. 4
( 10)


admits a positive involution, denoted by ,-, thus:
˜ =(p)-l˜*(p). This is the classical theorem o˜ Rosati
[10J. Taken together with the complete reducibility
theorem of Poincare' [8] ,[9], we obtain:
Johnson: Flat algebraic manifolds

and let
Theorem 2.1 be a subfield oT
Let K
(A,t) be an algebraic Riemann matrix over K.
(A,t) is isomorphic in to a product


where (Ai,t i ) (15i5m) are simple Riemann matrices over
K, and the isomorphism types (Ai,t i ) and multiplicities
ei are unique up to order. Moreover, EndK(A,t) ,the
algebra of K-endomorphisms OT (A,t), is a product


where Di EndK(Ai,t i ) is a positively involuted
division algebra over K. In particular, EndK(A,t) is a
positively involuted semisimple K-algebra.

We construct algebraic Riemann matrices in two ways

eM-algebras :
I :
By a CM-a˜.n we mean a quadruple (A,E,r,a) where
E is a totally real algebraic number field of
finite degree over Q;
(A,r) is a "finite dimensional positively'
involuted E-algebra such that rE = 1 E ;
aEA has the property that a 2 is a totally
E, in particular,
negative element of ·
= E(a) is a purely imaginary extension OT
reF) = F and r restricts to the nontrivial
element of Gal(F/E) .

In a eM-algebra (A,E,r,a), we do not assume that E is
centre A. With a eM-algebra (A,E,r,a), we

a,asoc i ate the t A@ER --+ A@ER
canonical complex structure
ax ® .1
p;iven by t(x®l)
Johnson: Flat algebraic manifolds

Proposition 2.2: Let (A,E,'T,a) be a CM-algebra with
canonical complex structure t. Then (A,t) is an
algebraic Riemann matrix over E.
ProoT It is easy to check that Ax A E
{J --+
de-fined by
P(x,y) TrE(ay'T(x) - &X'T(Y))
is a Riemann Torm Tor (A,t) IJ

For algebras of type III or IV, the -following
statement is tautologous; Tor algebras oT type II, it
is a restatement of an observation OT Shimura ([12],
p.153, Proposition 2 ).

Proposition 2.3: Let (A,'T) be a positively involuted
-finite dimensional division algebra OT type II, III
then there exists a subfield E C A, and
or IV over Q;
( e A such that (A,E,'T,() eM-algebra.
is a

II: The doubling construction:
Let K be a subTield oT R.To each Tinite dimensional K-
vector space V , we associate a Riemann matrix D(V) ,
the ˜˜kof V, over Ie thus; D(V) = (V$V,t@l) where
V$V ˜ V$V is the K-linear map = (-x2,x1).
Observe that D(V) is algebraic since

is a Riemann Torm Tor D(V), where (Ti)lSisn is a basis
Tor the K-dual oT V. The doubling construction can be
obtained from the CM-algebra construct ion as Tollows;
obtained -from the CM-
is the
D(K) Riemann matrix
Kn so that D(V) - D(K)'?
algebra K(v'-=1) • However, V -
Suppose that E, F are rings such that E C F C R. There
is an extension of scalars -functor 9,A ,
˜A,E -+ F
Johnson: Flat algebraic manifolds

g˜(A,t) (A˜EF ,t) (A,t) ˜EF)
= on making the
F˜ER ˜
identification R. When E is also a field ,
F E, we write
and is a finite algebraic extension OT

is a field imbedding
(J' :

F -+
(T :

E closure oT E.
where is the algebraic Suppose that,
in addi tion, is also real over E, in the sense that Tor
u(F) R. We may identiTY
E ˜E'
each C

˜E' ˜
(TR H .. iT
where, Tor each Thus, t is a

complex structure on A, there exists a natural complex

so that we obtain a functor
(A, (Tt)
; ˜F/E(A,t)
u E ˜E

constructions l!:˜,
The '3l. /
preserve algebraicity. Let
K be a Tinite algebraic extension OT Q. Applying the
construction to a Riemann matrix (A,t)
C!R,K/Q SAl('
enables us to construct Riemann matrix over Q. We
denote by A(t) the complex vector space obtained from
A®KR by means oT t, and regard A as being imbedded
in A(t) by means oT x x@l. A Riemann matrix (A,t)

over l clearly every
a complex torus A(t)/A
complex torus may be so described. An is a
abelian variety
complex torus which is algebraic, that is, which admits a
holomorphic imbedding into some Pn(C).. The Tollowing
r-esult Trom the classical theory OT theta-Tunctions
justifies the usage Halgebraic Riemann matrixH•
Johnson: Flat algebraic manifolds

Proposition 2.4: Let (A,t) be a Riemann matrix over Z.
Then the yollowing conditions are equivalent
(i) A(t)/A is an abelian variety
(ii) (A,t) is algebraic Riemann matrix over

is an algebraic Riemann matrix over Q.

Conversely, if (A,t) is an algebraic Riemann matrix
over Q, and , A 2 C A are Tree abelian subgroups
of maximal rank, then A(t)/A 1 is an abelian variety
isogenous to A(t)/A 2 , so that extension oT scalars
Z to Q yields a bijection

Q} .

