, all the computations of numbers of
reductions in w10 remain the same; so none of the applications of [FS] are
affected.
The adjoint bundle (fie of Lie algebras is defined by
~ E = {L~Hom(E, E)lLx~so(Ex) on each fiber E~ of E}.
The map E---,15E given by u ~ u •
where " x " is fiberwise cross-product
induces an isomorphism of bindles and " • turns E into Lie algebra bundle.
Define the involution z on L = P x 112 by z{p,v}={zp, F} where the
S0(2)
action of z on P is pulled back from S 5 via 2. Since for t~SO(2) one has zt
- t - 1 z, this involution is well-defined on L, and it is an action by bundle maps
covering the action of r on M. Next extend z to E = L G e by defining the
involution to be ( - 1 ) on e. Then z defines a Lie algebra bundle isomorphism
of E (and of ffiE).
Fix metrics on E and M in which ]l~ and z act by isometries. (I.e. fix DEsinvariant metrics.) As in [FS] our goal is to study the Z~-equivariant SO(3)gauge theory of the bundle E. It follows from Proposition 2.5 that the bundle
E has (up to orientation) a unique Z~-invariant reduction to an SO(2)-bundle.
Recall the invariant R(X) defined in w1.
(2.7) Theorem. Let X = S s / S 1 be the orbit space of a pseudofree Sl-action. Then
R(X)<0.
Proof. This follows immediately from [FS; Thm. 9.2].
[]
The point of the proof of this theorem in [FS] is that R(X) is the expected
dimension of the moduli space of Z~-invariant self-dual SO(3)-connections on
E modulo Z~-equivariant gauge equivalence. When this moduli space is positive dimensional, i.e. when R(X)>0, the moduli space must have an even
number of point singularities - which are in 1 - 1 correspondence with Z~invariant reductions of E to SO(2)-bundles (up to orientation).
0(2) actions on the 5-sphere
465
3. The anti-self-duality equations
Recall the action r on E described in the last section. It follows from [FS;
Prop. 5.3] that there is a reducible Z,-invariant self-dual connection on E so
that e defines the subbundle spanned by a nonzero covariant constant section.
So if we now average V with respect to z, we again obtain a reducible Z~invariant self-dual connection 170 which is also z-invariant. We note in passing
that 170 is actually Z,-equivariantly gauge equivalent to V because there is in
fact just one gauge equivalence class of reducible self-dual ~g,-invariant connections on E by [FS; Prop. 5.3].
The action of the Z~-equivariant gauge group on the Z~-invariant connections of E has a slice Cvo at Vo which consists of all connections V = 170+ A
where A is a Z,-invariant 1-form with values in (fie and satisfies 6V~
and
has small enough LZ3-norm. Here 6 v~ denotes the formal adjoint of dr~
f2~
where the superscript "cd' denotes Z,-invariant forms. Let
Vo have self-dual curvature RV~176
For 17~(9vo we wish to
consider the equation
RV = coo.
(3.1)
Proposition. The equation R v =co o on 6)Vo has only the solution V = Vo.
Proof Applying the Chern-Weil formula:
S blcoojl2=4n2pl(E)= S IIR~-112 -liRa-hi2"
M
M
Thus if 17 is a solution to R~+ =e)o, then ~ lIRa_It2=0; so e ~ = 0 ; i.e. 17is selfM
dual and RV= R~ = R~~= R v~ But this implies that V is a reducible connection,
for the Ambrose-Singer Theorem [AS] implies that V can be reduced to a
bundle with structure group whose Lie algebra is so(2), the holonomy algebra
of 17o at some point. Thus V reduces to S0(2) or 0(2). However, H1(M;Z)---0;
so any O(2)-bundle on M is actually an SO(2)-bundle. Thus V has an SO(2)reduction. However, 17o is the only self-dual reducible connection in @o" Hence
17=170-[]
On the other hand, solutions to RV+=co 0 on Cvo correspond to Aef2~((fi~)~
with small enough L23-norm which satisfy
(3.2)
~'dv~A + [A, A] + = 0
(6V~ =0.
