Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the...

Commun. Math. Phys. 153, 521-557 (1993) Communications in Mathematical

Physk �9 Springer-Verlag 1993

Chern-Simons Invariants of 3-Manifolds Decomposed along Tori and the Circle Bundle Over the Representation Space of T 2

Paul Kirk 1 and Eric Klassen 2

1 Department of Mathematics, Indiana University Bloomington, IN 47405, USA 2 Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA

Received June 1, 1992

Abstract. We describe a cut-and-paste method for computing Chem-Simons invariant of flat G-connections on 3-manifolds decomposed along tori, especially for G = SU(2) and SL(2, C). We use this method to make computations of SU(2) Chern-Simons invariants of graph manifolds which generalize Fintushel and Stem's computations for Seifert-fibered spaces. We also use this technique to give a simple derivation of a formula of Yoshida relating the flat SL(2, C) Chern-Simons invariant of the holonomy representation to the volume and the metric Chern-Simons invariant for cusped hyperbolic 3-manifolds.

1. Introduction

This paper is a continuation of [KK2]. In that paper we described a method for computing the Chern-Simons invariants of SU(2) representations of a 3-manifold obtained by surgery on a kno t / ( in a closed manifold M in terms of the image of the restriction R ( M - ts --~ R(T), where R ( X ) denotes the space of conjugacy classes of representations of the fundamental group of X in SU(2) and T is the boundary toms of M - K. The main purpose of this paper is to show how to compute Chern- Simons invariants of a closed manifold in terms of an arbitrary decomposition of the manifold along tori. Cutting a 3-manifold along tori is a useful procedure in 3-manifold theory. In addition to surgery on knots and links, this includes also decompositions along incompressible tori in the sense of Jaco-Shalen and Johannson [J]. This cuts a 3-manifold into simpler pieces, namely Seifert-fibered 3-manifolds and complete hyperbolic 3-manifolds. The basic idea is to define Chern-Simons invariants for a manifold whose boundary consists entirely of tori. We then show how to use these methods to explicitly compute Chern-Simons invafiants of various representations of 3-manifolds with toral boundaries, including many Seifert-fibered and hyperbolic manifolds.

Both authors acknowledge support from the NSF

522 P. Kirk and E. Klassen

The outline of the paper is as follows. In Sect. 1 we construct explicitly an S 1 bundle E ( T ) over R ( T ) when T is a torus or a union of tori and prove (see Sect. 2 for the precise formulation):

Theorem 2.1. I f X is a 3-manifold with O X = T, then the Chern-Simons invariant defines a lifting c x : R ( X ) -+ E ( T ) of the restriction R ( X ) ---+ R(T) . Moreover, there is an inner product ( ): E ( T ) x E ( - T ) --+ S 1 so that i f Z is a closed manifold decomposed along T, Z = X U T Y , then Cz(p) = (Cx(Q) , Cy(@)).

Although the existence of such a bundle was known (see [RSW]), the point of this result is that the construction we give of E ( T ) is totally explicit and makes computations very easy. This is especially true when combined with Theorem 2.7, which shows how to compute the difference between the Chern-Simons invariants of two representations which lie on a path of representations.

Theorem 2.7. Let X be an oriented 3-manifold with toral boundary O X = T l U. . .UTn and let o(t) : 7f iX --+ SU(2), t E [0,1] be a path of representations. Let ((~l(t), 131(t),. . . , (~( t ) , t3n(t)) be a lift o f o(t) Iox to I~ 2n. Suppose

C x ( L O ( t ) ) = [O~l(t), ~ l ( t ) , . . . , O~n(~), t3n(t); z(t)]

for all t C I. Then

z i z,0) '--exp o

\ a = l 0

In particular, i f 6(1) is the trivial representation (so that z(1) = 1 by Corollary 2.6) then [-

Cx(~O(O)) = ]OLI(0).~I(0),...,an(0), fin(O);

exp ~=lo ~o!~-~-t- - ~a - -~ - ) ) ) ] .

We refer to Sect. 2 for definitions of [O11(0), /~1(0),... ,O~n(0), fln(0),Z]. (The bundle E ( T ) is a quotient of the trivial bundle ~2n • S1 by a discrete group.) Theorem 2.7 implies that the lift c x is parallel with respect to the connection ~ o%dflk -i3kde~ k on E ( T ) (Corollary 2.8). Stated this way, this corresponds to the facts in [RSW] that there is a natural connection on E ( T ) whose curvature is the symplectic form and that the image R ( X ) --+ R(T ) is Lagrangian. The explicit formula for this connection makes computations easy.

We then extend the results to SL(2, C) representations. A / C * bundle Ec (T ) over the character variety of the toms Re(T) is constructed and the analogues of Theorems 2.1 and 2.7 are proven. The arguments for SU(2) do not carry over immediately to SL(2, C) because of the presence of non-diagonalizable reducible connections, and so the we carry out the necessary analysis to extend the results to the character varieties. We also make some comments about how to extend these results to SO(3) and P S L ( 2 , C) representations.

In Sect. 4 we then carry out explicit computations of the Chern-Simons invariants for several different types of representations of 3-manifolds with toral boundary. Theorem 4.1 computes the Chem-Simons invariants for abelian representations of X

Chem-Simons Invariants of 3-Manifolds 523

when H 1 (X; Z) is free abelian. We then compute the representation spaces and Chern- Simons invariants for F x S 1, where F is a punctured surface. Then we show how the results of Auckly [A] can be used to compute the Chern-Simons invariants of certain binary dihedral representations (the special representations of [KK2]) of punctured surface bundles over the circle. We also compute the Chern-Simons invariants of certain Seifert-fibered spaces, namely the complements of regular fibers in Seifert- fibered homology spheres (Proposition 4.4). (The method works with no substantial changes for any Seifert-fibered 3 manifold with boundary but the formulas are messier.) These computations can then be used to understand Chern-Simons invariants of closed 3-manifolds obtained by glueing together manifolds along tori by using the inner product of Theorem 2.1.

We then use these computations and Theorem 2.1 to compute the Chern-Simons invariants of certain closed 3-manifolds. First we consider circle bundles over closed surfaces; these manifolds arise as boundaries of the complements of surfaces in 4- manifolds. We prove:

Theorem 4.3. Let 0 : 7rlM(n ) ---* SU(2) be a representation of the circle bundle over a closed, oriented surface F with Euler class n. Conjugate Q so that the fiber is sent

k k 2 1 to e 2~i~. Then either/3 = -'n in which case cs(o) = ---,n or else n is odd and/3 = 2 '

in which case cs(Q) = - - . 4

We then compute the Chern-Simons invariants of certain graph manifolds, namely the manifolds Z = X U Y, where X and Y are Seifert-fibered but the identification OX ~ OY is not fiber preserving. This is the application which motivated the investigations of this paper. In an earlier paper [KKR] we showed how to compute the Floer Homology grading ([F]) of a representation of these manifolds in terms of Atiyah, Patodi, and Singer 0~ invariants and the Chern-Simons invariants. To finish the computation we needed to compute their Chern-Simons invariants. This is supplied by:

Theorem 4.5. Let ~9 : 71lZ~p ---4 SU(2) be a representation whose restriction to X and Y is non-abelian. The Chern-Simons invariant of Q is

e 2 c 2 p2u zok 2 - - e "~X __ E y - - (2p + z) M o d Z .

x 4a 4c 4v 4

The reader familiar with Fintushel and Stem's computations for Seifert-fibered homology spheres [FS1, FS2] will recognize the first two terms. They are "internal," depending only on the geometry of X (resp. Y) and the restriction of Q to X (resp. Y). The other two terms are "external" in the sense that they depend only on the gluing map OX --~ OY and the restriction of 0 to this torus.

We finish Sect. 4 with a computation which shows how to interpret Yoshida's formula [Y] relating the volume and metric Chern-Simons invariants on a hyperbolic 3-manifold obtained by hyperbolic Dehn surgery in terms of Theorem 2.1.

2. Manifolds with Boundary

Chern-Simons invariants of connections on manifolds with boundary are not gauge invariant, and there is a useful formalism described in [RSW] for dealing with them.


In that paper a certain complex line bundle over the space of flat connections on a surface F is constructed. We recall the definition of their bundle: Let F be an oriented closed surface and let P = F • SU(2) be the trivialized principal SU(2) bundle over F . Let J~ (F) be the space of connections on F and .~(F) the group of gauge transformations. The trivialization determines isomorphisms:

.A(F) ~ Y2~ | su(2) and ~'J(F) ---~ Maps(F, SU(2)).

If X is a 3 manifold and A E t2~c | su(2) a lie-algebra valued 1-form on X define its Cheru-Simons invariant to be:

' / CSx(A) = ~ Tr(dA A A + -~A A A A A).

x

Now let A C ~ ( F ) and g E ~'~(F). Extend A to A over some 3-manifold and extend g to ~ over X. Then define:

O(A, 9) = e2Cri(csx(O'A)-csx(A)),

where .~ acts on / i in the usual way; so as 1-forms .q �9 J[ = 04.0 -1 - d.q.0 -1. Then O(A, g) is well defined; this follows from the fact that on a closed manifold the Chern-Simons invariant is well defined mod Z.

Let ~ ( F ) act on the trivial circle bundle over .~(F) by:

g . (A, z) = (g . A, O(A, g)z).

To see that this defines a (topological) quotient bundle E ( F ) over 2 ( F ) = . A ( F ) / ~ ( F ) one checks that if g �9 A : A, then O(A, g) = l, i.e. that the fiber over fixed points is itself fixed.

Let ~ ( X ) denote the connections on a 3-manifold X and let ~ ( X ) denote the orbit space of ~ ( X ) under action of the gauge transformations. If OX = F , then it is a tautology that the map A H e 2~rics(A) defines a lifting of the restriction map 2 ( X ) --- J~(F) to the total space of E(F) .

We will construct this bundle explicitly over the space of flat connections modulo gauge transformations on a toms T by considering a 2-dimensional subspace of J~ (F) which maps onto the flat connections modulo gauge equivalence, and which is invariant under a certain discrete subgroup of ~ . The bundle we construct will be explicitly defined, and so after we define it we will have to prove that it is indeed E(T).

Let T be an oriented toms, and let R(T) denote the space of conjugacy classes of SU(2) representations of r q T into SU(2). As is well known, the holonomy defines a homeomorphism (in fact an analytic isomorphism) from the space of gauge equivalence classes of flat SU(2) connections on T to R(T). As a space R(T) is homeomorphic to $2; as a variety it has 4 singular points.

Let V(T) be the two dimensional vector space

V(T) = Hom(rrl(T), R).

Then the map V(T) ---* R(T) defined by

V e--+ ('ff ~ e 27r iv ( ' / ) )

is a branched cover. (Here we are identifying SU(2) with the unit quaternions. In what follows we will always use this notation, as well as identifying the lie algebra of SU(2) with the pure quartenions ~ i | Cj . )


The covering group is isomorphic to a semi-direct product of Z | Z and Z / 2 . To see this, fix an oriented basis #, A for 7rlT. Give G the presentation:

G = ( x , y , b [ [x ,y] = bxbx = byby = b 2 = 1).

