Hodge theory and the art of paper folding

Hodge Theory and the Art of Paper FoldingMichael Kapovich and John J. Millson �August 24 , 1996AbstractUsing Hodge theory and L2-cohomology we study the singularities and topologyof con�guration and moduli spaces of polygonal linkages in the 2-sphere. As aconsequence we describe the local deformation space of a folded paper cone inR3 .1 Introduction.This is a part of a series of our papers [KM2], [KM3], [KM4], [KM5] where we studyinterrelations between members of the following diagram:Con�guration spacesof geometric objects �����! Algebraic varieties& %Representation vari-eties of groupsExamples of \geometric objects" that we consider are: linkages in spaces of con-stant curvature, projective arrangements, folded pieces of paper. Representation va-rieties under consideration are: representation varieties of in�nite Coxeter, Shep-hard and generalized Artin groups, representation varieties of fundamental groupsof hyperbolic 3-manifolds, relative representation varieties of fundamental groups ofpunctured spheres. The arrows \Con�guration spaces" �! \Algebraic varieties"and \Representation varieties of groups" �! \Algebraic varieties" are obvious ones,since con�guration spaces and representation varieties have natural structures of al-gebraic varieties. The arrow \Con�guration spaces" �! \Representation varietiesof groups" is not obvious at all, it was introduced in [KM4]. For certain \Geomet-ric objects" (e.g. polygonal linkages in R3 and S3) the resulting algebraic varietieshave a complex-analytic structure and in fact coincide with moduli spaces of complexalgebraic objects.Our general goal to to see how properties of \Geometric objects" and \Groups"are re ected in local and global topology of \Algebraic varieties". In [KM2] we relate�This research was partially supported by NSF grant DMS-96-26633 at University of Utah(Kapovich) and NSF grant DMS-95-04193 the University of Maryland (Millson).1

con�guration spaces of n-gon linkages in R3 , relative representation varieties of thefundamental groups of the n-times punctured 2-sphere into the Euclidean group andMumford quotients of the n-fold product (C P1)n by PSL(2; C ).In [KM3] while studying the diagramCon�guration spacesof linkages in S2 �����! Projective va-rieties over Z& %Character varieties offundamentalgroups of hyperbolic3-manifoldswe construct compact hyperbolic 3-manifolds whose representation varieties have non-quadratic singularities. In [KM5] we consider the diagramCon�guration spacesof projective arrange-ments in CP 2 �����! Projective va-rieties over Z& %Character va-rieties of generalizedArtin groupsand show that the correspondence \Character varieties of (generalized) Artin groups"! \Projective varieties over Z" is essentially onto. In particular, representationvarieties of (generalized) Artin groups could have arbitrarily complicated singularities.In [KM5] we deduce from this that there exist in�nitely many Artin groups that arenot fundamental groups of smooth complex quasi-projective varieties.In this paper the \Geometric objects" are polygonal linkages in S2 and the simplestfolded pieces of paper in R3 , \Representation varieties" are relative representationvarieties of fundamental groups of punctured 2-spheres. The restriction to polygonallinkages in S2 results in restrictions in the global topology and local singularitiesof the corresponding \Algebraic varieties". Namely, the fact that our groups arefundamental groups of punctured spheres implies that their representation varietieshave some extra structure coming from the Hodge theory, which is the main technicaltool of the current paper.Let � be a (marked) geodesic n-gon on S2 with side-lengths r = (r1; :::; rn). Let Crbe the con�guration space of n-gon linkages with side-lengths r and Mr = Cr=SO(3)be their moduli space, see x3. It is immediate that Cr and Mr are the sets of realpoints of a�ne schemes over R, also denoted Cr and Mr respectively. Let [�] be theimage of � in Mr . Our �rst theorem determines the local nature of Cr and Mr .De�ne an n-gon linkage � to be degenerate if it lies in a great circle S1 of S2 .Suppose � is degenerate. We orient S1 and de�ne �i 2 f�1g to be 1 if the orientationof the i-th edge agrees with that of S1 and �1 otherwise. We say that the i-th edgeof � is a forward-track if �i = 1 and a back-track otherwise. Let f = f(�) be the2

number of forward-tracks and b = b(�) be the number of back-tracks, so f + b = n.De�ne the winding number w = w(�) bynXi=1 �iri = 2�wThen we haveTheorem 1.1 (i) dimCr = n, dimMr = n�3, where dimension is the Krull dimen-sion.(ii) If � is nondegenerate then Cr is smooth at � and Mr is smooth at [�].(iii) If � is degenerate with f; b; w as above, then the germ of Cr at � is analyticallyisomorphic to the germ of Z(Q) at 0, where Q is a quadratic form on Rn+1 of nullity3 and signature (f � 2w � 1; b+ 2w � 1), and Z(Q) denotes the null-cone of Q.(iv) If � is as in (iii) then the germ of Mr at [�] is analytically isomorphic to thegerm of Z(Q0) at 0, where Q0 is a nondegenerate quadratic form on Rn�2 of signature(f � 2w � 1; b+ 2w � 1).As a corollary we determine the (locally) rigid spherical n-gon linkages.Corollary 1.2 A linkage in S2 is rigid if and only if it is degenerate and eitherf = 2w + 1 or b = �2w + 1.Remark 1.3 If the perimeter r1 + ::: + rn � 2�, then the rigidity of the linkagesabove is obvious. If w 6= 0 only one of the two equations above is possible (dependingof whether or not w > 0, w < 0) and the rigidity of the linkages is less obvious.We apply Theorem 1.1 to determine the local deformation space of a folded papercone in the Euclidean 3-space. A mathematical model for a folded piece of paper is apair consisting of a graph Y in the Euclidean plane E2 such that the edges of Y are linesegments (possibly in�nite or half-in�nite) and a continuous map f : E2 ! E3 suchthat the restriction of f to each component of E2 � Y is a totally-geodesic isometricembedding. Thus we introduce dihedral angles along the edges of Y in such a waythat the cone angles around all vertices remain equal to 2�. Our mathematical modelallows the piece of paper to intersect or overlap itself. We let f0 : E2 ! E3 be thetotally-geodesic isometric inclusion (the unfolded piece of paper). Note that we donot divide out the space of paper-foldings by the action of the group of isometriesE(3) of the Euclidean space. In this paper we determine the local analytic structureof C(Y ) in the basic case when Y consists of n � 3 rays emanating from a singlevertex. We assume the angles between adjacent rays are all less than �.We prove that C(Y ) is quadratic at f0 . More precisely we prove the followingtheoremTheorem 1.4 Assume that all angles between adjacent rays are less than �. Thenthere is a neighborhood of f0 in C(Y ) which is real analytically equivalent to U � Vwhere U is an open ball in R6 and V is a neighborhood of O in the quadratic coneQ in Rn�2 given byQ = f(x1; :::; xn�2) 2 Rn�2 : x21 + ::: + x2n�3 � x2n�2 = 0g3

