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5. Gauge Theory for ˜(al' ... ,an)
The success of Donaldson's approach to 4-manifolds [DI-4] has motivated the
lise of similar techniques in the study of homology cobordism properties of homology
:˜-spheres. The approach taken in [FS3] is to study the mapping cylinder, W, of
t.he orbit map E(at, ... , an) --+ 8 2 , which appeared prominently in the last section.
Slippose, for example, that E(ai, ... , an) is the boundary of a simply connected,
positive definite 4-manifold U. IT we use the orientation on W which was described
ill the last section we can form the union W U U, obtaining a simply connected
()rhifold X. Even though X is not a manifold, it is a rational homology manifold,
uHd thus has a well-defined rational intersection form, with respect to which it is
positive definite. Let X o = Wo U U. There is a preferred class W E H 2 (X; Z) ˜
112 (Xo , axo;Z) which is represented by the 2-sphere orbit space in W, and w 2 = ˜.
As in §4, we form the 50(3) V-bundle L w E9 R over X, where L w is the 80(2)
\-' -bundle whose Euler class in H 2 (X o) is Poincare dual to w. In [FS3] we studied
I.lap moduli space M of self-dual connections on E w • For a fixed metric on X this
I˜; t.he solution space of the equation FA = *FA in Bx. In [FS3] we showed that
(I)(Orhaps after a compact perturbation) M is a compact manifold except at the
!iiu˜lc reducible self-dual connection, which has a neighborhood homeomorphic to
134 Fintushel and Stern: Invariants for homology 3-spheres

a cone on a complex projective space. Its formal dimension as computed by the
index theorem is

1ra k k k
n ai-l
˜2˜
2 1r. 2
=- - 3+n+ 1f
R(al' ... ' an;w) LJ -:- L....J cot(-2 ) cot(-. )sm (-. ).
a.
a i=t a. k=l ai a.

H this integer is positive, it will actually give the dimension of M. (Note that
since dimM = R(at, ... , an; w), the existence of the singular point in M implies
that the dimension R(at, · · · , an; w) is odd. This argument applied to W, consid- l'
ered as an orbifold with a noncompact end, rather than to X works independently
J
of the bounding properties of E(all • · . , an). In fact, it is shown in [FS3] that
I
R(al, ... ,an;w) is odd even when it is negative.) If R(al, ... ,an;w) > 0, this
means that the complex projective space which is the link of the unique singularity .J
of M is null-cobordant. In the case that we have a cp2m this is an immediate
contradiction, for cp2m has even Euler characteristic and thus cannot bound a
compact manifold. In the case of cp2m+l one can get a similar contradiction using
the so-called "basepoint fibration". Actually one has a slightly stronger statement:
5.1 ([FS3]). If R(al' .. ' ,an;w) > 0 then E(at, ... ,an) cannot bound a
THEOREM
smooth positive definite 4-manifold whose first homology has no 2-torsion.
Easy calculations show that R( 2, 3, 6k - 1) = 1 for all k > O.
5.2 ([FS3]). If R(al, ... ,an;w) > 0 then E(at, ... ,a n ) has infinite ,
COROLLARY
order in the homology cobordism group ef.
er,
Suppose a multiple m˜(al, ... , an) (connected sum) is trivial in then
PROOF:
it bounds an acyclic 4-manifold U. Since -U is also acyclic we may assume that
m > O. By attaching 3-handles to U we obtain a 4-manifold V whose boundary
is the disjoint union of m copies of E(aI, ... ,an), and with HI(V) = 0 = H 2 (V).
Recall that E(aI, ... , an) bounds a negative definite simply connected 4-manifold
- its canonical resolution, N. Now attach m - 1 copies of -N to the boundary
components of V to obtain a positive definite 4-manifold W with HI (W) = 0 and
with 8W = E(al, ... ,an). This contradicts Theorem 5.1.
Thus we see that the Brieskorn spheres E(2, 3, 6k -.1), k > 0 all have infinite
e:. In particular this is true for the Poincare homology sphere E(2, 3,5).
order in
For other interesting results concerning the moduli spaces of self-dual connections
over orbifolds one should read the work of Furuta [Ful],[Fu2].
Let us now return to the invariant T(E) defined in §3. Here we shall change
its definition in order to allow us to use SO(3)-representations and the orientation
conventions which we have adopted. Hence define

= min{-CS(a')la': 1t"}(E) -+ SO(3)} E [0,4).
r(E)
135
Fintushel and Stem: Invariants for homology 3-spheres

This may be interpreted as follows. Fix a trivial connection (J so that we may
view the Chern-Simons fWlction as integer-valued on AE. The application of a
gauge transformation 9 to an 80(3) connection changes its Chern-Simons invariant
by 4 · deg(g). So gauge equivalent to a given flat connection there is a unique
connection a whose Chern-Simons invariant CS«(), a) calculated with respect to 8
lies in the interval (-4,0]. We are minimizing -CS«(J,a) over the compact space
R(E). For Brieskorn spheres either of the two techniques for finding representations
described in §4 will determine all of the (finitely many) numbers e that occur in
Theorem 4.2, and thus give us a finite procedure for the calculation of T(˜(p, q, r)).
2
In fact Theorem 4.2 implies that T(E(p,q,r)) is the minimum of all peqr E [0,4)
where e is chosen as in that theorem. Note that pqrr(E(p,q,r)) E Z.
For example, for p and q relatively prime, r('E(p,q,pqk - 1)) = 1/(pq(pqk -
1»), for k 2: 1. One way to see this is to use one of the algorithms presented in
˜i4 to find a representation with associated Euler number e = 1. Another way
to see this is to consider the orbifold W which is the mapping cylinder of the

*in
orbit map E(p,q,pqk -1) ˜ 8 2 and study the moduli space M of asymptotically
the V-bundle E =
t.rivial self-dual 80(3)-connections with Pontryagin charge
L1 EB Rover W. (Here, and throughout the rest of the paper, we shall write W
t.o mean W U E X R+.) Then dimM = R(p,q,r;w) = 1 (e/. [FS3]), so that
(perhaps after a compact perturbation) M is a I-manifold with a single boundary
»oint corresponding to the unique reducible self-dual connection. The component
of M containing the reducible connection must then have a noncompact end. This
indicates that a self-dual connection "pops off" the end [Tl,§10]. That is, there
is a sequence of connections in M which limits to the 'union' of a nontrivial self-
(Inal connection Cover E X R which is asymptotically trivial near +00 and is
asymptotically a flat connection a' near -00 together with a self-dual connection
over W which is asymptotic to a'. Since C is self-dual and nontrivial, its charge
H˜2 fEXIJI Tr(Fc A Fe) > o. Now for any asymptotically trivial connection A in E
over W we have ˜ fw Tr(FA A FA) = e2 /(pq(pqk - 1)) = 1/(pq(pqk - 1)). Then
() < -C8(8,a') = ˜ fEXIJI Tr(Fc A Fe) ˜ l/(pq(pqk - 1)), so that -C8(8,a') =
I/(pq(pqk -1)).
e:
Our result of Corollary 5.2 was expanded by Furuta [Fu2], who showed that
is infinitely generated. Below is the proof of Furuta's result which we gave in [FS6]
IIsing the T-invariant. (Furuta's proof is similar.)
5.3 (FURUTA). Let p and q be pairwise relatively prime integers. The
'I'IIEOREM
('ollcction of homology 3-spheres {E(p,q,pqk - 1)lk 2: 1} are linearly independent
· aH .
Z ln 03 .
('Vt˜r


Fix k 2: 2 and suppose that E(p,q,pqk - 1) = E;=1 njE(p,q,pqj -1) in
PU()OF:
():˜/, where nj E Z and nk ˜ o. Then there is a cobordism Y between E(p, q,pqk-1)
Zlud the disjoint union 11;=1 njE(p,q,pqj - 1) with Y having the cohomology of a
( I +2: Injl)-punctured 4-sphcre. Now cap off the -nk copies of -E(p, q,pqk -1) by
136 Fintushel and Stem: Invariants for homology 3-spheres

adjoining to Y the positive definite manifolds -Nk bounded by -E(p,q,pqk -1),
where N k denotes the canonical resolution. Let V be the resulting positive definite
4-manifold, and, as usual, let W denote the mapping cylinder of the orbit map
˜(p, q,pqk - 1) -? 52. Finally, let X = W UE(p,q,pqk-l) V and consider the 80(3)
V -bundle E = L $ R over the positive definite orbifold X, where the Euler class
of L comes from the dual of w. For any asymptotically trivial connection A in E
IxTr(FA A FA) = l/(pq(pqk - 1)). The moduli space M of
over X we have ˜
asymptotically trivial self-dual connections in E has dimension R(p, q,pqk -1; 1) =
1, so that (perhaps after a compact perturbation) there is a component of M which
is an arc with one endpoint corresponding to the reducible self-dual connection. As
in the argument above, the noncompactness of M implies that there is a nontrivial
self-dual connection Cover Y = ±˜(p, q, pqj - 1) X Ii, for some j < k, which is
asymptotically trivial near +00 and is asymptotically a flat connection at near -00.
Also, there is a self-dual connection B over X which is asymptotically flat at the
Ix Ix
Tr(FBAFB) ˜ O. However, pq(p:k-l) = ˜ Tr(FAI\FA) =
ends of X; so ˜
Iy Ix Ix
˜ Tr(FcAFc)+˜ Tr(FBAFB) ˜ r(E(p,q,pqj-1»+˜ Tr(FBAFB) ˜
pq(P;i-l) ' a contradiction.