These are precisely the anti-self-duality equations. The solutions to the corresponding linearized equations comprise the kernel of the operator
a v~9 d~~ o l((fi~)~ -, o ~
9 ~ ((fi~r.
This is an elliptic operator since its symbol is
a(~)--icGP+ (~ ^ - )
466
R. Fintushel and R.J. Stern
where ir is interior product with ~ and P+ is projection O2(~E)~t-'-}~C~2(~JE) a.
(See e.g. [FU, p. 91].)
(3.3) Proposition. The index of 6V~
~ is 2 n - 9 - R ( X ) where n is the number
of singular points of X.
Proof This computation follows precisely the lines of [FS; Thm. 6.1]. The
operator fiVo~ dr has the same index as the twisted Dirac operator
D: F(V (M)| V + ( M ) ~ E ) ' - ~ F(V+(M)@ V+(M)~ ~bF)~.
Computing exactly as in [FS] we obtain
IndD=
- 2 - 6 + n - ~ 1 ~" 2"'-1
nkri cotnkSisin 2 ~k
~ cot
-.
O~
"=
i k= 1
ai
ai
ai
But
R ( X ) = 2 - 3 + n + ~ alike12
a'-1 cot r~kr/cotnksi sin 27zk_.
i= 1
Hence I n d D = 2 n - 9 - R ( X ) .
" =
ai
ai
ai
[]
Let G=D2~; it is also necessary for us to compute the index of Jv~176
restricted to the G-invariant forms:
(6Vo~dVo)G: 01(~SE)G_..O0(q~E)G~ 0 +2( ~ ) . G
(3.4) Proposition. The index of (6v~176 ~ is 89( 2 n - 9 - R ( X ) ) .
Proof Again as in [FS; Thin. 6.1],
Ind(6~~176 G = I
E L(g, D)
ZO~ g6G
where
D: F(V (M) | V+(M) | [bE)--*F(V+ (M) | V+(M) | ff~).
The Lefschetz numbers L(g, D) for 1 # g e Z ~ c G are exactly those computed in
Proposition 3.3. Now consider gEG-Z~; then g is an involution. Over the
fixed point set M g of g we have E = L ~ e where g acts on L by reflection
through an axis and as ( - 1 ) on e. Thus c hg(ff~E| C ) = - 1 + 1 - 1 = - 1. This
means that the Lefschetz number of g is L(g, D)= - L ( g , A) where
d: F(V (M) | V+(M)) ~ F(V+ (M) | V+(M)).
Now the G-equivariant index Ind Aa = 89(Z + a) [M/G] = 89()~+ a) [S 4] = 1, since
the index of the operator d is half the Euler characteristic plus the signature.
Thus
l = I n d d ~ = --1 ~ L(g, d).
2~t g~a
However,
1 E L (g, A) = Ind Az* = 89(X+ a) [M/Z,] = 89(X + a) [X] = 2.
0(2) actions on the 5-sphere
467
1
So 2aa g~z. L(g, A)= 1 = Ind AG. Hence we see that
1 ~
L(g, D)=0.
Now note that L(1, D) is precisely that given in the computation of Proposition
3.3 since the defect terms for g 6 G - T l vanish. Thus we have
Vo
Vo
1 ~
1
=~(2n-9-R(X)).
N o w set
h 1 = dim ker (6 v~• dr~ G
h ~ = dim ker (dV~ Qo (ffie)a ~ ~ 1(ff)E)~)
h 2 dim ker (fir~ 0 2 (~e) G~ f/1 (6)E)o).
=
Then clearly dim coker (6 v~+ dr+~~ = h ~ + h 2 ; SO Ind (6 v~+d~y) =h 1 - ( h ~ +hE).
Since Vo is a G-invariant reducible connection and h ~ is the dimension of
the space of covariant constant sections, h ~ 1. Furthermore, the splitting E
= L 9 e corresponds to a splitting Vo = D o @ 0 where 0 is the flat connection on
e. In turn we have a splitting
a~+o=a~+oed+ : a ' ( % ) ~ ~ a'(~)~-~ ~+ (%)~ ~ a +2(e)~
where d is the de Rham operator on M. The formal adjoint of d~_~ splits
similarly; so we have h2=h2(L)@h2(F..) where hZ+(F,) is the dimension of a
maximal subspace of H 2 ( M ; ~_,.)6 on which the intersection pairing is positive
definite. Since H2(M; R)6~HZ(M/G; R ) = H 2 ( Y ; R ) = 0 we have h 2 ( ~ ) = 0 .