{ isogeny classes of { Q-isomorphism classes of
algebraic Riemann matrices over
abelian varieties

13 Representations of finite groups
Throughout this section we fix a finite group .; let K
be subfield of The group algebra is
semisimple, by Maschke's Theorem, and has a positive
involution given by ;


By (1.1), there is an isomorphism involuted K-

(K [tt] ,r)

where each D i is a finite dimensional division algebra
over K, admitting the positive involution rie Let V be
K [˜]
a simple (left) module; for some unique (lS
iSm), V is isomorphic to a simple left ideal oT
Mni(D i ), and EndK[fI](V) Di - We are principally
= R.
K=Q; K
interested in the two cases
Johnson: Flat algebraic manifolds

= R,
When K each finite dimensional division a.lgebra
V ==
is isomorphic to R, H or C; an isotypic module
W(e) is said to be of type R , H or C according to the
type oT the division algebra EndR[.](\N). Similarly
V == Wee) is ascribed
when K = Q, an isotypic module
the type (I, II, IV)
III or oT the division algebra.
EndR[.](W). A complexstructure Tor the K[fI]-module V, is an
element t E EndR[.](V@KR) such that -1.

Proposition 3.1 : Let V be a Tinitely generated R[.]-
module: the following conditions on V are equiva.lent ;
(i) V admits a complex structure
(ii) each isotypic component of V admits a complex
structure ;
(iii) each simple summand oT type R occurs with even
multiplicity in V.

=> (ii)
Proof (i) Write V in its isotypic
decomposition V W 1 ED ... EDWm where each Wi is
= {O} i#:j.
isotypic, with HomR[.](Wi,W j ) -for
(˜orresponding to the Wedderburn decomposition oT R[.]
a product of simple two sided idea.ls, we may write

identity element of as a sum of central
t,he 1

+ + . . + en m ˜ n,
idempo!; elrtse l e2 with such that
i"# j , and
f˜ ˜ ˜
ie j 0 Tor Tor 1 i m,

w. A(j)

= =
where A(j) {v E V : ej.v 0 }. Let t be a complex
Ht˜ructure on V. Since tis R[.]-linear, it commutes
with e j.
each Henc\e t (A(j)) A(j) Tor each and
by (*) above, t(W i ) C Wi for each i. That is, each

iHotypic component OT V admits a complex structure.
LJ.E.D. (i) (ii).
Johnson: Flat algebraic manifolds

== V(n) be an isotypic R[.]-module
(ii) => (iii): Let W
R, A
with EndR[.](V) - so that EndR[.](W) - Mn(R).
complex structure t on induces a complex structure
t. on the real vector space EndR[.](W) thus; t*(f)
taL It Tollows that dimR(EndR[.](W)) is even,
Q.E.D. ˜
and hence n is even. (ii) (iii) .

- VR ED Vc ED VH where
(iii) => (i) V

Val E9 V ˜ E9 .ED V˜r
VR 1 2

.ED U˜
UbI U b2
Vc - 1 E9 2 E9

VH .$
1 E9 2 ED t

and where Vi' Uj Wk are all simple R[.]-modules, with

= =H.
EndR[.](V i ) EndR[.](U j ) C, and EndR[.](W k )
each U. admits a
Clearly, complex structure. Moreover,
C H, Wk
since each admits complex structure.
However, since complex
by taking
is even,
each ai
structures on admits a
doubles, each V
V is now a direct sum OT submodules each of
which admits a complex structure, and so admits a
=> (iii): D
complex structure. Q.E.D. (ii)

Recall that if W is a finitely generated Q[.]-module,
W contains a .-invariant I-lattice L, and that any two
such are commensurable. We say that a complex
structure t on W is a <!P-structure Tor W when Tor some (and
hence Tor any) .-invariant l-lattice, W(t)/L is an
abelian variety. We wish to give an analogue of (3.1)
Tor the existence of '-structures ; first we note
Johnson: Flat algebraic manifolds

V Q [.] -module; i of
Proposition 3.2: Let be a simple
I I , III IV V admits a ,-
V is oT type or then
structure ; iT is oT type then admits a

Proof: The second statement Tollows easi ly -from the
doubl ing construction OT §2. To prove the first, put D
= EndQ[.](V). (2.2), subTield E o-f
By there exists a
t.he division algebra D, and an element D such that
(D,E,T,() is a CM-algebra, where is the canonical

D (D,E,T,e)
involution on inherited Trom Q[.]. Then
has the canonicalcomplexstructure t : D@ER -+ D@ER defined thus;
= ex ® 1. . (D,t) is an algebraic Riemann
-I-e 2
nlgebraic over E. However, V is a vector space over D,
of dimension m, say, and also has a canonical complex
Ht.. ructure T:V®ER -+ V@ER defined by the same -formula
= (y ® .1
t , namely Clearly there is
== (O,t)m, so
nn isomorphism of Riemann matrices (V,T)
that (V,T) is also algebraic. Since E D, and the D-
nction on V commutes with that of . , it Tollows easily
t.hat T commutes with the .-action on V; that is, T is
e,,-structure for V. []

'I'heorem 3.3: Let W be a finitely generated
module. Then the following conditions on Ware
(˜qu i val ent ;
(i) each Q[.]-simple summand OT type I has even
multiplicity in W;
( i i) W admits a c:P-structure
(iii) W admits a complex structure.