Other non-cobordism relationships can be detected by the explicit computa-
tions of r(E(p,q,r). For example, r(E(2,3, 7)) = 25/42 and r(-E(2,3,7» =
4 - 121/42 = 47/42. Thus, the proof of Theorem 5.3 shows that E(2, 3,5) is not a
e:.
multiple of ˜(2, 3, 7) in Further uses of these r invariants will be given in §7.


6. TJ and p-Invariants
Atiyah, Patodi, and Singer introduced in [APSl-3] a real-valued invariant for fiat j




connections in trivialized bundles over odd-dimensional manifolds. These invariants
arose from their study of index theorems for manifolds with boundary. In this
section we shall describe these invariants and show how they have been used in low
dimensional topology.
Let B be a self-adjoint elliptic operator on a compact manifold M. The eigen-
values A of B are real and discrete. Atiyah, Patodi, and Singer [APSl] define the
function
1](8) = L(sign,x)IAI- S •
A˜O

In [APSl] it is shown that '1](8) has a finite value at s = o. Note that for a finite-
dimensional operator B, the TJ-function evaluated at 0, 7](0), cOWlts the difference
between the number of positive and negative eigenvalues of B.
Now let E be a 3-manifold with a flat connection a in a trivialized U(n )-bundle
n: d denote the space Lt(!l(E) 0 un), and consider the self-adjoint
over 2.:. Let
elliptic operator
Fintushel and Stern: Invariants for homology 3-spheres 137

Since the operator B a involves the * operator, it depends upon a Riemannian metric
on ˜ and changes sign when the orientation is changed. The importance of the .,,-
invariant to low-dimensional topology is due to the role that the .,,-invariant of B a
plays in the computation of twisted signatures of 4-manifolds with boundary.
Let X be a 4-manifold with boundary ˜ and let f3 : 7r"l(X) ˜ U(n) be a unitary
representation of its fundamental group. This defines a flat vector bundle V,s over
X, or equivalently a local coefficient system. The cohomology groups H*(X; V,s)
and H*(X,˜; Vp ) have a natural pairing into C given by cup product, the inner
product on V,s, and the evaluation of the top cycle of X mod˜. This induces
a. nondegenerate form on H*(X; V,s), the image of the relative cohomology in the
a.bsolute cohomology. On iI 2 (X; V,s) this form is Hermitian and the signature of
t.his form is denoted by sign,s(X).
Assume that the metric on ˜ is extended to a metric on X which is a product
ncar ˜ and that the restriction of f3 to E is a. It is shown in [APS3) that


LL(pd
=n - 77a(O)
signp(X)

where L(Pl) is the Hirzebruch L-polynomial of X. Thus 7Ja(O) may be viewed as
a. signature defect. However, it depends on the choice of a Riemannian metric
eH}˜. To resolve this dependency, Atiyah, Patodi, and Singer define the reduced
1/-function by
Pa(S) = 1]a(s) -1]o(s)
where 8 is the trivial U(n) connection. An application of the above signature
f()rmula to ˜ x I shows that Pa(O) is independent of the Riemannian metric on ˜
and is a diffeomorphism invariant of ˜ and a. It is denoted by Pa(˜). Furthennore,
if E = ax with a extending to a flat unitary connection (3 over X, then

Pa(˜)
(n.! ) = n · sign(X) - sign,8(X)

The Pa-invariants were made important in low dimensional topology via the
(˜a.sson-Gordon invariants for knots [CG). We shall next discuss these invariants
Bud indicate how gauge theory can enter into their considerations.
A smooth knot K in S3 is called slice if there is a smooth 2-disk D C B 4 with
1\" = aD. Oriented knots K o, K 1 are cobordant if there is a smoothly embedded
f)riented annulus C in S3 x I with ac = K 1 x {I} -1<0 x {OJ. Addition of cobordism
('Iasses of oriented knots is given by connected sum, resulting in the knot cobordism
I˜'.r()np e˜ in which slice knots represent the trivial element.
Suppose the knot K C S3 bounds an oriented surface F C S3. Thicken F to an
f'llloedding F x I C S3. Then given x, y E H1(F), let A(x, y) = linking number of
.r xU and yx 1. This defines the Seifertform, a bilinear form A : HI (F) xH1 (F) ˜ Z,
:iuch that A(X,y) - A(y,x) is the intersection number of x and y. It is called null-
('obordant if it vanishes on a subgroup of H}(F) of dimension! dimH1 (F». J.
Fintushel and Stern: Invariants for homology 3-spheres
138

Levine has proved (in all dimensions) that if K is slice, then any Seifert form for K
is null-cobordant. Such a knot is called algebraically slice. Furthermore, in higher
(odd) dimensions the analogous condition is necessary and sufficient for K to be
slice.
A knot K is called a ribbon knot if it bounds an immersed disc (ribbon) in S3
each of whose singularities consists of two sheets intersecting in an arc which is
interior to one of the sheets. Ribbon knots are slice, for one can push the interior
of the ribbon into B4 and then deform slightly a neighborhood of each arc. An old
problem of Fox, which is still unresolved, asks whether every slice knot is ribbon. In
[CG], Casson and Gordon present an invariant for detecting when an algebraically
slice knot fails to be ribbon and a modified version of this invariant that detects
when it fails to be slice. We discuss these ribbon invariants and indicate how gauge
theory makes them slice invariants.
Let L be the double branched covering of the knot K in S3. If K is slice, then
the double covering of B 4 branched over the slicing disc is a 4-manifold W with
H*{W;Q) = 0, and if the image of H 1 (L) in H1 {W) has order m, then IH1 (L)1 =
m 2 • Furthermore, if the slicing disc is obtained by deforming a ribbon, then 1I"1{L)
surjects onto 1r} (W).
Let X : H 1 (L) --+ U(l) be a representation with image the m-th roots of unity,
em. Then X is induced by a map L --+ K{C m , 1). Since 0 3 K(C m , 1) is torsion,
rL bounds a compact 4-manifold W over K(C m , 1), for some r > o. Thus the '/
representation X factors through H 1 (W) and induces a flat U(1) bundle Vx over W.
Then the Casson-Gordon invariant u(K, X) is defined by

=.!.(sign(W) -
(6.2) u(K, X) signx(W)).
r

Hence it follows from (6.1) that q{K, X) = Px(L).
Casson-Gordon invariants were originally applied to those knots K in S3 whose
double branched covering is a lens space L = L(p, q). This contains the collection
of 2-bridge knots. If a knot K in this class is ribbon, then 1fl (L) = Zm2 and
1fl(W) = lm. Using W to compute u(K, X), one gets that Ho(W; Vx ) = 0 if X is
non-trivial, and since the m-fold covering of W is simply-connected, H 1 {W; Vx ) =
H 3 (W; Vx ) = 0 (this is where the ribbon assumption is used). Also the Euler
characteristic of W with Vx coefficients is the same as its Euler characteristic with
l coefficients, namely 1; so that H 2 (W;·VX ) has dimension 1. Thus q(K, X) = ±1.
Calculations then show that there are algebraically slice 2-bridge knots K for which
there is a X with q(K, X) -# ±1; hence they are not ribbon. Casson-Gordon then
proceed in reG] to refine these invariants. By considering infinite cyclic coverings
they define invariants of (1<, X) that show that these K are also not slice in the case
that m is a prime power order.
At this point gauge theory can enter the picture to show that in fact if K is slice,
then q{K, X) = ±1 for any m. This was done in [FS4] as follows. Consider the
139
Fintushel and Stem: Invariants for homology 3-spheres

