. 5
( 10)


T xD A.

Viewed from the knot complement K, (,." A) defines sections of the bundle
1\.0, "1
;)I{ over two simple closed curves in 18KI = T2, through

= {l} x aD x {I}

= T x {I} x {I}.
106 Floor: Instanton homology, surgery and knots

This class of knots is acted upon by the group

r = Z˜ ˜ 812 Z
= (Diff(T 2 )j Diff 1 (T2» H 1 (T 2 , Z2)

of isotopy classes of equivariant diffeomorphisms of the bundle T 2 80a . In the
complementary picture,
= (toe-1,K).

e = (˜ Z˜ with Z2 = {a, I}, we have
(a, b) E S12Z)4

= {(tn, t P , p(at) It E T} ˜ a(nK" +pK
(eK)O 1r )

(eK)l = {(tm,tq,p(bt) It E T} ˜ b(m˜" +q˜7r).

1: The restriction

= Z2 ˜ Z ˜ Aut(T2
Aut(T x D x BOa) x 803 )

defines the subgroup r o of surgeries leaving K" = ta( {I} x T x {I}) invariant. Hence
it describes the subgroup of all surgeries on K, which do not change the topology
of the bundle A = K(K,,). In fact, r o is generated by the operator e changing the
orientation of K, the operator ˜ changing the framing of K and the operator q, of
order 2 changing the trivialization of A over the core of K,:

˜o = {e (˜ ˜)4 (O,b) I b, e E Z2, mE Z}.
2: The element
E (0 1)
-1 0
of order 2 in PSl(2,Z) corresponds to framed ("honest") surgery on the framed
(and lifted) knot K.
E together with r 0 spans r. However, there is a more canonical generating system

where Z2 is generated by E and Z3 by

(0 -1) (1 1 (-11 -01
- 0)= )
(2.3) : = Er.p = 0
1 1
Floer: Instanton homology, surgery and knots

(-1 -1) (-1 -1) _(0 1)
,=,2 _ _,::,-1
'-' - 1 0 1 0 - -1 -1 --
1 +a +S2 = 1 + a +a-I = o.
Hence ("pE)3 = id for any "p E H 1 (T 2 , Z2). If"p = (0,1)
changes the lifting
along the longitudinal, i.e. t/J E r o, then "pS corresponds to a framed surgery on
1',. Denote by X" the corresponding surgery cobordism. Replacing I', by t/JBK, and

repeating the process, we obtain a "surgery triangle" associated to a framed knot
1'\.. We abbreviate it as


=K 2
A" Ua (D T S03)

I'\.A U a (D T 503 ).

The attaching maps and & are determined by

a(8D x {I})
+ "'1.
a(8D {I}) =
X 1'\.0

The surgery triangle of K coincides with (2.4) if H 1 (K) = Z and 1',1 is canonical,
i.e. 1'\.1 = 8SK for some subbundle (SK, 8SK) C (K, oK) with two-dimensional base.
Therefore exactness of (2.1) follows from the following result
3. H all bundles M, K.M, and M" are in 50;[3], then (X, Y, Z) :=
(XK, YK , Z K) induces an exact triangle.
Because of the symmetry of (1.7) with respect to S, we only need to prove exact-
ness of the two homomorphisms
= kerX. C I.(A).
rrhe sequence property X. ˜ = 0 follows from a connected sum decomposition.
(This is simlar to the vanishing theorem for Donaldson's invariants.)
= o.
where P is the S03-bundle over Cp2 with Wl(P) = 1 and Pl(P) There exists
some s E p(X) sucb that the discrete part of M.,(X) is empty.
108 Floer: Instanton homology, surgery and knots

3. Instanton homology for knots.

All 3-manifolds are interrelated through surgery on knots, and are interrelated
in many ways. What is needed is a method to "decrease the complexity" of 3-
manifolds by surgery, such that the third term which arises is simpler, too. A
way of doing this is to consider the instanton homology of certain 3-manifolds as
"relative" homologies 1.(A, ,\) for a link (a disjoint collection of knots) in A.- Then
by surgery we can reduce the problem to I.(A) for a link in S3. At least in Sa there
is an organized scheme of simplification of links, by the skein move, where two parts
of A pass through each other. In knot theory, such a move often involves a third
link '\/1, which is described as follows.
3.1. A crossing"Y of a link ,x in A is an embedding, : D x I A such

that ,-1[,\] is given by the graphs of the functions

Then define 7'\ by replacing the graph of e± by the graph of

To define '\/7, replace e± by the sets [-1/2,1/2] x {O, I}.
Topologically, ,A is obtained by a ±l-surgery on the circle, defined by the
crossing, and which does not change K.. We will therefore use the operative notation

· ,x),
,(A,'\) = (, · A,'\) (A"

where A is omitted if it is the three sphere. Of course, AA differs from A by the ,
connected sum with the non-trivial bundle over T x 8 2 • One still obtains skein
relations between links in A, if one extends the concept of instanton homology for
links: For any 3-manifold QA with aQA = A x T, one can define invariants for
framed knots by "closing up" A \,\ x D by QA. It turns out that the following two
types of closures are needed.
T x D X BOa -+ A, we define A(K,) to be the '
DEFINITION 3.2. For a knot K, :

bundle obtained by (framed) surgezy on K,It Moreover, let 8 g ,6 be the oriented
surface of genus 9 with b boundary components, and let Uo, Ul denote the non- I

trivial BOa-bundles over 8 0 ,2 x T and 8 1 ,1 X T, respectively. (Uo, Ul are unique up
to automorphism). Let A[K] be obtained by gluing U1 to the complement of K, where
Floer: Instanton homology, surgery and knots

T is identified with the normal fibre. These definitions extend componentwise to
links A. Ifeis a link of two components in M and A is a b˜dle over its complement
which does not extend to M, then we define A(e) to be obtained by gluing Uo to
All gluings are performed according to the given framings, or more precisely
by using the parametrizations of the normal neighborhoods that we assume to be
given for every knot. We sometimes refer to a link of the type Definition 2 as
a "charged link in A", and consider A as a "discontinuous SOa-bundle over M". IT
K., A, and disjoint links, then we define

= I.(A(e))
I.(A, A) = I.(A[A]).

A general combination of these three constructions leads to homology groups

The reason for this definition is the fact that there exists a self-contained skein
theory for link homologies defined in this way.
Actually, I. does not depend on the framing of A, and it is entirely independent
of the parametrization of A near This will follow from the "excision" property
of Theorem 6. Note that the construction is natural with respect to "strict" link
cobordisms, which are cobordism of pairs of the form

(X,'\ x I) : (A,"x) (A', A').

= aX,
In fact, we can define for any closure U>. with au>.

This applies in particular to all cobordisms obtained by framed surgery on a link
disjoint from A, e.g. for the surgery triangle

A.oy U>. U>.
= A U"y>' U>.
AU>. U>. 1'A U>. U>.

surgery on a skein loop l' = 8D'Y. Now a redefinition of the top term of (3.1)

yields the following skein relations for I.:
110 Floor: Instanton homology, surgery and knots

4. Assume that "1 crosses K, and a component of a link A with boundary
conditions given by U>.., and let A+ K, be the connected sum along i and (K" A') =
"Y(K" A). Then we have an exact triangle

I.(A U>..+I' U>..)
I.(A(K,) U>.. U>..) I.(A(K,') U>.., U>..).

