Torsion and symplectic volume in Seifert manifolds
aa r X i v : . [ m a t h . G T ] O c t Torsion and symplectic volume in Seifert manifolds
L. Charles and L. Jeffrey ∗ October 9, 2018
Abstract
For any oriented Seifert manifold X and compact connected Liegroup G with finite center, we relate the Reidemeister density of themoduli space of representations of the fundamental group of X into G to the Liouville measure of some moduli spaces of representations ofsurface groups into G . For any Lie group G and manifold Y , the moduli space M ( Y ) of conjugacyclasses of representations of π ( Y ) in G , has natural differential geometricstructures. If Σ is a closed oriented surface, M (Σ) has a symplectic stucturedefined via intersection pairing [AB83], [Gol84]. More generally, if Σ is acompact oriented surface and u ∈ M ( ∂ Σ), the subspace M (Σ , u ) of M (Σ)consisting of the representations restricting to u on the boundary has a nat-ural symplectic structure. If X is a closed 3-dimensional oriented manifold, M ( X ) has a natural density µ X defined from Reidemeister torsion [Wit89].In this article, we relate these structures for X any oriented Seifert mani-fold and Σ a convenient oriented surface embedded in X . We will prove thatwhen G is compact with finite center, the subspace M ( X ) ⊂ M ( X ) of irre-ducible representations, is a smooth manifold covered by disjoint open sub-sets O α , such that each O α identifies with M (Σ , u α ) for some u α ∈ M ( ∂ Σ).Furthermore, on each U α the canonical density µ X identifies, up to somemultiplicative constant depending on α , with the Liouville measure of thesymplectic structure of M (Σ , u α ).Our main motivation is the Witten’s asymptotic conjecture, which pre-dicts that the Witten-Reshetikhin-Turaev invariant of a 3-manifold X has ∗ Supported in part by a grant from NSERC M ( X ). In the case where X is a Seifertmanifold, some of these amplitudes are actually function of the symplec-tic volumes of the moduli spaces M (Σ , u ), [Roz95], [Cha]. So a relationbetween Reidemeister and symplectic volumes was expected. At a more gen-eral level, it is known that the Chern-Simons theory on a Seifert manifoldcan be interpreted as two-dimensional Yang-Mills theory [BW05].Let us state our results with more detail and then discuss the relatedliterature. Statement of the main result
The Seifert manifolds we will consider are the oriented closed connectedthree manifold equipped with a locally free circle action. Any such manifoldmay be obtained as follows. Let Σ be an oriented compact surface with n > C , . . . , C n . Let D be the standard closeddisk of C . Let ϕ i be an orientation reversing diffeomorphism from ∂D × S to C i × S . Let X be the manifold obtained by gluing n copies of D × S to Σ × S through the maps ϕ i . We have [ ϕ i ( ∂D )] = − p i [ C i ] + q i [ S ] in H ( C i × S ) where p i , q i are two relatively prime integers. We assume that p i > i .Let G be a compact connected Lie group with finite center. For Y = X ,Σ, C i or S , we denote by M ( Y ) (resp. M ( Y )) the set of representations(resp. irreducible representations) of π ( Y ) in G up to conjugation. Since C i and S are oriented circles, we can identify M ( C i ) and M ( S ) with the set C ( G ) of conjugacy classes of G . For any u ∈ C ( G ) n , we denote by M (Σ , u )the subset of M (Σ) consisting of the representations whose restriction toeach C i is u i . Recall that M (Σ , u ) is a smooth symplectic manifold.For any ( u, v ) ∈ C ( G ) n +1 , we denote M ( X, u, v ) the subset of M ( X )consisting of representations whose restriction to each C i is u i and to S is v .Let P be the subset of C ( G ) n +1 consisting of the ( u, v ) such that M ( X, u, v )is non empty.