orbifold X =cone(L) UL W. The bundle Vx over W extends as a flat V-bundle over
X. The fiat connection detennined by Vx is both self-dual and anti-self-dual. Now
consider the moduli spaces M+ and M_ of self-dual and anti-self-dual connections
in the 50(3) V-bundle Ex = Vx EB R. Each contains the reducible flat connection
determined by X and is compact (since these fiat connections are representations
into a compact group). Using the index theorem one computes that the formal
dimension of M± is -2 ± q(K, X). As in the proof of Theorem 5.1, this is an
odd integer. If dimM± > 0, then a perturbation of the equations has a moduli
space that is a compact manifold of dimension dimM± with an odd number of
singularities of the form a cone on a complex projective space, and as in (5.1), an
odd number of complex projective spaces cannot bound in B. Thus the formal
dimensions of both M ±, are negative, and so q(K, X) = ±1.
This program was extended by G. Matic in [Mal and independently D. Ruberman
in [R]. They show that if L is a rational homology sphere which has an integral
homology sphere as a finite cover, and if L bounds a rationally acyclic 4-manifold W
which has a character X with image em, then the same conclusion holds. The main
new ingredient is the gauge theory for manifolds with ends developed by C. Taubes
in [Tll. Rather than coning off the boundary, the idea is to add an open collar
to W 4 and consider self-dual and anti-self-dual connections that are asymptotically
flat. With the results of [Tl] in place, the proof is formally the same.
Note that the only po-invariants that were used in the above set-up were those
associated with representations a : 1rl(L) --. U(2) that factored through a finite
p;roup. In general, there are many irreducible representations. What role do these
invariants play in the study of e˜? We should keep this question in mind during
t.he next few sections, where these irreducible representations play an essential role.



7. An Integer Instanton Invariant
Let ˜ be a homology 3-sphere, and let a' : 1rl (˜) --. 50(3) be an irreducible
(i.e. non-trivial) representation, which corresponds to a flat connection in the trivial
bundle over E. Fix a trivialization of the SO(3)-bundle over E; this gives us a fixed
t.rivial connection 8. The Chern-Simons invariant CS(8,a') of c/ is the Pontryagin
cha.rge of a connection on the trivial bundle over E x R that tends asymptotically
t.o a' near +00 and to 8 near -00. (Recall from §3 that this calculation takes place
in A E , not in BE.) Let a'(E) denote that connection gauge equivalent to a' with
CS(8,a'(E») E [0,4).
We can associate an integer to the representation a' as follows. Let M(a'(E),8)
(I( -note the moduli space of self-dual connections over E x R whose Pontryagin charge
i:˜ -CS(8,a'(E» and that are asymptotically a'(E) near -00 and (J near +00. Let
diluM(a'(E),8) denote its virtual dimension as predicted by the index theorem and
cIc-fiue

lea') = dimM(a'(E),8) + ho,
140 Fintushel and Stern: Invariants for homology 3-spheres

where ha , is the sum of the dimensions of Hi(E; VOt' ), i = 0, 1. This definition does
not depend on our original trivialization (), for if 8 is changed by a gauge equiv-
alence, then a'eE) will change by the same gauge equivalence. When considered
with appropriate boundary conditions (e.l. [APSl],[Tl]), the self-duality operator
(whose index gives the formal dimension of the above moduli spaces) restricts to the
boundary of a smooth 4-manifold as a self-adjoint elliptic operator. Let 1J(s) denote
its 7]-function. The Atiyah-Patodi-Singer index theorem can be used to compute
I(a'):

, =1 A(E x R)ch(V_)ch(g) - l718(0)) + 2(h a / (E)
A 2(h8 + l + 1/Ot (E)(O))
1(0: ) l

Exll

where the forms A(˜ x R) and ch(V_) are computed from the Riemannian connec-
tion on E x R (choose a product metric, say) and g is the 50(3) bundle over E x R
carrying the connection a'eE). Recalling that pa' = 1/01,(0) - 110(0) and noting that
h(J = 3 we have

A(˜ x R)ch(V_)ch(g) _ ˜ + ha,,(E) + Pa,,(E) .
=f
[(a')
(7.1)
2
lExR 2 2

The integral term of (7.1) is

A(˜xR)ch(V_)ch(g)=2 f 3
f f (.c.-e)
Pl(g)+-2
JEXfl JEXfl JEXfl

where .c and e are the L-polynomial and the Euler form of ˜ x R. Since g
is our SO(3)-bundle over E x R with connection a'(E), we have JEXfJI Pl(g) =
-CS(8,a'(E». We then get as in [APSl,§4] that the integral term is

3 3
= -2CS(6, a'(˜».
(7.2) -2CS(8, a/(˜)- 2[X(E x R)-u(E x R)]+2(1/8(0) -1/8(0»)

Combining (7.1) and (7.2) we have

˜ + hai E ) + Pai E )
-2CS(9,a'(˜»
I(a') =
(7.3) - E l.

We now have a relationship between the Chern-Simons invariants introduced in
§3 and the Per-invariants of the self-duality operator, namely

+ hOt' -
4CS(a') == POI' 3 mod 2l.

Like the Chern-Simons invariant, the integer invariant 1(0') is useful in detecting
when homology 3-spheres are not homology cobordant.
141
Fintushel and Stem: Invariants for homology 3-spheres


In [FS6] we computed 1(0:') for 0:' E n.(E(p,q,r». We now indicate the key
ideas behind the computation. As in §5 we have the integer

˜2˜ 1rak 1rk. 2 1rek
2
(7.4) R(at, ... ,a n ;e·w)=--3+m+ /-i-:- L...J cot (-2 )cot(-.)sln (-.)
a i=1 a. 1:=1 ai a. a.

=
(a at, ... , an), which is the virtual dimension of the moduli space of asymp-
t.otically trivial self-dual connections in the 80(3) V-bundle L e ."., EB Rover W. It
follows from Proposition 4.1 that hoI = 0 for each nontrivial representation of a
Drieskorn sphere. Combining (7.4) with the choices of e that arise from represen-
t.ations of 1rl(E(p,q,r» (see §4) and Theorem 4.2 yields the computation of 1(a');
for -CS(O, a'(˜» = pe:r - 4k E [0,4). Then l(a') = R(p, q, rj e) - 8k.

˜
7.5 (c.f.[FS6]). Suppose R(a1, ... ,an ;w) 1. IfE(al, ... ,a n ) is ho-
rrHEOREM
11lOlogy cobordant to a homology 3-sphere E, then
(1) T(E) ˜ T(E(a1, ... ,a n » =˜, and
(2) 1:5 l(a'):5 R(a1, ... ,an ;w) for some representation 0:' E neE) with
o:5 -CS(a') :5 :.
Fhrthermore, }J(at, ... , an) is not homology cobordant via a simply-connected ho-
IIlOlogy cobordism to any other homology sphere.
Let X be the union of the mapping cylinder W of E(al, ... ,an) and the
I)ROOF:
homology cobordism, and consider the 80(3) V-bundle E = LEf)R over x, where L
is the 80(2) V-bundle whose Euler class comes from the dual of the preferred class
I.v E H 2 (W; Z). Let M denote the moduli space of asymptotically trivial self-dual
connections on E with Pontryagin charge -:. We now apply the ideas of the proof
of Theorem 5.3 to M.
The moduli space M contains a single reducible connection, and so is nonempty.
/1'llUS M is a manifold of odd dimension R(a1, ... , an; w) ˜ 1, and has one singu-
larity, a cone on a complex projective space. As in (5.1) this complex projective
:-ipa.ce cannot be null-cobordant inside Bx . Therefore M has an 'asymptotic end'
as in (5.3). Since the smallest Pontryagin number of a bundle on the suspension of
;i
It l(˜ns space L(ai, bi) which admits a self-dual (V-) connection is > ˜, no instan-
tOil can pop off at a cone point. This means that there is a nontrivial flat 80(3)

('˜)Jlnection a' on ˜ and a sequence of connections in M which limits to the 'union'
(tf self-dual connections B over X and Cover E x R, where B tends asymptotically
1,0 H˜' and C is asymptotic to a' at -00 and to 8 at +00. The Pontryagin charge
••f C is -CS(8,a'); hence 0 :5 -CS(8,a') :5 .;. Notice that this also means that
,.'(E) = a'. Now if we apply the very same argument to W rather than to X we
\V ill find a fiat connection {3' on ˜(al, ... , an) with 0 :5 -esc8, {3') :5 :-. Since the

e: '
( ˜I J('rn-Simons invariants of flat connections on E( a1, ... , an) are all of the form
lilis proves the assertion (1).
142 Fintushel and Stern: Invariants for homology 3-spheres

To prove assertion (2), note that by the translational invariance of the self-duality
equation on E x R we get 1 $ ME(a', 8) for the component of the moduli space
containing the connection C. Furthermore, if Mx(a') is the component of the
moduli space containing B, then by the index theorem of [APSI]

=dimMx{a') +dimME(a',(J) + hal
R(al' ... ,an;w) = dimM
+hal = l(a') 2= 1.
˜ dimME(a', 6)

The last statement follows as in Proposition 1.7 of [Til. Let U be a simply-
connected homology cobordism from ˜(at, , an) to E, and let V be the simply-
,an) to itself obtained by doubling U
connected homology cobordism from E(at,
along E. We obtain a reducible (asymptotically trivial) self-dual connection on the
bundle E = LeEBR over the union X of the mapping cylinder W of E{aI, ... ,an) with
infinitely many copies of V adjoined. Let M be the moduli space of asymptotically
trivial self-dual connections on E. Now, since X has a simply-connected end, M is
compact ([TI]). As above, since dimM = R(al, ... ,an ) > 0, we can cut down to
obtain a compact moduli space with one end point, a contradiction.