In particular,

I.(A, A + K,)
I.(A(",), ./ I.(A(",'), A')
A) ---+

+ K,))
(lb) ./ '\.
I.(A(K,)\A) I.(A(",')\A')

H, crosses two components AO' Al of A, then we have an exact triangle


I*(A(A)) I*(A('Y. A))

H "Y is a crossing of a knot, and jf the lifting of "1 extends to D.." then (A( K,)).., =
A["'/i] and we have an exact sequence


with i. = (X(i)).- Finally, for a crossing between two components of a link A we
have an exact triangle

I.(A, A/I)
(3) ./ '\.
I.(A\A) I*(A\i· A)

There is no skein relation for a crossing of a single charged component. In fact,
it is unnecessary to unknot a charged component since whenever A+ and A_ are
separated by a two-sphere in A, I.(A\A) = o. (There are no flat connections on the
non-trivial bundle over S2.) It therefore suffices to consider crossings of A+ with
A_ or nonna! or framed components.
Floor: Instanton homology, surgery and knots

5. Set A = H·(T 2 ) ˜ Z4. Then in the situations of Theorem 1, we have
exact triangles
I.(A\,.,/'1) ® A
'Y- ./ "'Yo
I.(A,'Y· K,)
I.(A,"') --+

I.(A, A/'1) ® A
1'- ./ "70
I.(A, A) [.(A,'Y·A)
[.(A\(A/'Y)) ® A
(3') ./ "
= 'Y - A
1.(A\ Ai , A6); A'
I. (A \ AI, Ao) --+
The proof of Theorem 4 proceeds by elementary topology, showing that the top
term of the surgery triangle (4) of '1 is diffeomorphic to the top tenns in the triangles
of Theorem 4. The proof of Theorem 5 involves the Kiinneth formula of I._ Let
U denote the non-trivial S02-bundle over the two-torus. Then I. is multiplicative
with respect to connected sums along U-bundles in the following sense:
6. A, B be closed connected bundles containing two-sided homologically
non-trivial thickened U-bundles UA and UB. Define
= A+u B = A\UA UUB\UB-
Here, the identification is uniquely determined by requiring that the boundary of
W= «-00,0] x (AIIB)) U ([0,00) x C)
has boundary
aw = U x T = a(U x D).
Then the cobordism
W := WU (U x D) : A IT B C

induces an isomorphism
w:• : I *A ®gU I • B I ·(A +u B ),

= gUA a ® b +a ® gUB a .
where gu(a ® b)
NaturaJity with respect to surgery cobordisms and the canonical excision isomor-
phisms qo, ql determine the functor I. in a similar way as ordinary homology is
(letermined by the Eilenberg axioms: Let 80:[3] denote the category of bundles over
A over closed oriented 3-manifolds such that either HI(A/G) = 0 or w(A)[S] = 0 on
sOlne oriented surface in A/G. Let the morphisms in 80;[3] be given by arbitrary
S'03-bundles over 4-dimensional cobordisms.
112 Floor: Instanton homology, surgery and knots

7. Let J. be a functor on 80:[3] such tbat tbe surgeJY triangle induces
an exact triangle and such that W induces an isomorphism. Then any functor
transformation I. -+ J. which is an isomorphism on [.(53 ) = 0 and [.(U X T)/gu =
Z is a functor equivalence.

Note that 0: is an isomorphism I.(A(K.)\e, A) if K. U e U Ahave no crossing.
Assume that 0: is an isomorphism whenever K. U have less than k crossings, and
consider a link JJ with k-crossings. Then for any It = AU the 5-lemma applied to
the exact triangle

I.(8 3 \e u,\ U,\) I.(S3\e 1 u,\ U,\)
u,\ U,\)

corresponding to a skein between and ,\ implies that if a is an equivalence on
1.(S3\e u,\ U,\), then it is an equivalence on I.(S3\e 1 u,\ U,\) where 1 is obtained
eby e+ e-.
from passing through an arbitrary number of components of ,\ and Since
I. as well as J. are trivial if e+ is separated from A\e+ by a sphere, we conclude
that a is an isomorphism on 1*(83 (K.)\e,,\) for all
Similarly, a is an isomorphism [.(S3( K,),,\) if it is an isomorphism on 1.(S3("I), AI)
for (,,1 , ,\ 1) obtained from K., ,\ by skein. Since (K., A) is skein related to the trivial
link, this proves by induction that a is a functor equivalence. Since all 3-manifolds
can be obtained as A = S3 ( K,) for some framed link K. in S3, this also proves that
a: is an equivalence on I*(A\e,'\) for any link e U A with e = 0 or e = (e+, e-) in
3-manifold A. 0
Let us apply Theorem 4 to some simple examples. First, note that if is the

trivial knot, A the standard link, and 31 the trefoil knot, then

are U-bundle over T. Hence

This isomorphism is also represented in the skein triangles
Floer: Instanton homology, surgery and knots

./ I.(T)
1*(31 )
More generally, we obtain

0./ ,,˜
1*(S3\'\k) 1.(S3\Ak+l)

This is essentially the relation that was used by Casson to determine the knot

= '\(˜PS3) _ A(˜p-l S3)

= ˜:(1),

where is the Alexander polynomial.


[A] Atiyah, M. F., New invariants of3 and 4 dimensional manifolds, in "Symp. on
the Mathematical Heritage of Hermann Weyl," University of North Carolina,
May 1987 (eds. R. Wells et al.).
[AM] Akbulut, S. and McCarthy, J., Casson's invariant for oriented homology
3-spheres - an exposition, Mathematical Notes, Princeton University Press.
[C] Cassen, A., An invariant for homology 3-spheres, Lectures at MSRI, Berkeley
reS] Chern, S. S. and Simons, J., Characteristic forms and geometric invariants,
Ann. Math. 99 (1974),48-69.
[D] Donaldson, S. K., The orientation of Yang-Mills moduli spaces and 4 . . manifold
topology, J. Diff. Geom. 26 (1987), 397-428.
[Fl] Floer, A., An instanton-invariant for 3-manifolds, Commun. Math. Phys.
118 (1988), 215-240.
114 Floer: Instanton homology, surgery and knots

