topological methods to compute chern-simons invariantsdav/topmethcs.pdf · 2009. 5. 19. ·...

Math. Proc. Camb. Phil. Soc. (1994), 115, 229 2 2 9

Printed in Great Britain

Topological methods to compute Chern-Simons invariants

BY DAVID R. AUCKLY

University of Texas, Austin, TX 78712-1082 USA

(Received 21 September 1992; revised 3 June 1993)

1. Introduction

In this paper we develop a method which may be used to compute theChern—Simons invariants of a large class of representations on a large class ofmanifolds. This class includes all representations on all Seifert fiber spaces, all graphmanifolds, and some hyperbolic manifolds. I owe many thanks to Peter Scott, JohnHarer, Frank Raymond, Ron Fintushel, Paul Kirk and Eric Klassen, without whosehelp and support this paper could not have been written.

After covering some background material, we will compute the Chern—Simonsinvariants of all Seifert fiber spaces. The techniques we use are very natural forSeifert fiber spaces, but work equally well for many other manifolds after thetechniques are translated into the language of surgery. We finish by translating thetechniques into a sequence of moves which may be used to compute Chern—Simonsinvariants in a way similar to that in which skein relations may be used to computeknot invariants.

First, we will define the Chern-Simons invariant and list some related facts anddefinitions. Throughout the paper we will use the group of unit quaternions, denotedby Sp r Geometrically, Spj is a 3-sphere with + 1 at the north pole and — 1 at thesouth pole. Two elements of Spj commute if and only if they lie on a great circlerunning through + 1 . The conjugacy classes are 2-spheres perpendicular to thisfamily of great circles, and every element can be expressed as the commutator of twoelements.

The Chern-Simons invariant is defined for connections on the trivial Spx bundleover a 3-manifold. Let w be a connection on the trivial Spj bundle over a closed 3-manifold.

Definition. The Chern-Simons invariant of w is

cs((o) =-j—£ cr*Re(w A rfw + |w A w A w),*±n JM

where a is a section of the trivial bundle.

Fact. The Chern-Simons invariant only changes by an integer if the section ischanged or if w is replaced by a connection that differs by a gauge transformation.

Fact. Flat connections on M/Gauge equivalence = Hom(7T1(Jf), Sp^/AdSpx-By the Chern-Simons invariant of a representation, we will mean the Chern-Simonsinvariant of the corresponding flat connection.

Definition. A representation a: G-± Spx is called reducible if its image is containedin a great circle.

230 DAVID R. AUCKLY

Definition. Two representations, aM and aN, are called flat cobordant if there existsa compact orientable 4-manifold W with dW = M—N such that a: 7r1(W)-»-Sp1induces aM:7r1(M)-*Sp1 and aN:n1(N)^'Sjil.

Fact. The Chern—Simons invariants of flat cobordant representations are equal.

Fact. If a,/?eHom(77'1(M), Spj) are in the same path component, then cs(a) = cs(fi).

THEOREM (Kirk, Klassen). Let

and let ji}: S1 -+N(Aj: S1 -*-N) be the meridian (longitude) of the jth solid torus.

Suppose a): [0, l]->Hom(77'1(iV),Sp1) satisfies

Thenn n

cs(«(l))-cs(w(0)) = - 2 £ a'}{t)b}3-iJo

Kirk and Klassen prove this theorem by showing that the only difference betweenthe differential forms integrated to give these two invariants occurs in a collar of theboundary ofN, and then doing an explicit calculation on this collar. For more detailson this background material see [1], [2] and [5].

2. The Seifert fibered case

We will start our computations with Seifert fiber spaces because they wellillustrate and motivate the techniques we use throughout. A Seifert fiber spaceshould be thought of as an S1 bundle over a surface with singularities. Our methodof attack will be the same for all of the computations we do. We will replace a givenmanifold by a more complicated manifold in order to increase the size of therepresentation space. Once the representation space is large enough, we will be ableto compute the Chern—Simons invariant of the original representation from theChern—Simons invariant of a simpler representation on the new manifold.