For example, r(2,7,15) = T(2,3,35) and R(2,3,35;w) = 1. However, for the
unique a in 'R.(E(2, 7,15» with CS(a) = T(˜(2, 7,15», we have lea) = -5; so that
E(2, 7, 15) is not homology cobordant to E(2, 3, 35).
8. Instanton Homology and an Extension
We now return to our discussion in §2 of Casson's invariant and Taubes' interpre-
tation in terms of gauge theory. In the situation where all nontrivial representations 0




!
of 1rl(}J) in SU(2) are nondegenerate, recall that A(E) = EaER*(E){ _l)SF(Ci)
where SF(a) is the spectral flow of the operator Db of §2 as b varies from 8 to a.
Equivalently we can restate this for 80(3) representations with the orientation con-
l
ventions we have been using, i.e. A(}J) = ECi/E1l.*(E)(-l)SF(a',S) where SF(a',8)
is the spectral flow of Db (defined now for 80(3) connections) as b varies from a' to
the trivial 80(3) connection. Floer interpreted this sum as an Euler characteristic.
The idea is as follows.
H a flat SO(3) connection is changed by a gauge transformation g, its spectral
flow to a fixed trivial connection changes by 8 · deg(g). Similarly, if the choice of
trivial connection is changed, the spectral flow again changes by a multiple of 8.
This means that SF(a',8) is well-defined on n.*(E) as an integer mod 8. Floer
defines chain groups Rn(E) indexed by Zg as

= Z(a' E R*(E)ISF(a',8) == n
Rn{E) (mod 8))

! E˜=o( -l)nrank(Rn(E»).
(so A(E) = The boundary operator is given by

oa' = ˜ #M] (a', (3') · (3'
Fintushel and Stem: Invariants for homology 3-spheres 143

where M 1 ( a' ,,8') is the union of I-dimensional components of the moduli space of
self-dual connections on E X R which tend asymptotically to a' as t -+ -00 and to
{3' as t -+ +00, and "#"denotes a count with signs. Floer [F] shows that this defines
a chain complex. Its homology is the instanton homology I*(E) graded by la, and
again can be defined even when there are nondegenerate representations of 1rl (˜).
Since flat connections are the critical points of the Chern-Simons function, and
since it can be shown that the gradient trajectories of the Chern-Simons function
are exactly the finite-action self-dual connections on E x R, this theory becomes
quite analogous to Witten's calculation of the homology of a manifold from a Morse
function I'W]. Instanton homology is a very important new invariant and deserves
a much more complete description than we have given here. We refer the reader to
the original source [F] and to the excellent survey papers [At] and [Br].
In [FS8] we show how to extend this theory to one with integer grading. For
simplicity continue the assumption that all nontrivial (80(3» representations of
1rl (E) are nondegenerate, and also assume that there is no nontrivial representation
with Chern-Simons invariant congruent to 0 mod 4. (These restrictions are satisfied
I)y Brieskorn spheres and in general can be removed.) As in the previous section,
we fix a trivial 80(3) connection f) over E.
For any integer m, let nm(E) be the free abelian group generated by the a' E
'1l*(E) with I(a') = m. (Since we are assuming that all these representations are
Ilondegenerate, h a , = 0 here.) For n E Zs and s E l with s == n (mod 8), define the
free abelian groups
=L 1(.s+8j(˜).
FsRn(E)
j˜O

Then


Rn(E). Furthermore, it can be shown
is a finite length decreasing filtration of
(using the fact that the Chern-Simons functional is non-increasing along gradient
t.ra.jectories) that Floer's boundary operator IJ : FsRn(E) -+ Fs-1Rn-1(E) preserves
t.he filtration. Thus Floer's la-graded complex (R.(E), 8) has a decreasing bounded
filtration (FsR*(E), IJ). For nEls and s == n (mod 8) let Istn(E) denote the
)1( )ffiology of the complex

888 8
... ---+ F s+1 R n+ 1 (E) ---+ FsRn(E) ---+ Fs-1Rn-1(E) ---+ . . . .

w(˜ In(˜)
then have a bounded filtration on defined by


\vith


As usual, there is a hOlnology spectral sequence:
144 Fintushel and Stern: Invariants for homology 3-spheres

8.1 [FS8]. There is a convergent El-spectral sequence (E;,n(E),d r )
THEOREM
such that for n E Zs and s E Z with s == n (mod 8) we have



E˜,n(E)
Furthennare, the groups are topological invariants.

In particular for s E I the groups is(E) = E˜,n(˜) give an integer-graded version
of instanton homology. As in §7 this theory is easily seen to be independent of our
choice of bundle-trivialization. See [FS8] for more details.
».
In [FS6] we gave an algorithm for computing 1.(E(p, q, r We list some exam-
ples. The groups Ii are free over Z and vanish for odd i, so we denote the instanton
homology I*(E(p, q, r» of E(p, q, r) as an ordered 4-tuple (fa, fl, /2, f3) where Ii is
the rank of 12 i+l(E(p,q,r».


!:El for k odd
( !±1 k±l k±l)
={ k'˜'
I.(E(2, 3, 6k ± 1» k2 2 '-2-
( 2" '2' '2' '2 ) for k even
for k odd
(ll:E! 3k±1 ll:E! 3k±1)
3˜ 3 k ˜k 2 ,
2 2
={
I*(E(2, 5, 10k ± 1» 3k 2
(T' T' "'2' "2 ) for k even
(˜ ˜ ˜˜) forkodd
I.(E(2, 5, 10k ± 3» = { 3k˜2 ' 3k 3˜±22 3˜ 2
2
(-2-'2-' -2-' '2) for k even
I (E(2 7 14k ± 1» = { (3k 1= 1,3k ± 1,3k =F 1,3k ± 1) for k odd
* " (3k, 3k, 3k, 3k) for k even
I.(E(2, 7, 14k ± 3» = (3k, 3k ± 1, 3k, 3k ± 1)
I.(E(2, 7, 14k ± 5» = (3k ± 1, 3k ± 1, 3k ± 1, 3k ± 1)
(A!±! .Qil! A!±! ˜) for k odd
I.(E(3,4, 12k ± 1) = { 5k2 5˜ 52k ˜k 2 , 2
(T' '2' T, 2' ) for k even
for k odd
(˜ 5k-3 5k-l 5k-3)
I.(E(3, 4, 12k - 5» = { 5k˜2 ' 5k˜2 ' 5k˜2 ' 5k:2
(-2-' -2-' -2-' -2-) for k even
(4k,4k ± 1,4k,4k ± 1) for k odd
I * ( E (3, 5, 15 k ± 2» = {
(4k ± 1,4k,4k ± 1,4k + 1) for k even