Proposition. If 2 n - 9 - R ( X ) > = O there is a G-invariant metric on M for
which ker(5~176O2 (L)a--* Q I ( L ) a ) = 0 ; hence h 2 =0.
(3.5)
We shall prove Proposition 3.5 in the next section. Suppose now that 2 n - 9
-R(X)>__O and use a metric obtained from (3.5) on M. We have
h 1=Ind(6~~176176189
1.
However, we know from Theorem 2.5 that R ( X ) < 0 . So h ~ > 0 when n > 3 . This
means that the linearization of the equations (3.2) has a positive dimensional
solution space when n > 3 because each G-invariant solution is also a Z,invariant solution. Of course Proposition 3.1 implies that the nonlinearized
equations (3.2) have a unique solution in OVo- We shall see that the presence of
the involution z forces the above two facts to be contradictory.
The Kuranishi technique is the means by which the linear information
about h ~ is turned into nonlinear information about (3.2). Consider the map
~0: {AeOl(ff~e)alavoA =O} -~ 02 ((5~)a
given by ~b(A)=d~+~ + [A, A]+. Notice that any small enough neighborhood of
0 in the domain of ~h is contained in (9%. The Kuranishi technique provides a
468
smooth map
R. Fintushel and R.J. Stern
4': q/-o ker 3v~
(6~)) G
with 05(0)=0 where q / = ker d~+~c~ (gVo, and the zero set of if, i.e. the solution set
{A ~ O ~(ffiE)GI~3V~ = 0, de+~ + [A, A] + = 0}
of (3.2) is locally isomorphic at Vo to 4'-1(0). (See [AHS], [ F U ] , or ILl.)
Theorem. Suppose S 5 carries a pseudofree SX-action with n exceptional
orbits and orbit space X = S s / S 1. I f the action of S t extends to an action of
O(2), then
2n-9-R(X)<O.
(3.6)
Proof Suppose 2 n - 9 - R ( X ) > O and use the metric of Proposition 3.5 on M.
Since hz=O, the Kuranishi map is a map 4': q / ~ 0 , so 4 ' - ~ ( 0 ) = ~ . Since q/ is
an open subset of {A~Om(ff~E)613V~176
the dimension of q/ is h 1
= 89(2 n - 9 - R (X)) + 1. Hence if h ~ = dim q / > 0, then dim 4'- t (0) > 0 in q / = Cvo.
But this would contradict Proposition 3.1; so h~<0. This contradicts our
original assumption that 2 n - 9 - R ( X ) > O , i.e. that h ~ > l . []
As a corollary to Theorem 3.6 we have an affirmative answer to the
Montgomery-Yang Problem for pseudofree St-actions on S 5 which extend to
actions of 0(2).
Theorem. A pseudofree SX-action on S 5 which extends to an action of 0(2)
can have no more than 3 exceptional orbits.
(3.7)
Proof Let X = S s / S ~ and let n be the number of exceptional orbits. By Theorem 2.7, R ( X ) < 0 ; so O > 2 n - 9 - R ( X ) > 2 n - 8 .
Hence n < 3 . []
4. Generic metrics
We wish to show that for a generic D 2 ~ = G - i n v a r i a n t metric on M, h 2
=dimker(6V~ O2(6iE)G--~f2t((5~)a)=0. This amounts to a reworking of the
ideas espoused by Freed and Uhlenbeck in their genericity theorems [ F U ,
Theorems 3.4, 3.17, and 4.19]. Recall that f f a e ~ E ' ~ L O ~ and that Vo splits as
V0=D 0 ~ d . We are interested in the elliptic operator
D~ ~ D O+ : Or(L) G ~ O ~ (L) ~ ~ 0 2 (L) 6,
where D~ is the formal adjoint of D o with respect to a given base G-invariant
metric
on
M.