The Poincare homology sphere is a Seifert fiber space lying over the sphere withthree singularities. This is, in a sense, the hardest case we will encounter. Up toconjugation, it has exactly three representations into Spx: the trivial one and twoirreducible ones. The fundamental group of the Poincare homology sphere is

rr^M) =

Topological methods to compute Chern—Simons invariants 231

Fig. 1.

Visualize the cobordism that we are about to construct as this picture times S1.The cobordism is in fact an S1 bundle over this in the orbifold category. With S1 andD2 as the unit circle and disc in C, let V = S1xD2x [0,1]/ ~ with (A,z,0) ~ (A,z, 1).If M is any Seifert fiber space, let W = Mx[0,1] U S'xB> F where a tubularneighborhood of a regular fiber of Mx{l} c M x [0,1] is identified with a tubularneighborhood of (A, 1,1/2) in dV a V. This W is a cobordism between M and a Seifertfibre space with non-orientable base. The boundary of W has two components If andW.

IfM is the Poincare homology sphere, M' will be a Seifert fiber space over the Kleinbottle with three singular fibers and

[QltH], HAHA-\HBHB-\ Q\l

232 DAVID R. AUCKLY

where X and Y are chosen so that

[X,Y] = W(e,n1,...,ns)[g1,...,gs](Q;l...Q?Hb),gjeSVl,njeZ,e = O,i

If b + SJ_J /?j/a4 is zero these are all the representations we need, but when it is non-zero we will also need the representations defined below.

Definition.

H H+ exp f2m \ln0 + £ (nJocAllb + £ (A/a,)jlJ,

I f II i I \ f i l l

If M is a Seifert fiber space over a non-orientable base,

In this case, there are representations v{e, n1,...,ns)[g1,..., gs]: n^M) ->• Spx defined inthe same way as w(e,n1,... ,ns) [gf1;... ,gs] except Ag^-x, where

Just as in the case with an orientable base, when b+ H'i_1(fti/at) =t= 0, we need morerepresentations.

Definition.


Let N = M—N (singular fibers).

, k

234 DAVID R. AUCKLY

cobordism, we may add as many Klein bottles to the base orbifold as we like withoutchanging the Chern-Simons invariants. The formula for the remaining represen-tations is similar and is given in the following theorem.

THEOREM. If w = p(n0,... ,ns) or w = o~(n0,... ,ns), then

cs(w) = - i te %2 + "jK + c/2 + Sf-t nJaJKb + XU,

(n0 + c/2) (n0 + c/2 + Sf-i

where pt and

Topological methods to compute Chern-Simons invariants 235

Fig. 2.

Fig. 3.

is, the surgery framing should be the blackboard framing. The second step is to orientthe plane and each component of the link. A presentation for the fundamental groupof the manifold is now given by associating one generator to each arc in theprojection and relations for every crossing and every component. The relationassociated to a crossing is constructed by following a small oriented circle around thecrossing. Each time a strand is crossed, that generator is put down to the + 1 poweraccording to the orientation. The relation associated to a component is constructedin the same way except the component itself should be followed, and a crossingshould only count when it is travelled underneath. As an example consider themanifold depicted in Figure 2; call it M.

Component defined representations form a large class of representations. Redrawand orient this link as in Figure 3.

Then,

n^M) = t,u,v,w,x,y,z

ywxtvu xv x,t xu,tv, wzuxy 1

The relation corresponding to the knotted component is wzuxy 1. The relationcorresponding to the unknotted component is tv, and the rest of the relations comefrom crossings. This called the Wirtinger presentation of the fundamental group.

Definition. A representation a: 7r1(Af)^^Sp1 is called component defined if thereexists a surgery description of M, on a link L, such that any two generators associatedto different arcs on the same component of L have the same image under a.

Component denned representations form a large class of representations. In fact,we do not know of any representation which is not component defined. If such arepresentation exists, a good candidate might be any irreducible representation on— 1/2 surgery on the figure eight knot.

236 DAVID R. AUCKLY

Fig. 4.