We conclude by listing some explicit computations of the Poincare-Laurent poly-
».
nomials p(aI, a2, a3)(t) of the homology groups i.(E(p, q, r
p(2, 3, 5) = t + t S
p(2,3, 11) = t + t 3 + t5 + t 7
=
p(2, 3, 17) t + t 3 + 2t 5 + t 7 + t 9
Fintushel and Stem: Invariants for homology 3-spheres 145


p(2, 3,23) = t + 2t 3 + 2t 5 + 2t 7 + t 9
p(2, 3, 29) = t + t 3 + 3t5 + 2t 7 + 2t 9 + tIl
p(2,3,35) = t + 2t 3 + 3t 5 + 3t7 + 2t 9 + tIl
p(2, 3,41) = t + t 3 + 4t 5 + 3t7 + 3t 9 + 2t ll
p(2, 3,47) = t + 2t 3 + 3t 5 + 4t 7 + 3t9 + 2t tl + t l3
=t + t 3 + 4t 5 + 4t 7 + 4t 9 + 3tII + t l3
p(2, 3, 53)
p(2, 3,59) = t + 2t 3 + 3t5 + 5t 7 + 4t 9 + 3t ll + 2t 13
=
p(2, 3, 65) t + t 3 + 4t 5 + 5t 7 + 5t 9 + 4t l ! + 2t l3
p(2, 3, 71) = t + 2t 3 + 3t 5 + 5t7 + 5t 9 + 4t II + 3t 13 + t I5
p(2, 3, 7) = t- l + t 3
=
p(2, 3, 13) t- I + t + t 3 + t 5
p(2, 3, 19) = 2t- l + t + 2t 3 + t 5
p(2, 3, 25) = 2t- 1 + 2t +2t 3 + 2t 5
p(2, 3, 31) = 2t- 1 + 2t + 3t3 + 2t 5 + t 7
p(2, 3, 37) = t- l + 3t + 3t 3 + 2t 5 + 2t 7
p(2, 3, 37) = 2t- I + 3t + 4t 3 + 3t S + 2t7
p(2, 3,43) = t- I + 4t + 4t 3 + 4t 5 + 3t7
p(2, 3, 49) = 2t- I + 3t + 5t 3 + 4t 5 + 3t7 + t 9
p(3, 4, 13) = t- 5 + 2t- 3 + 3t- 1 + 2t + 2t 3
p(4,5,21) = 4t- 9 + 5t- 7 + 8t- 5 + 6t- 3 + 5t- 1 + t + t 3
peS, 6,31) = 2t- 15 + lOt- 13 + lIt-II + 15t- 9 + 13t- 7 +8t- 5 +5t- 3 +4t- 1 +t +t 3
For a' E n('E(2,3,6k-l)), CS(a') == 36˜˜6 mod 4 where e == 1 mod 6. Choos-
ing the largest e and computing R(2, 3, 6k - 1; e · w), it can be shown that for
a. given any positive integer N, there is a K with i 2N - 1 (E(2,3,6K - 1)) =F O.
Silnilarly, it can be shown that for any negative integer N, there is a K with
j˜N-I('E(K,K+ 1,K(K +1) + 1)) =F O.



u. Homotopy K3 Surfaces Containing E(2, 3, 7)
In this section we shall describe an application of the technology which we have
(Iiscussed in previous sections to the problem of computing the Donaldson invariants
()f smooth 4-manifolds homotopy equivalent to the I<:3-surface (i.e. homotopy K3-
˜;tltfaces). The Donaldson invariant may be briefly described as follows. Given a
:illlooth simply-connected 4-manifold, M, for a generic metric 9 on M, the moduli
:˜pace, Mk,M(g) C Bk,M of anti-self-dual SU(2)-connections (*FA = -FA) with
"'.˜ = k is a manifold of dimension 8k - 3(1 + b;). (It may have singular points
if bt = 0.) If b˜' is odd, then this dimension is even, say 2d. Donaldson has
dc-fined a homomorphism JJ : H 2 (M; Z) -+ H 2 (Bk,M; l) (see [D4]) and has shown
f.ha.t. if k > ˜(1 +.bt) then there is a well-defined pairing q(ZI, ... ,Zd) = P(Zl) U
.. ·ll(Zd)[Mk,M(9)]. This defines Donaldson's invariant q : tfJdH2(M; Z) ˜ Z. Our
I.)u\orcm is:
146 Fintushel and Stern: Invariants for homology 3-spheres

9.1 [FS7]. Iftbe Brieskorn sphere ˜(2, 3, 7) is embedded in a 4-manifold
THEOREM
which is homotopy equivalent to a K3-surface, then there are classes Zl, ••• ,ZlO E
H 2 (M;Z) such that q(Zl, ••• ,ZlO) == 1 (mod 2)
This result plays a role in an important result of S. Akbulut.
9.2 [Ak]. There is a compact contractible 4-manifold W which has a
THEOREM
fake relative smooth structure.
To prove this Akbulut gives a construction which produces a homotopy K3 sur-
face M containing E(2, 3, 7) and a compact, contractible 4-manifold W together
with a self-diffeomorphism f of oW which extends to a self-homeomorphism of W
such that either
(1) all of the Donaldson polynomial invariants of M are trivial, or
(2) there is no self-diffeomorphism of W extending f.
Theorem 9.1 then implies that (1) is false; so Akbulut's construction gives (2).
The proof of Theorem 9.1 proceeds via a degeneration argument. The intersection
form form of a K3-surface is 2Ea EB 3H, and E(2, 3, 7) embeds in the standard K3-
surface, splitting it into two submanifolds, X and Y, where X has intersection
form E s EB H and boundary ˜(2, 3, 7) and Y has intersection form E s E9 2H and
boundary -E(2, 3, 7). Using the work of Donaldson [D2], it is not difficult to see
that if ˜(2, 3, 7) embeds in any homotopy K3-surface, M, then it splits M as XU Y
in a similar manner. The idea is then to study the effect of letting the metric 9
degenerate to a metric which stretches a tube E(2, 3, 7) x (-1,1) in M until it has
infinite length.
For the homology classes Zl, ••• ,ZlO, we choose classes ZI, ••• ,Z4 E H 2 (X;l) =
EstJ)H such that Zl, Z2 E Es satisfy zi = zi = 2 and Zl ·Z2 = 1, and choose Za, Z4 E H
with zi = zl = 0 and Z3· Z4 = 1. Similarly choose Zs, ••. ,ZlO E H 2 (Y; Z) = E sff)2H
such that zs, Z6 form a pair in E s and Z7, Zs and Zg, ZlO form pairs in the two copies
of H. Suppose for a moment that q(Zl, .•. , ZIO) =f: 0 and follow the above-described
degeneration of metrics. In the process M is pulled apart into noncompact manifolds
X+ and Y_ (X+ = X U R+, and Y_ = R- U Y). A finite-action anti-self-dual
connection on X+ or Y_ is asymptotically flat, and the techniques of §4 can be
used to show that ˜(2, 3, 7) has only two gauge equivalence classes of nontrivial :fiat
connections.
IT q(Zl' ... ' ZIO) # 0 the degeneration process will produce nontrivial "relative"
Donaldson invariants which can be described roughly as qX(Zl, ... , Z4) = E p Jt(Zl)U
... U Jt(z4)[M˜(p)], where p is a :fiat connection on E(2, 3, 7), and qy(zs, ... , ZlO) =
EpJt(zs)U .. ·UJ.t(zlo)[M¥(p)]. One can now make counting arguments (see [FS7])
to the effect that only one possible p can appear in the above sums, and furthermore,
for this p we have

= qX(Zl, .. . , Z4)(p) · qy(zs, ... , ZlO)(p).
q(Zl, ... , ZlO)
147
Fintushel and Stem: Invariants for homology 3-spheres

The proof is now concluded by constructing the moduli spaces M˜(p) and
MV(p) whose appropriate relative invariants are odd. The philosophy is to ap-
ply the proofs of Donaldson's Theorems B and C of (D2]. These theorems prove
that there are no closed 4-manifolds which have the intersection forms of X or Y.
Applying the proofs to X and Y, rather than contradictions we obtain information
ahout the asymptotic behavior of certain moduli spaces. As in the proof of (5.3) we
J!:et anti-self-dual connections popping off the ends of M 2 ,x(6) and M 3 ,y(8) which
I('ave us the correct moduli spaces over X and Y (see [FS7]).




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[T2]



Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Department of Mathematics, University of California, Irvine, California 92717
On the Floer Homology of Seifert
fibered Homology 3-Spheres
CHRISTIAN OKONEK
Mathematisches Institut der Universitat Bonn
Wegelerstr.lO
D-5300 Bonn 1, F .R.G.



1. INTRODUCTION

This note contains a somewhat expanded version of my talk given at the Durham
Geometry Symposium. In this talk, which was based on a joint paper with S.Bauer
[B/O], I tried to explain how some fairly remote results in algebraic geometry -
like e.g. Deligne's solution of the Weil conjectures - can be used to calculate the
instanton homology of Seifert fibered homology 3-spheres.

Instanton homology groups In(˜),n E Z/8 have recently been defined by A.Floer
for every oriented 3-dimensional Z-homology sphere ˜[F].
They provide an important link between the geometry of 3- and 4-dimensional
11
lnanifolds [AI]. In fact, Donaldson has shown that his polynomial invariants
for a 4-manifold X [D2], which can be decomposed along a homology sphere E
into two pieces factor through the Floer homology of ˜ [AI]. A nice introduction
to the Donaldson-Floer theory can be found in Atiyah's survey articles [Al],[A2].