Our �nal theorem gives a \wall-crossing" algorithm to determine the topology ofthe moduli spaces Mr . Our results depend on a result of Galitzer [Ga] describingthe set Dn(S2) of n-tuples r = (r1; :::; rn) which are side-lengths of a closed n-gonin S2 . It turns out that Dn(S2) is a convex polyhedron which is discussed in x8.Let Qn be the space of all n-gons in S2 up to the action of SO(3). We then have amap � : Qn ! Dn(S2) which assigns to an n-gon � its side-lengths r = (r1; :::; rn).Thus the moduli spaces Mr are the �bers ��1(r). The set of critical values of � insideDn(S2) is a union of hyperplane sections of Dn(S2) called walls. The connectedcomponents of the complement of the union of walls in Dn(S2) are called chambers.Since � is proper the moduli spaces Mr are all di�eomorphic if r varies within achamber. It is easy to determine Mr for special values of r (e.g. if the perimeterless than 2� and one side is much larger than the other sides, then Mr �= Sn�3, c.f.[KM1]). Thus, if we can determine howMr changes when we cross a wall, then we cancompute the topology of Mr (though in practice formidable combinatorial problemsoccur).First we give Galitzer's description of the walls. For each subset I � f1; :::; ng welet �I be the complement of I, jIj be the cardinality of I and rI denotes Pi2I ri . If wis a nonnegative integer we let HI;w be the hyperplane in Rn given byrI � r�I = 2�wThen Galitzer proves HI;w \ Dn(S2)0 6= ; if and only if jIj � 2w + 2. Moreover, allwalls of Dn(S2) are of this form. We now state our wall-crossing formula, we will givea slightly more general version in Theorem 8.10.Theorem 1.5 Let L be an oriented line segment in Dn(S2) crossing the wall HI;wtransversally at r� and not meeting any other wall. Let r0; r00 2 L be the end-points.Then Mr00 is obtained from Mr0 by surgery of the type:(jIj � 2w � 1; n� jIj+ 2w � 1) or (n� jIj+ 2w � 1; jIj � 2w � 1)(depending on orientation of L).Remark 1.6 It is interesting to note that in our proof (see Lemma 8.8 and x8 forde�nitions and notation) the deformation theory of representations (the cup-productQ) determines the Morse theory (the Hessian Q�L = d2(rnjXL)u�).If we take polygonal linkages in S3 (instead of S2) as our \Geometric objects",then the resulting \Algebraic varieties" have a complex-analytic structure{ they arethe moduli spaces of rank 2 parabolically stable bundles over S2 of Mehta and Seshadri[MS]. The equivalence is obtained as follows. By [FS], x4, page 129 (see also [KK],Lemma 2.7), the moduli space of an n-gon linkage in S3 is isomorphic to the relativerepresentation variety Hom(�; S;SU(2))=SU(2), where � and S are as in x5 of ourpaper (in particular, � is the fundamental group of the n times punctured 2-sphere).By Simpson's generalization [S] of [MS] to the genus zero case, the above relative rep-resentation variety is isomorphic to the moduli space of parabolically stable bundlesover S2.In this case our Hodge theory implies that the singularities are complex-analyticallyisomorphic to the quadratic cones of [KM2], Lemma 2.5 . It is easy to see that4

Dn(Sm) = Dn(S2) (here Dn(Sm) is the set of possible side-lengths of closed geodesicn-gons in Sm). Accordingly, the result of Galitzer above is equivalent to Proposition4.4 of [BY]. As in the case of S2 the deformation theory determines the Morse theoryand we obtain the \wall-crossing" formula of [BH], Theorem 3.1. Similar results arealso contained in [KK]. Thus the algebraic varieties associated to the n-gon linkagesin R3 and S3 have a description as moduli spaces in algebraic geometry. Is there suchan algebraic description for the case of n-gon linkages in hyperbolic 3-space?We would like to thank Dick Hain and Mark Stern for help with the L2{cohomo-logy calculations of this paper and Amy Galitzer for allowing us to include resultsfrom her PhD thesis. We would also like to thank Robert Bryant for suggestions thatled to the \wall-crossing" approach used here and in [KM1]. Finally we thank DickHain for suggesting the title of this paper.2 Relative Deformation Theory.In this section we review the material of [KM4]. By the relative deformation theoryof a representation �0 we mean the following. Let � be a �nitely generated group, Gbe the group of real points of an algebraic group over R which will also be denotedby G and S = f�1; :::;�ng a collection of subgroups of �. We denote by G the Liealgebra of G. Let �0 : � ! G be a homomorphism such that the AdG{orbits of�0j�j in Hom(�; G) are closed, 1 � j � n. We then de�ne the space of relativedeformations of �0, to be denoted by Hom(�; S;G), to be the a�ne subvariety ofHom(�; G) consisting of those � such that �j�j is in the AdG orbit of �0j�j .Assume that we have realized � as the fundamental group of a smooth connectedmanifoldM (possibly with boundary) containing disjoint domains U1; :::; Un such that�j is the image of �1(Uj) in � after suitable choice of base-points and approach paths.Let H be a �nite group of di�eomorphisms acting e�ectively on M so that eachdomain Uj is invariant under this action. We set U = [nj=1Uj and let P be the atprincipal G-bundle overM associated to �0 . Let adP be the associated bundle of Liealgebras. We construct a controlling di�erential graded Lie algebra B�(M;U ; adP )of adP -valued di�erential forms on M for the relative deformations of �0 . Roughlyspeaking this means we can calculate the deformation space of �0 by solving theintegrability equation d� + 12[�; �] = 0in B�(M;U ; adP ). Precisely this means that the complete local R-algebra RB�0 asso-ciated to B�0 by the procedure of [M] is isomorphic to the complete local ring of thereal-analytic germ (Hom(�; S;G); �0). We de�ne B0(M;U ; adP ) to be the subalgebraof smooth sections of adP whose restrictions to Uj are parallel, 1 � j � m. For i > 0we de�ne Bi(M;U ; adP ) to be the subspace of smooth adP -valued forms that vanishon Uj; 1 � j � n. We de�ne an augmentation� : B�(M;U ; adP )! Gas follows. Let m 2M be a base-point chosen so that m =2 U . We de�ne� : B0(M;U ; adP )! G5

by evaluation at m and � : Bi(M;U ; adP ) ! G to be zero if i > 0. We letB�(M;U ; adP )0 be the augmentation ideal of B�(M;U ; adP ), i.e. the kernel of �.Then B�(M;U ; adP )0 is a controlling di�erential graded Lie algebra for the relativedeformations of �0 by Theorem 2.9 of [KM4].Remark 2.1 Let E be any at bundle over M . Then we may de�ne a complexB�(M;U ;E) by replacing adP in the above de�nition by E. We will use this notationthroughout this paper without further comment.We have an extension of groups1! �! �! H ! 1where � is the orbifold fundamental group of M=H. We assume that �0 extends to�, we retain the notation �0 for the extension. Let �j; 1 � j � n, be the orbifoldfundamental groups of Uj=H and de�ne R = f�1; :::;�ng. By Theorem 2.10 of [KM4],the algebra of invariants B�(M;U ; adP )H0 controls the germ (Hom(�; R;G); �0).We will need the following general result about controlling di�erential graded Liealgebras.De�nition 2.2 A subvariety S � Hom(�; R;G) is said to be a cross-section to theorbits of G if the map � : G � S ! Hom(�; R;G) given by �(g; s) = Ad(g)s is anisomorphism of varieties.De�nition 2.3 Suppose there exists a G-invariant open neighborhood V of �0 andan analytic subvariety S of V such that �0 2 S and the natural map � : G� S ! Vis an isomorphism of analytic spaces. Then we call S a local cross-section through�0 to the orbits of G.Theorem 2.4 Suppose a local cross-section through �0 exists. Then the algebra ofinvariants B�(M;U ; adP )H controls the germ (Hom(�; R;G)=G; [�0]).Proof: We let S be the local cross-section and V = GS. Clearly the germs(S; �0) and (Hom(�; R;G)=G; [�0])are isomorphic. We now prove that B�(M;U ; adP )H controls the germ (S; �0). Wewill use the notations of x2 from [KM4] freely.If A is an Artin local R-algebra and (X; x0) is an analytic germ we will use X0A todenote the set of A-points of (X; x0). We note that G0A = exp(G M) where M isthe maximal ideal of A. We will abbreviate Hom(�; R;G) to X and B�(M;U ; adP )Hto L�. We have an isomorphism (of functors of A)V 0A = X0A �= G0A � S0ALet (P; !0) be the at principal bundle over M associated to �0 . The assignment ofthe holonomy representation to a at connection induces the functorhol : F rA(!0)! X0A6