Fig. 5.

It is not possible to determine that a representation is not component definedfrom just one surgery description. As an example, consider the manifold depicted inFigure 4.

The Wirtinger relation coming from the one component in this picture is

3-1 3'2 "*"3 %i ^6 "̂ 4 ^5 = 1 •

If a representation is component defined with respect to this surgery description,each of the xt maps to the same element, say a, so the component relation is a = 1.This implies that the only component defined representation with respect to thisspecific surgery description is trivial, but that does not mean that the trivialrepresentation is the only component defined representation on this manifold. Thismanifold is the Poincare homology sphere. Another surgery description of it is shownin Figure 5, and any representation is component defined with respect to this surgerydescription.

It is apparent from Figure 5 that the Poincare homology sphere is what we call atree manifold, i.e. a manifold which has a surgery graph which is a tree. Any 3-manifold may be expressed as surgery on a link whose components are unknottedsuch that any pair of components is either unlinked or the Hopf link. This link mayalso be chosen to have the property that if any one component is unlinked from eachmember of a set of components, then that component is unlinked from that set ofcomponents. From such a link, we may construct a graph with one vertex for eachcomponent and edges between any two components which link. We will call such agraph a surgery graph for the manifold. When the surgery graph is a tree, everyrepresentation is component defined. To see why, consider Figure 6.


Fig. 6.

Fig. 7.

If a: 7r1(cM)->Sipi, then crossings 1, 5 and 7 imply

[a(v),a(w1)] = 1, [a(y),a(xs)] = l and [a(z),a(z3)]= 1.

Now, crossings 2, 6 and 8 imply a{wx) — a(w2), a(x2) = a(x3) and a(a;1) = oc(x2), that is,a is component denned. This shows that all representations on tree manifolds arecomponent denned. It will also follow that all reducible representations arecomponent defined.

LEMMA. Any reducible representation is component defined.

Proof. Let a: n^M) -*• Spx be reducible. Let x and y be two adjacent arcs in the samecomponent of any link in a surgery description of M. The crossing relation with x andy is of the form xzy~lz~^ = 1, so 1 = a(x)a(z)a(y~1)a(z~1) = a(x)oc(y)~1 because, bythe definition of reducible, the image of a is abelian. I

Now that we have many examples of component defined representations, it is timeto describe how to compute the Chern—Simons invariant of a component definedrepresentation. By using a surgery description of the flat cobordism used in theSeifert fiber space section, we will be able to replace component definedrepresentations with reducible representations in our computations.

238 DAVID R. ATJCKLY

Fig. 9.

Surgery is a way of describing a handlebody structure by looking just at theboundary. For more information on surgery and handle-body decompositions see [3],[4] and [7]. We will use two basic surgery descriptions in what follows. The first is theaddition of a 1-handle to a 4-manifold with boundary. The space constructed byadding a 1-handle to the 4-disc is W = D4 U S°XD* (D1

X^3)> w h e r e S°xD3 c+S° xD3 \J( - D 1 x S2) = d(Z)1 x D3) c> D1 x D3. There are two ways that S° x D3 may be mappedinto 3D4, but only one will create an orientable manifold. We have drawn a 2-dimensional analog of this in Figure 7.

We will draw the S° xD3 in 3D4 to denote this handle structure. See Figure 8.This W is an S1 x D3 and its boundary is an S1 x


U

Fig. 10.

Fig. 11.

X

O

Fig. 12.

240 DAVID R. AUCKLY

and

Fig. 13.

Fig. 14.

We can describe any representation on any 3-manifold by drawing a surgerypicture of the manifold and labelling each strand with the element of Spj in the imageof the corresponding Wirtinger generator. If we start with S2 x S1 and construct thecobordism W = (S2 xS1 xl) U siXD^ V, we will have a flat cobordism between the tworepresentations shown with blackboard framing in Figure 13.