'fhe definition of the groups In(˜) uses gauge theoretic constructions in dimension
3 and 4, so that it is difficult to compute them for a general homology sphere.
˜or the subclass of Seifert fibered homology spheres however, Fintushel and Stern
have found a way to calculate the instanton homology provided that a certain
conjecture holds [F/82].
This conjecture, which I will explain in a moment, has recently been settled by
Kirk and Klassen [K/K], so that the Fintushel-Stern program can be carried out.
\tVhat it comes down to is to understand the representation spaces

= Hom*(1rl (˜), SU(2)/conj.
R*(˜)

c.f conjugacy classes of irreducible SU(2)-representations of the fundamental group
7fl(˜).
rhe Floer homology of a Seifert fi bered homology sphere I: is determined by the
I



Betti numbers of the components of R*(E) and certain (explicitely computable)
integers associated to each component [F/52].
150 Okonek: On the Floor homology of Seifert floored homology 3-spheres

One way to study these representation spaces R*(˜) is to identify them - via Don-
aldson's solution of the Kobayashi-Hitchin conjecture [Dl] - with moduli spaces
of stable vector bundles over certain algebraic surfaces [0/V2]. In this way they
become complex projective varieties which turn out to be smooth and rational.
Moreover, these moduli spaces come with a stratification whose individual strata
can be described in terms of secant varieties of rational normal curves [B/O].
Using a trick due to Deligne-Illusie [D /1], one can assume that everything is de-
fined over an algebraic number ring, so that the Weil conjectures can be applied.
The explicit geometric description of the strata of the moduli spaces then allows
to determine the Zeta functions associated to the components and thereby to com-
pute their Betti numbers.
The final result is an algorithm which, in principle, could be implemented on a
computer.

2. FLOER HOMOLOGY

I recall very briefly the idea of the definition of Floer's instanton homology groups;
a much more detailed description can be found in Floer's paper [F] and the survey
article [BR].
Let E be an oriented Z-homology sphere of dimension 3. Consider the space 8 of
gauge equivalence classes of SU(2)-connections on the trivial SU(2)-bundle over
˜. The map A 1--+ FA, sending a connection A to its curvature, defines a natural
I-Form F on B which is locally exact. More precisely, F is - up to a constant -
the differential of the Chern-Simons function

f: 8 R/ 4Z·
-+


This function associates to a connection A the integral




where At := (1 - t)A +to is a path of connections from A to the trivial connection
f), thought of as a connection on :E x [0, 1].
The critical set of the Chern-Simons function, i.e. the zeros of F, can be identified
- via the monodromy representation - with the space

R(˜) = Hom(1rl(˜)' SU(2))/conj.
of conjugacy classes of SU(2)-representations of 7rl(˜).
Suppose now that all non-trivial critical points of I are non-degenerate (if not one
uses a suitable Fredholm perturbation); this means that R(I;) is finite and that the
Hessian of f (considered as an operator on the tangent space) is an isomorphism
at every critical point a = [A] in R*(E) = R(˜)\{[O]}.
Let S(O,a) E Z/s denote the spectral flow associated to a path of connections
Okonek: On the FIoer homology of Seifert fibered homology 3-spheres 151

from the trivial connection to A [BR]. Floer's instanton homology I*(˜) is then
defined as the homology of the following chain complex (R., 8):
The nth chain groupRn of R. = E9 Rn is the free abelian group generated by
· neZ/s
the elements 0: E R·(E) with S(8,0) = n.
The boundary operator 8 : Rn --+ R n - 1 is given by

=L m((t, 13)13;
80:
.8eR·(D)
8(8,,8)=",-1


here m(o:, {3) is the number of oriented I-dimensional components of the moduli
space M E (a,!3) of self-dual connections (relative to a product metric) on E x R,
which are asymptotic to a and (3 for t tending to ±oo [F]. Floer shows that
8 2 = 0 and proves that the homology of the instanton chain complex (R.,8) is
independent of the various choices (metric, perturbation).

3. SEIFERT FIBERED HOMOLOGY 3-SPHERES

Let 1r : ˜ --+ ˜/8 1 be a Seifert fibration of a homology 3-sphere I: with n excep-
tional orbits 1r- 1(Xi), i = 1, ... , n of multiplicities a1, ... , an- The multiplicities are
necessarily pairwise relatively prime and the orbit space 'E/Sl is homeomorphic
to (8 2 ; (Xl, a1), ... , (x n , an)) [N/R].
Conversely, given n pairwise coprime integers ai 2: 2, there exists a Seifert fibered
homology 3-sphere I: with these multiplicities; its diffeomorphism type is deter-
ruined by .a = (at, ... , an) [N/R].
Denote such a homology sphere by E(g) = I:(al' ... ' an). I will always assume that
the multiplicities are indexed in such a way that at most at is even. The links of
certain Brieskorn complete intersections provide standard models of Seifert fibered
hornology spheres [N/R]. The fundamental group of E(g) has the following repre-
sentation [F/82]:
Let a := at · ... - an and choose integers b, bi, i = 1, ... , n with
(1 (-b + L:˜=1 ˜) = 1. Then




(t 1 , ••• ,tn, h I h
(E(<<)) ˜ = h- bi , t1 • ••• • t n = hb )
central, ti i
1rl •


If n ˜ 3, then this group is infinite with center (h) ˜ Z, except for (E(2, 3, 5)) ˜
11"1
˜ Z/2. The quotient 11"1 (˜(.G)) / (h) is isomorphic to the
FS) with center (h)
.',1 L(2,

:! orbifold fundamental group ?rIb (E(<<)/S1) of the decomposition surface [F/S2].
I t. is isonlorphic to a cocompact FUchsian group of genus 0 with representation
Okonek: On the Floer homology of Seifert fibered homology 3-spheres
152

4. REPRESENTATION SPACES OF SEIFERT FIBERED HOMOLOGY SPHERES

= :E(al' ... ' an) with n ˜ 3 exceptional
Fix a Seifert fibered homology sphere ˜(g)
fibers and consider a representation

0: : 1fl (˜(<<») ˜ SU(2).
is irreducible, then the generator h of the center must be mapped to ±1 E
If Q

SU(2); the images a(ti) of the remaining generators are conjugate to diagonal
matrices
(w˜' wi';)
a(t;) ""

with Wi = exp (21rt=J) and certain numbers Ii E Z/ai.These numbers
(±It, . .. ,±In) are the rotation numbers of the representation Q.
Proposition 1 ([F/82],[B/O]) The representation R* (E(!!») 01 a Seifert fibered
homology sphere is a closed differentiable manifold with several components. The
rotation numbers of an irreducible representation Q are invariants of the connected
component of Q in R* (˜(g). A component with rotation numbers (±ll' ... ' ±In)
has dimension 2(m - 3), where m = U{i 1 21i =I OJ.
Furthermore, there exists at most one component in R* (˜(g)) realizing a given
set of rotation numbers [BfO].

In the special case of 3 exceptional orbits, Le. for Brieskorn spheres ˜(al' a2, a3),
the associated representation spaces are finite.
Fintushel and Stern have shown that the number of elements in R* (˜(al' a2, a3»
is equal to - ˜ times the signature of the Milnor fiber of the corresponding singu-
larities [F/82].
The Casson invariant of a Brieskorn sphere is therefore equal to ˜ times the sig-
nature of its Milnor fiber.
The latter result has recently been generalized by Neumann and Wahl (N/W].