Then via the isomorphism V ( T ) ~ I~ 2 defined by v H (v(#), v(A)) the action of G on V ( T ) = ~2 is

x(c~,/3) = (c~ + l,/3), y(a , /3) = (c~,/3 + 1), b(c~,/3) = (-c~, - /3) .

Let G act on the trivial S l bundle over V ( T ) in the following way, still using # and A to identify V ( T ) with ]~2 as above. Then let G act by:

x(o~,/3; z) = (a + 1,/3; ze27ri~3),

y(a, /3; z) = (a , /3 + 1; ze-2~i'~),

b(a,/3; z) = (-c~, - /3 ; z).

It is easy to see that this defines an action extending the action of G on lI~ 2. For example, [x, y](c~,/3; z) -- (c~,/3; ze 4~i) = (c~,/3; z).

I f g c G fixes (c~,/3) E ]R 2 then g = x~y~b for some integers r , s and

g" (0~,/3; Z) -~ (--0~ + r, --/3 + 8; ze 21ri(sa-r~+rs))

= (~ , /3 ; z ) .

So 9 also fixes the fiber over (c~,/3). Thus the quotient circle bundle E ( T ) over R ( T ) is defined:

E ( T ) = V ( T ) x S1 /G.

Although we have written down the action of G on V ( T ) x S 1 in coordinates determined by the choice of # and A, the bundle E ( T ) depends only on the orientation of T. To see this, suppose m = p p + q A and l = r # + s A is another choice of basis for ~rlT, where p, q, r and s are integers satisfying ps - qr = 1. Then xayb(v(m), v(1)) = (v(m) + pa + qb, v(l) + ra + sb). One then checks that

e27ri(v(1)(pa+qb)-v(m)(ra+sb)) ~ c27ri(v(A)a-v(tt)b).

We will fix a basis #, A for 7VlT for most of this section and just write elements of E ( T ) as [a,/3; z], where the square brackets indicate the orbit of G.

Notice that if the orientation of T is reversed, then E ( T ) is replaced by the inverse line bundle. As a smooth object this should be thought of as an orbifold bundle, double covered by an "honest" bundle over the toms V ( T ) / Z �9 Z.

We can define a natural bundle map from E ( T ) • E ( - T ) to {pt} • S 1, the trivial bundle over a point, (which we view as a bundle over R( r given by taking the pair ([a, /3; z], [c~,/3; w]) to zw E S 1.

The definition generalizes easily to the case of a union T 1 U . . �9 U T n of toil. Then R ( T 1 U - - . t2 Tn) = R(T1) • . . . • R(Tn) and we take the bundle E ( T 1 U . . . U T~) to be the "tensor product" of the E(Ti) . More precisely the product action of G '~ on (]K2) n extends to an action of G n on (~2)n • S 1 using the same formula as above, so that for example

Yk" (OZl' i l l ' ' ' ' ' O~n,/3n; Z) = (OZl./31,""" , ak,/3k "~- 1 , . . . , c%,/3n; ze--Z~ric~k)"


We denote the quotient bundle by E ( T 1 U. �9 �9 t3 T,O. Again we have a natural "partial" inner product map

(,) : E ( T 1 U . . . U T n) x E ( - T 1 U . . . U - T , O --~ E(T,~+I U . . . U Tn).

Given a 3-manifold X with OX a union of tori, let J ~ F ( X ) denote the SU(2)- connections on X which are flat near the boundary. The gauge group acts on these connections and we let ~ F ( X ) denote the orbit space. Notice that there is a restriction map

2 F ( X ) ~ R ( O X )

given by taking the holonomy of the flat connection on the boundary. Although the Chern-Simons invadant is not well-defined on ~ F ( X ) as an element of ~ / Z , the following result shows that we can define it as a section of this circle bundle. It is convenient to introduce the notation:

c x ( A ) = e27ricsx(A)

for A E ~2~c | su(2). So cx (A) is just a different way to express the Chern-Simons invariant.

~-+ e x ( A ) defines a lifting o f the restriction map 2.1. Theorem. 1. The map A J~F(X) ~ R ( O X ) to E (OX) :

E ( O X )

,22F(X ) > R(OX) .

2. Let X , Y be oriented 3-manifolds with toral boundaries, and let C = T 1 U . . . U T n be a collection o f tori. Suppose that we are given a diffeomorphism h x o f C with a part o f the boundary o f X and a diffeomorphism h y of - C with part o f the boundary o f Y . Let Z = X Uhvoh]l Y . I f A is a connection on Z which is flat near C, then

e z ( A ) = ( e x ( A i x ) , r

3. The bundle E ( T ) --~ R ( T ) has Euler class equal to - 1 . More generally i f C = T 1 t2 . . . U T n is a union oftori , then R ( C ) = • so that H2(R(C) ) = •kH2(R(Tk)) . In this case the Euler class is ( - 1 , - 1 , . . . , - 1 )

We will prove this later in this section. We first develop some of the ideas we will need.

Let X be a compact, oriented 3-manifold with toral boundary OX = T 1 U . . . U T,~. 1 1 2 For each k, choose an identification of T k with S x S . This identifies ~ with

the universal cover of T k via the covering map given by (x ,y ) H (eiX,eiY). It also determines a symplectic basis Pk, Ak E 7ra(T k) by letting # be the image of a horizontal line and A the image of a vertical line in ]~2 under the covering map. Also the forms dx and dy in ]I~ 2 factor through to give us corresponding forms in ~21(Tk), which we will still denote by dx and dy. Let T k z [0, 1] C X be a collar with T k identified with T k • 1. This allows us to define 1-forms {dx, dy, dr} on X near T k. We assume the orientation of T k as the boundary of X agrees with the orientation inherited from the c o v e r ]~2 ___> Tk ' SO that {dx, dy, dr} is an oriented basis of 1-forms

Chern-Simons Invariants of 3-Manifolds 527

on X. Thus we orient O X with "outward normal last" convention. Stokes' theorem says f dw = f w on a 3-manifold X with this convention.

X cOX With respect to the trivialization of the SU(2) bundle over X any connection A

can be written as

A = a d x + / 3 d y + T d r

in a neighborhood of T k, where

a,/3, 7 : Tk • I ~ su(2).

We will focus on the connections for which a and/3 are constants and "7 is zero:

2.2. Definition. We say A is a normal form if for each k there exist ak,/3k c R so that in a neighborhood of T k,

A = ia k dx + i/3 k dy.

Similarly, if A(t) is a path of connections on X we say that the path is in normal form if each A t is; in other words there exist functions ak,/3k : I ---* ~ SO that

A(t) = iak( t ) dx + i/3k(t) dy.

Using the local definition of the curvature F A = dA + A A A it is easy to see that if A = i a k dx + i/3 k dy near T a, then A is flat near T k. Furthermore, the holonomy representation of 7 t i t k (with respect to a base point near Tk) is given by

~ k ~ c27ri(~k~ "~Ir ~-4 e 27ri~k.

This can be computed directly from the definition of holonomy. Alternatively one uses developing maps as explained in [KK2]. The map D : ~2 _._. SU(2) given by (x, y) ~ e -2~ri(ax+f~y) is a developing map for this holonomy and A = - d D D -1.

Notice that the representations

~t a ~ e2~ia ~ ~ e2~:i~

and #a ~ e27riS~ ~ ~ e27ri/~

are conjugate in SU(2) if and only if there exist integers m, n and e E { • such that (~ = e i (a + m) and/3 -- ei(/3 + n). Since gauge-equivalent flat connections have

conjugate holonomy it follows that if i a k dx + i/3 k dy and i5 a d x + i~k dy are gauge equivalent then a, 6~ and/3, /3 are related as above.

2.3 Proposition. 1. Let A be a connection on X which is fiat in a neighborhood of OX. Then there exists a gauge transformation g so that g . A is in normal form. 2. Let A(t) be a path o f connections on X so that A( t ) is flat near O X for each t. Then there exists a path o f gauge transformations g(t) so that g(t) . A( t ) is in normal form

for each t. Furthermore, i f A(O) is already in normal form we can choose g(O) = Id. 3. Let A be any connection on X which is fiat near OX. Then for any choice o f ak,/3k E ~, k = 1 , . . . , n such that (up to conjugation)

(L)A(#k), PA( '~k ) ) -~- (e27ric~k e27ri/3k)


(where LOA denotes the holonomy representation near O X ) there exists a gauge transformation g supported near O X so that

g �9 A = i ~ k dx + i/3 k dy

near T k .

Proof. 1. First observe that this is a local question since if we can find a gauge transformation g near O X the obstructions to extending g over X lie in H i ( X , OX; 7ri_I(SU(2)) ) and these groups are all zero. So it suffices to show that any flat

connection on T 2 x I can be gauge transformed to this form. But this follows from the fact that the holonomy gives a homeomorphism between the set of flat connections modulo gauge equivalence and the space of conjugacy classes of representations of the fundamental group. 2. If g(t) is a path of gauge transformations defined in a neighborhood of OX, then g(t) can be extended to a path on X since H i ( X x I , O X x I ; 7r i_ 1SU(2)) = 0. If A(0) is already in normal form, we can take g(0) to be the identity since the obstructions then lie in H i ( X x I , O X x I U X x 0; 7ri_lSU(2)) = O.

Notice that we cannot in general gauge transform a path into normal form leaving both endpoints A(0) and A(1) fixed. The obstruction to doing this is usually non-zero. 3. Using the first part of this proposition we may put A in normal form by a gauge transformation. Suppose that A = ic~ dx + i/3 dy near T 1. We will show that there are gauge transformations gx, gy and gb equal to the identity outside a neighborhood of T 1 so that

9~ " A = i(c~ + 1) dx + i/3dy,

gy . A = ic~ dx + i(/3 + 1)dy,

and

gb " A = - i s dx - i/3 dy.

By composing these gauge transformations we can get any normal form on T 1, and similarly near the entire collection of toil OX .

Let h : S 1 ~ S 1 c SU(2) be the map e ix H e - i x . As a map into SU(2), h is nullhomotopic. Let ht , t c [0, 1] be a nullhomotopy which is constant for t near 0 or l, with h 0 constant at 1 C SU(2) and h I = h. Use h t to define a map f : S 1 x [0, l] ~ SU(2) which equals h near S t x 1 and is equal to 1 near S 1 x 0. The map gx is then defined in a collar of T 1 to be the composition of the projection T 1 = S 1 x S 1 ---+ S 1 onto the first factor with the map f . Then extend gx to be 1 outside this collar. Notice that d g x g x 1 = - i dx near T 1 x 1.

If we let B = gx �9 A, then near TI:

B = g A g 1 _ d g g - 1 = A + i dx.

We construct gy in the same way, taking the projection T 1 ~ S 1 onto the second factor.

To construct gb, repeat the construction starting with the map h : T 1 ---+ SU(2) which is the constant map at j . It will be convenient in what follows to assume that the extension to T 1 x I factors through the projection T 1 x I ~ I .

Since j i j -1 = - i , this has the effect of reversing the sign of c~ and/3. []

Remark. We will use the gauge transformations g~, gy and gb later. We note here that

dgx = O, dg____y_y = O, dgb = 0 dg----~b = O. dy dx dx dy


The following result shows that gauge equivalent connections in normal form have the same Chern-Simons invariants if they agree near the boundary.