The cobordism may be constructed even when there are additional surgeries on theS2 xS1. Thus, we may replace any 0-framed unknot with an 0-framed unknot withtwo more components without changing the Chern—Simons invariant. By a sequenceof Kirby moves, any component of a framed link may be unknotted and zero framed.This means that any strand in a surgery picture may be replaced by a strand withtwo loops wrapped around it, without changing the Chern-Simons invariant. Weshow a picture of this move in Figure 14.

The method we have in mind for computing Chern-Simons invariants would be touse a sequence of Kirby moves and flat cobordism moves to reduce the problem toone where Kirk and Klassen's theorem applies. The Chern—Simons invariants of thePoincare homology sphere can be found in this way. A picture of a representationwith the Sp! labels missing is shown in Figure 4. A sequence of Kirby moves willmake this look like Figure 5. A cobordism move will then transform it into Figure 15.The representation is now in the path component of a reducible representation, soKirk and Klassen's theorem may be used to compute its Chern-Simons invariant.

We will now show that this method will work for any component dennedrepresentation.


Fig. 15.

THEOREM. Any component defined representation is flat cobordant to a representationin the path component of a reducible representation.

Proof. Let a: 7TX(ikfJ-̂ -Spj be component defined, and pick a framed link Lrepresenting M so that a is component defined with respect to L. If every componentof L is sent to + 1 , we are done. Otherwise pick a component which is not sent to +1and conjugate the whole representation until that component is complex. Now labelall complex valued components with a 1. Next pick any unlabelled component andlabel every component with which it commutes, with a 2. Keep repeating this processwith unlabelled components and higher labels until all components have been labelled.Apply the flat cobordism move to every component which has more than one labeland extend the representation to send each of the new components to 1. It is this newrepresentation that we will show to be in the path component of a reducible one. Wewill proceed by induction on the labels. Assume that the representation is in the pathcomponent of a representation an which takes all components which have a label lessthan n + 1 to a complex number. Pick a path g: [0, lj-^-Spj, so that g0 = 1 and anycomponent labelled with a n n + 1 would be conjugated into the complexes by g1. Wewill determine a path of representations from an to a representation an+1 which sendsall components labelled less than n + 2 to a complex number. Let yt: n1(M')-^8]}1;

(a.n(x(i})) when x(i,j) is not labeled n + 1x(i i) ""*• 1

\g.a.tl(xun)q,1 otherwise.

When two components cross, the crossing relation is that the components commute.If neither component is labeled n + 1, yt is the same as

242 DAVID R. ATTCKLY

Given any reducible representation defined on a manifold, there is a path ofrepresentations from the reducible representation to the trivial representation on thecomplement of a link in a surgery description of the manifold. Kirk and Klassen'ssurgery theorem may, therefore, be used to compute the Chern-Simons invariant ofany reducible representation.

With a little linear algebra we can see that these invariants will always be rationalnumbers.

THEOREM. / / a: 771(ilf)->Sp1 is reducible, there exists /?: n1(M)^8p1 which isreducible with finite image and in the same path component of the representation spaceas a.

Proof. Assume M is surgery on a link L. Then n^M) has a Wirtinger presentation

where -X ĵ> is the generator associated to the jth strand of the ith component, cnis the relator corresponding to the n-th crossing and rm is the relator commuting fromthe wth component. The previous lemma implies ot,(x^ n) =


Fig. 16.

Fig. 17.

4. Examples and graph manifolds

Consider the representation shown in Figure 16 as examples of these theorems. Wehave underlined the framings to keep them separate from the Spx labels.

Let a; = i, y = zcosP

+7sin —— , and z = cos — +&sin — •J V V ) \p) \P)

It is easy to check that these are representations, and that these representations areflat cobordant to the representations in Figure 17. (In Figure 17 we have suppressedall of the framings.) The extensions of the representations in Figure 16 are in the pathcomponent of the representations where b = d = 1, x = y = i, z = exip{2ni/p} anda = c = exj>{ni(p + 2)/2p}.