[F2] , Instanton homology and Dehn surgery, Preprint, Berkeley (1989).
[FS1] Fintushel, R. and Stem, R. J., Pseudofree orbifolds, AIUl. Math. 122 (1985
[FS2] , Instanton homology of Seifert fibered homology 3-spheres,
[FS3] , Homotopy K3 surfaces containing E(2,a, 7), Preprint.
[FU] Freed, D. and Uhlenbeck, K. K., "Instantons and Four-Manifolds," Springer,
[G1l Goldman, W. M., The symplectic nature of the fundamental group of sur-
faces, Adv. in Math. 54 (1984), 200-225.
[G2] , Invariant functions on Lie groups and Hamiltonian flows
of surface group representations, MSRI Prep˜int (1985).
[H] Hempel, J., a--manifolds, Ann. of Math. Studies 86, Princeton University
Press, Princeton (1967). / .
[K] Kirby, R., A calculus for framed links in S3, Invent. Math. 45 (1978), 35-56.
[Ko] Kondrat'ev, V. A., Boundary value probleriM for elliptic equations in do-
mains with conical or angular points, Transact. Moscow Math. Soc. 16 (1967).
[M] Milnor, J., On the 3-dimensional Brieskorn manifolds M(p, q,r), knots,
groups, and 3-manifolds, Ann. of Math. Studies 84, Princeton University Press
(1975), 175-225.
[8] Smale, S., An infinite dimensional version of Sard's theorem, Amer. J. Math.
87 (1973), 213-221. '
[TI] Taubes, C. H., Self-dual Yang-Mills connections on non-self-duaI4-manifolds
J. DifF. Geom. 17 (1982),139-170.
[T2] , Gauge theory on asymptotically periodic 4-manifolds, J. Diff.
Geom. 25 (1987), 363-430.
[W] Witten, E., Supersymmetry and Morse theory, J. DifF. Geom. 17 (1982),
Instanton Homology
Lectures by

Department of Mathematics,University of California,Berkeley

Notes by

The Institute for Advanced Study, Princeton, and
Queens' College, Cambridge

(Editors Note.Floer gave three lectures at Durham; two on his work in Yang..
Mills theory and one on his work in symplectic geometry. As an addition to Floer's
010n contribution to these Proceedings, immediately above, Dieter K otschick kindly
(I,,qreed to write up these notes, which give an overview of Floer's three talks.)

'I'hese are notes of the lectures delivered by Andreas Floer at the LMS Symposium
ill Durham. After a general introduction, the three sections correspond precisely to
his three lectures. The reference for the first lecture is [Fl], and for the third [F2]
a.nd [F3]. For details of the second lecture see Floor's article in this volume.

'I'he unifying theme behind the topics discussed here is Morse theory on infinite
dilnensional manifolds. Recall that classical Morse theory on finite-dimensional
Illanifolds, as developed by M. Morse, R. Thorn, S. Smale and others, can be viewed
il.S deriving the homology of a manifold from a chain complex spanned by the critical

1)( lints of a Morse function, with boundary operator defined by the flow lines between
4'1'itical points. This was the approach taken by J. Milnor in his exposition of Smale's
on the structure of high-dimensional smooth manifolds [M].

'I'his point of view was described in the language of quantum field theory by E.
vVit.t.en [W]. To him, critical points are the groundstates of a theory, and flow lines

Supported by NSF Grant No. DMS-8610730.
FIoer: Instanton homology

between them represent tunneling by "instantons". These ideas form the back-
ground against which Floer developed the theories described in these lectures, in
which the manifold considered is infinite-dimensional. In the first of these theories
the manifold under consideration is the space of gauge equivalence classes of (irre-
ducible) connections on a bundle over a 3-manifold. The gradient lines are given,
literally, by the instantons of 4-dimensional Yang-Mills theory. In the second theory
the manifold considered is the loopspace of a symplectic manifold, and the gradient
lines or instantons are Gromov's pseudoholomorphic curves [G].

A common feature of these "Morse theories" is that there is no Palais-8male
condition satisfied by the Morse function. In each case the Hessian at a critical point
has infinitely many positive and negative eigenvalues, and a "Morse index" is defined
by a suitable renonnalisation. This is quite different from the classical infinite-
dimensional variational problems, such as the energy function on loops, for example.
In fact these instanton homology theories on infinite dimensional manifolds should
be considered as "middle dimensional" homology theories, as suggested by M. F.
Atiyah [A] and others.

Instanton Homology for 3-manifolds
Let M be an oriented 3-manifold. Considering finite-action instantons over M x R,
we want to construct invariants of M from moduli spaces. Let P be an 80(3)-
bundle over M, extended trivially to M R, and denote by Mg(M x R) the space
of gauge equivalence classes of connections on P M x R satisfying

J IlFII < 2
(1) 00

(2) F+*F = 0,
where (2) is the anti-self-duality equation with respect to the product metric (gx
canonical) on M x R.
The condition of finite energy (1) forces elements of Mg(M x R) to converge to
well-defined flat connections on the ends of the cylinder M x R. Denote by R(M)
the space of gauge equivalence classes of fiat connections on P -+ M. Then

Mg(M x R) M(A, B),
Floer: Instanton homology

where M(A, B) denotes the moduli space of finite-action instantons interpolating
between A and B.

We always assume that there are no nontrivial reducible flat connections on P --+- M.
This is the case in particular if M is a homology 3-sphere, and also if M has the
8 2 and W2(P) i= o. Moreover, just for the sake of exposition, we
homology of 8 1 X

assume that n(M) is discrete. (If it is not we can still carry out all the constructions
after replacing n( M) by the solution ˜pace of a suitable perturbation of the flatness
equation, cf. [FI].)

Now all the moduli spaces M(A,B) can be oriented in a coherent fashion by consid-
ax =
(˜ring extensions to a 4-manifold X with boundary M, and using Donaldson's
t.heory [D].

'rhe action of R by translation on M x R induces an action on each M(A, B),
and we denote by M(A, B) the reduced moduli space M(A, B)/R. Uhlenbeck's
compactness principle [U] has the following consequences:

(a.) O-dimensional components of M(A,B) are compact, and

(h) if a sequence in a I-dimensional component of M(A, B) has no convergent sub-
sequence, then some subsequence decomposes asymptotically into instantons in
M(A, C) and M(C, B), where C is a nontrivial flat connection.

With all these technicalities in place we can define instanton homology. Let R*(M)
1)(' the set of gauge equivalence classes of nontrivial flat connections on P --+- M.
,I'hen counting, with signs given by the orientations, the number of O-dimensional
nnnponents of M(A, B) gives a linear map on the free abelian group generated
the elements of R,*(M) (via (a) above). Moreover, (b) can be used to prove
iJ;; == o. Thus

I)(˜finition [Fl]: The instanton homology of M is

'1'11(' two cases we are interested in are when M is a homology 3-sphere, in which
(·H˜˜(' there is only one SO(3)-bundle over M (the trivial one), and when H*(M, Z) =
118 Floor: Instanton homology

H.(8 1 X 8 2 , Z) and P -+ M is the unique SO(3)-bundle with W2 =1= O. In each case
there is a grading on the instanton homology, which we have indicated by writing
* E ls, and in the second * E Z4.
I •. In the first case
The whole construction underlying the definition of instanton homology has certain
functorial properties. Most importantly, if X 4 is a cobordism between M3 and
N 3 , then counting points in zero-dimensional moduli spaces over X gives a chain
homomorphism 'R*(M) · l -+ 'R*(N) . Z. Applying this to M x R with a "twisted"
metric gives homomorphisms

which can be seen to be inverses of each other. This proves that I*(M) is, in fact,
independent of the metric g.