of [KM4], x2. We let S0A be the inverse image of S0A under hol and G0(PA) be thedeformed gauge group of [KM4], x2. We assume we have chosen p 2 P . In [KM4] wede�ned �p : G0(PA)! G0Aby F (p) = p�p(F ). Then hol(F �!) = �p(F )hol(!). It is immediate that the aboveproduct decomposition induces a decompositionF rA(!0) �= G0(PA)�ker �p S0AWe recall that the complete local ring RL is a hull for the functor IsoC(L;A). ByProposition 2.7 of [KM4] the functor IsoC(L;A) is isomorphic to F rA(!0)=G0(PA).But clearly we have isomorphisms (natural in A):F rA(!0)=G0(PA) �= S0A= ker �p �= S0AHence OS;�0 pro-represents IsoC(L;A) and consequently is a hull for IsoC(L;A). Sincehulls are unique we have OS;�0 �= RL . 23 Con�guration spaces of Spherical PolygonalLinkages.In this section we will begin our study of the con�guration spaces of polygonal linkagesCr and moduli spaces Mr . Here r = (r1; :::; rn) is an element of the n-fold Cartesianproduct In, where I = (0; �). We now give necessary de�nitions.De�nition 3.1 An n-gon � = (�1; :::; �n) is an n-tuple of oriented geodesic arcs �j(in S2) of lengths between 0 and � (inclusive) such that the end-point of �i�1 is equalto the initial point of �i, 1 � i � n+ 1 (the indices are taken modulo n).De�nition 3.2 A free linkage with n vertices is an an n-tuple of oriented geodesicarcs �j (in S2) of lengths between 0 and 2� such that the end-point of �i�1 is equalto the initial point of �i 1 � i � n.We let ri be the length of �i in the spherical metric. The arcs �1; :::; �n will becalled the edges of �. We will use u = (u1; :::; un) to denote the set of vertices of �,that is the set of initial points of the edges �i . In case 0 � ri < � the polygon � isdetermined by its vertex set u and we will write � = (u1; :::; un). We will sometimeswrite u instead of �.De�nition 3.3 Let r 2 In . The con�guration space Cr of (marked) n-gon linkageson S2 with side-lengths r = (r1; :::; rn) is the set of all n-gons u = (u1; :::; un) such thatthe distances d(ui; ui+1) in the spherical metric satisfy d(ui; ui+1) = ri, 1 � i � n.It is immediate that Cr is the set of real points of the a�ne scheme over R de�nedby (ui; ui+1) = cos ri; 1 � i � nwhere (�; �) denotes the scalar product in R3 . The group SO(3) acts on Cr accordingto g � u = (gu1; :::; gun); u 2 Cr ; g 2 SO(3)7

De�nition 3.4 The moduli space Mr of n-gon linkages on S2 with side-lengths r =(r1; :::; rn) is de�ned to be the quotient scheme of Cr by SO(3).In fact there are no di�culties in passing to the quotient in this case because thereexists a cross-section Sr to the SO(3)-orbits in Cr . Indeed, we de�ne Sr to be thesubvariety of Cr such that u1 coincides with the �rst standard basis vector �!e1 of R3and u2 lies on the half-equator in S2 de�ned by(�!e3 ; u2) = 0; and (�!e2 ; u2) > 0Lemma 3.5 The variety Sr is a cross-section to the SO(3)-orbits in Cr .Proof: It is obvious that Sr is a set-theoretic cross-section. To see that Sr is a scheme-theoretic cross-section we embed Cr into eCr, the con�guration space of the linkageconsisting of n vertices u1; :::; un and a single edge of length r1 joining u1 and u2. Welet eSr be the subvariety of eCr such that u1 and u2 are as described above. ClearlyeCr = SO(3)� (S2)n�2 so eCr �= SO(3)� eSr as schemes. Now Sr is the pull-back of thecross-section eSr under the equivariant embedding Cr ! eCr. The reader will verifythat the pull-back of a scheme-theoretic cross-section under an equivariant morphismis a scheme-theoretic cross-section. 2We may then identify the quotient scheme Mr with the subscheme Sr � Cr . Wewill compute the real-analytic germ (Cr ;�) of Cr at � = u = (u1; :::; un) for anyr and u as above. Our goal is to relate (Cr ;�) and a germ of a certain relativerepresentation variety.Let � be the free product of n copies of Z=2, � = h�1i�:::�h�ni where �j correspondsto the vertex uj of a spherical n-gon �. Let j = �j+1�j . Let � = �� : �! SO(3) bethe representation which assigns to the generator �i the rotation sui of 180 degreesaround ui . We denote by �i the subgroup of � generated by the involutions �i; �i+1,we put R = f�1; :::;�ng. Now we pick a particular n-gon �� in a con�guration spaceCr . Theorem 3.2 of [KM4] implies the followingTheorem 3.6 The map � 7! �� gives an analytic isomorphism of germs(Cr ;��) �= (Hom(�; R;SO(3)); �)Corollary 3.7 The map � 7! �� induces an analytic isomorphism of germs(Mr ; [��]) �= (Hom(�; R;SO(3))=SO(3); [�])Proof: Since the isomorphism in Theorem 3.6 is SO(3)-equivariant, it induces anisomorphism of quotient germs. 2We say that a spherical polygon � is degenerate if it is contained in a greatcircle of S2 . As we shall see, degenerate polygons are precisely the singular pointsof the con�guration spaces Cr and moduli spaces Mr . Let's assume that � 2 Sris a degenerate polygon, in particular it lies in xy-plane and the edge �1 of � hascounterclockwise orientation. 8