= V o) EB where, for T = I,
PI-oof: Write W E9

III,IV, VeT) a direct sum of simple Q[.]-
Johnson: Flat algebraic manifolds

modules OT type T.
(i) => (ii) : By (3.2), V(ll)ED V(ll) ED V(IV) admits a ,-
structure; by hypothesis, each simple summand oT V(I)

has even multiplicity, so that, again by (3.2), V(I)

admits a '-structure. Hence admits a '-structure.
=> Obvious .
(ii) (iii):

˜ == V˜el) V˜m)
Let W
(i i i) (i): be the il80typic
@ @

decomposition oT with Vi simple le£t ideal oT
W, a
Mn1 (D t ) x . .
Mni(D i ), where Q[.] - is the
x Mnm(Dm )
Wedderburn decomposition oT On writing the
identity element oT Q[.] as a sum oT primitive
central idempotents 1
see that

o #: j
if i
1V . j
= {O} when
Hom R [fI] (V i ®QR, Vj®QR) ;/=
Hence i On
extending scalars, we obtain

{ #=
if i j
= =j
1 V .@R iT i

Since fj®l is a sum oT primitive central idempotents in
R[.] , no R[.]-simple summand of Vj is isomorphic to any
R[.]-simple summand Vi' and hence
Hom R [fI] (V i ®QR, Vj®QR) = 0 Tor i j. Now suppose tha.t
Vi is OT type I; that is, D i is a totally real
algebraic number field with d i = dimQ(D i ). Then

are isomorphically distinct simple R[.] -modules Tor
we may write
R x .x
Johnson: Flat algebraic manifolds 87

UtED U2 ED • • ED U˜
a.nd identiTY
Mn . (R) Mn . (R) where U r is the
of x
1 1

Hubspace OT consisting oT
the copy oT Mn(R)
matrices concentrated in the "first column. Clearly
(,:&ch Ur is R[.]-simple, and Tor r s, Ur and Us
correspond to distinct simple "factors ofR [.], and so
are not R[.]-isomorphic.
We assume that W admits a complex structure; that is,
W®QR admits a complex structure. Since
R[.](Vi˜QR, Vj®QR) #=
{O} "for i j, it Tallows
t.hat the multiplicity oT each U r in is the same
u.s the multiplicity o"f Vi in W ,namely ei. Now each
R-summand OT W˜QR,
Ur is a type so that, by (3.1),
(˜i is even when Vii s a type I summand. []

§˜ Flat Klhler manifolds and Tlat algebraic manifolds
Ilccall that a closed flat Riemannian manifold X is
= G\E(n)/O(n) where
i Hometric to one of the form X
is the group oT Euclidean motions aT Rn ,
I (0) O(n) is
.. he isotropy group o"f the origin, and G '1"1 (X) is a
.. orsion free discrete cocompact subgroup oT E(n).
Moreover, there is a natural exact sequence
0--. A 1
˜G--.. ---+
in which -t, the holonomy group OT X, is "finite, and A
Zn is the trn.nslation subgroup oT G. Conversely,
K i ven any such extens ion, G imbeds as a.
torsion (ree

= G\E(n)/O(n)
cI iscrete cocompact subgroup oT E(n), and XG
a compact flat Riemannian manifold. For these
cI(˜tai Is, we re"fer the reader to [2] [4], [13J.
K˜hler a. closed
Since the condition is purely local,
Riemannian maniTold oT real dimension 2n which
,ulmits a compatible complex structure is automatically
I(;˜hler. We denote by ˜flat the class OT groups which
occur as the fundamental group of some compact flat
Johnson: Flat algebraic manifolds

˜-flat the
Klthler mani-fold, and subclass OT S"flat
consisting -fundamental groups o-f smooth flat complex
projective varieties. We may represent a compact Tlat
= G\H(n)/U(n) where
K£hler mani-fold X in the -form X
= C ><l D(n) is the
H(n) group o-f "Hermitian"
isometries o-f CD, and G is a torsion -free discrete
c:PTlat ,
cocompact subgroup o-f H(n). The classes S'flat
may be characterised in the -following way, which "for
S'flat is already known (see [6] ).

Theorem 4.1 '-flat (resp. S-flat) consists precisely
of those torsion -free groups G which occur in an

o --+ Z2n --+ G --+. --+ 1

in which the operator homomorphism GL 2n (Z)
p •
admits a '-structure (resp. complex structure), and in
which. is Tinite. Given any such pair (S,p), there is
a smooth -flat complex projective variety (reap.
compact, complex flat KKhler manifold) X whose
-fundamental group is G, and whose holonomy
representation is p.

Proo-f We give the proof -for smooth projective
prooT Tor K˜hler mani-folds
varieties: the is slightly
simpler [6].
Let X be a smooth -flat projective variety oT complex
X oT X is
dimension n; the universal covering
C n . Considering X as a
holomorphically equivalent to
Hermitian manifold, 71(X) = G occurs in an extension

o --+A --+G--+c1 --. 1

in which • is -finite; A, the kernel of the holonomy
Johnson: Flat algebraic manifolds

r-epresentation of "'1 (X), is isomorphic to Z2n and acts
C n as a group of translations. Let X be the

finite covering of X with A. Then CnlA is
a complex torus which, being a finite holomorphic
covering of a smooth projective variety, is algebraic.
'rhe operator homomorphism p GL 2n (Z) extends to

t.he holonomy representat ion GL 2n (R) OT the
p ---+

Riemannian manifold X. However, the Hermitian metric
is preserved by the holonomy representation so