The definition of I*(M) given above does not bring out completely the variational
origin of the whole theory. There is a function (the "Chern-Simons functional")
on the space of connections on P -+ M 3 , such that R(M) is the critical set of the
function, and the M(A, B) parametrize gradient lines between the critical points A
and B. We will come back to this point of view in the third lecture, to explain the
analogy with constructions in symplectic geometry.

Instanton Homology and Dehn Surgery
The definition of instanton homology given in the previous lecture is enough to
allow one to calculate it completely in simple cases. Indeed, such calculations have
been carried out by several people for the Seifert fibered homology 3-spheres, with
the most complete results in [FS].

Leaving aside the study of examples, we want to describe a more systematic ap-
proach to such calculations by producing an "exact triangle" relating the instanton
homology of a homology 3-sphere M with that of M', obtained from M by Dehn
surgery on a knot. This is in the spirit of the usual exact sequences for standard
homology theories. It gives, in principle, a universal computational tool, because
every homology 3-sphere, or, more generally, every orientable 3-manifold, can be
obtained from S3 by a sequence of Dehn surgeries [L].

Let K, be a knot in M. We think of this as an embedded solid torus K,: 8 1 X D 2 M.

=M D 2 ) with boundary oK = T 2 , a 2-torus.
The knot complement is K " K(Sl X
FIoer: Instanton homology

k the "closure" of K obtained by gluing in 81 x D 2 interchanging
We denote by
= H*(SI 8 2 , Z).
parallels and meridians on T 2 • Then H*(K, Z) X

We want to do (+1)-Dehn surgery on 1C. To this end we take out 1C(8 1 x D2) and
glue it back in mapping the meridian to the diagonal in T2. This gives another
homology 3-sphere M'.

Now for any such surgery we have a surgery cobordism obtained as follows: take M x
[0,1] and attach a 2-handle D 2 XD2 to M x {I} using IC. The resulting 4-manifold X
has two boundary components; one is M, and the other is the surgered manifold. As
(˜xplained in the previous lecture, each cobordism gives a homomorphism between
K related
the instanton homologies of the ends. We apply this to the triple M, M',
K we use a Zs-graded double cover 1*(K) of the l4-
I>y surgery cobordisms. For
˜raded I*(K). The result is:

'rheorem (Floer): The triangle

;8 exact.
It. turns out that this can be proved by analyzing the effect of Dehn surgery on
tile representation space of the fundamental group. No detailed understanding of
illstantons on tubes is needed.

It'irst, to see that for two consecutive maps a, f3 in the triangle ker f3 ima holds,
= Hom(7rl (.), 80(3»)/80(3) for all the
looks at the representation spaces R(·)

Illanifolds involved. (For K the condition W2 i= 0 has to be imposed.) Now each of
til( ˜ manifolds M, M', K contains K :::> 8K = T 2 • Thus for each of these manifolds
= T2 /Z2.
c R(K) C R(8K) It can be shown that a suitable perturbation of
1\ ( .)
1,'( M') splits into R(M) and R(K) and that this implies ker f3 C im a.

'1 '0 prove iroa c ker (3, i.e. f3 a = 0, one uses a connected sum decomposition

l.r the 4-manifold defining f3 0 a, in the spirit of Donaldson's vanishing theorem.
example, one could use one of the arguments in [K], §6. The point is that this

Illanifoid splits off Cp2 as a connected summand, and that the relevant 80(3)-

I˜tltl(lle has =1= 0 on CP2. This means that any O-dimensional moduli space on
120 Floer: Instanton homology

f3 0 a =
the 4-manifold giving (3 a must be empty, which implies For a more

detailed account of Floer's proof of this theorem, we refer to his article in these

Finally, we note that the above theorem ties in nicely with Casson's work, described
in [AM]. Casson defined a numerical invariant A(M) for homology 3-spheres M by
counting, with suitable signs, the number of nontrivial representations of 7rl(M) in
SU(2). In fact, A(M) is one half this number, and Casson showed that A(M) is
always an integer. Casson gave the following formula for the change of his invariant
under Dehn surgery:
A(M') = A(M) + A'(K:),
where A'(IC) is a knot invariant extracted from the Alexander polynomial.

The relation of Casson's invariant with instanton homology is simply that the Euler
characteristic,X(I.(M» = E(-l)idim Ii(M), is 2A(M), as proved by Taubes [T],
[FI]. The mystery of the integrality of Anow becomes the mystery of the evenness
of X(I.). However, the above theorem, and the fact that going around the triangle
once gives a shift in the grading by -1, show

= X(I*(M» + X(I*(K»
= X(I.(M» + 2X(I*(I(».

Now working inductively from 8 3 to any given Musing [L], we find that X(I.(M» "
is always even, as X(I*(S3» = 0, and Dehn surgery changes the Euler characteristic .
= X(I*(K»).
by 2X(I.(K». Note that this proves A'(IC)

Symplectic Instanton Homologies
There are deep analogies between gauge theory on 3- and 4-dimensional manifolds
as used in the previous two lectures, and the theory of J-holomorphic curves in
symplectic manifolds, as initiated by Gromov [G] and developed in [F2,3]. We
summarize some of these analogies in the following table, in which X is a smooth
oriented 4-manifold, and V is a symplectic 2n-manifold with symplectic form w.

80(3) -. p X

1) A choice of conformal struc- A choice of almost complex
structure gives a a-operator
ture gives the ASD equation
Floor: Instanton homology

for Ulaps

C a Riemann surface.
for connections on P.
If C = 51 X R, the
If X = M X R, the ASD 8-
equation is
equation is

8A + Jau = o.
(M) au
&f+*MFA =0.
In fact, *MF˜M) is the grad- Similarly, J ˜˜ is the grad-
ient of the symplectic action
ient of the Chern-Simons
defined below.

This setup for M 3 X R leads Here we obtain an "index
cohomology" which turns
to instanton homology.
out to be H*(V, Z2)'

Itoughly speaking, the space of connections on M 3 on the left hand side is replaced
the right by the loop space LV of V. Point loops correspond to flat connections,

V to instantons on M 3
and holomorphic maps 51 X R R. The variational

I»roblem underlying the symplectic theory comes from the symplectic action function
defined as follows. Fix a loop Zo E LP, and set

a(z) = u*w,
8 1 X[o,l]

\vll(˜re u: V interpolates between z and zo.
51 [0,1]
X --+

l)oth of these theories the PDEs considered are elliptic, we have Fredholm equa-

and finite-dimensional trajectories. Moreover, the spaces of trajectories have
11itnilar compactness properties.

N()w let us look at some applications. Let H: 51 V --+- R be a time-dependent

Iialuiltonian function on P. Then the critical points of

a(z) +
122 FIoer: Instanton homology

are solutions of

a˜t) = JV'H(t,z(t»,
i.e. they are the fixed points of the exact symplectic diffeomorphism defined by H.