2.4. Theorem. Suppose that X is a 3-manifold whose boundary components are tori: OX = T 1 U . . . U T n. Let A and t3 be connections on X in normal form with respect to OX and suppose that 1. A and 13 are equal near the boundary. 2. A and 13 are gauge equivalent. Then cs(A) and cs(13) coincide Mod Z.

Proof Let 9 be a gauge transformation such that 9 " A --- B. We first assume that 9 is trivial near OX. In this case, the connections A u A and A U 13 on ( - X ) U X are gauge equivalent by the gauge transformation 1 U 9. Since cs is additive over unions, we may cancel C s x ( A ) to conclude that CSx(A) = CSx(13).

Now drop the assumption that g is trivial near OX. Since A -- 13 near OX, we know that

ia k dx + i3k dy = g( ia k dx + i/~ k dy)g -1 - dgg -1

Og near each T k. We conclude that ~rr = 0. This implies that g is constant in the r

direction, and fixes A as a connection on T k. The group of those gauge transformations on T k which fix As r k is isomorphic to the centralizer in SU(2) of the image of the

holonomy of AIT k. Because 7Q(Tk) is abelian, this centralizer is either S 1 C SU(2)

or all of SU(2). In any case, it is connected. Hence for each k we may choose a path gk(t) of gauge transformations on T k from gk(O) = 1 to gk(1) = grTk such that for all t, gk(t) fixes AIT k. (Assume each path is constant near each end.) View each gk as a gauge transformation on T k x I , where T k • I is a collar of T k on which A and B are in normal form. Define a gauge transformation ~ on X by setting ~ = g outside of the collars T k x I and ~0 = gk(t) on T k x {t}, for t E I . Let C = ~ �9 A. By the first paragraph of this proof, we know that cs(C) = cs(A). We will now show that cs (C) = cs(B).

Note that on the complement of the collars the two connections are identical, so we need only consider their Chem-Simons integrals over the collars. For each s E I , define a gauge transformation h s on T k x I x {s} by setting hs = ga(1 - s + st) on

T k x {t} x {s}. Consider the flat connection -48 = h s . A on T k x I x {s}. Define a

connection A on T k x I x I to be the union of the / ]~ . Since A is actually a path of flat connections on T k x I , it follows that

0 = / " T r (F A A F A ) . , ]

T k x l x I

By Stokes ' theorem, however, this integral is equal to the sum of a Chem-Simons integral over T k x 01 x 1 and one over T k x I • OI. The former integral vanishes (because it is a constant pa th of connections over Tk), while the latter is equal to

/ 2 / T r ( d A A A + g A A A A A ) - Tr(dCAC+~CACAC)

TkXI TkXI

which proves these latter two integrals are equal and completes the proof of the theorem. []


The next result shows how the Chern-Simons invariant changes when a flat connection is altered near the boundary. A consequence of this theorem is that the bundle E ( T ) is the same as the bundle constructed in [RWS].

2.5 Theorem. Let X be a manifold whose boundary is a union o f tori O X = T 1 O �9 . . U T n and let A and 13 be gauge-equivalent connections in normal form. Suppose that near T k,

A = i(~ k dx + il3 k dy

and

B = %(i((~ k + ink) dx + i03 k + nk )dy )

f o r some collection o f integers m k, n k, k -- 1 , . . . , n and signs e k E {• Then

n c s ( B ) - cs(A) = Z m k / ~ k - - nkOlk M o d Z .

k=l

Proof. Suppose 9 is a gauge transformation so that g �9 A agrees with B near the boundary. Then we have seen that cs(9 �9 A ) = c s ( B ) .

The proof therefore reduces to showing that cs(gx . A ) = e s ( A ) + ~1, cs(gy �9 A ) = c s ( A ) - cq, and cs(9b �9 A ) = cs (A) , where 9x, 9 u, 9b are the gauge transformations constructed in the proof of Proposition 2.3.

The difference cs(9~ �9 A ) - c s ( A ) is the integral of a function supported in S 1 x S 1 x I , since 9x is the identity outside a collar. Moreover, the Chem-Simons integrand vanishes for A on S 1 X S 1 X I since d A = 0 and A A A A A = 0. Thus we must show that if C = 9~ " A on S 1 • S 1 • I , then

1 Tr d C A C + - ~ C A C A C =/31. 87r 2

S 1 x S I x I

We do this computation directly. This is similar to the argument in the proof of Theorem 4.2 of [KK]. Write 9 = 9x. First notice that g �9 A is flat on S 1 x S ~ x I so that

1 Tr(C A C A C). Tr A C + A C C) =

Using the remark after the proof of Proposition 2.3,

C = 9" A = 9 A 9 -1 - d9g -1

~--- (ffiOLlff--'-- O-~ffg-l~dx-]-ffi/~lff-ldy- Offff-ldr o x / Or "

A bit of manipulation yields:

_ _ rr,,-,o, 31 Tr(C A C A C) =/31 Tr \ L Ox

We can now integrate out the y variable since the form is constant in the y direction and since we are integrating over a product manifold. So

1 /, S l x S l x I S l x I


Let w = Tr(i 9 - ld9). Then

[[ - l O g g - lOg] i )d rd x d w = T r ~ [ g Ox OrJ

on S 1 X I . So we need to compute f dw, which by Stokes' theorem equals f w - f w.

sl• S l • S i x 0 (A bit of care must be taken to see that the signs are correct.) Since g is the constant map at 1 c S U ( 2 ) n e a r S i x 0 , f w i s z e r o . N e a r S 1 • 1, g(e i z ) = e -ix and so

S1xO

Tr(ig - 1 dg) = 2 Re(ie ix ( - i) e- ix dx) = 2 dx.

(For v, w C su(2), Tr(vw) = 2 Re(vw) when viewed as unit quartemions.) Therefore, the integral of co over S 1 • 1 is 47r. Hence:

Tr (dC A C + ~C A C A C) =/31"

A similar argument applies to g y . The opposite sign arises from or ien ta t ion

considerations. The argument involving gb is simpler. Since Ogb vanishes, equation

( .) shows that the Chern-Simons integral vanishes on the collar, completing the proof. []

We have the following corollary:

2.6 Corollary. Let X be a compact, oriented 3-manifoM whose boundary is a union of tori. Suppose A is a flat cohnection in normal form with respect to OX so that the holonomy representation of A is trivial. Then

CSx(A) c Z.

Proof. The connection A is gauge-equivalent to the trivial connection, whose Chern- Simons invariant is O. Thus

cs(A) = ~ m k / ~ k - - nka k ModZ. k

but the a k and flk are integers, since the holonomy representation is trivial. []

We can now prove Theorem 2.1.

Proof of Theorem 2.1. Given [A] E 2 F ( X ) , choose a representative A in normal form. If A = ic% dx + i~k dy near T k let

Cx([AD = [0/1' ]~1' " ' ' ' an,/3n; e2~ricsx(A)]"

Theorem 2.5 shows that ex([A]) transforms properly. The second assertion of the theorem is now obvious. To prove the third assertion, observe that the region [0, 7r] x [0, 27r] is a fundamental

region for the action of G on ]R 2. The identifications of the boundary are indicated in the next figure. Split the region into two pieces: A = [0, T r] x [0, T r] and B = [0, 7r] x [Tr, 27r]. The space R(T) is homeomorphic to S 2 and A and B map to hemispheres. The trivial S 1 bundle over N 2 restricts to trivializations over A and B.


Fig. 1

(0,2n)

( (o,o)

\

\ xyb

/ A

/ =~'-'-" ~,0)

Y

Thus we need to compute the degree of the clutching function OA --+ S 1. Using the figure one can see that this degree is - 1 . []

The invariant c X (6) is an invariant of rel boundary flat cobordism. More precisely, if M is a 4-manifold with boundary X o U OX o • I U - X 1 and Q : 7riM ~ SU(2) is a representation then Cxo(O ) = Cxl(Q). This can be seen as follows. Choose a flat connection A on M with holonomy ~ and in normal form near OX o • I. Then the identity

2 Tr (F A A F A) = dTr (dA A A + 7A A A A A) .

2 Stokes ' theorem, and the observation that Tr (dA A A + 7A A A A A) vanishes when

A = i a d x + i~ dy imply that

1 S 0 = ~ T r (F A A F A) = CSxo (A) - CSx~ (A).

X •

We turn now to our main tool for computing the Chern-Simons invariants. This result generalizes the main theorem of [KK2]. It should also be viewed as an extension of the observation of the previous paragraph. Rather than considering a flat connection on X • I , we consider a path of flat connections on X which we view as a connection on X • I which is "flat except in the t direction."

2.7 Theorem. Let X be an oriented 3-manifold with toral boundary OX = T 1 L3 �9 .. U T n and let o(t) : 7f iX ~ SU(2) , t E [0, 1] be a path o f representations. Let (a l ( t ) , /31(t),.. . , an ( t ) , /3n(t)) be a lift o f o(t) to ~2n. Suppose

c(0(t)) = [a l ( t ) , 3~(t),..., a,(t), 3n(t); z(t)]

for all t E I. Then

( = e x p


In particular, if 6(1) is the trivial representation (so that z(1) = 1 by Corollary 2.6) then

F

e(~(0)) = ]oq(o), 3~(o),... ,c~n(o), 3n(o); [ _

n l exp( "a ' . '

Proof. Choose a path A(t) of flat connections on X in normal form so that near T~,

A(t) = iaa(t) dx + i/3~(t) dy

such that the holonomy of A(t) is ~(t). Orient X x I near OX x I by taking {dr, dx, dy, dr} to be an oriented basis. The path A(t) determines a connection A over X • I which satisfies Tr(F A A F A) = 0, since F A has only a dt component.

As explained in the paragraph preceding this theorem:

: ( . ) 1 Tr d C A C + ~ C A C A C 0 = cs(A(1)) - es(A(O)) - 87r~ O X x I

where C is the 1-form over OX x I given by C(z, y, t) = A(t). O n O X x I , C A C A C = O a n d

d/3k - da k \ Tr(dC A C) = 2 Z c%--d[- - /Jk--~-)

Therefore:

cs(A(1))_cs(A(O))= 1 / ( 2 ) 87r2 Tr dCAC-+- ~ C A C A C

O X x I

1

= ~ k ~ - - ~k - ~ �9

k=l 0

Since e27rics(A1)e -27ticS(A~ = z ( 1 ) z ( 0 ) - 1 , the theorem follows. []

We next give a more abstract restatement of the previous theorem, together with some additional differential geometric information about E(T).

2.8 Corollary. 1. The connection 1-form

A = -27ri Z akd~k --/3kdak k

on the principal U(1) bundle 1R 2n x U(1) descends to give an orbifoM connection on E(T) = ]R 2n • U(1)/G n. Given a 3-manifold X with boundary a union of tori, the lift c X : R (X) --+ E(T) of the restriction map is parallel with respect to the connection A. 2. The curvature of A is F A = -47ri ~ dot k A d/3 k. This is a real multiple of the

k symplectic form on the symplectic orbifold R(T). []


Observe that by Chern-Weil theory the Euler class is - 2 ~ dakdfl k. Restricting k

to each factor _R(Tk) c R(OX) we get -2dakdfl k. Finally integrating over the

fundamental domain for the action [0, 1] • [0, 1] we see that the Euler number is -1.