We recall that we associated numbers f; b; w and �j, 1 � j � n to � in theIntroduction. We set �j = �jrj, whencenXj=1 �j = 2�wWe will see that the numbers f; b and w will determine the singularity of Cr at �.Let � := �� be representation associated with the polygon �. We claim a basise1; e2; e3 for so(3) can be chosen so that e3 = [e1; e2], e3 is �xed under ad(�( j)),1 � j � n and Ad(�0( j))e1 = cos(2�j) � e1 + sin(2�j) � e2Indeed, take ej be the images of the vectors �!ej of the natural basis of R3 under thecanonical isomorphism ad : R3 ! so(3) which is given byad�(�) = � � �4 Spherical Polygons and Paper-folding.Let Y � R2 be union of n geodesic rays emanating from the origin, we shall assumethat none of the complementary components R2�Y contains a half-plane, let r1; :::; rnbe the angles of the complementary regions. Intersections of the rays in Y with theunit sphere S2 centered at the origin determine a collection of points u01; :::; u0n 2 S2 .These points are the vertices of the degenerate spherical polygon �0, with the side-length d(ui; ui+1) = ri . If f : R2 ! R3 is a paper-folding with pleating locus Y , thenf is uniquely determined by the restriction of f to the components of Y � f0g. Wehave assumed that ri < � for all i. Let Cr be the con�guration space of �0 .Theorem 4.1 C(Y ) �= Cr � R3 as algebraic varieties.Proof: Recall that ui 2 R3 , 1 � i � n are vectors tangent to the rays in Y . Any mapf 2 C(Y ), is di�erentiable along the rays in Y , hence we have well-de�ned vectorsvi = df0(ui) 2 Tf(0)(R3)The unit vectors vi satisfy the property:(vi; vi+1) = cos(ri); 1 � i � n+ 1 ( mod n) (1)It is clear that ordered collections of unit vectors vi 2 Tx(R3); x 2 R3 which satisfythe above equation 1-1 correspond to elements f 2 C(Y ).Note however that the same equation (1) de�nes the variety Cr as well. Theoremfollows. 2Remark 4.2 With extra care one can generalize the above theorem to the case whenone of the angles ri is equal to �, but we are not going to discuss this case here.Our problem of determining (Cr ;�0) � R3 �= (C(Y ); f0) then amounts to com-puting the real-analytic germ of Cr at �0 for the case in which r1+ r2+ :::+ rn = 2�and �0 is as above. 9

5 A Controlling Di�erential Graded Lie Algebrafor the Deformations of a Spherical PolygonalLinkage.In this section we will describe controlling di�erential graded Lie algebras for thegerms (Cr ;�) and (Mr ; [�]). We let � and R be as in x3. Observe that the sphericalpolygon � gives rise to the representation �� = �0 : � ! SO(3) which sends �i tothe Cartan involution sui on S2 at ui . Now we use [KM4] to construct a di�erentialgraded Lie algebra of forms on the n times punctured 2-sphere that controls (Cr ;�).Let � � � be the subgroup of words of even length in �1; :::; �n and put 1 = �2�1 ; 2 = �3�2 ; :::; n = �1�nThen � is generated by the elements 1; :::; n, subject to the single relation n n�1 ::: 2 1 = 1Thus � is the fundamental group of the n times punctured sphereM = S2 � fm1; :::; mngWe observe that exact sequence1! �! �! Z=2! 1splits. If we split this sequence by sending the generator � of Z=2 to �1 then we obtainthe following action of Z=2 on �:�( i) = �1 i�1 = ( �11 � : : : � �1i�1) �1i ( i�1 � : : : � 1)We let P be the at principal SO(3)-bundle associated to �0j� and adP be the as-soicated Lie algebra bundle. Take Ui be the disjoint punctured disc neighborhoods ofmi, 1 � i � n, and put U = U1[:::[Un . Then by the discussion in x3, the di�erentialgraded Lie algebra B�(M;U ; adP )0 is a controlling di�erential graded Lie algebra forthe deformations of �0j� relative to the collection of subgroups S = f�1; :::;�ng, where�j is the cyclic subgroup generated by j . Note that �j = �j \ �.We now may realize M as the standard round sphere with the points m1; :::; mnremoved from the equator. The action of H := Z=2 on the round sphere given byre ection in the equator carries M into itself. We choose the discs Ui to be invariantunder this action. In particular H acts on �1(M) and we may form the semidirectproduct := H n �1(M). By de�nition is the orbifold fundamental group of theorbifold M=H. The following proposition is central in what follows.Proposition 5.1 and � are isomorphic.Proof: We replace S2 by C [ f1g and the re ection in the equator by re ection inthe x-axis. We take 0 2 C as the basepoint and 1; 2; ::; n as punctures. We let �kbe the loop that proceeds from 0 to k in the upper half plane, encircles k once inthe counterclockwise direction, then returns to 0 in the upper half plane. Clearly10

�1; :::; �n generate �1(M) subject to the relation �1:::�n = 1. The reader will verifythat �(�k) = (�1:::�k�1)��1k (�1:::�k�1)�1; 1 � k � nThus we obtain the required isomorphism by sending �k to �1k , 1 � k � n, and � to�. 2Thus we may apply Theorem 2.10 of [KM4] to deduce the followingTheorem 5.2 The subalgebra of Z=2-invariants B�(M;U ; adP )Z=20 is a controllingdi�erential graded Lie algebra for the relative deformations of �0 in SO(3).Corollary 5.3 The algebra B�(M;U ; adP )Z=20 is a controlling di�erential graded Liealgebra for the germ (Cr ;�). 2As a consequence of Theorem 2.4 we deriveTheorem 5.4 The di�erential graded Lie algebra B�(M;U ; adP )Z=2 controls the germ(Mr ; [�]).Proof: We have only to produce a cross-section S to the orbits of SO(3) on Cr . Thiscross-section S = Sr however was constructed in Lemma 3.5.We can now determine the singular points of Cr . Our goal is to prove thatnon-degenerate polygons are smooth points of Cr . First recall a general result guar-anteeing smoothness of the complete local k-algebra RL . We sketch a proof for theconvenience of the reader.Theorem 5.5 Suppose L� is a di�erential graded Lie algebra over a �eld k of char-acteristic zero. Suppose H2(L) = f0g. Then RL �= k[[H1(L)]] is smooth.Proof: Let MH2 be the maximal ideal corresponding to f0g 2 H2(L). By Theorem3.9 of [GM2], there is a formal map (the Kuranishi map) F : H1(L) ! H2(L) suchthat RL is isomorphic to the complete local k-algebrak[[H1(L)]] =F �MH2(L)2 We can now prove our smoothness theorem for points in Cr .Theorem 5.6 If � is nondegenerate then Cr is smooth at �.Proof: By Artin's Theorem, see Theorem 3.1 of [GM1], it su�ces to prove that thecomplete local ring ofCr at � is isomorphic to R[[H1(L)]], where L = B�(M;U ; adP )Z=20 .Thus by the previous theorem it su�ces to show that H2(L) = 0. ButH2(B�(M;U ; adP )Z=20 ) = H2(B�(M;U ; adP )Z=2)11

since the di�erential graded Lie algebras involved di�er only in degree zero. ByLemma 2.16 of [KM4] we haveH2(B�(M;U ; adP )Z=2) �= H0(�; so(3) �)Here so(3) is the Lie algebra of SO(3) and � : � ! f�1g is the signum characterde�ned by �(�i) = �1, 1 � i � n. ButH0(�; so(3) �) �= H0(�; so(3))�where the � denotes the �-isotypic subspace for the induced action of Z=2 on H0(�; so(3)).If � is nondegenerate the action of � is irreducible and H0(�; so(3)) = f0g. 2Thus we will assume henceforth that � is degenerate and is contained in theequator of S2 and equivalently that the image �0(�) is contained in SO(2) (which weidentify with the subgroup of SO(3) �xing the third coordinate vector of R3). Thuswe have a decomposition adP = E0 � Ewhere E0 is the trivial one-dimensional at bundle and E is the 2-dimensional atbundle corresponding to an irreducible representation from � into SO(2). Of coursewe have E C = L� L_where L is a at complex line bundle and L_ is its dual.We conclude this section with some remarks on the cup-product (or bracket)Q : H1(B�(M;U ; adP ))! H2(B�(M;U ; adP ))Lemma 5.7 H2(B�(M;U ; adP )) �= RProof: We write adP = E0 � E as above and observe by Poincare duality thatH2(B�(M;U ;E)) �= H0(M;E) = f0gH2(B�(M;U ;E0)) �= H0(M;E0) = R2 Thus Q is a scalar-valued quadratic form. We observe that Q is induced by thetensorial bilinear formb : T �(M) adP � T �(M) adP ! �2T �(M) adPgiven by b = b1 b2 where b1 is the wedge product and b2 is the Lie bracket.12