Assume '1r2(V) = o. Then it turns out that the cohomology of the complex generated
by the solutions of (*), with boundary operator defined by the flow lines of the
modified function, is isomorphic to H*(V, Z2). This gives the following

Theorem [F2]: Let V be a closed symplectic manifold with '1r2(V) = 0, and 4> an
exact diffeomorphism of V with nondegenerate fixed points. Then the number of
fixed points is at least the sum of the Z2-Betti numbers of V.

This result was previously conjectured by V. I. Arnold, and special cases were proved
by Conley and Zehnder, Hofer and others. A more general theorem, covering the
case of degenerate fixed points, was obtained in [F3].

In fact, the above theorem can be recovered from a more general relative version
for Lagrangian submanifolds LeV. This says that if 1l"2(V, L) = 0, then

IL n 4>(L)1 ˜ Ldim H i (L,Z2).

To deduce the fixed-point theorem above from this Lagrangian intersection result,
one considers V and the graph of a symplectomorphism as two Lagrangian subman-
ifolds of the product V X V. To prove the Lagrange intersection result FIoer used a
homology constructed with a boundary operator defined by "holomorphic strips";
i.e.pseudo- holomorphic maps from [0, 1] x R to V which map one boundary, {O} X R,
to L and the other, {I} R, to ¢(L).These can be regarded as the gradient lines of
a function on the space of paths in V beginning on L and ending on <p(L), and the
critical points of the function' are the constant paths , mapping to the intersection
points of L and ¢(L),

Among the various interesting questions raised by these results, we mention
the following :

(1) Is it possible to remove the assumption on '1r2 ? Evidence that this can be done in
some cases comes from Fortune's proof [Fo] of the above theorem for V = CpR,
Floer: Instanton homology

On the other hand the "relative theorem" is false for a nullhomologous small
circle on a surface (the Lagrangian condition is vacuous here). The methods
described in this lecture have been pushed through in the case that [w] is zero
in 1r2(V), or more generally if, on '1r2(V), w is a multiple of the first Chern class
of V (defined by the almost-complex structure).

(2) What can be said for symplectic diffeomorphisms which are not exact, i.e. not
deformations of the identity? Similarly, can one estimate IL nL' I by something
better than the intersection number of the homology classes of L, L' if L' i:
</J(L) ?

(3) Let 8 be a surface and V the representation space of 1r1 (8) in a compact
semisimple Lie group. The representations coming from a handlebody HI with
boundary S make up a Lagrangian submanifold L 1 C V. Thus the representa-
tion space of a 3-manifold M is identified via a Heegard splitting with L 1 nL 2 •
This is the approach that Casson took to define his invariant. It was sug-
gested by Atiyah [A] that this should extend to give a link between instanton
homology in the symplectic and 3-manifold cases. Can this be made rigorous?

(11) If, in answer to (2), instanton homology for Lagrangian intersections in the
non-exact case is a new invariant, are there exact sequences etc. which can be
used to calculate the homology groups?


lAM] S. Akbulut and J. McCarthy, Casson's Invariant for Oriented Homology 3-
spheres - an exposition. Princeton University Press (to appear).

11\) M. F. Atiyah, New Invariants for 3- and 4-Dimensional Manifolds. Proc.
Symp. Pure Math. 48 (1988) 285-299.

II >\ S. K. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold
topology. J. Differential Geometry 26 (1987) 397-428.

IIe' I ) A. Floer, An Instanton-Invariant for 3-Manifolds. Commun. Math. Phys.
118, 215-240 (1988).

IJe':˜) , Morse theory for Lagrangian intersections. J. Differential Geometry
28 (1988) 513-547.
124 Floer: Instanton homology

, Cuplength Estimates on Lagrangian Intersections. Commun. Pure
Applied Math. XLII 335-356 (1989).

, Witten's complex and infinite dimensional Morse theory. J. Differ-
ential Geometry 30 (1989) 207-221.

B. Fortune, A symplectic fixed point theorem for cpn. Invent. Math. 81
(1985) 29-46.

M. Furuta and B. Steer, Seifert Fibred Homology 3-spheres and the Yang-
Mills Equations on Riemann Surfaces with Marked Points. (preprint)

M. Gromov, Pseudobolomorpbic curves in symplectic manifolds.
[G] Invent.
Math. 82 (1985) 307-347.

bI =
[K] D. Kotschick, SO(3)-invariants for 4-manifolds with 1. (preprint)

W. B. R. Lickorish, A representation of orientable combinatoria13-manifolds.
Ann. of Math. 76 (1962) 531-538.

J. W. Milnor, Lectures on the h-Cobordism Theorem. Princeton University
Press, 1965.

C. H. Taubes, Casson '8 invariant and gauge theory. (preprint)

K. K. Uhlenbeck, Connections with LP Bounds on Curvature. Commun.
Math. Phys. 83, 31-42 (1982).

E. Witten, Supersymmetry and Morse theory. J. Differential Geometry 17
(1982) 661-692.
Invariants for Homology 3-Spheres


L. Introduction
In this paper we survey our work of the last few years concerning invariants for
homology 3-spheres. We shall pay special attention to the role of gauge theory, and
we shall try to place this work in a proper context. Homology 3-spheres exist in
a.bundance. Let us begin with a list of some examples and constructions.
i. The binary icosahedral group I is a subgroup of SU(2) of order 120. The
quotient of its action on S3 is the Poincare homology 3-sphere p3 , and so 1rl (P3) =
I. In fact, it is known that the only finite non-trivial group that can occur as the
fllndamental group of a homology 3-sphere is I, and it is still unknown if p3 is the
(˜nly homology 3-sphere E with fundamental group I. (But of course this depends
011 the 3-dimensional Poincare conjecture.)

it The Brieskorn homology 3-sphere E(p, q, r) is defined as {zf +z˜ +za = O} nss
where p, q, r are pairwise relatively prime. It is the link of the Brieskorn singularity
of type (p,q,r) [BJ. In fact E(2,3,5) = p3, and if lip + l/q + l/r < 1 then

If I (E(p, q, is infinite.
iii. The Seifert fibered homology 3-spheres E = E(al, ... , an) (see [NR]) are a
rather ubiquitous collection of homology 3-spheres. These 3-manifolds possess an
,<,f'-action with orbit space S2. If E =1= S3, then the S1-action has no fixed points and
Ilas finitely many exceptional orbits (multiple fibers) with pairwise relatively prime
)f'( lcrs al, ... , an. If n ˜ 3, then E -:F S3 and the orders classify E = B(aI, ... , an) up

diffeomorphism. If n :5 2, then E = S3. As our notation predicts, the Brieskorn

nphcre E(p, q, r) is Seifert fibered with 3 exceptional orbits of orders p, q, and r. In
fact, the Seifert fibered homology 3-spheres are the links of singularities in Brieskorn
t't Huplete intersections [NR]. Also one can show that for E = ˜(a1, . .. ,an) we have

= h- bo >.
=< xl, ... ,xn,hlh central; Xii = h-bi,i =
(1.1) 7rl(E) 1, ... ,n;x1·· .X n

II c're bo, b1 , ••• , bn are chosen so that

L ˜ )= 1
a(-bo +
i=l ai

where a = al · · · an. We say that E has Seifert invariants {bo;(aI' b1), . .. ,(an, bn )}.
(' I'lt(˜se are, of course, not unique.)

