ñòð. 5 |

-+

X

I\.

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}

˜o

= T x {I} x {I}.

"'1

106 Floor: Instanton homology, surgery and knots

This class of knots is acted upon by the group

r = Z˜ ˜ 812 Z

(2.2)

= (Diff(T 2 )j Diff 1 (T2Â» H 1 (T 2 , Z2)

)4

of isotopy classes of equivariant diffeomorphisms of the bundle T 2 80a . In the

X

complementary picture,

= (toe-1,K).

8(t,K)

:)4

e = (˜ Z˜ with Z2 = {a, I}, we have

(a, b) E S12Z)4

For

= {(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

EXAMPLE

= 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

EXAMPLE

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

107

Floer: Instanton homology, surgery and knots

satisfying

(-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

(2.4)

where

=K 2

A" Ua (D T S03)

X X

=K

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

X X

The attaching maps and & are determined by

Q

= VJK,I

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) :=

THEOREM

(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

I.A,,˜I.A˜I.(K.A).

= kerX. C I.(A).

imY.

rrhe sequence property X. ˜ = 0 follows from a connected sum decomposition.

(This is simlar to the vanishing theorem for Donaldson's invariants.)

2.1.

LEMMA

XY=Z˜P

= 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

DEFINITION -t

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

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

To define '\/7, replace eÂ± by the sets [-1/2,1/2] x {O, I}.

Topologically, ,A is obtained by a Â±l-surgery on the circle, defined by the

crossing, and which does not change K.. We will therefore use the operative notation

Â· ,x),

,(A,'\) = (, Â· A,'\) (A"

Rj

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 â€¢ 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

109

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

A.

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

ein

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

eare

K., A, and disjoint links, then we define

= I.(A(e))

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

e.

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>.

./

(3.1)

= 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)

()f

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

THEOREM

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>..)

"

(1)

./

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

--+

In particular,

I.(A, A + K,)

"

(Ia)

I.(A(",), ./ I.(A(",'), A')

A) ---+

+ K,))

I.(A\(A

(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/'Y))

"

(lc)

./

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

(2)

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\iÂ· 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.

111

Floor: Instanton homology, surgery and knots

5. Set A = HÂ·(T 2 ) ˜ Z4. Then in the situations of Theorem 1, we have

THEOREM

exact triangles

I.(A\,.,/'1) Â® A

'Y- ./ "'Yo

(I')

I.(A,'YÂ· K,)

I.(A,"') --+

"Y.

I.(A, A/'1) Â® A

1'- ./ "70

(2')

I.(A, A) [.(A,'YÂ·A)

--+

7.

[.(A\(A/'Y)) Â® 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

THEOREM

non-trivial thickened U-bundles UA and UB. Define

= A+u B = A\UA UUB\UB-

C

Here, the identification is uniquely determined by requiring that the boundary of

W= Â«-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:â€¢ : I *A Â®gU I â€¢ B I Â·(A +u B ),

--+

= gUA a Â® b +a Â® gUB a .

where gu(a Â® 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

THEOREM

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.

PROOF:

eU,\

Assume that 0: is an isomorphism whenever K. U have less than k crossings, and

e,

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,\)

I.(S3\eo

e+

corresponding to a skein between and ,\ implies that if a is an equivalence on

e

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

e.

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

T

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

113

Floer: Instanton homology, surgery and knots

1*(53\'\1)

"

./ I.(T)

˜

1*(31 )

More generally, we obtain

1*(r)

0./ ,,˜

1*(S3\'\k) 1.(S3\Ak+l)

----+

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

invariant

= '\(˜PS3) _ A(˜p-l S3)

,\1(˜)

=XÂ«(I.(S3(˜))/g)

= ˜:(1),

˜,,(t)

where is the Alexander polynomial.

REFERENCES

[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

(1985).

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

),335-346.

[FS2] , Instanton homology of Seifert fibered homology 3-spheres,

Preprint.

[FS3] , Homotopy K3 surfaces containing E(2,a, 7), Preprint.

[FU] Freed, D. and Uhlenbeck, K. K., "Instantons and Four-Manifolds," Springer,

1984.

[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),

661-692.

Instanton Homology

ANDREAS FLOER

Lectures by

Department of Mathematics,University of California,Berkeley

DIETER KOTSCHICKl)

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].

\v()rk

'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.

I)

FIoer: Instanton homology

116

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

X

of gauge equivalence classes of connections on P M x R satisfying

-+

J IlFII < 2

(1) 00

MxR

(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

U

=

Mg(M x R) M(A, B),

(A,B)E'R(M)2

117

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

a9

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

I)y

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

(Â·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 â€¢. 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) Â· 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.

L-7

=M D 2 ) with boundary oK = T 2 , a 2-torus.

The knot complement is K " K(Sl X

119

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 â€¢ 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.

c

It'irst, to see that for two consecutive maps a, f3 in the triangle ker f3 ima holds,

= Hom(7rl (.), 80(3Â»)/80(3) for all the

looks at the representation spaces R(Â·)

one

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 â€¢ 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

0

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], Â§6. The point is that this

Jt()r

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

-I

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

W2

120 Floer: Instanton homology

o.

f3 0 a =

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

0

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

Proceedings.

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.(MÂ» = 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*(MÂ» + X(I*(KÂ»

X(I.(M'Â»

= X(I.(MÂ» + 2X(I*(I(Â».

Now working inductively from 8 3 to any given Musing [L], we find that X(I.(MÂ» "

is always even, as X(I*(S3Â» = 0, and Dehn surgery changes the Euler characteristic .