Im(p) p
t.hat is contained in U(n). Thus admits a
complex structure, which, since CnlA is algebraic, is
also a
0 ˜ Z2n ˜ G ˜ 4) ˜ 1
Conversely, if is a torsion free
(˜xtension in which the operator homomorphism p admits a
˜-structure and where ˜
t Z2n®ZR Z2n®ZR, is

finite, let i denote the inclusion, i Z2n C

Z2n®ZR We have a corresponding inclusion i GL 2n (Z)
GLR(V), and induced maps i*: H*(<<t,y)
C ---+

˜2n y)
where (reap. denotes the Z[«IJ (reap. R[.J)
p ip).
module in which 4) acts by (resp. Up to
congruence, G is classified by the pair (i,c), where c
H2(˜,˜2n) is the characteristic class oT the
extension defining G. Let L(G) be the extension
o --+ L (G) ---+ ˜ 1
---+ ---+

classified by the pair i*(c». Then there is a
morphism of exact sequences

Z2n 4) 1
0 G
---+ --+ --+ ˜

·n II
• 1
0 Y L(G)
---+ --+
--+ ˜

which cocompact
G is imbedded as a

Hubgroup of the Lie group L(G). The identity component
of connected
is and has finitely many

•• is
components, of Since
1,h(˜ (˜1
indexed by cmcnts
Johnson: Flat algebraic manifolds

"finite and V is divisible, i*(c) 0, so that L(G)
= ˜
splits as a semidirect product L(G) V ••
Write G\L(G)/˜.
= Then X is a compact "flat Riemannian

X = Z2n \V(t) which,
manifold having a finite covering
since p admits a c:P-structure, is a complex algebraic
torus, and on which • acts "freely by complex analytic
di"ffeomorphisms to give Thus is also a smooth
X. X
flat complex projective variety ([11] pp. 395-398). By
= G,
construction, and the holonomy
representation of is p.

Thus we obtain

Theorem 4.2 Let be a smooth compact flat
Riemannian manifold; i"f admits the structure oT a.
flat KKhler manifold, then also admits the
structure of a flat smooth complex projective variety.

= (0 ˜ Z2n ˜ ˜l(X) ˜ • ˜ 1) be the exact
Proof: Let S
sequence defining the fundamental group of with
GL 2n (I).
holonomy representation Since X
p .. --+
admits a flat structure, admits a complex
structure. By admits a 9-structure, and, by
(3.3), p
(4.1), X admits the structure of a smooth flat complex
projective variety.

Corollary 4.3 are
The classes Sflat and c:P"flat
identical .

By the Bertini-Lefschetz Theorem , we obtain

Corollary the
4.4 element is
Every of STlat
"fundamental complex
group compact
o"f smooth
algebraic surface.
Johnson: Flat algebraic manifolds

[1]: A.A.Albert; Involutorial simple algebras and
real Riemann matrices: Ann. OT Math.
36 (1935) 886 - 964 .

[2]: L.Bieberbach; Uber die Bewegungsgruppen der
Euklidischen Raume I : Math. Ann.
70 (1911) 297 - 336.

[3]: N.C. Carr; Ph.D. Thesis, Univ.oT London,
(in preparation ).

[4]: L.S. Charlap; Compact flat Riemannian
manifolds I : Ann. of Math. 81 (1965) 15-30.

[5]: F.E.A. Johnson and E.G. Rees ; On the
Tundamental group of a complex algebraic
manifold: Bull. L.M.S. 19 (1987) 463 - 466.

[6]: F.E.A. Johnson and E.G. Rees ; Kahler groups and
rigidity phenomena : (to appear) .

[7]: F.E.A. Johnson and E.G. Rees ; The fundamental
groups of algebraic varieties : (to appear in
Proceedings of the International Conference on
Algebraic Topology, Poznan, 1989).

[8J: S. Lang; Introduction to algebraic and abelian
functions : Graduate Texts in Mathematics ,
vol.89: Springer-Verlag, 1982.

[9]: D.Mumford; Abelian varieties: Oxford
University Press, 1985.

[10]: C.Rosati ; Sulle corrispondenze fra i punti di
una curva algebrica Annali di Matematica,
25 (1916) 1 - 32 .

[11]: I.R. Shafarevitch Basic algebraic geometry
Springer-Verlag .

[12]: G.Shimura ; On analytic families oT polarised
abelian varieties and automorphic functions
Ann. of Math. 78 (1963) 149 - 192.

[13]: J.A. Wolf; Spaces of constant curvature
MCGraw-Hill, 1967.

The seminal ideas of Andreas Floer have attracted a great deal of interest over
til(· past few years. In his work Floer has developed a number of important new
ilasights, notably through the novel use of ideas from Morse theory. His work also
Ii t.s perfectly into an overall theme of these Proceedings by illustating the paral-
I.-Is between symplectic geometry and gauge theory in 3 and 4 dimensions. On
til(· symplectic side a decisive achievement of Floer's programme was his proof of
Arnol'd's conjecture on fixed points, a conjecture which had its origins in Hamilto-
.. inn mechanics and the work of Poincare. See also the introduction to the section
uH symplectic geometry in Volume 2. Floer's lecture on this work is described in

t.ll(˜ notes by Kotschick in this section.