'1·llrtitl.lly supported by NSF Grant DMS8802412
'I·nl·t.ia,lly supported by NSF Grant DMS8703413
126 Fintushel and Stern: Invariants for homology 3-spheres

iv. Given a knot K in 8 3 one can perform a l/n , nEZ, Dehn surgery on K to
obtain a homology 3-sphere. The homology 3-spheres E(p, q, pqn ± 1) are obtained
by ±l/n surgery on the (p, q) torus knot; the homology 3-spheres E(p, q, r, s) with
qr - ps = ±1 are obtained by a ±1 surgery on the connected sum of the (q, r)
and (-p, s) torus knots. It is not known if every irreducible homology 3-sphere
can be obtained by a Dehn surgery on some knot in 8 3 • However, every homology
3-sphere can be obtained as an integral surgery on a link in S3. Recently, Gordon
and Luecke [GL] have shown that nontrivial Dehn surgery on a nontrivial knot
never yields S3. Furthermore, any homology 3-sphere obtained by Dehn surgery on
a knot is irreducible.
v. Given a knot K in S3 one can take the n-fold cyclic cover of 8 3 branched over
K, denoted K n . This is a homology 3-sphere when In?=l ˜(wi)1 = 1, where ˜(t)
is the Alexander polynomial of K normalized so that there are no negative powers
of t and has non-zero constant coefficients and w = eJ.;-i. The homology 3-sphere
E(p, q, r) is the r-fold cover of the (p, q) torus knot and every E(al, ... , an) is the
2-fold cover of S3 branched over a rational knot (see [BZ]). Not every homology
3-sphere is a cyclic branched cover [My]. However, every 3-manifold is an irregular '
branched cover of the figure eight knot [HLM].
Oriented homology 3-spheres E 1 and E 2 are said to be homology cobordant if there I

is an oriented 4-manifold W with aw = E 1 II -E 2 and such that the inclusions
˜i ˜ W induce isomorphisms in integral homology. Equivalently, ˜l and ˜2 are
homology cobordant provided (-E 1 )#E 2 bounds an acyclic 4-manifold. The set of
equivalence classes of oriented homology 3-spheres under this relation is denoted
e¥. With the operation of connected sum "#", ef is an abelian group, the
homology cobordism group of oriented homology 9-spheres. The additive inverse in I

ep is obtained by a reverse of orientation.
Until recently the only fact known concerning the group ep was the existence
e: --+ 12 defined as JL(˜) =
of the Kervaire-Milnor-Rochlin homomorphism p, :
sign(W )/8 (mod 2) where W is a parallelizable 4-manifold with boundary E. The
proof that p(E) is independent of the choice of W 4 and depends only on the class
of E in e¥ utilizes Rochlin's theorem, which states that the signature of a closed
spin (i. e. almost-parallelizable) 4-manifold is divisible by 16. Since ˜(2, 3, 5) is the
Es-singularity, JL(˜(2,3,5» = 1 and J.L is a surjection.
The group 0: has a distinguished history in the study of manifolds. Its structure
is closely related to the question of whether a topological n-manifold Mn, n ˜ 5, is
a polyhedron. In [GS] and [Mat] it is shown that Mn is a polyhedron if and only
if an obstruction 'TM E H5 (Mn; ker(p, : e˜ ˜ Z2») vanishes, and that if TM = 0
there are IH5(Mn; kerJl)I triangulations up to concordance. Furthermore, TM = 0
for all M if and only if there is a homology 3-sphere E with /-leE) = 1 and such
that E˜E bOllllds a smooth acyclic 4-manifold. At the time that these papers were
er = Z2, so that kerJ-t = O.
written (circa 1978) a reasonable conjecture was that
To date, the existence of a homology sphere with the above properties is unknown.
Fintushel and Stem: Invariants for homology 3-spheres

However, in §5 we shall utilize techniques from gauge theory to show that the group
fo)f is infinite and, in fact, infinitely generated.
Another importance of arises in 4-manifold theory. One can study 4-manifolds
by splitting them along embedded homology 3-spheres. In §9 we shall give an ex-
ample of this approach. In the other direction, one can attempt to construct 4-
Inanifolds by studying the bounding properties of homology 3-spheres, for example
e¥. IT a homology 3-sphere ˜ bounds the 4-manifold U with inter-
t.heir image in
section form Iu, and if -˜ bounds V with intersection form Iv, then X = UUV has
intersection form 1uE9Iv. Conversely, if the intersection form of a closed 4-manifold
˜\ decomposes as II E9 12 , then there is a homology 3-sphere ˜ in X splitting it into
t.wo 4-manifolds WI and W 2 with intersection forms II and 12 respectively [FT].
'fhis has been useful in constructing exotic 4-manifolds and group actions. For ex-
a.tople in [FSl] it is shown that E(3,5, 19) bounds a contractible manifold W 4 and
t.hat the double of W 4 Ut W 4 along the free involution t : :E ˜ ˜ contained in the
5i 1 -action on ˜ is S4 with a free involution 'T (obtained by interchanging the copies
W 4 ) that is not in any sense smoothly equivalent to the antipodal map. Thus

S"4 / T is a smooth homotopy Rp4 that is not s-cobordant to RP4. Other related
constructions are given in [FS2].
Prior to the 1980's there were few invariants for homology 3-spheres. One had
t.llc Kervaire-Milnor-Rochlin invariant discussed above and the TJ and p invariants
introduced by Atiyah-Patodi-Singer [APSl-3] and discussed in §6. Another, not so
w(˜ll-used, invariant is the Chern-Simons invariant discussed in §3. As we shall see,
it. is the Chern-Simons invariant that motivates many of the exciting new insights in
:˜'lnanifold topology. Of course the fundamental group also plays an important role
ill 3-manifold topology. However, its role was not underscored until the introduction
of Casson's invariant in 1983.

2. Casson's Invariant
It is natural to study 11"1 (˜) to obtain invariants of a homology 3-sphere ˜. One
way to do this is via representation spaces. Consider the compact space R(E)
of conjugacy classes of representations of 1I"1(E) into SU(2). "Generically" this
i:i a finite set of points which can be assigned orientations. Casson's invariant,
.\(E), is half the count (with signs) of those points corresponding to nontrivial
n'presentations. (Of course the construction of this invariant when 'R.(˜) is not
lillite is considerably more difficult. See [AM] for an exposition.) Casson showed
t.ha.t A(E) == JL(˜) (mod 2) and used this new invariant to settle an outstanding
prohlem in 3-manifold topology; namely, showing that if ˜ is a homotopy 3-sphere,
f hP11 1L(E) = O.
'The natural correspondence