5. THE INSTANTON CHAIN COMPLEX OF SEIFERT SPHERES

In this section I recall the relevant results of Fintushel and Stern's paper [F/S2].
Again, fix a Seifert fibered homology sphere ˜(G) = ˜(al' ... ' an), n ˜ 3. For any
integer e let

2e (1rak) cot (1rk) sin (7rek)
1
2
2 n
R(!!,e)=--3+m+L˜Lcot -˜
ai- 2
-. -. '
a al at at at
i=l k=l


where m = U{i , e ˜ O(mod ai)}. This number is the virtual dimension of a certain
moduli space of instantons; it is always odd [F/SI].
Okonek: On the FIoer homology of Seifert fibered homology 3-spheres 153

(˜(a)) with
Theorem 1 ([F/S2]) The spectral flow 8(9, a) of an element a E R*
rotation numbers (±ll' ... ,±In) is given by
= -R({!,e) -
8(6,a) 3(mod8),
E˜11i!i(mod2a).
==
ife satisfies e
If now every representation a E R* (E(g» is non-degenerate, then the grading in
the instanton chain complex is always even, so that the boundary operator must
vanish. This assumption holds if n == 3, Le. for Brieskorn spheres E(al' a2, a3).
Explicit examples of instanton homology groups 1* (˜(al' a2, a3) can be found in
[FjS2].
In the general case n > 3 the elements of R* (˜(g)) are usually degenerate, so
that one has to perturb the Chern-Simons function. Fintushel and Stern use
a Morse function on R* CE(aJ) to produce a perturbation with non-degenerate
critical points.
Theorem 2 ([F/82]) Let 9 : R* (˜(<<»Q ˜ R a Morse junction on the compo-
nent 01 a in R* (˜(g)). The critical points of 9 are basis elements oj the instanton
chain complex.
A critical point t3 E R* (˜(g)o: 01 9 with Morse index IJg(f3) has grading
8(8, a) + j.tg({3).
In order to make explicit computations possible Fintushel and Stern show how
R* (˜(q) can be described as a configuration space of certain linkages in 8 3 [F/82].
With this method they find copies of 8 2 as 2-dimensional components. On the
hasis of these examples they make the following

Conjecture Every com,ponent of R* (˜(<<)) admits Morse functions with only even
index critical points.
Note that this conjecture implies that the instanton chain complex of ˜(g) is
concentrated in even dimensions, the boundary operator vanishes and the Floer
homology can be read off from the rotation numbers and the Betti numbers of the
cornponents of R* (h(g).
6. THE ALGEBRAIC GEOMETRY OF THE REPRESENTATION SPACES

B* (˜(q» Every representation a : 1r1 (˜(q)) 8U(2) induces a representation
----+
H: 1rrb(˜(g)jS1) ----+ PU(2) s.t. the following diagram commutes:
˜ 8U(2)
(1:(<<»
?r1

1/(h) 1/z/2
˜ PU(2).
(˜(ll)/ 8 1
?rIb )

'I'his correspondence yields an identification of R* (E(aJ) with the representation
sl)a,ce
Okonek: On the Floer homology of Seifert fibered homology 3-spheres
154

of the orbifold fundamental group in PU(2). Recall that

(˜(ll)/8 1) ˜ (tt, ... , t n Itf' = 1, t 1 • ••• • t n = 1) ·
1rfb


In the sequel I shall denote this group by r(ll).

Theorem 3 ([B/O]) The representation space Hom· (r(g),PU(2» Icon;. admits
the structure a 01 smooth complex projective variety whose components are rational.

The proof has two essential steps:

i) Interpretation of Hom· (r(g), PU(2)) / con;. as moduli space of stable vector
bundles:
Consider a rational elliptic surface over pi - defined by a generic pencil of
plane cubic curves - and perform logarithmic transformations of multiplicities
aI, ..• , an along smooth fibers over Xl, .•. ,Xn E pI [B/P IV]. The resulting ellip-
tic surfaces X(g.) = X(at, ... ,an) over pI are algebraic with fundamental group
(X(a)) ˜ r(a) [U]. .
11"1
(The algebraic structure of X(g) depends on the choices which are involved in the
logarithmic transformations, but the COO-type does not [U]). Choose a (sufficiently
nice) ample divisor H = H(g.) on X = X(g) and let M˜(Cl' C2) denote the moduli
space of H-stable rank-2 bundles over X with Chern classes Cl,C2 [O/S/S].
Using Donaldson's solution of the Kobayashi-ffitchin conjecture [DI] and some
simple arguments [O/VI] one obtains an identification of
Hom· (1rl (X(g) ,PU(2») / elm;. with the differentiable space underlying the disjoint
union M˜(O,O) IIM˜(K,O). Here K = -Cl(X) is the canonical class of X.

ii) Description of the moduli space of stable vector bundles:
The moduli spaces M˜(O, 0) and M˜(K, 0) can be handled by similar methods;
but a little trick allows to avoid computing the latter. Indeed, the homomorphism



sending a generator ti of r(2al' a2, . .. ,an) to the corresponding generator in
real, a2,· .. , an) induces an isomorphism

r· : M˜(O, 0) II M˜(K, 0) ˜ M1<0, 0)

with the moduli space Mf(O,O) of stable bundles over an elliptic surface X of
type X(2at, a2, . .. ,an).
e
Consider now a stable 2-bundle £ over X with trivial Chern classes. admits a
unique representation as an extension
Okonek: On the Floer homology of Seifert fibered homology 3-spheres 155

dF + Ei=l diFi is a vertical divisor with 0 ˜ di < ai.
of line bundles, where D f'J


Here F denotes a generic fiber of X over pI, aiFi F, i = 1, ... , n the n multiple
"J


fibers over the points Xl, •.. X n E pl.
The line bundles O(D) which occur in such extensions form a finite subset I C
NS(X) in the Neron-Severi group of X.
Conversely, for every line bundle O(D) E I and every (non-trivial) extension
class [f] E P (Ext l (O(D),O( -D») one obtains a simple 2-bundle e given by the
extension
0 --+ o( -D) ˜ E --+ O(D) --+ O.
f:

Denote by Mx(O,O) the moduli space of simple 2-bundles over X with triv-
ial Chern classes. This is a locally (but not globally) Hausdorff complex
space containing M˜(O,O) as (Hausdorff) open subspace [O/V2]. Let P(D) =
P (Ext l (O(D), O( -D») and define Zariski open subsets U(D) C P(D) by
U(D) = P(D) n M˜(O, 0). Then one has a cartesian diagram
˜ Mx(O, 0)
IfP(D)
u
U
M˜(O,O)
IfU(D)

and a stratification II U(D) of M˜(O, 0) by locally closed subspaces U(D), each
I
sitting as a Zariski-open subset in its 'own' projective space. In order to prove
the smoothness of M˜(O, 0) as a complex algebraic variety, one shows that the
coefficients di of a divisor D determine the rotation numbers of the component
which contains U(D).
More precisely, if a representation Q : 1r1 (X(<<» -» SU(2) corresponds to a vector
bundle e given by a class [f] E U(D) with D dF + Ei=l diFi , then a has the
f'J


rotation numbers (±ll' ... ,±In).
The rationality of all components of M˜(O, 0) now follows immediately.

'fhe next point is to understand the projective varieties P(D)\U(D) parametrizing
(simple but) unstable bundles.
A bundle e given by [e] E P(D) is unstable if and only if X contains a vertical curve
J˜ with O(D - E) E I s.t. [e] is contained in the projective kernel P (Ker(eE» of
t he multiplication map

eE : Extl(O(D),O(-D)--+Extl(O(D-E),O(-D).
I E I)) over the linear system I E I
,I'hcse kernels form a projective bundle P (Ker(e
which admits a natural map

"pIEI: P(Ker(e, E I)) ˜ P(D)

the projective space P(D). The image of "pIEI is the subvariety Dest(1 E I) of
to
huudles which arc destabili˜ed by curves in IE I.
Okonek: On the Floer homology of Seifert floored homology 3-spheres
156

Recall that the join W l * ... *Wit of subvarieties WI, ... , W k C pN is the smallest
closed subvariety of pN containing span(Wi, ... ,Wk) for every tuple (WI, ... ,Wk) E
WI X _. - X W k - The secant variety Seck(W) = W *... *W (k times) is a particular
case.
Proposition 2 ([B/O]) Suppose that dim P(D) > o. There exists a natural em-
bedding of (PI; Xl, ••• , X n ) as a rational normal curve N(D) c P(D) with marked
points x j, such that the following holds:
i) Every destabilizing subvariety Dest(1 E I) 01 P(D) is a join
Seck(N(D» * {Xii} * ... * {Xii} of a secant variety of N(D) and some of the
points Xj.
ii) P(D)\U(D) is a finite union 01 destabilizing subvarieties Dest(' E I).
iii) The intersection Dest(1 E I) n Dest(1 E' I) of two destabilizing subvarieties in
P(D) is a finite union of other destabilizing subvarieties.