6 Formality Via L2{cohomology andHodge Theory.We now give M a complete hyperbolic metric and letA�(2)(M; adP )be the di�erential graded Lie algebra of smooth adP{valued forms � on M such that� and d� are square integrable for the hyperbolic metric onM and the parallel metricon adP . Since M has �nite area we have an inclusionj : B�(M;U ; adP )! A�(2)(M; adP )We then have the following theorem.Theorem 6.1 The inclusion j is a quasi-isomorphism.Proof: The proof of the theorem will be contained in the next three lemmas. We leti :M ! C P1 be the inclusion. It is easy to see that there is a cover of C P1 by convexopen sets Vi , 1 � i � N , such that each puncture mi is contained in at most one ofthe Vi's. Furthermore, there is a partition of unity f'i; 1 � i � Ng subordinate tothe Vi's and such that 'i is constant in a neighborhood Uj of each mj , 1 � j � n.Now let B� and A�(2) denote the complexes of sheaves on M corresponding toB�(M;U ; adP ) and A�(2)(M; adP ) respectively. We obtain the corresponding \push{forward" complexes i�B� and i�A�(2) on C P1 . Then it is immediate that for any opensubset D � C P1 and spaces i�B�(D) and i�A�(2)(D) are closed under multiplication by'j , 1 � j � N . Thus we have the following lemma.Lemma 6.2 The complexes of sheaves B� and A�(2) are complexes of �ne sheaves. 2We let gadP denote the locally constant sheaf associated to the at bundle adP .The following lemma is obvious.Lemma 6.3 The inclusion i� gadP ,! i�B�is a quasi-isomorphism of complexes of sheaves. 2The next lemma is not obvious but follows from [Z].Lemma 6.4 The inclusion i� gadP ,! i�A�(2)is a quasi-isomorphism of complexes of sheaves.Proof: We recall that adP = E0�E where E0 is a trivial one-dimensional unitary localsystem and E is irreducible. Then the above inclusion is a direct sum of two inclusions.The inclusion corresponding to E is a quasi-isomorphism by [EV], Theorem D2. Theinclusion corresponding to E0 is a quasi-isomorphism by [Z] , Proposition 6.6. 2.Theorem 6.1 now follows by the standard double complex argument, see [G],Theorem 4.6.6. 2Theorem 6.1 has a large number of consequences which we now enumerate.13

Theorem 6.5 The augmented di�erential graded Lie algebra B�(M;U ; adP ) is for-mal.Proof: By Theorem 6.1 it su�ces to prove that A�(2)(M; adP ) is formal. But byTheorem 2.7 of [Z] we have the expected relations among the Laplacians associatedto @; �@ and d, namely �@ = ��@ = 12�dThe ddc{ lemma follows by the usual argument, [DGMS], Lemma 5.11. We thenobtain the standard quasi-isomorphism(B�(M;U ; adP )) � (Ker dc; d) �! (Ker dc=Im dc; 0)The details may be found in [GM1], Section 7. 2In fact we need the following consequences of the above theorem.Theorem 6.6 The augmented di�erential graded Lie algebra of Z=2-invariantsB�(M;U ; adP )Z=2is formal.Proof: The above quasi-isomorphism induces a quasi-isomorphism of Z=2-invariants.2Corollary 6.7 The germ of the relative representation variety(Hom(�; R;SO(3)); �0)is quadratic for the representation �0 : � ! SO(3) associated with any degeneratepolygon �0 .Proof: The corollary follows from [GM1], Theorem 3.5. 2Corollary 6.8 The germ (Cr ;�0) is quadratic for any degenerate polygon �0 .The corollary follows from Theorem 6.6 and [GM1], Corollary 3.6. 2Corollary 6.9 The germ (Mr ; [�0]) is quadratic for any degenerate polygon �0 .Proof: The corollary follows from Theorem 6.6 and [GM1], Corollary 3.6. 2It remains to compute the quadratic equations de�ning (Cr ;�) and (Mr ; [�0]).Thus we must compute the cup product Q from H1(B�(M;U ; adP )) toH2(B�(M;U ; adP )). This will be done in the next section.14

7 Calculation of the Cup Product From theHodge-Riemann Bilinear Relations.In what follows we will use the notation H�(2)(M; adP ) for the cohomology of thecomplex A�(2)(M; adP ). Thus we have as a consequence of Theorem 6.1 a naturalisomorphism H(j) : H�(B�(M;U ; adP )! H�(2)(M; adP )Before stating our �nal consequence of Theorem 6.1 we need a de�nition.De�nition. A Hodge complex is a pair of complexes A� = (A�R; (A�C ; F )) and aquasi-isomorphism A�R C ! A�C such that:(1) Hk(A�R) is �nite dimensional for all k.(2) F � is a decreasing �ltration of A�C of �nite length.(3) The di�erential d of A�C is strict with respect to F � (i.e. an element of F pA�Cis exact in A�C if and only if it is exact in F pA�C ).(4) (Hk(A�R);Hk(A�C ); F p) is a Hodge structure of weight k (see [DGMS], Paragraph5.19).We de�ne a �ltration F � on A�(2)(M; adPC ) as follows.(i) F 0A�(2)(M; adPC ) = A�(2)(M; adPC ),(ii) F 1A�(2)(M; adPC ) = A1;0(2)(M; adPC )� A1;1(2)(M; adPC ),(iii) F jA�(2)(M; adPC ) = 0; j � 2.The following theorem is a consequence of [Z], Section 7, see also [EV], AppendixD.Theorem 7.1 The complex(A�(2)(M; adP ); A�(2)(M; adPC ); F �)is a Hodge complex.As a consequence of Theorem 6.1 and 7.1 the cohomology groups ofB�(M;U ; adPC ) admit a Hodge structure. It is clear thatH0(B�(M;U ; adPC )) = H1;1(B�(M;U ; adPC )) = CIn order to understand the Hodge structure on H1(B�(M;U ; adPC )) we observe that itis the direct sum of the Hodge structures on H1(B�(M;U ;L)) and H1(B�(M;U ;L_)).Thus we have a direct sum decompositionH1(B�(M;U ; adPC )) = H1;0(B�(M;U ;L))� H1;0(B�(M;U ;L_))�H0;1(B�(M;U ;L))� H1;0(B�(M;U ;L_))Remark 7.2 The trivial local system E0 has no cuspidal cohomology of degree 1, i.e.H1(B�(M;U ;E0)) = f0g. We will abbreviate Hp;q(B�(M;U ;L)) (resp.Hp;q(B�(M;U ;L_)) ) to Hp;q(2)(M;L) (resp. Hp;q(2)(M;L_) ).15