= representations of 7r} (E) into SU(2) flat SU(2) connections over E
R(E) f-+

conjugation gauge equivalence
Fintushel and Stern: Invariants for homology 3-spheres

indicates that there should be a differential-geometric approach to the definition of
.A(E), which was discovered by Taubes [T2]. ("Gauge equivalence" means equiv-
alence under the action of the automorphism group of the (trivial) SU(2) bundle
supporting these connections.) First fix a Riemannian metric on˜. Then from this
point of view, one computes .A(E) by counting equivalence classes of nontrivial fiat
connections with sign given by the parity of the spectral flow of the elliptic operator:

given by (a, (3) ˜ (d: (3, dbll + *dbf3) as the connection b varies along a path from
the trivial connection (J to the given flat connection a, and db denotes the covariant
derivative corresponding to the connection b and db is its formal adjoint. The
spectral flow is the net number of negative eigenvalues of Db which become positive
as b varies along the path. (Since D(J has three zero eigenvalues, one must fix
a convention for dealing with them.) At a flat connection, a, the kernel of the
operator D a measures the dimension of the Zariski tangent space of'R.(E).
Let AE be the space of all SU(2) connections over E, and let BE be the quotient
of AE modulo gauge equivalence. An appropriate Sobolev norm on A E turns BE
into a Hilbert manifold (with a positive codimensional singular set which meets
'R.(E) only in the trivial connection). The tangent space to AE at a point a is
n˜ ®.6u(2), and the normal space to the orbit of a under gauge equivalence may be
identified with the solutions of the equation d:f3 = o. Thus, loosely speaking, we
may view the map sending a connection a to the Hodge star of its curvature *Fa as
a vector field on AE whose critical set consists of the flat connections. In the next
section we shall see that this is actually the gradient vector field of a function on
AE. The Hessian of this function at a is thus *da , and at a critical point it preserves
the equation d:(3 = o. It is easy to see that for a nontrivial flat connection a, the
kernel of *da on {d:f3 = O} may be identified with the kernel of D a • Since D a is
self-adjoint, we can add a compact perturbation term so that the corresponding
vector field has a zero-dimensional critical set. This explains the statement above
that R(E) is generically a finite set of points.

3. Chern-Simons Invariants
Let E be a homology 3-sphere. Each principal SU(2)-bundle P over E is trivial,
i.e. is isomorphic to ˜ x SU(2). As we have alluded in the last section, given a
trivialization, one can identify the space of connections AE of Sobolev type Lt with
the space Lt(n1(E) ® aU(2» of I-forms on E with values in the Lie algebra sU(2)
in such a way that the zero element of AE corresponds to the product connection
(J on E x SU(2). The gauge group of bundle automorphisms of P can be identified
with g = L1+t(E,SU(2)) acting on AE by the nonlinear transformation law
Fintushel and Stem: Invariants for homology 3-spheres 129

We shall assume that k+ 1 > 3/p so that 9 consists of continuous maps. The group
9 is not connected; in fact 1ro(9) = l given by the degree of 9 : ˜ ˜ SU(2). The
quotient BE = AE/Q can be considered as an infinite dimensional manifold except
near those connections a for which the isotropy group

= {g E Qlg(a) = a}

is larger than {±id}. Such connections are called reducible. For example, the trivial
eonnection 8 is reducible since its isotropy group consists of all constant maps
fJ : ˜ ˜ SU(2). Irreducible connections form an open dense set BE in BE- The set
()f flat connections is invariant under Q.

Given any connection a, we can take a path I : I = [0, 1] ˜ AE from the trivial
eonnection fJ to a. This path determines a connection Ai' in the trivial SU(2) bundle
()ver ˜ X I. Let
= 8\ lEX] Tr(FA-y A FAy).
'fhis definition is independent of the choice of path I because AE is contractible.
lIowever, the function CS(8,·) : AE ˜ R does depend on the trivialization of P.
If ()' is the trivial connection with respect to another trivialization, and " is a
path in AE from 8' to a, we can glue the connections A" and A-y' together over
˜˜ x {OJ and along ˜ x {I} via a gauge transformation to obtain a connection A in
nn SU(2)-bundle E over ˜ X 8 1 and

= 8 12 f Tr(FA A FA) = c2(E)
CS(8,a) - CS(8',a)

Q then CS(fJ,g(a» = deg(g) +
au integer. A similar argument shows that if 9 E
(˜S(B,a), so that CS(B,.) descends to a function CS : BE ˜ R/l, independent of
t.he choice of trivialization. It has an L2-gradient given by a 1-+ *Fa ; hence it is a
fUllction on 8(˜) whose critical set is n.(˜). At an a E n.(˜), the Hessian is *da •
'I'his R/l invariant can be regarded as a (mod l) charge of the connection A", for
,)'r{ FA.., 1\ FA-y) is the Chern-WeiI integrand. .
Chern-Simons invariants were overlooked by low dimensional topologists since it
was shown in [APSl-3] that the Pa invariants discussed in §6 (well-defined as real
1l1lInbers) were congruent to CS(a) mod I. The only utility of CS(a) appeared
t.o be that it determined the nonintegral part of Pa. As it turns out, it is the
(:hcrn-Simons functional that plays a central role in the modern understanding of
Il<Hnology 3-spheres. As a simple starting point, noting that n(˜) is compact, we
= min{CS(a)la E neE)} E [0,1).
\;V(˜ Hhall see in §5 that coupled with the techniques of [FS3] these invariants are
x t.relnely useful.
Fintushel and Stem: Invariants for homology 3-spheres

4. Representations of E(al , .•. , an)
A Seifert fibered homology sphere E = E( aI, ... ,an) admits a natural 5 1 -action
whose orbit space is B2. Orient E as the link of an algebraic singularity or equiva-
lently as a Seifert fibration with Seifert invariants {bo;(ai, bi), i = 1, ... , n} as in §1.
With this orientation E bounds the canonical resolution, a negative definite simply
connected smooth 4-manifold. Let W = W(al, ... ,a n ) denote the mapping cylin-
der of the orbit map. It is a 4-dimensional orbifold with boundary oW = E and
W has n singularities whose neighborhoods are cones on the lens spaces L(ai, bi)
(see [FS3]). H we orient W so that its boundary is -E its intersection form will
be positive definite. Let W o denote W with open cones around the singularities
removed. Then

='!rl(E)/ < h > =T(al'
?rl(WO) ,an )
= 1, ...,n;
=< xl, = 1 >.
,xnlxi i = 1,i ··X n

When n = 3 this is the usual triangle group and in general it is a genus zero Fuchsian
group. The element h E ?rl(E) is represented by a principal orbit of the 5 1 -action.
It is central, and for any representation a of '!rl(E) into 8U(2) we have a(h) = ±1.
Thus a gives rise to a representation of 'Kl(WO ) into 80(3). Conversely, any flat
80(3) bundle over Wo restricts to one over ˜, and there it lifts to a Hat SU(2)
bundle since ˜ is a homology sphere. Thus 8U(2)-representations of 'Trl(E) are in
one-to-one correspondence with SO(3)-representations of '!rl(WO).
Given it E n(E), let VQ denote the fiat real 3-plane bundle over Wo determined
by a. When V a is restricted over L(aj, bj) c awo it splits as La,; EB R where R
is a trivial real line bundle and La,i is the flat 2-plane bundle corresponding to
the representation 1rl(L(aj,bj) -. Zaj of weight Ii, where a(xi) is conjugate in
SU(2) to e1rilj/aj. (The presentation (1.1) shows that a(xi) is an ajth or 2ajth
root of unity.) The preferred generator of 1rl(L(aj, bi» corresponds to the deck
(z, w) (bj W )

of S3 where ( = e2frj/aj. Thus La,i is the quotient of 8 3 x R2 by this Zaj -action.
The bundle La,i extends over the cones cL(aj, bj) as (C 2 x R2) El1 R, an 80(3)-