Since
L
is
a
nontrivial
SO(2)-vector
bundle,
ker (Do: O~ ~ ~ 01 (L) ~) = 0. Let h 2 = dim ker (D~: 0 2 (L) ~ --, O t (L)~). We have
seen that h 2 =hL;
2 so to prove Proposition 3.4 we must exhibit a G-invariant
metric for which h2=0. In fact, we shall show that this is true for an open
dense subset of the space of all G-invariant metrics on M.
Let c~ = Ck(GL(TM)~) be the space of all G-invariant Ck-automorphisms of
the tangent bundle of M. The space c~ is a Banach manifold (cf. [ F U ; p. 60]),
and if g is our given G-invariant metric on M, then every other such metric
0(2) actions on the 5-sphere
469
may be written as 4)*g where 4)~cg. As in [ F U ] we consider the map
Q : (f21 (L) a - {0}) x cg__. Qo (L)~ ff~ 0 2 (L),~
(A, r -~ (D* ((4)- 1), A), P+ ((4)- 1), Do A))
where D o is defined implicitly as the self-dual solution relative to the Ginvariant metric 4)*g and P+ is projection onto self-dual 2-forms with respect
to g.
Suppose that Q(A, 4))=0 and (v, 4))~coker6Q(a.~ ) where 6Q denotes the
differential of Q. We have
(~1 Q(A,*~(a)= (D*((4)-1)*a), P+ ((4)-1), Doa))
where D* is the formal adjoint of Do
(v, (P)~coker 6Q(A, r = coker fi ~ QCA,4,)'we have
in
the
metric
g.
Since
0= j" (o*((4)-1)*a),v)~= S ((4)-')*a, Dov)~
M
M
for all aef21 (L)L Hence
(4.1)
Dov=0.
Since ker (Do: f2 ~(L)~ --, 0 1(L) ") = 0, this implies that v--- 0. Similarly,
O= ~ (P+ ((4)- ')* Doa), ~b)g= ~ (Doa, 4)*q~),.g= ~ (a, D ~ ) * * g
M
M
M
where q~= 4)* q~. Thus
D*$=0.
Let F be the curvature of D o. Infinitesimal variations r~c, the Lie algebra
of ~, satisfying r ' F = 0 fix D o to first order. For these r:
32 Q(A,0) = (O* ((4)- 1). (r* A)), P+ ((4)- 1), (r* DA))).
(See [ F U ] , p. 86J.) Then since (v, q))~coker 62 Q(A,, )
0 = ~ <P+(4)- ')*(r*OA, eh)g= ~ <r*OA, c~)r
M
M
for all re~ with r*F=O. Using G-invariant cut-off functions we see that this
implies
(4.3)
(r* DA, (~)~**=0 (pointwise)
for all r with r ' F = 0 . (It is important here that we are dealing with
AeO~(ffJE) 6, i.e. with G-invariant forms, in order that the standard argument
which uses cut-off functions to make the above integrand pointwise nonnegative (equivariantly l) actually works.)
Now F = a |
where aef22+ and uEf2~
I[ul[-=l. It follows from the
Bianchi identity that da=Dou=O. Now [ F U ; Lemma 4.16] implies that where
470
R. Fintushel and R.J. Stern
D o A = 0 a n d F # 0 , A=Dow for some wef2~
But D * ( ( $ - X ) * A ) = 0 (since
Q(A, $ ) = 0 ) a n d ( ~ - 1 ) * A = ( $ - 1 ) * D o w = D o ( ( $ - l ) * w ) ; so ( $ - 1 ) * A = 0 . Hence A
= 0 o n any open set where D o A = 0 and F # 0 . By unique c o n t i n u a t i o n [ F U ;
Prop. 6.38] A does not vanish on open sets unless A - 0 . But A-Oef21(L) ~ is
excluded from the domain.