'I'lte other papers in this section are related to Floer's work in gauge theory. The
id.-as here are closely related to those involved in the section above, on the use of
Yang-Mills instantons in 4-manifold theory. The Floer instanton homology groups
()f a. 3-manifold Y are defined by instantons on the cylinder Y x R, interpolating
Ju-t,ween fiat connections at the two ends. The space of fiat connections over Y, or
n'presentations of the fundamental group, occupies a central place in the theory,
nlld Floer's groups can be regarded as a refinement of this space of representa-
t,i()llS. Roughly speaking, if one tries to extend an argument for instantons over
c'I()sed 4-manifolds to instantons over a 4-manifold-with-boundary, one finds that
t1l(˜ new phenomena that arise, which have to do with connections which are flat
hilt not trivial over the 3-dimensional boundary, are captured by the Floer homol-
•)I!.y groups of the boundary. Similarly, if a 4-manifold X is split into two pieces by
u. 3-dimensional submanifold Y, then the instantons on X can be analysed in terms
()f those on the two pieces and the Floer homology of Y.
For some time after Floer's work first appeared there was a dearth of explicit ex-
Iunples on which to test the theory, due to the difficulty of performing calculations.
'I'his picture is now beginning to change, with progress on a number of fronts which
is well-illustrated by the papers in this section. On the one hand, in the new de-
vc'lopments which he described in his lectures in Durham, Floer has found exact
s.'quences for his homology groups which open up the possibility of making system-
u.t,ic calculations from a Dehn surgery description of a 3-manifold. (The development
()f a similar programme in one dimension higher, for the Yang-Mills invariants of

-I Inanifolds, also forms an important goal of current research.) On the other hand,
feH" a particular class of 3-manifolds - the Seif˜rt fibred manifolds- special geometric
f('atures have been used to calculate the Floer homology groups. Two approaches
f.() this are described in the contributions of Fintushel and Stern and of Okonek

I)(·low. The simplifying feature here is that, as shown by Fintushel and Stern, one
ca.n obtain the Floer groups directly from the representations, without considering
illstantons. The technique of Okonek brings the problem of describing the represen-
t.ations into the realm of algebraic geometry, and holomorphic bundles on a complex
algebraic surface: a natural problem is to see if algebro-geometric techniques can be
I)rought to bear on other calculations of the Floer groups (and of the "cup products"

which can be defined in Floer homology). The paper of Fintushel and Stern also
describes an application of the calculations for Seifert manifolds to obtain a result
on 4-manifolds (a result which has in turn been used by Akbulut to detect exotic
structures on open 4-manifolds). Here one should refer also to the contribution of
Gompf in the preceding section.
Before Floer introduced his more general theory, ideas which can now be seen as
going in a similar direction were developed by Fintushel and Stem, Furuta and oth-
ers. Here the 3-manifolds concerned are lens spaces and, instead of 4-manifolds with
boundaries, one can consider compact 4-dimensional orbifolds. These developments,
and their striking topological applications, are described in the paper of Fintushel
and Stern. Yang-Mills theory over orbifolds has had a number of other applications:
for example Furuta and Steer have developed a theory for 2-dimensional orbifolds,
generalising the work of Atiyah and Bott, which gives another, rather complete,
description of the Floer homology of Seifert manifolds.
The contribution of Furuta deals with some more geometrical aspects. As we have
mentioned, Floer's homology groups depend in general on knowledge of the instan-
tons over cylinders, and these are normally quite inaccessible. The work of Atiyah,
Drinfeld, Hitchin and Manin (ADHM) gives a complete description of the instantons I

over the 4-sphere in terms of matrix data. This can then be used to describe instan-
tons on quotients of 8 4 , and in particular (by conformal invariance) on the cylinder
with a lens space as cross-section. This is at present the only kind of example where
one can obtain such explicit information. There are many different reductions of
the ADHM description which can be made in this fashion and the investigation of
these is at present an active area of research, with work by Furuta, Braam, Austin,
Kronheimer and others. These descriptions involve an attractive blend of geome-
try, representation theory and matrix algebra. In his paper below Furuta gives a I

number of applications of these ideas, including an analogue of Floer's homology
groups for lens spaces in which the groups can in principle be computed completely
in terms of matrix algebra, via the ADHM description.
lnstanton Homology, Surgery, and Knots
AIldreas Floer
I)epartment of Mathematics
tJuiversity of California
Il(˜rkeley, CA 94720

We describe a long exact sequence relating the instanton homology of two homol-
op;Y 3-spheres which are obtained from each other by ±l-surgery. The third term
iH a Z4-graded homology associated to knots in homology 3-spheres.

I. Instanton homology.

Let M be a homology 3-sphere, i.e. an oriented closed 3-dimensional smooth
or topological manifold whose first homology group Ht(M,l) vanishes. Poincare,
who first conjectured that M would have to be the standard 3-sphere, the first
IH)ntrivial example, now known as the Poincare sphere. Its fundamental group is
c)f order 120. Since then, many other examples were found, all of which have an
ilJfinite fundamental group. For example,

= {x E C 3 11xl = 1 and xi +x˜ +x˜ = O}

is a homology 3-sphere if p, q, and r are relative prime. In this case, M(p, q, r) is
conlled a Brieskorn sphere. Properties of Brieskorn spheres were studied e.g. in [M].
I{(˜cently, Donaldson's theory of instantons on 4-manifolds applied successfully to
t.he study of 3-manifolds. First, Fintushel and Stem [FS] proved that the Poincare-
Hphere has infinite order in "integral cobordism". Pursuing the same approach
J1ltrata [Fu] proved that all manifolds M(2,3,6k - 1) are linearly independent for
allY kEN (see also [FS2]). There is a strong feeling that instantons have more
to say about 3-manifolds, even though the above results rely very much on special
I)foperties of the Brieskom spheres and of their fundamental groups. In [Fl] and
ill the present paper, we therefore approach the problem from the other side, by