So this gives an alternative argument for the 3 rd part of Theorem 2.1. Consider the following situation. We are given a closed oriented 3-manifold Z and

a collection or tori: T 1U.. �9 UT n C Z. Let ~ : 7r 1Z -~ SU(2) be a representation so that if X 1 U--. U X ~ denotes the components of Z cut along C, the restrictions ~ = Qlx~ lie on a piecewise smooth path of representations &(t) to the trivial representation. (Notice that each toms appears twice in the boundary of U~Xc) Then, making use of Theorems 2.1 and 2.7 we can compute the Chem-Simons invariant of p. Theorem 2.7 is used to compute the Chem-Simons invariants of the pieces, and Theorem 2.1 to compute the effect of gluing the pieces together. In Sect. 4 we will describe how to compute the representations and Chem-Simons invariant for a number of 3 manifolds.

One last remark is needed for computations: For a given representation of a 3- manifold X there is usually a convenient choice of basis of 7r 10X determined by the topology of X and the representation. (For example, if p is an abelian representation and OX = T 2 it is convenient to take A to be the generator of Ker H~T --~ H~X since then A is sent to 1 along any path of abelian representations. For this choice, then, fl(t) can be taken to be the constant path at 0.) However, in glueing X to Y along T we must choose the same normal forms for Aix and Aiy to compute

(c(Aix) , c(Aiy) ). Any linear change of coordinates of T takes connections in normal form to connections in normal form. More precisely, if

then M determines a change of basis of the 1-forms

d X = p d x + r d y , d Y = q d x + s d y ,

and so if A is in normal form with respect to dx, dy it remains in normal form with respect to dX, dY. Of course, the Chern-Simons invariant does not depend on the choice of local coordinates. The point is that the vector space V(T) is intrinsic, but to do explicit computations requires fixing a choice of basis for 7qT.

We can now outline the "algorithm" we will use to compute Chern-Simons invariants in the simplest case of a decomposition Z = X t2 Y along one torus. If g : 7r 1Z ~ SU(2) is a representation then let Px and py denote the restrictions to X and Y. Choose convenient bases {#x, )~x} for 7rl(OX ) (resp. {py, )~y} for 7r~(OY)). Suppose there exist paths ~x(t) and #y(t) of the restrictions to the trivial representation. Then with respect to the bases we obtain paths C~x(t ), fix(t) and c~y(t), fly(t). We now compute the expression of Theorem 2.7 to get

c(Aix) = [ax(0) , fix(0); Cx]

and c(Aly) -- [C~y(0), fly(0); Cy].

Next we express {#y, Ay} in terms of {#x, Ax} using a matrix


Since the representations 8x and gy agree on the boundary, it follows that there are integers m, n such that

~x(0) ] Now Theorem 2.1 can be used

so that

(:)) ~y(o) ] + to adjust c(A[y) to get

c( Aiy ) = [ay (0), t3 v (0), Cy ] = [a x (0),/3 x (0); Cy e z~i(~zx (o)- n~x (o))]

Cz(9) = CxCye2~'i(mJ3x(O)-n~

3. S L ( 2 , C)-Representations

The results for SU(2) generalize to the larger group SL(2, C). Even for some SU(2) computations it is convenient to consider SL(2, C) representations since Hom0r, SL(2, C)) is an algebraic variety containing the real subvariety Hom(yr, SU(2)). In particular, a given SU(2) representation may not lie on a path of SU(2) representations to the trivial representation, but there may nevertheless be a path to the trivial representation in the space of SL(2, C) representations. There are some subtleties which arise in going to the SL(2, C) representations which we describe in this section. The essential point is that all the results of the previous section extend to SL(2, C) representations provided we work with the character varieties instead of the representation spaces.

Consider a (trivial) principal SL(2, C) bundle P = X • SL(2, C) over a 3 manifold X. As before the trivialization allows us to identify the space of all connections in this bundle with the sl(2, C)-valued 1 forms on X. The Chem-Simons invariant of A c Y2 x O s/(2, C) is defined as before, but notice that this time the expression

Tr (dA A A + ~A A A A A) is a complex-valued 3-form on X. Hence

1 ;Tr ( / = e A /, A + A A A A 87r z J

x

is in general a complex number. Changing A by a gauge transformation preserves csx(A)modZ when X is closed. We will construct a C* bundle over the space of SL(2, C) representations of a surface mod conjugacy in the same way as the SU(2) case.

Notice that SL(2, C) is a non-compact group, and the space

Hom(yr, SL(2, C))/conjugation

is often badly behaved, e.g. non-Hausdorff. There is, however, a natural further quotient of Hom0r, SL(2, C)), the character variety, which is an algebraic variety. Our immediate task is to show that the results for SU(2) representations of Sect. 2 extend to these character varieties. A good reference for these varieties is [CS].

Recall that the character variety of a finitely presented group 7r into SL(2, C) is the variety whose ring of functions is the functions on Hom(Tr, SL(2, C)) which are invariant under the conjugation action of SL(2, C). It can be explicitly realized as follows. There is a finite set -y j , . . . , -y,,~ of elements of 7r so that the functions

rTi : Hom(Tr, SL(2, C)) --~ C


defined by

~-7,(0) = Tr(0(70)

generate the ring of invariant functions. Then the image of the map

t : Hom(Tr, SL(2, C)) ---* C m

taking 0 to the n-tuple (0(3'1), . . . , Q(7,~)) is closed, and is the character variety. We denote this variety by Rc(Tr) or R e ( X ) when 7r = 7rlX. Two representations 01 and 02 have the same image in the character variety if and only if Tr(Ol(3")) = Tr(02(3')) for all 3' C 7r. In particular conjugate representations have the same character.

An SL(2, C) representation is irreducible if the natural action on C 2 has no invariant subspaces. If 01 and 02 have the same character and 01 is irreducible, then 01 and 02 are conjugate (see, for example [CS]. Proposition 1.5.2). Thus Rc(Tr ) is a quotient of the space of conjugacy classes of representations in which certain (non-conjugate) reducible representations are identified.

Suppose 00 and cO1 are representations with the same character on a closed 3- manifold. If one of them is irreducible, then they are conjugate and so their Chern- Simons invariants coincide Mod Z. Suppose they are both reducible and 01 is diagonal. By conjugating we may assume 00 is upper triangular. Define a path Pt by the formula

@t(3")= a - 1 whenever 0 0 ( 7 ) = 0 a -1 "

It is easy to check that this gives a path of representations from 00 to 01. Thus the Chern-Simons invariant of an SL(2, C) representation on a closed 3-manifold depends only on the character of the representation. It therefore makes sense to try to extend the results of the previous sections to the case of SL(2, C) representations, substituting character varieties for the spaces of conjugacy classes of representations. We first construct the bundle where the Chern-Simons invariants take their values.

3.1 Lemma. The map C 2 ~ Re(T) taking the pair (c~, ~) to the character of the representation defined by

0) 0) C--2rri,3

is the composition of an analytic branched cover with covering group G and a 1-1 and onto algebraic map.

Proof. Any representation 0 : Z �9 Z ~ SL(2, C) is reducible since Z (9 Z is abelian. Let L c C 2 be the invariant line. By conjugation we may assume L = {(z, 0)} so that 0(3") is upper triangular for all 3".

But then the map 0' : Z | Z --* SL(2, C) defined by

whenever

(o ~ ) 0'(7) = a_ l

is a representation with the same character as O. Thus the map is onto.

(;b) ~(~) = a_ 1


The map t : R c ( T ) ~ C 3 given by O ~ (Tr(o(#)), Tr(0(A)), Tr(o(#A))) is an algebraic embedding of Re(T) . The composite

C 2 ~ Hom(Z | Z, SL(2 ,C) ) t R c ( T ) C C 3

is given by (a,/3) H (2 cos(2zra), 2 cos(27r/3), 2 cos(27r(a +/3))

which is analytic. It is easy to see that the action of G (extending the action on N 2 defined in Sect. 2) is analytic and has quotient homeomorphic to Rc(T) . However, C2/G and R c ( T ) are not isomorphic as analytic varieties. This may be seen by comparing the dimensions of their Zariski tangent spaces at the singular points corresponding to central representations. The map C2/G ~ R c ( T ) is algebraic, 1-1 and onto. []

Construct the bundle E c ( T ) over R c ( T ) by the same formulas as the definition of E(T). This extends to the case when T is a union of tori. We also get an inner product

(,) : E c ( T ) x E c ( - T ) ~ C*

defined in the same way, and a partial inner product E(T U S) x E ( - S ) ---* E(T) when S and T are collection of tori.

3.2 Theorem. The Chern-Simons invariant defines a lifting c x : R c ( X ) ---* Ec(O(X)) of the restriction map from the character variety of X to the character variety of OX :

Ec(OX)

R c ( X ) , Rc(OX) such that if X E Rc(Y), Y = X 1 I J X 2 , and Xi denotes the restriction of x to Xi, then

exp(27ricsy (X)) = (e(x1), C()~2))"

Furthermore, the statement of Theorem 2.7 continues to hold with C 2n replacing N 2n.

The proof will occupy the rest of this section. Our first task is to extend the notion of normal form to an SL(2, C) connection. Since 7rl(T 2) is abelian, any representation

: 7rl(T 2) --~ SL(2, C) is conjugate to an upper triangular representation, and furthermore it is conjugate to a diagonal representation unless p is non-central and Tr(Q(7)) = -4-2 for all "7 G %(T) . Now let ~ : % ( X ) --* SL(2, C) be a representation. We consider the two cases when 0 is or is not diagonalizable on the boundary. Let #, A be a fixed basis for 7tiT. Case 1. If o) o)

~0(#) = 0 e -27ri~ and 0(A) = e_2~i~

for complex numbers a , /3 , then as before we define a flat connection A with holonomy conjugate to p to be in normal form if

o) - i a -i/3 dy


near the boundary. In this case, we define

CX(~) = [O~,/~; e27ricsx(A)],

where A is in normal form. Case 2. If Tr(o(7)) = +2 for all "7 C 7rl(T), then

( 1 1 ) (10 ~ ) 6(#) = ( -1 ) ~ 0 and 6(),) = ( - 1 ) v

for some complex numbers a, b and integers u, v. We find a connection with this holonomy using developing maps, as in [KK2]. The map D : ]I( 2 ---+ SL(2, C) defined by:

D(x, y) =

satisfies

e x P ( 2 ( u x T v y ) )

0

(ax + by) { i . .~ \ exp

e x p ( - ~ ( u x + v y ) ) )

D(x + 27rm, y + 2rrn) = D(x, y)Q(#m/kn)-l.

From Proposition 2.1 of [KK2] it follows that A = - d D D -1 is a connection 1-form with holonomy 6. One computes:

A = 2 27r dx + 2 27r exp(z ux + vy)) �9 dy.

0 ~u zv ] y o

We will say that a flat connection on T is in normal form if it is in the above form for some u, v, a, b. Notice that (as in the case of diagonal representations) the condition of being in normal form is independent of the choice of basis p, ~ for 7rl(T) (although the specific normal form does depend on the basis). A connection on a 3-manifold X with toral boundary will be said to be in normal form if it is in normal form near each boundary component. Write

Then define:

A = B(u, v, a)dx + C(u, v, b)dy.