In order for the representations to be completely determined, and in order tocompute the Chern—Simons invariants, we need to specify some orientations. Toorient the Wirtinger generators, pick the usual orientation in the projection planeand orient all of the components counter-clockwise. These 3-manifolds all inheritorientations from the usual right hand orientation of U3. We can now compute theChern-Simons invariants with Kirk and Klassen's theorem. First, remove the

244 DAVID R. AUCKLY

Fig. 18.

components labelled x, y, and z, to get ax(t) = \t, bx(t) = t, ay(t) = \t, by(t) = t, az{t) =(l/p)t and bz(t) = t. Finally,

2 V

Thurston has shown that the link in Figure 16 is hyperbolic and therefore all butfinitely many of these manifolds are hyperbolic [8].

We will consider graph manifolds for our next example. Almost all representationson graph manifolds are component defined. The representations which are notobviously component defined are close enough to component defined representationsthat the same techniques still apply. Before we can define a graph manifold we needto discuss plumbing (see [6]).

Plumbing is a way of gluing disc bundles together. Let E1-*-F1 and E2^F2 be twodisc bundles over surfaces. Let hi:D

2xD2^Ei be local trivializations and leti: D2 x D2 -+D2 x D2; (x, y) i-> (y, x) be the natural involution. The manifold obtainedby plumbing Ex and E2 together is E1 U D'XD2^2' where h2(D

2 xD2) is identified withhl(D

2xD2) by AjOtoAj1. See Figure 18.A septic system is a connected graph with a disc bundle at each of the vertices

where the disc bundles are plumed together whenever there is an edge between them.A graph manifold is the boundary of a septic system.

The fundamental group of a graph manifold may be computed with Van Kampen'stheorem, if we describe the graph manifold as a union of pieces. The manifold iscomposed of an S1 bundle over a surface with boundary at each vertex, say Mk, andthese S1 bundles are joined whenever there is an edge between them. To compute thefundamental group, let T c G be a maximal tree and let

E =

be a free group. Then if

| there is an edge from mt to m3

Qki iA

where lk n is an edge between Mk and Mn, the fundamental group of the graphmanifold will be

TT^M) = (*M7T1(Mk))*E/«LijLji!LijHjLjiQijy*

Topological methods to compute Chern-Simons invariantso o o

245

Fig. 19.

can construct surgery descriptions of any disc bundle over any surface. As anexample in Figure 19, we can give a surgery description for the Euler-class threecircle bundle over a surface of genus two.

Euler class 5 Euler class 3

Fig. 20.

The dotted circles represent typical fibers in these surgery descriptions. The graphmanifold obtained from plumbing two disc bundles together may be obtained bycutting out a tubular neighbourhood of a regular fiber in each S1 bundle andidentifying the corresponding boundaries. I t is easy to construct surgery descriptionsof this type of graph manifold. We give an example in Figure 20.

This generalizes to any graph manifold whose underlying graph is a tree, but ittakes a bit more work when the graph is not a tree.

246 DAVID R. ATJCKLY

Fig. 21.

Euler class 4over

Euler class 1

over ( 2

A maximal tree

Euler class 2

over

Fig. 22.

Fig. 23.

Before describing the surgery picture of a general graph manifold, we need to givea surgery description of the boundary of the manifold obtained by identifying twounknotted, unlinked solid tori in S3 c Z)4. This manifold is the boundary of the 4-manifold D* U s^D'xaD1 S1 X-D3- As depicted in Figure 21, S1 xDs is the union of a1-handle and a 2-handle.

To get a surgery description of an arbitrary graph manifold, we will begin with asurgery description for a manifold corresponding to a maximal tree and modify it toreflect the pluming from the other edges. As an example, consider the graph manifoldindicated in Figure 22.

A surgery description of the manifold associated to the maximal tree is given inFigure 23. To get the graph manifold, we only need to cut out tubular neighborhoodsof the dotted circles and identify the boundaries in the appropriate way.

Notice that just identifying tubular neighborhoods of the dotted circles andlooking at the boundary will not give us what we want because the fibers will beidentified. We want these identifications turned sideways. If we cut out a tubularneighborhood of the dotted circle on the left, turn it sideways and glue it back in (i.e.perform 0-framed surgery on it), then identifying it with the solid torus on the rightwould give us the graph manifold in question. Thus, the boundary of the 4-manifoldcreated by identifying the dotted solid tori in Figure 24 is the graph manifold in


Fig. 24.