V-bundle whose rotation number over the cone point is Ii (with respect to the
preferred generator). So Va extends to an SO(3) V-bundle over W. In [FS6] we
determine which (11 ,12 ,13 ) can arise for representations of 'Trl(WO), n = 3, thus
determining n(E) for all Brieskorn spheres E.
Here is another way to think about representations a : 'Trl (E) --+ SU(2). After
conjugating in SU(2), we may assume that a(xl) = e1rilt/at E S1 C SU(2). For
j = 1, ... , n let 8j be the conjugacy class of efri1j /aj. This is a 2-sphere in SU(2)
which contains a(xj). So a(XtX2) lies on the 2-sphere a(x1) · S2, and generally,
n(x1 ... Xj+l) lies on the 2-sphere a(xl ... Xj). 5i+1- Finally, since a(h) = ±1,
Fintushel and Stem: Invariants for homology 3-spheres

the presentation (1.1) implies that a(x1 ... x n ) = ±1. Thus a corresponds to a
Inechanicallinkage in SU(2) with ends at e7rilt/at and a(x1 ... x n ) = ±1 and with
arms corresponding to radii of the spheres a(x1' ... ' xi) -8i+1- (See Figure 1, where

Figure 1

Representations 7rl(E) --+ SU(2) thus correspond to choices of (11' ... ' In) such
t.hat a linkage from e7rilt/al to a(xl ... x n ) = ±1 exists; numerical criteria for this
are given in [FS6]. The connected component of any a E R.(E) is the correspond-
iug component in the configuration space of mechanical linkages modulo rotations
«>aving 8 1 invariant.
For example, consider a Brieskorn sphere E(at, a2, a3). A representation corre-
HI>onds to a linkage as in Figure 2.

Figure 2

This linkage is rigid modulo rotations leaving 51 fixed. Thus R( E(at , a2, a3»
consists of a finite number of isolated representations. More generally we have:
132 Fintushel and Stern: Invariants for homology 3-spheres

4.1 [FS6]. Let E = ˜(al, ... ,an). If a : ?r1(E) -+ SU(2) is a rep-
resentation with a(ai) =F ±1 for i = 1, ..., m, and a(ai) = ±1 for i = m + 1, ... , n,
nOt R(˜)
of a in the space
then the connected component is a closed manifold of
dimension 2m - 6.

In [FS6] we conjectured that any connected component of R(˜) has a Morse
function with critical points only of even indices. This was proved to be the case
in [KK1] using the mechanical linkages discussed above. Furthermore, these com-
ponents were also shown to be rational algebraic manifolds in [BO] and Kahler
manifolds in [FoS].
One can show further that, as the critical set for the Chern-Simons function,
n(˜) is nondegenerate_ That is, the Hessian *da of the Chern-Simons function is
nondegenerate normal to R(˜). Our next goal is to compute the Chern-Simons
invariant of a representation a : 1rl(E(al' _. _, an) -+ SU(2). Our technique will
work with the corresponding SO(3)-representation a'. H Ais an SU(2)-connection
which interpolates from the trivial SU(2)-connection to a, and if A' is an SO(3)-
connection interpolating from the trivial SO(3)-connection to a', then the Chern-
Simons invariant of a is obtained by integrating the Chern form of A, i. e. CS(a) =
fEXIC2(A) E R/I; whereas CS(a') = JEx/PI(A') is obtained by integrating the .
(SO(3)) Pontryagin form of A'. We have CS(a') E R/41 since PI(A') = 4c2 (A) if
A is a lift of A' (c.!. [HH]).
An SO(2) V-vector bundle Lover W is classified by the Euler class e E H 2 (WO) ˜
Z of its restriction over Wo = W - (neighborhood of singular points). Let L e denote
the V-bundle corresponding to the class e times a generator in H 2 (Wo ,I), and let
B be any connection on L e which is trivial near oW. Then the relative Pontryagin
number of L e is e: = fw PI(B) where a = al · _. an- Let A be the SO(3)-connection
on V a over W U (˜ X R+) ˜ W which is built from the flat (V-) connection a' over
W and from a connection A' over ˜ x R+ which interpolates from a' to the trivial
connection. The rotation numbers Ii of the representation a' depend on choices
of generators for the fundamental groups of the lens space links of W. These are
then determined by the Seifert invariants of E. We shall suppose that ˜ has Seifert
invariants {bO;(al,b1), ... ,(an,bn )} with bo even. (This can always be arranged.)
If one of the a˜s is even, assume it is aI, and arrange the Seifert invariants so that
the bi, i =F 1, are even. This specifies rotation numbers Ii for a'. Let e'= E?=l Ii :,
(mod 2a), and let L e be the corresponding SO(2) V-vector bundle. Then stabilize
to get an SO(3) V-vector bundle L e El7 R with connection A e which is trivial over
the end ˜ X R+.
Truncate W by removing neighborhoods of the singular points, leaving Wo , and
let 6Wo = awo \ ˜ (a disjoint lillion of lens spaces). Let E be the 50(3) vector
bundle over Wo carrying the connection A, i.e. E is the restriction of V Ot. Also let
Ee"o be the restriction of E e over W o, and let YO = TVo U6Wo Woo
Over Yo we can construct the SO(3) vector bundle E U E by gluing by the
Fintushel and Stem: Invariants for homology 3-spheres

identity over 6Wo, and we obtain the connection "A U A". The relative Pontryagin
number of E U E is fyo PI (A U A) = 0, since orientations get reversed. The number
e is chosen so that it is possible to form the bundle E U Ee,o over Yo, gluing by
a connection-preserving bundle isomorphism over SWo• In [FS6] it is shown that
w2(E U E) = w2(E U Ee,o). Then, since the connections A U Ae,o and A U A are
asymptotically trivial, we have (just as we would for a closed 4-manifold) fyo PI (A U
A) == fYoPI(A U Ae,o) (mod 4Z). Thus
f f == f
0= Pl(A) - Pl(Ae ) Pl(A) - e (mod 4Z).
Jwo Jwo Jw a

f =f +f =0- CS(a').
Pl(A) Pl(a') Pl(A')
Jwo Jwo Jr.XR+ .
Hence we have
4.2. Let E = E(al' ... ,a n ) and let a be a representation of 7r1(E) into
SU(2) with associated representation at into 80(3). Then eS(a') = - ea mod 4Z
= - :: mod Z where e == E:=11i (mod 2a).
and CS(a)
Another very interesting technique for computing Chern-Simons invariants is
discussed in [KK2]. It applies to homology spheres which are obtained from surgery
on a knot.


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