Cx(6) = [ u v ] 2 ~ 2; e27ricsx(A) "

What must be shown is that cx(6) is well-defined and depends only on the character of 6. We will first show that it depends only on the conjugacy class of 6. In what follows we will assume for notational convenience that there is only one component of the boundary. Everything extends in the obvious way to the case when the boundary has several components.

When 6 is diagonalizable on the boundary the proof that c X (6) depends only on the conjugacy class of 6 is identical to the arguments given in Sect. 2. We will therefore consider the second case, when L) is parabolic on the boundary.

3.3 Lemma. If A and A I are two gauge equivalent connections which are in normal form and equal near the boundary, then cs(A) = es (X) Mod Z.


Proof There is a gauge transformation g which takes A to A t. Suppose first that 9 is the identity near the boundary. Then A U - A and A U - g " A are connections on the double of X which are gauge equivalent. However A U - A has zero Chem-Simons invariant and so A and A t = 9 " A have the same Chem-Simons invariant rood Z.

Og Since g �9 A = 9A9 -1 - d99 -1 near the boundary, it follows that Orr = 0, where

r is the inward normal. Thus gl T is a gauge transformation of T which fixes A. A computation shows that near T, 9 must be of the form

g(e~X' eiU) ---- • 1

for some complex number c. We may assume the + sign holds since if not, we may replace 9 by - g while preserving the fact that g . A = X . It follows (by varying c) that there is an arc of gauge transformations on T, all fixing AIT, joining g to 1. The rest of the proof is exactly like the proof of Theorem 2 .4 / so we omit it. []

Notice that conjugating an upper triangular matrix by the matrix

(o ~ w - l /

has the effect of multiplying the upper right-hand entry by 21) 2. The following lemma shows that this kind of conjugation does not affect the Chern-Simons invariant:

3.4 L e m m a . I f A and A t are gauge equivalent connections such that

A = B(u, v, a)dx + C(u, v, b)dy

and

A ~ = B(u, v, w2a)dy + C(u, v, w2b)dy

near OX then cs(A) = cs(A' )mod Z.

Proof. The outline of this proof is the same as that of 2.4 and 3.4. Let '7 : [0, 1] --~ C* be a path from w to 1, constant near the endpoints. Use 7 to define a path of matrices g given by

0 ) 9 t = 0 " 7 - 1 ( t ) '

constant near the endpoints. We extend 9 to be a gauge transformation 9 : X SL(2 , C) by using a collar of the boundary and extending it to be equal to 1 outside this collar. Now use 9 to define a path of gauge transformations 9~ from 9o = the trivial gauge transformation to 91 = 9, as indicated in Fig. 2


s aX

mallat

Fig. 2

Let A s = g, " A. Notice that each A s is in normal form. Since c s ( A 1 ) : cs(A') by 3.4 and cs(Ao) = cs(A), it will suffice to show that cs(Ao) = cs(A1). Since these connections agree outside of a collar, we need only to compare their Chem-Simons integrals on T 2 • I . Let A be the connection on T 2 x I x I which is the union on the A s. Then

1 f Tr (F A A F A) 0 - 87r2

T 2 x I x I

: C S T 2 x l x i ( A 1 ) - - C S T 2 x l x i ( A o )

/ ( ) 1 Tr d A s A A s + 3 s 871. 2 A A s A A s .

OT2 x {O,1} x I

It remains to prove that the integral vanishes. Since this is obvious over T 2 • 1 x I , we concentrate on (I '2 • 0 • I = cgX x I. Since A s A A s A A, = 0, we must show that f Tr(dA s A A s) vanishes.

O X • Now

A s = B(u , v, 72(s)a)dx + C(u, v, 72(s)b)dy

near the boundary. Then A s A A s A A s = 0 and

Tr(dA s A As) = Tr \ as - B dx dy ds

- - 0 . []

Let g~ and gv be the gauge transformations constructed in Sect. 2. Notice that near the boundary,

g~g~ " (B(u , v, a)dx + C(u, v, b)dy) = B ( u + 2m, v + 2n, a)dx

+ C(u + 2m, v + 2n, b)dy.


(Notice that conjugating by

J - - - ( - - 1

takes an upper triangular matrix to a lower triangular matrix. Perhaps to include the action of gb it would have been more elegant to consider upper and lower triangular matrices in the definition of normal form. This makes no difference in the end.)

3.5 Lemma. If A is a fiat connection with holonomy 0 in normal form and with

e x ( A ) = - ' 2 ' z ,then

] cs(g~ gy �9 A) = - ~ + m, - 2 + n; ze ~i(un-vm) .

Proof. One first computes that if A = B(u, v, a)dx + C(u, v, b)dy on OX x I, then

gx " A = B(u - 2, v, a)dx + C(u - 2, v, b)dy

in a smaller collar of the boundary OX. Since 9x is the identity outside the collar OX x I, just as in the proof of Theorem 2.5 we must show that

1 / T r ( d E A E + 2 E ) 87r2 -~ A E A E = v, O X x I

2 where E = g~ �9 A. Write g = gs, and since E is flat, dE A E + 5E A E A E =

1 1 f Tr(E A E A E). - 5 E A E A E. Therefore we need to compute - 5

Since E = gAg -1 - dgg -1, a small calculation shows

- 1 - T r ( E A E A E ) : T r ( C ( [ g - 1 0 9 ' 3 L Or B-g-lOgOxl) ) '

Og where B = B(u, v, a) and C = C(u, v, b). (The fact that ~y = 0 is used here.)

Wef lrs t showthat fTr (C([g- l~-rr ,B]) = 0. The reader can check that since g

isindependentof y, T r ( C ( [ g - l ~ r r , B ] ) is the product of ei(Uz+~Y) with a function

h(x, r) which is independent of y. So

f Tr ( C ( jig- ) f l ~ B] = h(x,r)ei~(fe~VYdy) dxdrO.= S 1 X1 S 1

( ( [ - ' O g g - ' 0 9 1 ) ) i s handled as follows. Write The other term Tr ,. , , / J C [g Or' OxJ '

p(x,y) ~ [g-109,9 -1 091 We know from the proof of Theorem 2.5 that if [ Or OxJ"

,=(; o)


then

Now

So

1/ 871-2

O X x I

Tr( Ip( x, y) ) dx dy dr -- 1.

(~ ~ C = v I + e ivy ~ . 0

= v / T r ( I p ) d x d y d r + /e iVy(some function of x and r )dxdydr

= 871-2V. []

This completes the proof that the map c defines a lifting from the conjugacy classes of representations of X to E c ( T ). It remains then to show that the result depends only on the character of the representation. If two representations have the same character and one is irreducible, then it is shown in [CS] that they are conjugate. Thus it suffices to consider the case when two reducible representations have the same character. First notice that two diagonal representations with the same character are conjugate. Furthermore, if Q : 7rlX ~ SL(2, C) is a representation, the path of representations Os defined by

whenever

LOs(~/) : a - 1 ]

is a path of representations to a diagonal representation. Thus it suffices to show that for this path, cs(Oo) = cs(ol) M o d Z .

Since the upper triangular matrices U form a subgroup of SL(2, C), there is a developing map D : X -+ U with holonomy 6. But then

whenever (ab) D ( x ) = 0 a -1

is a path of developing maps with holonomy ~s. From this it follows that we have a path of flat connections A s in normal form with A s -- B(u, v, (1 - s)a)dx + C(u, v, (1 - s)b)dy near OX. Thus

O = c s ( A o ) _ c s ( A l ) i f T r ( d A s A A s + 2 A A A s A A s ) 87r 2 3 s �9

O X x I

= ( a ( 1 - s ) b ' ~ D~(x) \o a -1 ;

(ab) 0 ( 7 ) = 0 a -1

Chern-S imons Invar ian ts of 3 -Mani fo lds 543

Clearly A s A A s A A s = 0. When we compute dA s A A s, we find its diagonal entries and hence its trace to be 0, just as in the proof of 3.4. This completes the proof of Theorem 4.2. []

The results of Sect. 1 and 2 apply also to SO(3) = PSU(2) and PSL(2 , C) representations with some modifications. We outline one way to do this. It is perhaps not the most general way to extend the results, but it suffices for our applications.

Let R(T, t) denote the conjugacy classes of PSU(2) = SO(3) representations of Z | Z which have trivial second Stiefel-Whitney class, in other words so that the associated fiat bundle is trivial. Thus R(T, t) is the image of the SU(2) representations under the map induced by the homomorphism SU(2) --~ PSU(2).

Let H = ( X , Y , b I IX, Y] = X b X b = Y b Y b = b 2 = 1).

So H is isomorphic to G. Think of G as a subgroup of H via the map x ~-+ X 2, y ~ y2 , and b ~-+ b. (So ( H / G = Z / 2 | Z/2.) The action of G on N 2 extends to H via

1 X . (a,/3) = (a + [ , /3), Y . (a,/3) = (a,/3 + �89 b- (a,/3) = ( - a , -/3).

Extend this action to an action of H on I~ 2 x S 1 by

1 1 4rri~ ~ x . (~ ,9 ;z )= (~+~,/3;ze-4~i9), y.(~,/3;z)= (~,/3+~;ze ), b. (~,/3; z) = ( - a , - /3; z).

The quotient bundle E(T, t) is well defined. If R(X, t) denotes the PU(2) representations of a 3-manifold X which have trivial w 2 on the boundary, then c x defined a parallel lift of the restriction map R(X , t) --+ R(T, t). We remind the reader that PSU(2) = SO(3) Chem-Simons invariants are based on the first pontrjagin class, and the map H4(BSO(3)) --+ H4(BSU(2)) takes Pl to - 4 c 2, hence the sign difference on the action of H on IR 2 x S x.

Similarly replacing ]R 2 by C 2 in this discussion leads to a C* bundle over the character variety of PSL(2 , C) representations having trivial w 2.

Finally suppose we are given a manifold X with toral boundary and an arc Qt of SL(2, C)-representations of rq(X), and we wish to calculate the difference between the Chem Simons invariants of p o ~1 and p o O0, where p : SL(2, C) ~ PSL(2 , C) is the quotient map. We may calculate this difference using an integral just as in Theorem 2.7, excjept that we must introduce a multiplicative factor of - 4 because of the formula relating Pontryagin classes and Chern classes given above.

4. Chern-Simons of Manifolds with Torai Boundary

In this section we describe the representation spaces of several 3-manifolds with toral boundary in enough detail to use the results of the previous sections to compute their Chern-Simons invariants.

Our first result concerns abelian representations of X when H 1X is torsion free.

4.1 Theorem. Let X be a 3-manifold with boundary OX = T 1 U . . . U T n. Assume H1X is torsion free. Choose syrnplectic pairs #k, Ak for H1T k. Let x i, i : 1 , . . . , m be a basis for H1X and write

P k = ~ ak jx j and A k = ~ bkjx j .


Suppose that 0 : 7rlX ~ SU(2) is an abelian representation and let ~/j E ]R so that O(xj) = e izj �9 Then."