Fig. 25.

figure 22. By using the surgery description in Figure 21 with the 1-handle replacedby a 0-framed 2-handle, we will get a surgery description of the graph manifold.See Figure 25. This surgery description may be simplified with Kirby moves to theframed link in Figure 26.

This picture generalizes in the obvious way for any graph manifold. Namely, to geta surgery description of an arbitrary graph manifold, draw the standard surgerypicture for each disc bundle with two linked together when there is an edge betweenthem, and add one 0-framed 2-handle around each edge that is not in the maximaltree.

We are now ready to compute the Chern-Simons invariants of graph manifolds.The method used for component defined representations works just as well here.First, label the vertices of the graph according to which great circle in Spx containsthe image of the fiber from that vertex. After a possible flat cobordism, we can

248 DAVID R. AUCKLY

Fig. 26.

Fig. 27.

construct a path of representations which conjugates each differently labelled regioninto the same great circle. Thus, we may assume that any representation sends thefibers of each of the circle bundles into the same great circle. As a concrete example,consider the representation in Figure 26, which sends H1i-> 1, H2\-ê

i7ri/2, H3ê8ni/2,

Li-^j,A-l\-^j,B1^ê~6ml1, A2*-> l,i?2i-s*l. Kirk and Klassen's theorem applied to all of

the components labelled with an H will now give the Chern—Simons invariant. In ourconcrete example, a2(t) = §


Fig. 28.

Fig. 29.

Two manifolds with boundary which are closely related toil/ are shown in Figure 28.These are the manifolds created by doing the surgeries indicated on the framedcomponents and subtracting the interior of a tubular neighborhood of the unframedcomponents. We compute

and

i) = {t,u,v\[t,u],[v,u],tau,vcu-1}

j = (p,q,r\\jo,ql[r,q],pdq-1,rbqy.

The boundary of both Ifj andilf2 is T2 with n^T2) = . This is included

in M1 byif : nx(T

2) -+ TT^MJ ; IH> r 1 v,Mw- VT1

and in M2 byif: ; l^q,

The union ofikfx andM2 along this T2 isM. This decomposition will help us compute

the Chern-Simons invariants of M.Let a: n^M) -> Spx. If a(pr

-1) and

250 DAVID R. AUCKLY

and

Finally, let

Af', M"2

Fig. 30.

=


of S1 xD2 are properly chosen. In addition, a 2-handle may be added to any circle inthe kernel of the representation and 1-handles may be added almost at will. All ofthese flat cobordisms translate into moves which change the framed link but donot change the Chern-Simons invariant.

Question. Does there exist a collection of moves which would be sufficient tocompute the Chern-Simons invariants of any 3-manifold ?

Finally, since none of these examples have irrational Chern—Simons invariant, weask:

Question. Are all Chern—Simons invariants rational ?

REFERENCES[1] D. AUCKLY. Computing Secondary and Spectral invariants, Ph.D. Thesis, University of

Michigan (1991).[2] S. CHERN and J. SIMONS. Characteristic forms and geometric invariants. Ann. of Math. 99

(1974), 48-69.[3] J. HARER, A. KAS and R. KIRBY. Handlebody decompositions of complex surfaces. Mem. Amer.

Math. Soc. 350 (American Mathematical Society, 1986).[4] R. KIRBY. A calculus for framed links in 83. Inv. Math. 45 (1978), 35-56.[5] P. KIRK and E. KLASSEN. Chern-Simons invariants and representation spaces of knot groups.

Math. Ann. 287 (1990), 343-367.[6] P. ORLIK. Seifert manifolds. Lecture Notes in Math. 291 (Springer-Verlag, 1972).[7] D. ROLFSEN. Knots and Links. (Publish or Perish, 1976).[8] W. THURSTON. The geometry and topology of 3-manifolds. Princeton Lecture Notes.

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