We will now compute the dimensions of the four summands in the above Hodgedecomposition. We have decomposed the complexi�ed local systemadPC = E0 C � L� L_Recall that E0 is a trivial local system, L is the one-dimensional unitary local systemwith monodromy representation � given by�( j) = exp(i2�j) ; 1 � j � nand L_ is the dual of L. We note that the equation �j = �jrj, 1 � j � n, of x3 holds.Let L be the holomorphic line bundle over M corresponding to L and L be the sheafassociated to L. We next de�ne �j by �j = �j=�, whence �1 < �j < 1. ThennXj=1�j = 2w and nXj=1 �j = 2�wWe now de�ne the canonical extension Lcan of L to a holomorphic line bundle onC P1 �M . We let O denote the sheaf of germs of holomorphic functions on C P1 . Wede�ne a sheaf Lcan of O-modules byLcan(U) = � �(U; L); if U � M�2(U \ M; L); otherwiseHere �2 denotes the space of square integrable holomorphic sections of L over U \M .In what follows we use K to denote the canonical sheaf of C P1 .Lemma 7.3 The sheaf Lcan is locally free and deg(Lcan) = �b� 2w.Proof: Let � be the canonical multi-valued at section of L. We shall denote by thesame letter a lift of � to the universal cover of ~M of M . We may identify the totalspace of L with the quotient ~M ��1(M) Cand let [ ~m; x] denote the equivalence class of ( ~m; x), where ~m 2 ~M;x 2 C . Then�( ~m) = [ ~m; 1]We see that �( j ~m) = [ j ~m; 1] = [ ~m; �( �1j )1] = exp(�i2�j)�( ~m)Now let mj be a puncture (i.e. an element of C P1�M) and Uj be a neighborhood ofmj on C P1 (as in Section 5). Let z be a holomorphic coordinate vanishing at mj . Wede�ne sj 2 Lcan(U \M) bysj(z) = � z�j�j(z); 0 < �j < 1z1+�j�j(z); �1 < �j < 0Note that sj is a single-valued nowhere vanishing holomorphic section of L overU \M . Now suppose s 2 Lcan(U). We may write s = fsj with f 2 O(U \M). We16

have jsj2 = jf j2r2�j . We recall that the volume element for the Poincare metric onM is rdrd�r2 log2 rthus s 2 Lcan if and only if f is regular at mj . Thus Lcan is locally free and it remainsto compute its degree.We identify C P1 with C [1 in such a way that m1; :::; mn are identi�ed with the�nite complex numbers z1; :::; zn . De�ne �j = 1 if �j < 0 and �j = 0 if �j > 0. Thens(z) = nYj=1(z � zj)�j+�j�is a global meromorphic section of Lcan which has no zeros in C . Clearly s has a poleof order Pnj=1(�j+�j) at1. Since �1+ :::+�n = b and �1+ :::+�n = 2w, the lemmais proved. 2The proof of the next lemma is analogous to that of Lemma 7.3. We let (L_)can bethe sheaf constructed as above using the at line bundle L_ on M dual to L (insteadof L).Lemma 7.4 The sheaf (L_)can is locally free and deg((L_)can) = �f + 2w. 2We can now compute the Hodge pieces Hp;q(2)(M;L) and Hp;q(2)(M;L_). Following[EV], Appendix D, we de�ne a graded sheaf ~� by~0 = O; ~1 = (D)here D = Pnj=1mj, so degD = n. Then by [EV], Proposition D.4, we have(i) Hp;q(2)(M;L) = Hq(C P1 ; ~p Lcan);(ii) Hp;q(2)(M;L_) = Hq(C P1 ; ~p (L_)can);The following theorem gives the dimensions of the Hodge summands.Theorem 7.5 (i) dimH1;0(2)(M;L) = f � 2w � 1,(ii) dimH0;1(2)(M;L) = b + 2w � 1,(iii) dimH1;0(2)(M;L_) = b + 2w � 1,(iv) dimH0;1(2)(M;L_) = f � 2w � 1,Proof: Use the previous formulas for the Hodge structure and the Riemann-Rochtheorem for C P1 . 2We now extend the quadratic form Q of Section 5 to a Hermitian form H on thecomplexi�ed �rst cohomology. Recall that Q is induced by the pointwise bilinearform b = b1b2 on T �(M)adP . We extend b1; b2 to a vector-valued skew-hermitianforms h1; h2 on T �(M) C and adP C respectively by the following formulas� ^ �� = h1(�; �); [u; �v] = h2(u; v)Here �; � 2 T �(M) C and u; v 2 adPC . Note that the standard �ber of L isspanned by � = e1 � ie2 and the standard �ber of L_ is spanned by �� = e1 + ie2 .Then [�; ��] = 2ie3 . Thus h1 and h2jEC take values in trivial line bundles over M .We identify these forms with scalar-valued Hermitian forms using the bases vol for�2T �(M) and e3 for E0 . The following lemma is immediate.17

Lemma 7.6 (i) The subspaces (T �)1;0(M) and (T �)0;1(M) are orthogonal for h1 .(ii) The form ih1 is positive de�nite on (T �)1;0(M) and negative de�nite on (T �)0;1(M).(iii) The subspaces L and L_ are orthogonal for h2 .(iv) The form �ih2 is positive de�nite on L and negative de�nite on L_ . 2Corollary 7.7 The tensor product h = h1 h2 is positive de�nite on(T �)1;0(M) L and (T �)0;1(M) L_ and negative de�nite on (T �)0;1(M) L and(T �)1;0(M) L_ . 2We now de�ne a hermitian form H on H1(2)(M; adPC ) byH(�; �) = ZM h(�; �)The pointwise results of Lemma 7.6 imply the following lemma.Lemma 7.8 (Hodge-Riemann bilinear relations)(i) The four Hodge summands are orthogonal for H.(ii) The restriction of H to each of the four summands is de�nite, H is positivede�nite on the two summands of dimension f � 2w � 1 and negative de�nite on thetwo summands of dimension b + 2w � 1. 2Corollary 7.9 The form H is nonsingular of signature (2(f�2w�1); 2(b+2w�1)).2 We next compute the signature of the restriction of the form H to the subspaceH1(2)(M; adPC )Z=2of Z=2{invariants. Recall that the Z=2 action is induced by the action of � on M andthe action of �0(�) on the coe�cients of the (complexi�ed ) monodromy action. Werecall that we have chosen�0(�) = �0(�1) = 0B@ 1 0 00 �1 00 0 �11CAWe �nd that the action of � on coe�cients of E = L� L_ is given by�(�)� = ���; �(�)�� = ��Thus �(�) realizes the duality on H1(2)(M; adPC ), it interchanges the two (f �2w�1)-dimensional components and the two (b + 2w � 1)-dimensional components. Thus,H1(2)(M; adPC )Z=2 = (H1;0(2)(M;L)� H0;1(2)(M;L_))Z=2� (H1;0(2)(M;L_)� H0;1(2)(M;L))Z=2We obtain the following lemma.Lemma 7.10 The induced hermitian form onH1(2)(M; adPC )Z=2has signature (f � 2w � 1; b+ 2w � 1). 218