T h e curvature F of the G-invariant connection 17o is a G-invariant form; so
the set U I = { F # 0 } is a G-invariant open subset of M, a n d U 1 is dense because
F - 0 c a n n o t vanish on open sets. W e have just seen that the open subset Uz of
U 1 on which DoA+O is also dense. Set V = U1-U2; since U1 is G-invariant,
G(V)c U1. T h e n U = U 1 - G ( V ) = ( ] U 1 - g ( V ) = ( ] g(U2) is a finite intersecg~G
g~G
tion of open dense sets; so U is a n open dense G-invariant subset of M on
which F#:0, DoA#O. Since the principal orbits of G o n M also form an open
dense subset of M [B; IV.3.1], we m a y assume that G acts freely on U.
As in [ F U ; p. 61] if we are given a 4-dimensional oriented vector space V
with oriented o r t h o n o r m a l basis {ek} for V*, let aiJ=p§
eS)EA2 V* and aij
=P_(ei^ eS)~A2_V*. Then {aij;j=2, 3,4} is a n o r t h o n o r m a l basis for A2+ V*
a n d {a~s;j=2, 3, 4} is an o r t h o n o r m a l basis for A 2- V*.
Consider a point x~ U. F o r a small e n o u g h n e i g h b o r h o o d N(x) of x in U,
G(N(x))= U g(N(x)) is a disjoint union since G acts freely on U. So as in
geG
[ F U ; L e m m a 3.5] we can choose a G-invariant frame at x in which a=o ~3. In
this frame
F=
o-13 ( ~ u
4
DA= ~ c~li|
i
i=l
,4
q,= y~ alJ|
j=l
Since DA+O on U, there is a small open n e i g h b o r h o o d N(x) of x as above
on which some w i, i = 2 , 3 , 4 , is nonzero. In [ F U ; (3.8), (3.10)] there are defined
local frame changes rl, r 2, r a such t h a t
r*~12
,
= 2~
12
r20~ = 2 a
r,~12=20.12,
13
,
v * e~v 1 3 ----~f *l u 1~4
"1
=0
r*ry13__~*~14
.2 ~
--.2 ~
/1
--u
,
13
,
14
r 3 ~x
= r 3 ~z = 0
r , ~,13 _ r , a l 3 _ ~ , t r l 3 _ f ~
1 ~
- - - 2 TM
--.3 ~
--~,.
Since G(N(x)) is a disjoint u n i o n of copies of N(x), one copy for each geG, we
can extend rl, r 2, r 3 G-equivariantly; i.e. we m a y suppose r~, r2, r3~c. Furthermore, r~F=r*F=r~F=O since each r ' a 1 3 = 0 . Thus from (4.3)
O=(r~DA, 6>4,,g = 2 <w2, $ 4 )
Similarly, applying r* a n d r~ we see that (w2, $3> = 0 a n d (w 2, $2> = 0 . Thus if
w24:0 in N(x), then tb has r a n k _<_1 at x.
0(2) actions on the 5-sphere
471
If w 2 vanishes in N(x), then w j + 0 in N(x) where j = 3 or 4. We can change
frame using a skew symmetric reso(3)_cso(4)=so(3)+@so(3)_ such that
r*cd2=cd j, r*cdJ=c0 2, and r*cdk=od k, k~2,j. As above r can be extended Ginvariantly so r~c. Since F is self-dual r*F=F, and applying the frame
changes r ' r * , we find as above that (w~, ~bm)=0, m = 2 , 3, 4. We thus see that
has rank _<_1 at each point of U, hence so does q3=~b*(~). Since U is open and
dense in M, this means that ~ has rank _<_1 o n M .
On the open subset { ~ + 0 } of M we may now write ~ = p |
for p~f22+
and IIw]l=1. By (4.2), D * ~ = 0 and so since q~ is self-dual, D o ~ = 0 . We have
O=Do~)=dp|
(4.4)
Since Ilwll = 1, 0 = 8 9
p /,,Dow.
w)=(Dow, w). Taking the inner product of (4.4) with
W:
dp= - p /x (Dow, w)=O.
So again by (4.4), pADow=O. By [ F U ; L e m m a 3.5] we can choose a coframe
{0 i} (at a point) such that
Ipl
0 2 +03 A 04).
p=~(O~A
If Dow=Y~ Oi|
then
0 = p A Dow=2(01
A 02 A 03 ( ~ u . ) 3 ~ - O 1
A 02 A 04 (~) (D4~- 03 A 04 A 01 (~)0) 1
-~ 03 A 04 A 02 (~ (D2).