Theorem 1.1. M ( X ) is a smooth manifold, whose components may havedifferent dimensions. For any [ ρ ] ∈ M ( X ) , the tangent space T [ ρ ] M ( X ) is canonically identified with H ( X, Ad ρ ) where Ad ρ is the flat vector bun-dle associated to ρ via the adjoint representation. Furthermore, P is finiteand for any ( u, v ) ∈ P , M ( X, u, v ) is an open subset of M ( X ) and therestriction map R u.v from M ( X, u, v ) to M (Σ , u ) is a diffeomorphism. ρ of π ( X ) in G , the homology groups H ( X, Ad ρ ) and H ( X, Ad ρ ) are trivial. By Poincar´e duality, H ( X, Ad ρ )is the dual of H ( X, Ad ρ ). So the Reidemeister torsion of Ad ρ is a non van-ishing element of (cid:0) det H ( X, Ad ρ ) (cid:1) − well-defined up to sign. Consequently,the inverse of the square root of the torsion is a density of H ( X, Ad ρ ). Since H ( X, Ad ρ ) identifies with the tangent space of M ( X ) at ρ , we define inthis way a density µ X on M ( X ).For any u ∈ C ( G ) and [ ρ ] ∈ M (Σ , u ), the tangent space T [ ρ ] M (Σ , u )is identified with the kernel of the morphism H (Σ , Ad ρ ) → H ( ∂ Σ , Ad ρ ).The symplectic product of T [ ρ ] M (Σ , u ) is induced by the intersection prod-uct of H (Σ , Ad ρ ) with H (Σ , ∂ Σ , Ad ρ ). We denote by µ u the correspond-ing Liouville measure of M (Σ , u ).As a last definition, let ∆ : C ( G ) → R be the function given by∆( u ) = (cid:12)(cid:12) det H g (Ad g − id) (cid:12)(cid:12) / where g is any element in the conjugacy class u and H g is the orthocomple-ment of ker(Ad g − id). Equivalently, let t be the Lie algebra of a maximaltorus of G , R ⊂ t ∗ be the corresponding set of real roots and R + ⊂ R be aset of positive roots. Then for any X ∈ t ,∆([ e X ]) = Y α ∈ R + ; α ( X ) =0 | sin( πα ( X )) | Theorem 1.2.
For any ( u, v ) ∈ P , we have on M ( X, u, v ) µ X = n Y i =1 ∆( u r i i ) p (dim G − dim u i ) / i ! R ∗ u,v µ u where R u,v is the restriction map from M ( X, u, v ) to M (Σ , u ) and foreach i , r i is any inverse of q i modulo p i , and u r i i ∈ C ( G ) is the conjugacyclass containing the g r i for g ∈ u i . Several definitions require an invariant scalar product on the Lie algebraof G : the symplectic structure of M (Σ , u ), the Poincar´e duality between H ( X, Ad ρ ) and H ( X, Ad ρ ) and the Reidemeister torsion of Ad ρ . Ourimplicit convention is to choose the same invariant scalar product each time.During the proof, we will prove interesting intermediate results:- for any irreducible representation ρ of π ( X ) in G , the cohomologygroups H ( X, Ad ρ ) and H ( X, Ad ρ ) both identify naturally with thekernel of the restriction morphism H (Σ , Ad ρ ) → H ( ∂ Σ , Ad ρ ).3 by these identifications, the intersection product of H ( X, Ad ρ ) with H ( X, Ad ρ ) is sent to the intersection product of H (Σ , Ad ρ ) with H (Σ , ∂ Σ , Ad ρ ).- the Reidemeister torsion τ (Ad ρ, X ) is equal to C − det ψ where ψ : H ( X, Ad ρ ) → H ( X, Ad ρ ) is the map induced by the previous iden-tifications and C is the factor appearing in Theorem 1.2.This results are respectively proved in Sections 4, 5 and 6. Theorem 1.2 isproved in Section 7 and Theorem 1.1 in Section 3.2. Related results in the litterature