We now restrict to the real subspace H1(2)(M; adP ) of H1(2)(M; adPC ) and obtainthe following theorem.Theorem 7.11 The quadratic form Q on H1(2)(M; adP )Z=2 has signature (f � 2w �1; b+ 2w � 1).Proof: We will prove that the above splitting of H1(2)(M; adPC )Z=2 is de�ned over R.Indeed, forms in the �rst summand may be represented by f(dz � + d�z ��) andforms of the second summand by g(dz ��+ d�z �) (recall that we identify M withC [ f1g� fz1; :::; zng. Both spaces of forms are clearly de�ned over R. 2Corollary 7.12 The quadratic form Q on H1(B�(M;U ; adP ))Z=2 has signature (f �2w � 1; b+ 2w � 1).Let Z(Q) be the null-cone fQ = 0g � H1(B�(M;U ; adP ))Z=2 .We can now prove Theorem 1.1 of the Introduction. We claimH0(B�(M;U ; adP ))Z=2 = f0gIndeed, H0(B�(M;U ; adP )) �= Re3 � so(3) and the action of Z=2 on this space iseasily seen to be given by multiplication by �1. By Theorem 5.2, the augmentationideal (B�(M;U ; adP ))Z=20is a controlling di�erential graded Lie algebra for the deformations of �0 . We nowapply Theorem 3.5 of [GM1] noting the cone QH(L) of that theorem is the cone Z(Q)discussed in the above corollary.Theorem 1.1 is now immediate from Theorem 5.4, Corollary 6.9 and Corollary7.12.It is immediate from the above that a spherical polygonal linkage is (locally) rigidif and only if the quadratic form Q is de�nite, which is equivalent to: f � 2w� 1 = 0or b + 2w � 1 = 0. We thus obtain Corollary 1.2 from the Introduction.8 The Topology of the Moduli Space of a Spher-ical Polygonal Linkage.In this section we combine our results in x7 with results of A. Galitzer [Ga] to give a\wall-crossing" algorithm for computing the topology of the moduli spaces Mr .Let Pn denote the space of all n-gons � = (�1; :::; �n) in S2 (see the de�nitionin x3). Thus Pn �= (S2)n . Let Qn denote the quotient of Pn by SO(3) and let� : Qn ! Rn be the map given by �(�) = r = (r1; :::; rn), where ri is the lengthof the geodesic arc �i, 1 � i � n. We let Dn(S2) denote the image of �. In [Ga]A. Galitzer has described Dn(S2). We will need some notation to describe her results.If I � f1; 2; :::; ng we let �I denote the complement of I, jIj be the cardinality of Iand rI = Pi2I ri . De�ne a polyhedron Kn � Rn by the system of inequalities0 � ri � �; 1 � i � n; and19

rI � r�I + (jIj � 1)�; I � f1; 2; :::; ng; with jIj oddThen Galitzer provesTheorem 8.1 Kn = Dn(S2) .In addition she proves that the codimension 1 faces of Dn(S2) are given by theintersections of the hyperplanes corresponding to the above inequalities with Kn, i.e.the above representation of Kn is irredundant.The space Qn is di�cult to work with since it has singularities corresponding to�xed points of subgroups of SO(3). To remedy this we let P0n denote the open subset ofPn corresponding to those n-gons such that successive vertices do not coincide and arenot antipodal. We let Q0n denote the quotient of P0n by SO(3). Then Q0n is naturallya smooth manifold of dimension 2n� 3. Indeed, Q0n is naturally di�eomorphic to thesubmanifold S � P0n consisting of those n-gons with the vertex set u = (u1; :::; un)satisfying u1 = �!e1 ; (u2;�!e3 ) = 0; (u2;�!e2 ) > 0Note that �(Q0n) � int(Kn) = K0n . We will henceforth replace � by its restriction toQ0n .We shall see shortly that the set of critical values of � inside K0n is the union ofa collection of hyperplane sections of K0n . We call these hyperplane sections walls ofKn . Connected components in K0n of the union of walls are called chambers. In [Ga],Galitzer determines the walls of Kn . We again summarize her results.Let I � f1; :::; ng be any non-empty subset. For each nonnegative integer w letHI;w denote the hyperplane in Rn de�ned by the equationrI � r�I = 2�wWe then have the following lemma of GalitzerLemma 8.2 HI;w \K0n 6= ; () jIj � 2w + 2.We now prove that the hyperplanes HI;w with jIj � 2w + 2 are the walls of Kn .Lemma 8.3 Let r 2 K0n . Then r is not a regular value of � if and only if r 2 HI;wfor some I; w � 0 with jIj � 2w + 2.Proof: Suppose �rst that r is on HI;w with jIj � 2w+2. We will show that ��1(r) issingular, which would mean that r is not a regular value of �. Since r is on a wall thereis a subset I � f1; :::; ng such that rI�r�I = 2�w with jIj � 2w+2. We can constructa degenerate n-gon u in ��1(r) by taking �I to be the set of indices corresponding tothe back-tracks and I to be the set of indices corresponding to forward-tracks. (Note:we do not assume here that u belongs to the cross-section Sr .) Since rI � r�I = 2�wthe resulting degenerate linkage closes up. By Theorem 7.11 we �nd that u is asingular point on Mr because the germ of Mr at u is isomorphic to the germ of aquadratic cone of signature (f � 2w � 1; b+ 2w � 1) at 0. Thus ��1(r) is singular.Conversely, suppose that r is not on a wall of Kn . This implies that Mr containsno degenerate polygons. Let u 2 ��1(r). Now the kernel of d� : Tu(Q0n)! Tr(Rn) isthe Zariski tangent space Tu(Mr) of Mr at u. By Theorem 5.3 we haveTu(Mr) �= H1(B�(M;U ; adP )Z=2)20

Since u is nondegenerate the corresponding representation � is irreducible anddim H1(B�(M;U ; adP )Z=2) = n� 3Hence dim Im(d�u) = (2n� 3)� (n� 3) = n and d�u is onto. 2Since � is proper it is a �bration over each chamber and the topology of the �bersdoes not change with a chamber. We now compute how the topology of the �berschanges when we cross a wall.Suppose that r� 2 K0n lies on the intersection of the wallsHI1;w1; HI2;w2; :::; HIp;wpChoose u� a degenerate linkage with �(u�) = r� . Let L � K0n be the line segmentde�ned by ri = r�i , 1 � i � n� 1 and �� � rn � �. Here � is chosen so that L doesn'tintersect any wall except at r� . Let XL = ��1(L).Lemma 8.4 XL is a smooth submanifold of Pn of dimension n � 2. The inclusioni :Mr� ,! XL induces an isomorphism of tangent spacesi� : Tu�(Mr�)! Tu�(XL)Proof: A point in XL is a closed n-gon where the lengths of the �rst n� 1 sides areprescribed to be r�1; r�2; :::; r�n�1 but the length of the n-th side is not determined. Theoperation of forgetting the n-th side gives an isomorphism to the moduli space of thefree linkage where the underlying map is obtained by deleting the n-th side of then-gon. Clearly the moduli space of such a free linkage is the product of n� 2 circles.Since dim(Tu�(Mr�)) = n � 2, dim(Tu�(XL)) = n� 2, and i� is an injection, it isnecessarily an isomorphism.Remark 8.5 In the above Tu�(Mr�) is the Zariski tangent space of Mr� . Note thatdim Tu�(Mr�) = dim Mr� + 1Every in�nitesimal deformation of u� in Mr� is tangent to a curve in XL .Thus we have reduced the problem of �nding a wall-crossing formula to computinghow the level sets of rnjXL change when we pass from �� to �. Our desired formulawill be a consequence of the next three lemmas.Lemma 8.6 If u 2 XL is a critical point of rnjXL then u is degenerate.Proof: Our arguments essentially repeat the proof of Lemma 8.3. Let u 2 XL andV = ker d(rnjXL)u . Then V � Tu(XL). By de�nition V is the Zariski tangent spaceof u to the �ber of rnjXL through u. HenceV �= H1(B�(M;U ; adP )Z=2)Therefore if u is nondegenerate then dim(V ) = n � 3 and dim (Im d(rnjXL)u) = 1.2 21