Hence (Dl=O)2=(D3=604=0, I.e. D0w=0.
This means that under the assumption that coker3QIA.o~+0 we have produced a covariant constant section w of L over {q~#0}. We shall now repeat
an argument of [ F U ] to show that { ~ 0 } is connected, so that the covariant
constant section w of L extends over all of M. This is a contradiction since L is
a nontrivial SO(2)-vector bundle. So suppose that { ~ + 0 } is not connected.
Then on a component V, the self-dual ~ is in the kernel of the self-adjoint
operator DoD*+D*D o and vanishes identically on ~?V. Thus the first eigenvalue of DoD* +D~Do on the larger domain M is negative. This is a contradiction
because DoD* +D*D o is a positive operator.
The upshot of our argument is that 0 is a regular value of Q; hence Q-1 (0)
is a manifold. Freed and Uhlenbeck show that Q-1(0)__, ~ is Fredholm and its
index is the index of the restriction of Q to (f21(L) G - {0})x {q~} for any fixed
~bEff (cf. [ F U ; p. 71]). This means that by Proposition 3.4 the projection
Q - l ( 0 ) ~ c g has index 89176
2 ( ~ ) -__1
h+
7 ( 2 n - 9 - - R(X))+I.
The Sard-Smale Theorem now implies that the set of regular values of
Q - I ( 0 ) ~ ~ is dense in ~. This gives a dense collection of G-invariant metrics
on M in which h~= 89
and h2=0.
The metrics are only Ck-metrics. However, as in [ F U ; L e m m a 4.15], one
sees that the above set of metrics is also open. This means that we can choose
a C ~ G-invariant metric on M for which h2=0. This proves Proposition (3.5).
472
R. Fintushel and R.J. Stern
5. Two bridge knots and links
A 2-bridge or link in S 3 is one which is obtained from gluing together two
copies of (B3;/1, I2) (Fig. 1) by a diffeomorphism of the boundaries which sends
~3(I1u I 2 ) to c3(I1 ~J12). (See [Sch].) A 2-bridge knot or link can also be described
as shown in Fig. 2 (cf. [CG]). The double branched cover of S 3 with this as
branch set is the lens space L(p, q) where q/p has the continued fraction expan1
1
1
sion - - + - - + . . . + - - .
If p is odd, this is a knot and if p is even, it a two
C1
C2
Cn
component link. This gives a 1 - 1 correspondence between 2-bridge knots and
links and lens spaces. The 2-bridge knot or link corresponding to L(p, q) is
usually denoted (q/p).
Two-bridge knots and links entered our analysis in the proof of Proposition
2.1, basically because the Dz,-action on S 3 whose quotient by 7.,cD20 is
L(a; r, s) induces the involution z on L(a; r, s) with L(a; r, s)/z ~-S 3 with branch
set (rs-1/a). The idea of this section is that one can work backwards from the
orbit space to get O(2)-actions on rational homology 5-spheres; so that our
invariants of w167 become cobordism-type invariants of 2-bridge knots and
links.
(5.1) Theorem. Let L be the disjoint union of 2-bridge knots or links (qi/pl)cS 3,
i= 1.... , n, where the p~ are pairwise relatively prime. Let c~=pl ... p,. I f ~ is
odd, suppose L bounds an embedded punctured F..IP2 in S 4 - U B~ (where S 3
i=I
=r
I f ~ is even, suppose that L bounds an embedded punctured S 2 in S 4
Fig. l
-Cz
C crossfngs
-C~
-Cn
C3
Fig. 2
0(2) actions on the 5-sphere
473
- ~) B~. Then there is a pseudofree S~-action on a rational homology 5-sphere
i=1
Q5 with orbit space X = Q S / S x whose singularities are cones on L(pi, qi), i
= 1..... n. Furthermore, 2 n - 9 - R ( X ) < O , and if/~(e)---1 (mod2), then R ( X ) < 0 ;
so n<3.