(OOl1structing instanton-invariants on 3-manifolds which can be defined rather gen-
˜·ra.lly, leaving computations and applications (some luck provided) to the future.
'I'he invariant, as defined in [Fl], takes on the form of a graded abelian group I.(A)
.l!:raded by ls. This as well as the definition of I. suggests that one should consider
FIoer: Instanton homology, surgery and knots

it as a homology theory, and we will in fact refer to it as instanton homology. It is
the purpose of the present paper to expose further properties of I. to justify this
Since we will need a slight extension of instanton homology, we briefly review the
construction. Let A be a general S03-bundle over an oriented closed 3-manifold M.
Instanton homology is a result of applying methods of Morse theory to the following
(infinite dimensional) variational problem: Consider the space A(A) of L1-Sobolev :
connections on A. Choosing a reference connection (which we will always assume
to be the product connection (J if A is the product bundle), we can identify A(A)
with the space L1(Q 1 (A» of Sobolev I-forms with values in the adjoint bundle I

ad(A) = (A X 803)/503. The Sobolev coefficients are fixed rather arbitrarily to
ensure that each ˜nnection is actually continuous on M. A(M) is acted upon by
the gauge group
g = L˜(A XAd 803 ).
We rather want to restrict ourselves to the subgroup

(Note that 50a is the group of inner automorphisms of SU2 as well as of 80a.)
The double covering S03 = SU2 /Z 2 defines an extension


where HI (M, Z2) has the usual additive group structure. The homomorphism 1] can
be described topologically as the obstruction to deforming 9 to the identity over the
1-skeleton of M. In fact, we can define {is as the set of all gauge transformations in
M which are homotopic to one of the "local" transformations which map the exterior
of some 3-ball B a in M to the identity. We therefore have a natural isomorphism
1['o(gs) ˜ Z, through the degree deg(g) of the map M/(M - B 3) ˜ SU2 ˜ S3. '
The quotient space 8(A) = A(A)j{is is then a finite covering of the space of gauge

equivalence classes of connections on M, with covering group H 1 (M, Z2).
To define the Chern Simons function (see reS]) note that TaA(A) = Lf(Ql(A),
so that the integral of tr(Fa 1\ a) over M, for (a, a) E T A(M) defines a canonical
one-form on A(M). It turns out to be closed; in fact, there exists a function £i on
A(A) such that

(1.2) d.G(a)a . a).
Floer: Instanton homology, surgery and knots

It is almost gauge invariant in the sense that

= s(a) + deg(g).

Hence it defines, up to an additive constant, a function a : B(A) --+ R/l. It can
also be described 88 the "secondary Pontrjagin class". Recall that on a principal
bundle X over a closed 4-manifold, the first Pontrjagin class is represented by the
4-form PI(a) = tr(Fa A Fa) for any connection a. Here, Fa = da + a A a is the
curvature 2-form. (For forms with values in ad(X), the exterior product is here
(˜xtended by matrix composition in the adjoint representation of 803.) It follows
JPI (a) is an integer and is independent of A. H the boundary
t.hat the integral
ax = A is not empty, then this is generally not true any more, but JPI(a) modulo
the integers depends only on the restriction of a to M and is given by s. This can
a.ctually be used to define s, since every B03-bundle M can be extended to some
S03-bundle X as above.
By definition, the critical set of .6 is the set of flat connections on M, which we
will denote by R(A). It is well known that the holonomy yields an injective map

n,(A) Hom(7rI(M), B03 )/ ad(S03).

Flat connections are therefore sometimes referred to 88 representations (of the fun-
damental group). Conversely, for each representation one can construct an 803 -
I)undle with a flat connection whose holonomy is prescribed by the representation.
It is n,(A) which will become the set of simplices in instanton homology. To
nnderstand this, recall the following statement of (finite dimensional) Morse theory.

(Thorn-Smale-Witten). Let f be a function on a closed manifold B with
uondegenerate critical set C(/). Let Cp(/) denote the free abelian group over all
.r. E C(/) with Morse index p. Then there exist homomorphisms

/11 fact, if9 is a metric on B such that the gradient field on f induces a Morse-Smale
flow on B, then the matrix elements (8a, b) with respect to the natural basis can
Floer: Instanton homology, surgery and knots

be defined as the intersection number of the unstable manifold of x and the stable
manifold ofy in arbitrarily level sets between f(a) and f(b).
In particular, it follow.s that ICp(f)1 ˜ dimHp• It is the defining property of
Morse-Smale flows that the intersections above are transverse. The manifolds in-
volved can all be given natural orientations, so that the matrix elements are integers.
They would define homomorphisms with any Z-module (i.e. any abelian group) 88
coefficients, and the same is true for instanton homology. We will, however, restrict
ourselves to coefficients in Z for the sake of brevity.
We want to apply a similar procedure to the Chern Simons function. To define the
gradient flow, note that the set 8*{A) of irreducible (i.e. nonabelian) connections
is a smooth Banach manifold with tangent spaces


see e.g. [FU] or [FI]. Note moreover that for any metric q on the base manifold,
ftr{a) := *Fa E V{n 1 ® SU2) satisfies d:fiT{a) = 0 due to the "Bianchi identity"
daFa = o. Finally, the gauge equivariance of Fa implies that fa{g(a» = gftr{a)g-l,
so that f tT is in fact a section of the bundle (, whose fibres La are obtained by
replacing Lt by L 4 in (1.4). Even though it is not a tangent field over 8* in the sense
of Banach manifolds, it has properties similar to vector fields on finite dimensional
manifolds. The reason is that the How trajectories of f tr , Le. the solutions of the
''flow equation"