The following picture illustrates a typical 2-dimensional situation




N(D)




7. THE BETTI NUMBERS OF THE MODULI SPACES

The way in which the stratification of-M = M˜(O,O) has been defined makes it
difficult to describe the normal bundles of the various strata.
To circumvent this difficulty one can use an approach which has been ·applied by
Harder and Narasimhan in a similar situation [H/N]. Their idea was to calculate
the Betti numbers of a moduli space by 'counting points over finite fields and then
use the Weil conjectures'. In the situation at hand the space M is not a priori
defined over a number ring. However, one can find an extension of M over the
spectrum of a subring ACe and a closed point in Spec (A) with residue field F q
of positive characteristic p, so that there exists a good prime l i= p with



for all i.
Consider now the Zeta function of M F q ,
Okonek: On the FIoer homology of Seifert fibered homology 3-spheres 157

where vk(M-q ) counts the number of points in MF9 k. By Deligne's solution of
F
the Weil conjectures Z(M Fq , t) can be written in the form

= Pi · · · · · P2n- i
Z(M'F ,t)
po····· P2n
q


with polynomials ˜ of degree dim Hjt(M Fq ; '21), whose zeros all have absolute
value p-i [D]. In our situation we have to determine the Zeta functions of the strata
U(D) or, equivalently, of the varieties P(D)\lI(D) of unstable bundles. Using the
explicit description of these varieties as joins over secant varieties of rational normal
eurves, it is possible to calculate these Zeta functions. One finds that associated
t.o each component of M, there are natural numbers bo = 1, b2 , ••• ,b2 (m-3h so that
the counting function Vk of the component has the form
2(m-3)
L
Vie = b2i (qk)i.
i=O

course, this implies that the Zeta function of this component has the product
()f
decomposition


More precisely:

Theorem 4 ([B/O]) The odd Betti numbers of the moduli space M˜(O, 0) are
zero. The even Betti nu.mbers can be determined by a numerical algorithm.

Explicit formulas can be found in [B/O].

8. FINAL REMARKS

As I already mentioned in the introduction, the conjecture of Fintushel and Stern
has been shown to be true. In fact, there are at least three different (announce-
lnents of) proofs.
'[,he first Olle - by Kirk and Klassen [K/K] - uses the concept of linkages in S3
to construct directly a Morse function with only even index critical points.
!\. second proof has been announced by FUruta and Steer [FIST]. Their starting
point is the observation that the representation space
Ilom· (1rrb(:E(g)/81),PU(2») Iconj. can be interpreted as moduli space ofequivari-
ant Yang-Mills connections over a suitable covering surface of ˜(g)/8 1 • Extending
the Atiyah-Bott method [A/B] to this equivariant setting they give formulas for
t.h(\ Poincare polynomials of the instanton homology.
'('here is still another interpretation of Hom· (7rfb(˜(.a)18 1 ), PU(2)) Iconj., going
ha.ck to Mehta and Seshadri [M/S]. These authors show that representation spaces
of cocompact Fuchsian groups can be identified with moduli spaces of parabolic
vpctor bundles 011 nHuked _es.
_IDR˜˜˜˜˜,
Okonek: On the Floer homology of Seifert fibered homology 3-spheres
158


Using this description of R· (˜(G)) Bauer [B] gives a third proof of the conjecture.
Finally, I like to mention another point of view which might be interesting. The
fundamental group 1r1 (˜(g)) of a Seifert fibered sphere is isomorphic to the local
fundamental group of a corresponding Brieskorn complete intersection singularity.
Thus unitary representations of this group should give rise to reflexive modules
over the local ring of such singularities; equivalently, they should define vector
bundles over their minimal resolutions. It might be useful to consider the relevant
singularities as group quotients of cone singularities [Pl.

REFERENCES

[AI] Atiyah,M.F.(1988) New invariants of 3- and 4-dimensional manifolds.
In: The Mathematical Heritage of Herman Weyl, Proc.Symp.Pure Math.48,
Arner. Math. Soc..

[A2] Atiyah,M.F.(1989) Topological quantum field theory. Preprint, Ox-
ford.

[A/B] Atiyah,M.F.,Bott,R.(1982) The Yang--Mills equations over Riemann
surfaces. Phil.Trans.R.Soc. London A, 308,523-615.

[B/PIV] Barth,W.,Peters,Ch.,Van de Ven,A. (1985) Compact complex
surfaces. Erg.der Math.(3)4. Springer: Berlin, Heidelberg, New York.

Bauer,S.(1989) Parabolic bundles, elliptic surfaces and SU(2)-
[DA]
representation spaces of genus zero Fuchsian groups. Preprint 67, MPI Bonn.

[B/O] Bauer,S.,Okonek,Ch.(1990) The algebraic geometry of representa-
tion spaces associated to Seifert fibered homology 3-spheres. Math.Ann..

[B] Braam,P.(1988) Floer homology groups for homology three-spheres.
Preprint, Univ.of Utrecht.

[D] Deligne,P.(1974) La conjecture de Weil,I.Publ.Math.IHES 43,273-307.
[D /1] Deligne,P.,Illusie,L.(1987) Relevements modulo p2 et decomposition
du complex de de Rham. Invent.math. 89,247-270.

[Dll Donaldson,S.K.(1985) Anti-self--dual Yang-Mills connections over
complex algebraic surfaces and stable vector bundles. Proc.London
Math.Soc.(3), 50,1-26.

[D2] Donaldson.S.K.(to appear) Polynomial invariants for smooth four-
manifolds. Topology.

[F/SI] FintusheI,R.,Stern,R.(1985) Pseudofree orbifolds. Ann.of Math.
122,335--364.
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[F/S2] Fintushel,R.,Stern,R.(1988) Instanton homology of Seifert fibered
homology 3-spheres. Preprint, Univ.of Utah, Salt Lake City.

[F] Floer,A.(1988) An instanton invariant for 3-manifolds. Com-
mun.Math.Phys. 118,215-240.

[F/ST] Furuta,M.,Steer,B.(1989) The moduli spaces of flat connections on
certain 3-manifolds. Preprint, Oxford.

[BIN] Harder,G.,Narasimhan,M.S.(1975) On the cohomology groups of
moduli spaces of vector bundles over curves. Math.Ann. 212,215-248.

[K/K] Kirk,P.A.,Klassen,E.P.(1989) Representation spaces of Seifert fibered
homology spheres. Preprint, California Institute of Tech., Pasadena.

[MIS] Mehta,V.B.,Seshadri,C.S.(1980) Moduli of vector bundles on curves
with parabolic structure. Math.Ann. 248,205-239.

[N/R] Neumann,W.,Raymond,F.(1978) Seifert manifolds, plumbing, J.L-
invariant and orientation reversing maps. In: Algebraic and Geolnetric Topo-
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[N/W] Neumann,W.,Wahl,J.(to appear) Casson invariants of links of sin-
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Za-invariant SU(2) instantons
over the Four Sphere
MIKIO FURUTA
The University of Tokyo, Hongo Tokyo 113, Japan and
The Mathematical Institute , Oxford




1. INTRODUCTION
The purpose of this article is to give a classification of invariant SU(2)-instantons
on 8 4 for some equivariant SU(2)-bundles over S4 and to give some applications.
We use the term instanton for anti-self-dual connection here. It is well known that
t.he moduli spaces of SU(2)-instantons on the standard four sphere 8 4 are smooth
tuanifolds. When a group r acts on 8 4 isometrically and P is a r-invariant SU(2)-
bundle, the moduli space M(P) of r-invariant instantons on P is defined as the
quotient of the space of f -invariant instantons divided by the f -equivariant gauge
t.ransformations. Then M(P) has a natural smooth structure as well. Instantons
on 8 4 are classified by the ADHM-construction [2] or the monad description [5],
so, in principle, we have a description of invariant instantons. On the other hand,
ror some group actions the invariant instantons have some geometric interpretation:
M. F. Atiyah pointed out as an important example that when f is the rotations
around 8 2 in 8 4 , the r -invariant instantons are interpreted as hyperbolic monopoles
[1]. In this article we consider subgroups of the maximal torus of SOC4) as r
a.nd f-equivariant SU(2)-bundles over 8 4 for which the moduli spaces of invariant
illstantons are one-dimensional, in particular when r is a finite cyclic group Za of
order a.
/\. crucial observation is as follows. Suppose r is a cyclic group and the r -action
is semifree with fixed point set {O, oo}. The quotient s3 /r is called a lens space.
Il(˜cause (S3 If) x R is conformally equivalent to (S4 \ {O, 00 } ) If, a r -invariant
illstanton on 8 4 induces an instanton on (S3 If) x R. Conversely Uhlenbeck's
n˜lnovable singularity theorem [19] implies that an instanton on (S3 If) x R with
JJ2-bounded curvature comes from a f-invariant instanton on 8 4 •
III Section 2 a classification of invariant instantons is described for some equiv-

ariant bundles over 8 4 • This is a special case considered by D. M. Austin [3]
and Y. Hashimoto and the author [13]. (The results of [13] are an extension of

˜;t1prorted by the U.1(. Science and Engineering Research Council
162 Furuta: Z/a-invariant SU(2) instantons over the four-sphere

Hashimoto's MSc thesis (University of Tokyo, 1987)). As a byproduct the Euler

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