This theorem can also be used to compute numerical obstructions to L
bounding a punctured RIP 2 (orodd) or punctured S 2 (aeven) when n < 3 as
follows. In order to compute R(X) we must be able to determine the slice types
(p~; r~, s~) of the S~-action on Qs. Permutation (p~; r~, sl) ~(p~; s~, r~) and inversion
(p~; r~,s~)~(pi; -r~, -s~) clearly do not affect the computation of R(X). Recall
from [FS] that associated to the pseudofree Sl-action on Q5 there is a
pseudofree Euler class e~HZ(X; Q) which has a self-intersection number e2eQ.
We shall see in the proof of Theorem 5.1 that e = - i for the action which we
construct. So R(X) is given by the Eq. (1.1).
ct
(5.2) Lemma. Let S 1 act pseudofreely on Q5 with orbit space X as above. (So
the singular points in X have as neighborhoods the cones cL(p i, qi), i= 1.... , n.)
Then up to permutation the slice types of the exceptional orbits are (p~; ri, si)
where
2s 2-2ctq* (mod pi),
Pi
ri=qisi(modpi)
where qiq* = 1 (mod pi).
Proof Since L(Pi;ri, si)~L(Pi;ri, s*) and L(pi, qi)-~L(Pi, q*) we have r~si=qi or
q* (modp~). Now
g(x)=2-a+n+
~ ~(pj; rj, sj)
j=l
where
6(pj; rj, sj)= - 2r*s* (modZ)
Pj
(see [TL]).
Thus ~ 6(pj; rj, s j ) = R ( X ) + 3 - n - 2 / c t ; so
j=l
J=, ~ 6 ( p j ; rj, s j ) = ~ ( R ( X ) + 3 - n )
2
Pi
So we have Pi~--6(P~;r ~ ' s l ) = -(2m ~
This means that 2r's*
(~)~- 2 ( m o d p i )
and thus p-~ \~I(-2r*s*l=~-r, (modZ).
2~
i.e. 2ris~=~..(modp~). Now if ri -
2 *
=
siqi(modpi), then 2s i2 - - - (aqi
modpi).
If ri_siq
,, (modpi), then si-qiri(modPi)
r2 2~tq*
Pi
and 2 i = - - ( m o d p ~ ) ,
and the solutions are permutations of those already
Pi
obtained. []
474
R. Fintushel and R.J. Stern
Fig. 3
In practice, to check whether a disjoint union L of 2-bridge knots or links
(ql/Pi) as in the statement of (5.1) can bound embedded punctured ~-dp2's
(~ odd) or punctured S2's (~ even) one must check all possible values of R(X)
using the slice types possible according to Lemma 5.2.
For example, consider the 2-bridge knot (16/33) in S 3, which is equivalent,
up to orientation, to the 2-bridge knot (2/33), which bounds a genus 2 nonorientable surface in S 3 as illustrated in Fig. 3:
That this is the minimal genus of a nonorientable surface that (16/33) can
bound follows from (5.1) and (5.2). For if (16/33) bounded a M6bius band in
B 4, then by (5.2) the possible slice types of the corresponding 0(2) action are
(33; 4, 25) and (33; 7, 19) with the corresponding R being - 7 and - 9 ,
respectively. By (5.1) this is impossible.
Proof of Theorem 5.1. Let D denote S 4 - 0
B4
and suppose L bounds a
i=1
punctured RIP E or S z (according as to whether e is odd or even) in D. Let A
denote the punctured surface and let A * c S 4 be the corresponding RIP 2 or
pinched torus obtained by coning each knot (qjp~)cS~ in B~. Then using
Alexander duality we have
HI(S4-A*; Z)~
'
c~ even
;
so the Mayer-Vietoris sequence arising from S 4 - A * = ( i ) - A)c~ U (B 4 - c(qJPi))
i"
shows that
c~ even"
Let m denote the nontrivial element of H i ( D - A ) Z2)-~Hom(H~(D
- A ; Z ) , Z 2 ) ~ Z 1.