Witten [Wit91] proved that for S a closed oriented surface, the canonicaldensity µ S of M ( S ) defined from Reidemeister torsion, is the Liouvillemeasure of the natural symplectic structure of M ( S ). He also extendedthis result to surfaces with boundary. We tried to deduce Theorem 1.2 fromthis by expressing the torsion τ (Ad ρ, X ) in terms of τ (Ad ρ, Σ), withoutany success. Our actual proof does not use Witten’s result.Witten also computed explicitely the volumes R M (Σ ,u ) µ u , cf. [Wit91],Formula 4.114. For G = SU(2) and non central conjugacy classes u i , Park[Par97] adapted the Witten’s method to compute R M ( X,u,v ) µ X , X being ourSeifert manifold. Computing the volume of M ( X, u, v ) with Theorem 1.2and Witten’s formula, we can extend Park’s result to any compact connectedLie group G with finite center and any conjugacy classes u i .McLellan [McL15] proved a result similar to Theorem 1.2 for G = U(1).To do this, he introduced a Sasakian structure on X and used a computationof the corresponding analytic torsion [RS12]. We will explain in Section 8how we can recover McLellan’s result by adapting our method, providing anelementary proof. X Let g, n, p , q , . . . , p n , q n be integers such that g > , n > ∀ i, p i , q i are coprime and p i > . (1)To such a familly we associate the following manifold X . Let Σ be a compactoriented surface with genus g and n boundary components denoted by C ,4 . . , C n . Let D be a closed disc and for any i , let ϕ i : ∂D × S → C i × S bean orientation reversing diffeomorphism such that we have in H ( S × C i ),[ ϕ i ( ∂D )] = − p i [ C i ] + q i [ S ] . (2)where ∂D and C i are oriented as boundaries of D and Σ respectively. Then X is obtained by gluing n copies of D × S to Σ × S along its boundarythrough the maps ϕ i , X = (Σ × S ) ∪ ϕ ∪ ... ∪ ϕ n ( D × S ) ∪ n . (3)By construction Σ × S is a submanifold of X . In the sequel we often considerΣ and S as submanifolds of X by identifying Σ with Σ × { y } and S with { x } × S , where x and y are some fixed points of Σ and S respectively.The above definitions are all what we need for this article. Nevertheless,it is interesting to understand this in the context of Seifert manifolds. First,if X is obtained as previously, we can extend the S -action on Σ × S to X ,so that for any i , the action on the i -th copy of D × S is free if p i = 1 andotherwise it has one exceptional orbit with isotropy Z p i . Conversely, considerany three dimensional closed connected oriented manifold Y equipped withan effective locally free action of S . Then choose n > O , . . . , O n of Y including all the exceptional ones. Let T , . . . , T n be disjointsaturated open tubular neighborhoods of the O , . . . , O n respectively. LetΣ be any cross-section of the action on Y \ ( T ∪ . . . ∪ T n ). For any i , set C i = ( ∂ Σ) ∩ T i and define p i as the order of the isotropy group of O i and q i so that [ C i ] = q i [ O i ] in H ( T i ), where C i is oriented as the boundaryof Σ and O i by the S -action. Let X be any manifold associated to thedata Σ, ( p , q ), . . . , ( p n , q n ) as in (3). Then Y is diffeomorphic to X , cf.[JN83], Theorem 1.5 or the Section 1 of [NR78] for more details. We caneven choose the diffeomorphism between Y and X so that it commutes withthe S -action and fixes Σ. The collection( g ; ( p , q ) , . . . , ( p n , q n ))is called the unnormalised Seifert invariant of Y .5 Character space of a Seifert manifold
Notations
Let G be a Lie group. For any connected topological space Y , we denote by M ( Y ) the set of conjugacy classes of representations of π ( Y ) into G . Arepresentation ρ : π ( Y ) → G is said to be irreducible if the centraliser of ρ ( π ( Y )) is reduced to the center of G . We denote by M ( Y ) the subset of M ( Y ) consisting of conjugacy classes of irreducible representations.If Z is a subspace of Y , there is a natural morphism j ∗ from π ( Z ) to π ( Y ) and consequently a natural map from M ( Y ) to M ( Z ), sending [ ϕ ]into [ ϕ ◦ j ∗ ]. For any representation ρ : π ( Y ) → G , we call ρ ◦ j ∗ therestriction of ρ to Y . M ( X ) From now on, X is the Seifert manifold introduced in Section 2. Recall thatwe view S and Σ as submanifolds of X . Proposition 3.1.