Cx(0) = [ E a l j T j , E b l f f / j , ' " , E a ~ j T j , E b n j ~ / j ; 1 ] �9

Proof. Since the representation is abelian, the formula:

Q(t)(xj) = ei(1-t)'YJ

defines a path of representations from 0 to the trivial representation. On the boundary toil this path takes the form:

o(t)(/z k) = exp (i(1 - t)( Z akj~/J))' 0(t)(Ak) = exp (i(1 -- t)( E bkff/J))-

So we can choose the paths a(t) and/3(t) to be:

= (1 - t ) ( ~ - ~ (~k(t) a kj"/ j )

and

i lk ( t )= ( 1 - - t ) ( E b k j " , / j ) .

But then OZkd/3 k -- /3kda k = O. The theorem now follows from Corollary 2.6 and Theorem 2.7. []

For example, if X is the complement of a knot in a homology sphere, let #, A denote the natural meridian and longitude. If 0 : 7rlX ---* SU(2) is an abelian representation such that 0(#) = eia then

Cx(O) = [a, O; 1].

The next examples we consider are 3 manifolds of the form X = F x S 1 for a punctured surface F . By cutting along a toms we can consider the two cases: 1. F is a once-punctured surface of genus n. 2. F is a planar surface.

Consider first the case of a once-punctured surface F . We can describe the representation space. Write 7r = 7r~(F x S ~) = @1, Y l , . . . , xn, Yn) x (A). The curves # = l-[[xi, yi] and A generate ~I(OX). We take the orientation of A so that # and A

i form an oriented basis. Since the centralizer of any non-abelian subgroup of SU(2) is {-4-1}, any non-abelian representation of 7r must send A to 4-1. Moreover, any representation which restricts to an abelian representation of 7r I F is abelian.

Thus if ~ : 7rlX ~ SU(2) is any representation, either 1. ~ is abelian and Q(#) = 1; or, 2. p is non-abelian and p(A) = 4-1.

In the first case, let/3 c N so that 0(A) = e 2~i~. Then from the previous result

Cx(O) ---- [0,/3; 11.

T h e second case is slightly more complicated. First note that a homomorphism from a free group to SU(2) is determined by its values on the generators and so Hom(zqF, SU(2)) can be identified with SU(2) 2n via the map 0 H (O(Xl), 0(Yl) , . . . , O(Xn), O(Yn))" This space is path connected so there exists a path 0(t), t E [0, l] from ~ to the trivial representation which fixes A. We can conjugate


this path so that p(t)(#) = e 2trice(t). Along this part of the path fl is unchanged. We can

take 3 to be either 0 or �89 depending on whether O(A) = 1 or - 1 . The representation

Q(1) is abelian, and hence has Cx(O(l)) = [0, 3, 11. Since Q(1)(#) = 1, a(1) is an integer. Therefore we can re-write

ex(O(1)) = [0, 3, 11 = [a(1), 3; e2~iflc~(1)] �9

The path 3(t) is just the constant path at 3 and so:

1 1

i '~ -.I'~ a(t) - fl-jt- = -~- = 3 ( 0 4 0 ) - a(1)).

0 0

Using Theorem 2.7 we can compare Cx(O(O)) and ex(O(1)): If Cx(Q(O)) = [a(O), fl; z], then

e27riflc~( l) z - 1 = e27r i f l ( a (0 ) - - a (1 ) ) .

Write a = a(0). Then

Z = e 27 r / ( - c~a+2f l c f f l ) ) = e -2~rifl~

since cffl) is an integer and 3 E Z[�89

Therefore:

{ [ a , 0; 1] if L)(A)= 1, Cx(~) = [a, 3; e-2'~ifi~l = , 1. _ , ~ 1

tce, g ,e 1 i f o ( A ) = - l .

We turn now to the case of a planar surface F crossed with S 1. If F is a planar surface then its fundamental group is again free. This time we write 7rl(F x S 1) = ( # l , . - . , #~1#1 . . . #~ = 1) x (A), where the #k are loops following the k th boundary component of F . The symplectic pairs #~, A generate the fundamental group of the ith boundary component. Let Q be a representation. Suppose first that the restriction of 0 to F is abelian. Then ~ is abelian. Choose a k and 3 so that 0(A) = e 2~i~ and Q(#k) = e2'~i'~k for k = 1 , . . . , n - 1. Since #n = ( # l . - . # n - 1 ) - l ) , we can take

n - - I

O~ n z - - E O~k- i

The generators # 1 , . . . , # n - l , A form a basis for H 1 X and so by Theorem 4.1:

C X ( Q ) = O L 1 , 3 , ' ' ' , C t n - l , 3 , - - 0 ~ i , 3 ; 1 . i = 1

Consider now the case when ~ is non-abelian. Choose ak, 3 as before, except that now O(#k) is only conjugate to e 2~i~k. Since A is central, 0(A) = + l , and

so 3 C Z[1] . Since the fundamental group of F is free we can find a path of

representations t~8 from Q to a representation whose restriction to F is trivial, and so that A is fixed along the path. We may also choose continuous functions ak(s ) so that for each s, t)s(#k) is conjugate to e 2'~iok(s). For this path,

n n


Notice that c~k(1 ) E Z for all k, and ak(0 ) = c~ k. Since Q1 is abelian, and trivial on 7r 1 (F), Theorem 4.1 implies that

Cx(O~) = [0, f l , . . . , O, r ; 11

= [ a l ( 1 ) , f l , . . . , c t n ( 1 ) , f l ; e x p ( 2 z r i / 3 ~ a i ( 1 ) ) ] .

We now apply Theorem 2.7 to the path Os. Using the fact that 2ak(1) fl is an integer for each k.

ex(~)= [OZl,~,...,an,/~;e--2~ri~kak 1.

Let F be a k times punctured genus n surface. By decomposing F into a planar surface and a once punctured surface and applying the inner product E ( T 0 S) x E ( - S ) ~ E(T) , we obtain the following result:

4.2 Theorem. Let F be a genus n surface with k punctures and let X = F x S 1. Write

7rlF : (Xl,Yl,' ' . ,xn,yn,#l,' ' . ,#k I H[xj,Yj]#I.''~k ~- 1 )" J

Give ~rlOX the bases # j x * and A = * x S 1 (pick a path from the base point to each torus in O X to view these in IrlX). Let O : 7rlX --+ SU(2) be a representation. Let a j , fl E IR be defined by O(A) = e 2~ri~3 and O(#j) = e2~i~Y (i.e. for each torus Tj C O X there is an element of SU(2) which conjugates O to this form on Tj). Then k [ ] 27r~ %

Cx(O)= ~ l , / 3 , . . . , c ~ k , 3 ; e . []

Another class of examples for which computations are possible are the Chem- Simons invariants of certain representations of surface bundles over the circle which we called special representations in [KK]. D. Auckly has computed the Chern-Simons invariants of special representations of any (closed) surface bundle over S 1 in [A]. Using Auckly's computations, Theorem 2.1, and the computations for the solid toms given by 4.1 one can compute the Chem-Simons invariants of special representations of surface bundles when the surface has one boundary component. We sketch the idea, leaving the details to the reader.

Recall that a special representation of a surface bundle is one whose restriction to a fiber is abelian. Let b : F -+ F be a homeomorphism of a once punctured surface to itself which is the identity on the boundary and let M = F • B SI be the mapping toms. The fundamental group of M is 7riM = ( x l , . . . , x2n , t ] t x i t -1 = b,(xi) ). Thus the boundary of M has basis

# = t

and n

A = H [ X 2 j _ I , x2j]. j=l

Notice that since b is the identity on the boundary, it extends to the closed surface obtained by capping off the boundary. Moreover, the representation also extends since


p(/z) = 1. Let M denote the closed surface. Then ~ / = M U X, where X = D 2 x S 1. We give the boundary of the solid torus the basis A x ----- OD 2 x {1} = ), and # x = {1} x S 1 = #. By Theorem 4.1,

Cx(Q) = [0, 1; 1].

The computation of c s~ (g ) is in [A]. We refer to that paper for the precise formula since it involves some notation which we do not repeat here. Writing c~(Q) = exp(27rics~(o)) we conclude from the inner product of Theorem 2.1 that

CM(Q) = [0, 1; e2rrics~(O)].

As an example, since j is a 4 m root of unity any special representation extends to a representation of the closed manifold obtained by Dehn filling so that the curve p4k)~p is killed. In particular this gives the Chern-Simons invariants of some representations

4k of - - surgeries on fibered knots.

P We next turn to a computation of the Chern-Simons invariants of two types

of closed 3-manifolds. The first are circle bundles over oriented surfaces. These manifolds are the boundaries of tubular neighborhoods of embedded surfaces in 4- manifolds. The second type of manifold we consider are graph manifolds obtained by glueing two Seifert-fibered manifolds together along their boundary tori. This second application is needed in [KKR] in the computations of the spectral flows between flat connections on these manifolds. These examples show how to use the inner product in Theorem 2.1 to compute the Chern-Simons invariants of closed manifolds from the knowledge of the pieces.

Let F be an oriented surface with one boundary component and let _~ be the closed surface obtained by glueing a disc to OF. The circle bundle M ( n ) over/> with Euler class n is obtained by gluing F • S 1 to D 2 • S 1 using the map ~ : OF • S 1 ---+ OD 2 x S 1 given by (z, w) ~-+ (z, znw), where we have identified OF and OD 2. Figure 3 shows the images of R ( F x S 1) and R ( D 2 x S 1) in R(T) . The coordinates are OF • �9 and �9 X S 1 .

R(Fx$1) ~ . . ~ R(FxS ) g ( ~ sb

Fig. 3 n e,,en n odd

(Note that i f /~ = S 2, then the only representations of F x S 1 lie on the left edge of R ( T ) since there are no non-abelian representations of D 2 x S 1.)

One concludes from the figure that there are 3 types of representations: the first are representations which factor through the projection % M ( n ) ~ % F; these have their image in the lower left corner. The space of such representations is homeomorphic to


the space of representations of 7rl/W. The second type are the representations whose restriction to 7r 1F are abelian and non-central. These correspond in the figure to the

intersections along the left edge of R(T). The space of such representations has [2 ]

components; each component is homeomorphic to (S1)2g/(Z/2), where g is the genus of F and the generator of Z /2 acts by conjugation in each factor. The third type, corresponding to either the top left or top right corner in the figure depending on the parity of n, are representations whose restriction to 7rlF send OF to - 1 ; the space of such representations is homeomorphic to the SO(3) representations of 7r 1/~ which have non-trivial second Stiefel-Whitney class.

From these observations one can already conclude that the representations of the first type have zero Chern-Simons invariants, since they extend flatly over the D 2 bundle over/~. We can also conclude that the representations of the third type

have Chern-Simons invariants in Z [�88 since the corresponding SO(3) representations

extend over the D 2 bundle over/~. W r i t e X = F x S 1 and l e t Y = D 2 x 5 '1 and l e t p x = 0 F x * , A X = , x S 1,

# y = OD 2 • , , and Ay = * • S 1. The glueing map is then defined by # x ~ # v + n A v and A x ~ A v.

Let O : M ( n ) --~ SU(2) be a representation which satisfies O(#x) = 1 and

Q ( A x ) = e 2 ~ i ( ~ ) f o r s o m e k = l , . . . [ 2 1 . T h u s Q i s a r e p r e s e n t a t i o n o f t h e s e c o n d

type, or n is even and ~ is a representation of the third type. In either case we may assume the restriction to F • S 1 is abelian; this already holds for the second type of representations and for the third type (with n even) there is a path of representations of 7r IM(n) from 0 to an abelian representation.