+ f...(a(T)) = 0
are in 1-1 correspondence to self dual connections a on the infinite cylinder M x R '
with vanishing r-component and with aIMx{r} = a(T). One can also show (see
[FI]) that they "connect" two critical points if and only if their Yang Mills action J

IIFAII˜ is finite. Three problems arise if we try to fit this gradient flow into the
framework of the above theorem. First, we have mentioned above that the Chern ˜
Simons function is well defined only locally. Surprisingly, it turns out that we can
simply ignore this point, since not only the function, but also the Morse index is
ill defined along nontrivial loops in 8(A). Second, since flat connections are not
necessarily nondegenerate as critical points of .6, we perturb .6 by a function of the

FIoer: Instanton homology, surgery and knots

where h is a character of G, and K9(a) the parallel transport along a thickened knot
T D 803 --+ A.
K, :

Ilcre, D is the two disc and T = aD the I-sphere. The measure dp,(8) can be
assumed to be smooth and supported in the interior of D. Let us denote by s a
t.riple s = «(1, A, h), where (f is a metric on A and (A, h) a disjoint collection of
knots labeled by characters of G. It defines a perturbation .6 of the Chern-Simons
ftlnction, with L 2 -gradients

where s' is a smooth section of TB*(A) and in this sense a compact perturbation
()f *F • The critical points and trajectories of Is in B*(A) are now given- by

Rs(A) = {a E B*(A) IIs(a) = O}

= {a: R -+ A(A) I Baa(r) + f,,(a(r» = 0 and lim a(r) E'R,,(A)}.

Analytically, we consider Ms(A) as a perturbation of the space of self-dual connec-
tions on R X A. In fact, the temporal gauge (see [Fl]) defines a bijection

c {a E A(R X A) IFa + *Fa = s'(a) and IIFa + s'(a)1I2 < oo}/Q(R X A)
= Uo ,.8Ell.(A)a+ + f3- + Ll(n˜d(A»

,\.lid for each a E ns(A), Q± are chosen such that for A A,
R -+
1r :

= 1r* a on R± X A

a of a.
for some representative The flow equation is then given by a non-linear
x A)-equivanant map

Is : A(R x A) U(n-(R x A»

˜(Fa +*Fa) + s'Ca).
f,,(a) =
w(˜ call s stable if the operators .

= d;;,s + s"(a) : L˜(n˜d(R
f˜(a) L2(n˜(R x A»
A) -+

surjective for all a E Ms(A). This implies (see [Fl]) that Rs(A) is non-
t I(\˜cnerate as the critical set and contains no non-trivial reducible representations.

IJy eompactness, it is then also finite. The set of stable parameters plays the role of
t.1l(> Morse-Smale gradient flows, and is denoted by SeA). Then we proved in [Fl]:
102 FIoer: Instanton homology, surgery and knots

1. SeA) is not empty and contains for any metric (J' on A elements (0-, h)
such that
IIhll = Llh,,1 2
is arbitrarily small. Then 'Rs(A) is finite and Ms(A) decomposes into smooth
manifolds of nonconstant dimensions satisfying
= p.(a) -
dimMs(a,,B) p.(,B)(mod8)
for some function J.l. = 'R,s(A) ˜ ls. There exists a natural orientation on M s which
is well defined up to a change of orientation of a E Rs(A) (meaning a simultaneous
change of the orientations on M(a,,B) and M«(J,a) for a11(J in 'Rs(A») and which
has the following property: Denote by R p, p E Zs, the free abelian group over the
elements a ofn s with p.(a) = p. Define the homomorphism

8 s ,p : Rp Rp -
-+ 1

8s ,p(a) = o(a),B,

where o(a) = 0 for dima Ms(a,,B) =1= o. Then if A allows no non-trivial abelian
connections, we have OpOp+l = 0, and the homology groups
[p(A,s):= kerop/imop+l
E SeA).
are canonically isomorphic for any S

The only point in the proof of Theorem 1 that differs from the case of trivial SU2-
bundles is the question of orientations. This is not a property of the moduli space
alone, but follows from a global property of the operator family DA over B( M). To :
be more precise, define the determinant line bundle A of D as the real line bundle .
with fibers
A a = detD a = det(kerD a ) ®II det(cokD a ).
Local trivializations can be defined as follows. For any finite dimensional subspace '
E c La such that cokDa maps injectively into Hom(E,R), define the projection
"IrE : La -+ La/E. Then det ker "IrEDa can be identified naturally with det(DA), and I
can clearly be extended smoothly on a neighborhood of A ,in B. If now in addition
fs(A) E E, then the set
ME = {a + e,eE TaB and fs(a + e) E E}

is a smooth manifold locally at A, and its orientation is determined by an orientation
of A and E. In particular, any orientation on A defines an orientation on M.
Floer: Instanton homology, surgery and knots

1.1. Let X be an SOa-bundle over a compact 4-manifold X, and let Gs(X)
denote the group ofassociated 8U2 -gauge transformations defined as in (1.1). Then
A is orientable on A(X)/gs(X).