= X(I*(KÂ»).

by 2X(I.(KÂ». 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.

(˜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

121

Floor: Instanton homology

for Ulaps

u:C˜V,

F+*F=O,

C a Riemann surface.

for connections on P.

If C = 51 X R, the

If X = M X R, the ASD 8-

2)

equation is

equation is

8A + Jau = o.

(M) au

&f+*MFA =0.

at

as

In fact, *MF˜M) is the grad- Similarly, J ˜˜ is the grad-

3)

ient of the symplectic action

ient of the Chern-Simons

defined below.

function.

This setup for M 3 X R leads Here we obtain an "index

4)

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,

011

V to instantons on M 3

and holomorphic maps 51 X R R. The variational

X

--t

IÂ»roblem underlying the symplectic theory comes from the symplectic action function

defined as follows. Fix a loop Zo E LP, and set

fI.

J

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-

III

and finite-dimensional trajectories. Moreover, the spaces of trajectories have

tions

11itnilar compactness properties.

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

X

Iialuiltonian function on P. Then the critical points of

JH(t,z(t))dt

a(z) +

Sl

122 FIoer: Instanton homology

are solutions of

a˜t) = JV'H(t,z(tÂ»,

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).

i

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 Â¢(L).These can be regarded as the gradient lines of

X

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 Â¢(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,

123

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 â€¢

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?

Itcferences

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

[F3]

Applied Math. XLII 335-356 (1989).

, Witten's complex and infinite dimensional Morse theory. J. Differ-

[F4]

ential Geometry 30 (1989) 207-221.

B. Fortune, A symplectic fixed point theorem for cpn. Invent. Math. 81

[Fo]

(1985) 29-46.

M. Furuta and B. Steer, Seifert Fibred Homology 3-spheres and the Yang-

[FS]

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.

[L]

Ann. of Math. 76 (1962) 531-538.

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

[M]

Press, 1965.

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

[T]

K. K. Uhlenbeck, Connections with LP Bounds on Curvature. Commun.

[U]

Math. Phys. 83, 31-42 (1982).

E. Witten, Supersymmetry and Morse theory. J. Differential Geometry 17

[W]

(1982) 661-692.

Invariants for Homology 3-Spheres

RONALD FINTUSHEL l AND RONALD STERN 2

J.

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

rÂ»

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

C

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

t.()

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;x1Â·Â· .X n

II c're bo, b1 , â€¢â€¢â€¢ , bn are chosen so that

b.

n

L ˜ )= 1

a(-bo +

i=l ai

where a = al Â· Â· Â· an. We say that E has Seifert invariants {bo;(aI' b1), . .. ,(an, bn )}.

(' I'lt(˜se are, of course, not unique.)

'1Â·llrtitl.lly supported by NSF Grant DMS8802412

'IÂ·nlÂ·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 Â± 1) are obtained

by Â±l/n surgery on the (p, q) torus knot; the homology 3-spheres E(p, q, r, s) with

qr - ps = Â±1 are obtained by a Â±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 â€¢ 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

eÂ¥. 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, :

4

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 eÂ¥ 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(˜(2,3,5Â» = 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˜ ˜ Z2Â») 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.

127

Fintushel and Stem: Invariants for homology 3-spheres

However, in Â§5 we shall utilize techniques from gauge theory to show that the group

fo)f is infinite and, in fact, infinitely generated.

ef

Another importance of arises in 4-manifold theory. One can study 4-manifolds

by splitting them along embedded homology 3-spheres. In Â§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

eÂ¥. 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

()f

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 Â§6. Another, not so

w(˜ll-used, invariant is the Chern-Simons invariant discussed in Â§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

128

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˜ Â®.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 â€¢ 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) Â® aU(2Â» 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}

Qa

is larger than {Â±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).

f

CS(fJ,a)

1r

'fhis definition is independent of the choice of path I because AE is contractible.

lIowever, the function CS(8,Â·) : 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)

JEXSl

1r

Q then CS(fJ,g(aÂ» = 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 â€¢

'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 Â§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

flc'fine

= min{CS(a)la E neE)} E [0,1).

T(E)

\;V(˜ Hhall see in Â§5 that coupled with the techniques of [FS3] these invariants are

x t.relnely useful.

f'

Fintushel and Stem: Invariants for homology 3-spheres

130

4. Representations of E(al , .â€¢. , 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 Â§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 Â·Â·X n

XlÂ·

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) = Â±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, biÂ» corresponds to the deck

transformation

Â«(z,

(z, w) (bj W )

i-+

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)-

IGj

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) Â· S2, and generally,

n(x1 ... Xj+l) lies on the 2-sphere a(xl ... Xj). 5i+1- Finally, since a(h) = Â±1,

131

Fintushel and Stem: Invariants for homology 3-spheres

the presentation (1.1) implies that a(x1 ... x n ) = Â±1. Thus a corresponds to a

Inechanicallinkage in SU(2) with ends at e7rilt/at and a(x1 ... x n ) = Â±1 and with

arms corresponding to radii of the spheres a(x1' ... ' xi) -8i+1- (See Figure 1, where

5.)

n=

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 ) = Â±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

Â«>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, a3Â»

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-

PROPOSITION

resentation with a(ai) =F Â±1 for i = 1, ..., m, and a(ai) = Â±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 Â· _. 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

133

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 SWoâ€¢ 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

2

f f == f

0= Pl(A) - Pl(Ae ) Pl(A) - e (mod 4Z).

Jwo Jwo Jw a

o

But

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

THEOREM

2

SU(2) with associated representation at into 80(3). Then eS(a') = - ea mod 4Z

:j

= - :: 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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