A nonorientable Sl-bundle over D - A is classified by its first StiefelWhitney class w ~ E H I ( D - A ; Z 2 ) and its twisted Euler class e . E H Z ( D - A ; Z t)
where Z' denotes Z-coefficients twisted by the homomorphism H I ( D - A ; Z )
~ Z z corresponding to wl. We choose wl=~o. Let ( D - A ) ~ - - > D - A be the
2-fold cover arising from 09. This extends to a 2-fold branched cover D X ~ D
with branch set A, and once more to the 2-fold branched cover X ~ S 4 with
branched set A*. Since the branch set in $3= OD is (qJp~), i= 1, ..., n, X is a
pseudofree orbifold whose singularities have as neighborhoods cones on the
lens space L(Pi, q~), i= 1..... n. Now ~((X)+x(A*)=2x(S 4) = 4; so z ( X ) = 3.
0(2) actions on the 5-sphere
475
W h e n c~ is odd, Ha(D-A;TF)=7Z 2 so it follows as in [ M ] that
Ha((D--A)~;Z2)=O; when ct is even H I ( D - A ; Z ) = T Z and the proof that
HI((D-A)~; Z 2 ) = 0 is given in [CG, L e m m a 2]. Using the Mayer-Vietoris sequence for D X = ( D - A ) ~ =N(A) where N(A) is a tubular neighborhood of A
and (D-A)~c~N(A) is an Sl-fiber bundle over A, we get that Hl(DX;12)=O
and so H I ( X ; Z 2 ) = 0 . Since z ( X ) = 3 , bl(X)=b3(X)=O, a n d b 2 ( X ) = l . Thus
H z ( x ; 7~)~-TZ0)odd torsion, and from yet a n o t h e r Mayer-Vietoris a r g u m e n t
H2(DX; 7~)~7Z G o d d torsion.
Next consider the " G y s i n " sequence:
Ha(D-A; 7Z) ~o, H2(D - A ;
,Zt) p* , H Z ( ( D - A ) ~ ; 7~) r , H 2 ( D - A ; 7f)
where ~ H I ( D - A ; I t) is the characteristic class of the double cover. (Note
here that n 1 ( D - A ) ~ ~ n l ( D - A ) ~ Z 2
is the zero map, so H2((D-A)~;7I t)
= H 2 ( ( D - A ) ~ ; Z).) The m a p ~b is just integration along the fiber; so if T denotes
the involution on X, then for each class w~HZ((D-A)~;Z) with z ' w = - w ,
we have q~(w)=0. In particular, consider i*:H2(DX;7~)~H2((D-A)~;Z). We
k n o w t h a t H 2(DX ; 7~) ~ Z @ odd torsion a n d since H 2(DX ; R ~ 2= H 2(D; R) = O,
T* acts nontrivially on the Z s u m m a n d of H2(DX;Z). So this s u m m a n d is
killed by ~bi*.
N o w construct the principal Sa-bundle over DX whose Euler class e is
(1,O)~1@(oddtorsion)=H2(DX;Z). We have seen that dpi*e=O~H2(D
- A ; 7/); thus i*e is the image of an element in H 2 ( D - A ; 7Z~).This means that
the restriction of our principal bundle to ( D - A ) ~ is the pullback of a nonorientable Sl-bundle over D - A with w a =o9.
Since the image of e in each H2(L(p~,ql);Z)~7Zp, is a unit, the portion of
the S l - b u n d l e over L(p, qi) is S 3 x S ~. So in a s t a n d a r d linear fashion the S ~bundle can be completed to a pseudofree S~-action on a rational h o m o l o g y
sphere Q5 over x .
Recall that ~ is the involution on X which is the covering translation of the
b r a n c h e d cover X--,S 4. The principal Sl-bundle over ( D - A ) ~ has an involution compatible with r, a n d its action over a n o r m a l circle to ~ is (x, y ) ~ ( r x , y)
in (normal circle x St). This involution extends to the part of the Sa-bundle over
using the above reflection in the fiber. N o w the involution extends over all
of Q5 compatible with the involution r on X. O u r p r o o f of T h e o r e m (3.6) in
this situation goes t h r o u g h exactly as before. F r o m [FS; T h e o r e m 9.2], if p(e)
is odd, then R ( X ) < 0 .
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