Since gs contains only the SU2-gauge transformations, orientability can
')e proved in the same way as Proposition 3.20 and Corollary 3.22 of [D]. 0
The canonical isomorphism of the instanton homology groups can be understood
I)est in the following functional framework.
1.1: 80a [3] is the category whose objects are principal SOa-bundles
A, B over closed oriented 3-manifolds and whose morphisms X : A ---+ B are princi-
l)al80a bundles over oriented smooth 4-manifolds together with an oriented bundle
isomorphism gX : AU( -B) -+ ax. In this case we also write A = X+ and B = X_.
Note that everyequivariant diffeomorphism 9 of M defines an "endomorphism"
Zg = (M x [0,1],9) of M, where 9 : M U!VI -+ M x {O, I} is the union of go(x) =
(!I(x),O) and 91(X) = (x,!). We will restrict ourselves to the (full) subcategory
S10;[3] consisting of bundles M which do not admit a nontrivial abelian connection.
'I'hat is, the objects in 80;[3] are either trivial bundles over homology 3-spheres,
or nontrivial bundles with no reducible fiat connections. We define a functor 1*
from 80;[3] into the ls-graded abelian groups by means of the following auxiliary
Htructures on M. Consider conformal structures u on

which are equivalent to product metrics on the ends, and on a collection A X D x R
()f thickened cylinders in X oo • Let h be a time dependent character of G associated

t.o the components of A, and define

= {a E A(X) IFa + *tTFa = s'Ca) and IIFa + s'(a)1I2 < oo}/g(X)

SeX) the set of all such (u, A, h) such that Ms(X) is stable,
as in (1.5). Denote by
aud which define stable limits s± E S(X±).

2. SeX) is nonempty and for s E SeX),

R*(A,SA) R*(B,SB)
VX,s : ---+

vx,s(a) o(a)f3
104 Floer: Instanton homology, surgery and knots

has the following properties
(1) vXtSos = os+vxts.
(2) Let (X,s): (A,SA) -+ (B,SB) and (Y,sy): (B,SB) -+ (C,sc) be two stable
cobordisms. For p E R+ large enough define on the bundle

A) U X U ([-p, p] B) U Y U (R+ C) : A C.
X#pY = (R_ -+

Then for p large enough, the obvious perturbed parameter 1rX#p1rY on X#pY
is regular, and induces the composite

= vY,Sy 0 vx,s.

(3) The homomorphism

(X,s)* : I*(A,sA) I*(B,sB)

does not depend on the choice of sx E S(X).

These are three properties that allow us to consider 1* as a functor on the category I

80;[3] rather than a functor on the category of pairs (A,s). (The same argument
is used e.g. in the definition of algebraic homologic theories through projective
resolutions.) 1* has a cyclic ls-grading, meaning that ls acts freely on I. by
increasing the grading. H M is a nontrivial bundle, then there usually does not exist
a canonical identification of the grading label p with an element of ls. However,
if H 1 (M) = 0 then the gauge equivalence class of the product connection 8M is
a nondegenerate (though reducible) element of R. In this case, we can therefore
define a canonical ls-grading of I*(M) through the convention that

p(a) = dimM(a,8M).

2. Dehn surgery.

For a knot '" in a homology 3-sphere M, denote by "'¥ the homology 3-sphere
obtained by (+l)-surgery on "'. We want to show that I.(M) and I.(K.M) are related
by an exact triangle (a long exact sequence) whose third term is an invariant of K..
In fact, it will turn out that it depends only on the knot complement K of "', so that
it is the same for all exact sequences relating ",'1M and ",'1+ 1 M. We may say that
"integral closures" K(a) with Hl(K(a» = 0 is an affine Z-family of manifolds. The
FIoer: Instanton homology, surgery and knots 105

only closure of K which is canonical is K := K(K), which is homology-equivalent
to 8 2 X 51. Although there exist, consequently, reducible representations of 11"1 (K),
none of them can be represented by flat connections in the 803 -bundle P over [(
with nonzero W2(P) E H 2 (K,Z2). In fact, restriction yields a 1-1 correspondence

In fact, the elements of this set do not extend to 5U2-representations on K, but
their induced S03-representations do. Since ,., is trivial in H l (K), all abelian rep-
resentations of 1I"1(K) would assign it the identity 1 E SU2 • It is easy to see that
if ,,(g) 1= 0, then g2 generates 1ro(gs(P)). Our knot invariant now takes on the
following form:
2.1. The instanton homology of a knot It in a homology 3-sphere with
complement ,., is defuJed as the Z4 -graded abelian group

where a({1} x 8D x 1) is non-trivial in the nbre but trivial in HI (K). The surgery
triangle of I\. is the triple


of surgery cobordisms, where w is trivial relative to the boundaries and with the
(˜nds identified in such a way that its total degree is -1.

The extensions of the framed surgery cobordisms to B03-bundles is unique de-
t.ermined by the prescription in Definition 2.1. That the total degree of (2.1) is
equal to -1 modulo 4 follows from an index calculation. The construction becomes
perfectly symmetric in the three maps of (2.1) if one formulates the surgery problem
for general knots ("', A), which are principal BOa-bundle A over a closed oriented
3-manifold M = A1803 , together with an equivariant embedding

˜ 803


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