[ -k ; l l . N o t i c e t h a t ~ ( # y ) = e - 2 ~ k = l a n d Then by Theorem 4.2 Cx(~) = O, n

27ri~ - ' 1 = - k , - k ; e . The last equality 0(Av) = e n. Thus cg(0) = 0, n n

follows from the definition of the action of G on N 2. We can now take the inner --27ri k~

product since we have chosen compatible lifts. Thus CM(n)(p) = 1 �9 e , and hence the Chern- Simons invariant is

k2 CSM(n)(O) -- n

This leaves the representations of type 3 when n is odd. From the discussion of F • S ~ we know +,]

Cx(O)= , ~ ; e .

Also, g(py ) = 1 = e 2~ril~'~ and 0(Ay) = - 1 = e 2"i~. Thus

[- 1 1 . 1 - n ~;e2~i cy (Q)= 0, 2 ' 2 '

Taking the inner product we get:

1 1 - n n CSM(n)(~) = - -4 + T -- 4


We summarize:

4.3 Theorem. Let Q : 7rl(M(n)) --~ SU(2) be a representation o f the circle bundle over a closed, oriented surface F with Euler class n. Conjugate 0 so that the fiber is

k k e sent to e 2'~i~. Then either/3 = - , in which case cs(p) - , or else n is odd and

n n 1 n

/~ = -~, in which case es(Q) = - - ~ . []

We now compute the Chern-Simons invariants of certain graph manifolds. See [FS1, FS2, and KK1] for background on the representation spaces of Seifert-fibered manifolds. We consider the following situation: X and Y are two Seifert fibered manifolds over the disc with H 1 X = • = H 1 Y and 9) : O X --~ OY is a homeomorphism. For simplicity we will assume that X and Y are the complements of regular fibers in Seifert-fibered homology spheres or, equivalently, that the Euler class of the Seifert fibration is equal to 1. This restriction is merely for convenience: the formulas are easier to write down. (These examples are the examples for which the Floer homology grading were computed in [KKR].) The general case of a manifold decomposed along a union of tori into Seifert fibered pieces can be handled similarly. Glue X to Y using 99 to get the closed manifold Z~. We will describe the representation spaces and Chern-Simons invariants of X and Y and then apply Theorem 2.1 to calculate the representation spaces and Chern-Simons invariants of

We start first with the computation of Cx(O) for a non-abelian representation 0. Write:

7 r l X = ( X l , . . . , x m , h I h central, x ~ i h bi = 1).

Our hypothesis on the Euler class implies that the integers a i, b i can be chosen so that

m

i=1 i

Write a = a 1 . . . a n .

The pair # x = x l " " X r , and "~x = h forms a basis for 7rlOX. Assume the orientation of X is chosen so that this basis is oriented in the induced boundary orientation. If 0 : 7rlX --* SU(2) is a non-abelian representation, then since h is central it must be sent to -4-1. The relations x~*h b* = 1 then force 0(x r) to be conjugate to e "~t~/a" for some integers lr. Conjugate p so that Q(Xl... Xr,) = e 2 ~ for some a C R.

Consider the free linkage (cf. [KK1]) in SU(2) consisting of the geodesic segments joining 0(Xl . . .x r ) to p(xl . . .x~+l) . So the r th strut of this linkage has length cos(vrlr/%). This linkage determines the representation p and moreover the space of such (free) linkages maps onto R ( X ) in the obvious way.

Consider the following 1-parameter family of free linkages L t with L 0 corresponding to p. As t increases, pull the free endpoint of the linkage along the circle e is until the linkage winds around this circle in a monotone way. (That this is possible follows from the argument on p. 82 of [KK1], namely, that the distance from 1 E SU(2) is a Morse function whose only critical points occur when the free linkage lies entirely on the circle.) Define the path of representations p(t) by leaving h fixed along the path and sending x I �9 �9 �9 x~ to the r - i st endpoint of L t. Notice that the endpoint p(1) is abelian. Furthermore p(t)(#) is just the free endpoint of L t. Write e 2~i~(t) for this


endpoint. Then

and

a(O) = a

r = l

1 Along this path, fl(t) is constant since h is fixed at ~1. Choose /3 to be 0 or

according to whether p(h) = 1 or - 1 . Thus

l

a d / 3 - / 3 d a = / 3 a - ~ . (*) 0 r = l

We next need to compute c X (0(1)). Since 0(1) is abelian we can use Theorem 3.1. Notice that # generates H1X since (using additive notation)

a# = a E x,. = - Z ~Tb,~h = - A T

and since HIOX ---+ H1X is onto. Using Theorem 3.1 we see that

[ ~ Ir ' - a E lr ' r za,. Cx(0(1)) ---- a-7-_" 1 .

Write

e = a E 1---- r ar

T

lg, We consider first the case when/3 = 0. Then - a ~ ~ E Z and so

I 11 Cx(~~ = ~ a ' 2 ' = ~aa'O;e

Applying ( ,) (with/3 ----- 0) and Theorem 2.7 we conclude

a, 0; e -2"i'~ ] Cx(~O) =

l 1 r 1 Now suppose 9 = ~ . Then - a V E 2a-7 - ~ ~ Z and so

c x ( # ( 1 ) ) = 2 a ' 2 2 + 2 ; 1 = , ~ ; e

This time the integral in (,) equals - - - - e oL

4a 2 and so by Theorem 2.7,

�9 e 2 o~


In either case we can write the exponent as -27ri . From the covariance

of the line bundle and the fact that 2/3 E Z one can easily show that the equivalence

-27ri is of the choice of and/3, i.e. class c~,/3; e ~ + ; ~ independent

[o 1 Abelian representations can be handled using Theorem 3.1. Thus:

4.4 Proposition. Let X be the complement o f a regular f iber in a Seifert f ibered homology sphere, with

71-1X = ( X l , . . . , X m , h l h central, x a i h b~ = 1)

f o r integers a i, b i chosen such that ~ abi = l, a = a 1 . . . a m. Let p : 7fiX -+ SU(2) ai

be a representation taking x 1 �9 �9 �9 x m to e 27ria and )~ to e 27ri~3.

(i) I f p is non-abelian, (so that/3 �9 Z [�89 then

with respect to the basis # = x 1 . . . x m and A = h f o r % O X . (ii) I f Q is abelian, then

Cx(~) = [a , - ac~ ; 1]. []

Now let Y be another Seifert-fibered space over D 2 with

7 q Y = ( Y l , " ' , Y n , k ] k central, yCihdi = 1),

and ~ cd-A = 1 where c = q . . . c n. We glue X to Y using a diffeomorphism i ci

~o : O X ---+ O Y to obtain Z~. One remark about orientations is needed: The orientation of Z~o induces an orientation of X and Y which in turn orient their boundaries O X and OY. Now it may happen that these orientations are not the same as the orientations given by the ordered bases { x l . . . X m , h} and {Yl " ' ' Y n , k} .

So we assume Z~ is given an orientation and that e x and ey are signs so that

Z~ = e x X U~o e y Y .

We choose the bases (#x , Ax) = (xl "'" Xm, h~ ) and (#y , Ay) = (Yl " " Yn, k~). Let ~ : O X ~ O Y be an orientation-reversing homeomorphism given in these bases by ~ ( # x ) = U # y + W A y and ~(A X) = VlZy + ZAy. So u z - v w = - 1 .

Let ~ : % Z~ ---+ SU(2) be a representation. Assume that the restriction of ~ to both X and Y is non-abelian. The case when one or both restrictions are abelian is handled in the same way and is easier.

The restrictions to X and Y define real numbers c~ X and C~y, half-integers /3x and /3y , and integers e X and ey as above. Specifically:

O(Xl " " X m ) = e2rc iax , ~(Yl "'" Yn) = e 2 ~ i a r ,

o(h) = e 2rri/3x , and Q(k) = e 2rci/3Y .

552 P. K i rk and E. K las sen

Taking the signs into account:

-2~i~x t , ~ + ~ %xX(O) = a x , e x g x ; e

and

{ --2rcieV ~ 4c +~YaY ]. Cevv(O) =- Lay, EY3Y; C

According to Theorem 2.1 e 2~ic*(~) is the inner product of these two. We need to evaluate this inner product. This requires fixing compatible lifts to IR 2.

Fix lifts a v and/3y. Then

a x = u a v + e v W 3 y (*)

and e x 3 x = r a y + e v Z 3 v (**)

are compatible lifts. This has the consequences:

1. To decide whether or not there is a representation of % Zr which restricts to some given representations of X and Y one first checks that the ( ,) and (**) hold modulo

the action of G on ]~2. Since/3 x and/3y lie in Z [�89 It follows that a x and a v are

rational numbers with denominator 2v. (If v = 0 then Z~o is Seifert fibered and the computations are easier. We will assume v # 0.) 2.

-~- ]UOgy -J- g y W 3 y , V a y q- g y Z ~ y ;

e 2

Write p /g

a y = E , /~Y=7 for some integers p and ~. Then

�9 e 2 p

C / J .

We can now take the inner product:

) ) ( e 2 "J- g x ( U a y q- eyW/~y ) (Vay q- g y Z ~ y ) -- e y k, 4C "j- OLy/~y cs&~ = - e x \ 4a

g2 e2 _ _ f f ~ X _ C '~Y -- x 4a Y 4e - a Y 3 Y ( e Y ( 1 + uz q- vw)) - oe2 uv -/~2yWZ

e2 C2 -- --C ~X _ ~ ~ P 2u Wk2(2eyp + Z). - - x 4 a Y 4c 4v 4


Notice that ~eyp = pM odZ . We have shown that first part of:

4.5 Theorem. Let p : 7r 1Z~o --* SU(2) be a representation. 1. I f the restriction of O to X and Y is non-abelian, then the Chern-Simons invariant o f p is

g 2 e 2 --e '~X _ e ~y p2u W k 2 x 4a v 4c 4v 4 (2p + z) Mod Z.

2. I f the restriction of O to X is abelian, and the restriction to Y is non-abelian, then suppose Q(Px) = e2~i'~x, O(Ax) = e 2 ' ~ x . Then the Chern-Simons invariant o f 0 is

e 2 - e ~r + a2x(ZV _ e x a ) _ 2exvwCexl3 X _ t32xWU.

Y 4c

3. I f the restriction of both X and Y is abelian, then suppose ~(#y) = e 2 ~ r , o()'v) = e2~izv. The Chern-Simons invariant o f Q is

--OLy( g yCO~y + s X /~y ).

The proof for the second and third cases are similar to the proof of the first, using Proposition 4.4. []

The methods of Proposition 4.4 and Theorem 4.5 apply to any Seifert-fibered spaces or graph manifolds with few modifications. In light of standard results in 3-manifold theory, in particular the torus decomposition theorem, in addition to computations for Seifert-fibered 3-manifolds it would be useful to understand the representation spaces and Chern-Simons invariants of hyperbolic 3-manifolds. In the examples outlined above, the images R ( X ) ---+ R ( T ) are lines (more precisely their preimages in 1R 2 are straight line segments) but for a general 3-manifold, in particular for a hyperbolic 3-manifold with toral boundary, the image of the restriction map R ( X ) ~ R ( T ) can be a quite complicated curve. The polynomials of [CCGLS] are essentially the defining polynomials for the variety Im(Rc(X) ---+ Rc(T)) for a knot complement X. In principle one can use their polynomials to parametrize the image Rc(X) ---+ Rc(T) and then apply Theorem 3.2 to compute Chem-Simons invariants. To get started one needs to understand what happens at one representation and then one can apply Theorem 3.2 to see what happens at other representations. There is a natural representation on a hyperbolic 3-manifold, namely the holonomy representation of the complete hyperbolic structure.