Lemma 8.7 rnjXL has exactly p critical points.Proof: We apply Lemma 8.6 and observe that any critical point of rnjXL must be adegenerate linkage v lying in ��1(r�). Let J be the set of forward-tracks and w bethe winding number of v. Then rJ � r �J = 2�w. Hence there exists i between 1 and psuch that J = Ii, w = wi . But a degenerate linkage in Mr� is determined by its setof forward-tracks. 2Let u�1; u�2; :::; u�p be the set of critical points of rnjXL . Let u� be one of thesecritical points and let Q�L be the Hessian of rnjXL at u� . Recall thatQ : H1(B�(M;U ; adP )Z=2)! Rdenotes the cup-product. By Theorem 5.4 we have a commutative diagram (withthe horizontal arrows � and isomorphisms and the vertical arrows � and �0 thecanonical projections):Tu�(Mr�) ��! H1(B�(M;U ; adP )Z=2)�x?? x??�0T (2)u� (Mr�) �! T (2)�0 (Hom(�; R;SO(3))=SO(3)Here T (2) denotes the 2-jet bundle, i.e. the equivalence classes of analytic curvesup to order three contact (we recall that Mr has at worst quadratic singularities).The image of �0 is the null-cone Z(Q) of Q by Theorem 5.4, since the cup-product Qis the obstruction to lifting a tangent vector to a 2-jet, see [GM1], x4.4.Combining the isomorphism � above with the isomorphism i� of Lemma 8.6 weobtain a canonical isomorphism� : Tu�(XL) �= H1(B�(M;U ; adP )Z=2)The following Lemma gives the critical link between Morse theory and deformationtheory.Lemma 8.8 Under the isomorphism � the null cone Z(Q�L) of Q�L is carried onto thenull-cone Z(Q) of Q.Proof: Suppose that � 2 Tu�(XL) is annihilated by Q�L . Let a(t) be a curve inXL such that a(0) = u� and a0(0) = �. Then ri(a(t)) = r�i , 1 � i � n � 1 andrn(a(t)) � r�n (mod t3). Thus a(t) induces a 2-nd order deformation of the linkage u�in Mr�, i.e. an element � 2 T (2)(Mr�) such that �(�) = �. But by [KM4], Theorem3.2, the image of � under is a 2-nd order deformation of the representation ��corresponding to u� . Since �0( ) = �(�) we have �(Z(Q�L)) � Z(Q). Conversely,let � 2 Z(Q). Choose 2 (�0)�1(�) and put � = �1( ). Since XL is smooth andT (2)(Mr�) � T (2)(XL), the 2-jet � is represented by a curve a(t) in XL preserving rnup to a term of order 3. Let � = �(�). Then � 2 Z(Q�L). But � = ��1(�). 2Corollary 8.9 Q�L is nondegenerate. 22

Proof: We have seen that Q is nondegenerate. Hence the projectivization of Z(Q)is smooth. Hence the projectivization of Z(Q�L) is smooth. But a quadratic form isnondegenerate if and only if the projectivization of its null cone is smooth. Hence Q�Lis nondegenerate. 2Since Corollary 7.12 determines Z(Q) and Z(Q) = Z(Q�L) we obtainTheorem 8.10 rnjXL is a Morse function with a �nite collection of critical pointsu�1; :::; u�p, all located on the critical �ber Mr� . The critical point u�j corresponds toa degenerate n-gon linkage in Mr� with fi forward-tracks, bi back-tracks and windingnumber wi . Then the signature of the Hessian of rnjXL at u�j is either (fi � 2wi �1; bi + 2wi � 1) or (bi + 2wi � 1; fi � 2wi � 1) depending on the orientation of L. 2References[BH] H. Boden, Y. Hu, Variations of moduli of parabolic bundles, Math. Annalen,301 (1995) 539{559.[BY] H. Boden, K. Yokogawa, Moduli spaces of parabolic Higgs bundles andparabolic K(D) pairs overs smooth curves: I, preprint.[DGMS] P. Deligne, P. Gri�ths, J. Morgan and D. Sullivan, Rational homotopy typeof compact Kahler manifolds, Inv. Math., 29 (1975), 245{ 274.[EV] H. Esnault, E. Viehweg, Logarithmic de Rham complexes and vanishing the-orems, Inv. Math., 86 (1986), 161{ 194.[FS] R. Fintushel, R. Stern, Instanton homology of Seifert �bered homology threespheres, Proc. of London Math. Soc., (3) 61 (1990) 109{137.[Ga] A. Galitzer, Ph.D. Thesis, University of Maryland.[G] R. Godement, \Theorie des fasceaux," Hermann, 1973.[GM1] W. Goldman, J. J. Millson, The deformation theory of representations offundamental groups of compact Kahler manifolds, Publ. of IHES, 67 (1988)43{ 96.[GM2] W. Goldman, J. J. Millson, The homotopy invariance of the Kuranishi space,Ill. Journal of Mathematics, 34 (1990) 337{367.[KM1] M. Kapovich, J. J. Millson, On the moduli space of polygons in the Euclideanplane, Journal of Di�. Geometry, 42 (1995) N 1, 133{164.[KM2] M. Kapovich, J. J. Millson, The symplectic geometry of polygons in Euclideanspace, Journal of Di�. Geometry (to appear).[KM3] M. Kapovich, J. J. Millson, On the deformation theory of representations offundamental groups of hyperbolic 3-manifolds, Topology (to appear).[KM4] M. Kapovich, J. J. Millson, The relative deformation theory of representationsand at connections and deformations of linkages in constant curvature spa-ces, Compositio Math, to appear.23

[KM5] M. Kapovich, J. J. Millson, On representation varieties of Artin groups,projective arrangements and fundamental groups of smooth complex quasi-projective varieties, preprint.[KK] P. Kirk, E. Klassen, Representation spaces of Seifert �bered homology spheres,Topology, 30 (1991) 77-95.[MS] V. Mehta, C. Seshadri, Moduli of vector bundles on curves with parabolicstructures, Math. Annalen, 248 (1980) 205{ 239.[M] J. J. Millson, Rational homotopy theory and deformation problems from al-gebraic geometry, Proceedings of ICM 1990, Vol. I, 549{ 558.[S] C. Simpson, Harmonic bundles on noncompact curves, Journal of AMS, 3(1990) 713{770.[Z] S. Zucker, Hodge theory with degenerating coe�cients: L2 cohomology in thePoincare metric, Annals of Mathematics, 109 (1979) , 415{ 476.Michael Kapovich: Department of Mathematics, University of Utah, Salt Lake City,UT 84112, USA; [email protected] J. Millson: Department of Mathematics, University of Maryland, College Park,MD 20742, USA; [email protected]

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