We will show how a formula of Yoshida [Y] (see also [NZ] and [H]) relating the volume and Chem-Simons invariant of the metric connection on a hyperbolic 3-manifold relates to our cut-and-paste approach for Chem-Simons invariants of SL(2, C) representations. From this formula and a knowledge of the Dehn-surgery space for a cusped hyperbolic 3-manifold X one can obtain information about the Cheru-Simons invariants of flat connections which lie on the path component in Rc(X) which contains the holonomy of the complete hyperbolic structure. This can be useful even in computing SU(2) Cheru-Simons invariants since there might be paths joining SU(2) representations in SL(2 , C) but not in SU(2).

Let X be a complete hyperbolic 3-manifold, and let O0 : r qX --+ P S L ( 2 , C) be the holonomy representation. In general, given a representation 0 : % X --+ G let P(O) denote the associated fiat G bundle P(O) = X x~lX G; its fiat connection is

induced from the trivial connection on X x G. (Here X denotes the universal cover of X.)


We construct a map from the oriented orthonormal frame bundle of X to P(Q0) as follows. Fixing a frame over a point in hyperbolic 3-space H identifies the frame bundle of ]HI with PSL(2 , C). Let p : PSL(2 , C) ---+ H denote the projection. Then the map

p • id : PSL(2 , C) -+ H x PSL(2 , C)

is equivariant with respect to the action of 7rlX and descends to give a map of principal bundles q : F ( X ) ~ P(Qo) covering, the identity map of X. (In particular, P(~0) is trivial since F ( X ) is and if s is a section of F(X) , q o s is a section of P(00)-)

Let M be a closed hyperbolic 3-manifold. Let Vol(M) denote the volume of M and cs(M) the Chem-Simons invariant of the Levi-Civita connection on M. Then it follows essentially from Lemma 3.1 of [Y] that

i ESM(QO ) ~- cs(M) - -~ Vol(M).

Although Yoshida does not state it exactly this way, he defines a complex valued 3-form C on PSL(2 , C) with

= ~ d v o l + i c s + dT, C

where dvol is the pullback of the volume element on H, cs is the Chem-Simons form for the Levi-Civita connection on PSL(2, C) = F ( H ) and d7 is some exact form. To obtain the formula above, one takes the projection r : H • PSL(2 , C) --~ PSL(2 , C) and computes that r*(C) is just i times the Chern-Sirnons form for the trivial (flat) connection in the principal bundle H x PSL(2, C) ~ ]HI. The form C is invariant with respect to translation in PSL(2 , C). Thus, if s c F(F(M)) , and ~ = q o s E F(P(L)0)), then ~*r*(C) = s*(C), so that:

cs(Oo) = - i g* r* (C) = - i s* (C) = cs(M) - -~5 Vol(M).

M M

Now suppose X is a cusped (complete, finite volume) hyperbolic 3-manifold, and let D denote the Dehn surgery space - the space of deformations of hyperbolic structures on X (see [Th]). Let d o E D be the complete hyperbolic structure. For d E D let gd denote the holonomy representation of the (incomplete) hyperbolic structure d. Yoshida shows that there is an analytic function f : D ~ C so that if d E D corresponds to an incomplete hyperbolic structure which completes to give a closed hyperbolic manifold M(d) then

f ( d ) = e x p ( 2 V o l ( M ( d ) ) § k

where 7k are the geodesics added to X to complete the k th cusp. We claim that f (d) is just Cx(Qd) (up to a constant) and that this formula is

equivalent to the formula of Theorem 2.1,

CM(d)(~d) = (Cx (Qd)' CD2 X S 1 (Qd))"


To see this, first of all fix a meridian and longitude pair #k, Ak, k = 1 , . . . , n for each cusp. There are coordinates ~ : D ~-~ C n such that ~(d0) = 0 and if d is an incomplete structure, Pa(#k) is conjugate to

( e2~i~k(a) E -27ric~k(d) ) '

where c~k(d ) is the k th coordinate of c~(d). (See [Th] and [NZ] for details.) The image of c~ is the intersection of an analytic variety with a neighborhood of 0. Let /3(d) = (/31(d),... ,/3n(d)) be defined by

~d(Ak) (e27rit~k(d) V_27ri~k(d ) )

together with the stipulation that/3k(d0) = 0 for all k. Then/3k is the product of c~ k and some analytic function ~-k : D ~ C. The map taking d E D to the character of ~o d is an analytic embedding D ~ R(X , PSL(2 , C)). So we can think of (c~,/3) : D ---+ C 2n as a lift of the restriction map R(X, PSL(2 , C)) ---* R(OX, PSL(2 , C)), where C 2n --~ R(OX, PSL(2 , C)) is the map defined in Sect. 3. (See also the remark at the end of Sect. 3.) Since we have this lift, we can define a function z : D ---* C* by the formula

CX(~d) = [ . . . , c~k(d), /3k(d),.. . ; z(d)].

Keep in mind that this is the PSL(2 , C) Chern-Simons invariant, defined using the first pontryagin class.

Let g : D ~ C be the map

d

k do

where the integral is taken over any path from d o to d. Since c~ k and/3k are analytic, g is analytic. Moreover, by Theorems 2.7 and 3.2, if d, d ~ E D then

z(d') g(d') z(d) g(d) '

since g(do) = l, and z(d) = z(do)g(d ). A point d E D corresponds to a closed hyperbolic dehn filling if there are relatively

1 prime integers (Pk, qk) for each k so that pkc~k + qk/3k = 2"

Suppose this is the case, and let r k, s k be integers so that pksk -- qkrk = 1.

e ote losed = w h ere

P q the meridian is O(D 2 x *)k = #kAk and the longitude is (* x Sl)k = #~A~. The representation 0a is abelian and diagonalizable on each solid torums so by Theorem 4.1,

CD2• d) = [ . . . , 0, rk~ k + Sk/3k,... ; 1]

with respect to the meridians and longitudes. This in turn is equal to

I ( )] . . . , ~, rkc~ k + Sk/3k,... ; exp -- 47ri ~_,(rkc % + Sk/3k ) k

using the covariance of the bundle.


Taking the inner product we have:

CM(d)(Qd) = Z(d)exp ( 47ri Z (rk~k + Sk/3k) ) k

( ( )) We know the left side is equal to exp 27ri cs(M(d)) - Vol(M(d)) . So

( ( s ))Ilexp(47ri(rkak+sk/3k)). z(do)g(d ) = exp 27ri cs(M(d)) - -~ Vol(M(d)) k

The representation on the k th solid toms takes the core geodesic 7k to

This isometry of hyperbolic 3-space leaves the geodesic through 0 and cx~ invariant (in the upper-half space model) and computing with the hyperbolic metric one sees that the translation length is 1 k = Re(47ri(rka k + Sk/3k)) and the rotation angle is O k = Im(47ri(rka k + sk/3k)). Therefore:

e47ri(rk~ ~_ e(lk+iOk).

We then get the following version of Yoshida's formula (see also Hodgson's thesis [H]):

d

z(do)exp(--87rif~akd/3k--/3kdak) do k

= exp 27ri cs(M(d)) - Vol(M(d)) 1-I exp(/k + iOk)" k

Moreover, z(do)=exp(2Vol(X)+27rics(X)),wherecs(X)isdefinedbyMeyer -

h~176176176176 1-Iexp(lk)k

for d E D a surgery point. As d ~ d o, Vol(M(d)) ~ Vol(X) and 1 k ~ 0 [Th].

Thus,z(do)9(d)l---~exp(2Vol(X)) asd--+do. However, g(do)=l, so,z(do)l=

exp(2Vol(X)).Also, Meyerhoffshowsthat27rcs(M(d))+~OK-~27rcs(X).) Plugging this in and taking the log of both sides gives us:

- 8 7 r i ~ f akd/3k-/3kda~:2(gol(m(d))-Vol(X))

+~-~lK+i(27v(es(M(d))-cs(X))+~Ok)

To compare this with Theorem 5.7 of Hodgson's thesis, we need to note that instead of a and/3 he uses u = 47via and v = 47ri/3. (There is a slight error in constants in [H]; the observant reader will notice that the coefficient of the integral on the left in [H] differs from ours by a factor of i, and the coefficient of the Chern-Simons term in [H] differs from ours by a factor of 2.)


References

[A] [CCGLS]

[CS]

[F]

[FSI]

[FS2]

[H]

[J]

[KK1]

[KK2]

[KKR]

[Me] [M] [NZ]

[RSW]

[Th] [Y]

Auckly, D.: Thesis, U. of Michigan, 1991 Cooper, D., Culler, M., Gillet, H., Long, D., Shalen, P.: Plane Curves Associated to Character Varieties of 3-Manifolds. Preprint Culler, M., Shalen, P.: Varieties of Group Representations and Splittings of 3-manifolds. Ann. of Math. 117, 109-146 (1983) Floer, A.: An Instanton invariant for 3 manifolds. Commun. Math. Phys. 118, 215-240 (1989) Fintushel, R., Stern, R.: Instanton Homology of Seifert-Fibered Homology Spheres. Proc. Lond. Math. Soc. 61, 109-137 (1990) Fintushel, R., Stern, R.: Invariant for Homology 3-Spheres. In: Geometry of Low-Di- mensional Manifolds. Donaldson, S.K., Thomas, C.B., (eds.), Lond. Math. Soc. LNS 150, Cambridge: Cambridge U. Press, 1990 Hodgson, C.: Degeneration and regeneration of geometric structures on 3-manifolds. Princeton thesis, 1986 Jaco, W.: Lectures on 3-manifold Topology. AMS CBMS regional conference series number 43 Kirk, P., Klassen, E.: Representation spaces of Seifert-Fibered homology Spheres. Topology 30, 77-96 (1991) Kirk, P., Klassen, E.: Chern-Simons Invariants and representation space of Knot groups. Math. Ann. 287, 343-367 (1990) Kirk, P., Klassen, E., Ruberman, D.: Splitting the Spectral Flow and the Alexander Matrix. Preprint Meyerhoff, R.: Thesis, Princeton U., 1981 Mrowka: Personal communication Neumann, W., Zagier, D.: Volumes of Hyperbolic 3-manifolds. Topology 24, No. 3, 307- 332 (1985) Ramadas, T.R, Singer, I., Weitsman,J.: Some Comments on Chern-Simons Gauge Theory. Commun. Math. Phys. 126, 409--420 (1989) Thurston, W.: The geometry and Topology of 3-manifolds. Princeton lecture notes Yoshida, T.: The ~/-invariant of Hyperbolic 3-manifolds. Invent. Math. 81,473-514 (1985)

Communicated by N.Yu. Reshetikhin

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