Let ρ be a representation of π ( X ) into G . Then ρ isirreducible if and only if its restriction to Σ is irreducible. Furthermore, if ρ is irreducible, then ρ ( S ) is central. Finally, for any i , ρ ( C i ) p i is conjugateto ρ ( S ) q i . In the statement we slightly abused notation by applying ρ to orientedcircles of X . Since any loop γ of X is homotopic to an element of π ( X )unique up to conjugation, the conjugacy class of ρ ( γ ) is uniquely defined. Proof.
By Van Kampen theorem, the natural morphism π (Σ × S ) → π ( X )is onto. So ρ : π ( X ) → G is irreducible if and only if its restriction to π (Σ × S ) is irreducible. Since ρ ( π (Σ)) ⊂ ρ ( π (Σ × S )), if ρ | Σ is irreducible,then ρ | Σ × S is irreducible. Conversely, assume that ρ | Σ × S is irreducible.Since π (Σ × S ) ≃ π (Σ) × π ( S ), t = π ( S ) is in the centraliser of π (Σ × S ), and consequently ρ ( t ) is central. This implies that ρ ( π (Σ × S ))and ρ ( π (Σ)) have the same centraliser. So ρ | Σ is irreducible.By Equation (2), ρ ( C i ) p i and ρ ( S ) q i are conjugate.Let Z ( G ) be the center of G and C ( G ) be the set of conjugacy classes. If u ∈ C ( G ) and p is an integer, u p ∈ C ( G ) is defined as the conjugacy class of g p where g ∈ u . Let P be the subset of C ( G ) n × Z ( G ) consisting of the pairs A representation of π ( Y ) into G is a group morphism from π ( Y ) to G . Two repre-sentations ρ, ρ ′ are conjugate if there exists g ∈ G , such that ρ ′ ( h ) = gρ ( h ) g − , ∀ h ∈ G . u, v ) such that for any i , u p i = v q i . Then by the last part of Proposition3.1, M ( X ) = [ ( u,v ) ∈P M ( X, u, v ) (4)where M ( X, u, v ) consists of the [ ρ ] ∈ M ( X ) such that ρ ( S ) ∈ v and ρ ( C i ) ∈ u i for any i . Denote by R u,v the restriction map R u,v : M ( X, u, v ) → M (Σ , u ) , [ ρ ] → [ ρ | Σ ] (5)where M (Σ , u ) is the subset of M (Σ) consisting of the classes [ ρ ] suchthat for any i , ρ ( C i ) ∈ u i . Proposition 3.2.
For any ( u, v ) ∈ P , the map R u,v is a bijection.Proof. It is a consequence of the fact that π (Σ × S ) = π (Σ) × π ( S ) andthat the kernel of the surjective map π (Σ × S ) → π ( X ) is the normalsubgroup normally generated by the ϕ i ( ∂D )’s. From now on, assume that G is compact and has a finite center. As explainedin appendix A, for any compact connected manifold Y , M ( Y ) has a naturalHausdorff topology and M ( Y ) is an open subset. Lemma 3.3.
The set P is finite. For any ( u, v ) ∈ P , M ( X, u, v ) is anopen subset of M ( X ) .Proof. By identifying C ( G ) with the quotient of a maximal torus by theWeyl group, we easily see that for any v ∈ C ( G ) and p ∈ Z , the equation u p = v has only a finite number of solutions. This implies that P is finite.We deduce that the M ( X, u, v )’s are open by applying the following fact:for any compact connected manifold Y , for any x ∈ π ( Y ), the map from M ( Y ) to C ( G ) sending [ ρ ] into [ ρ ( x )] is continuous.By Appendix A, M ( X ) has a natural open subset M s , ( X ) which is amanifold. Furthermore, it is known that the spaces M (Σ , u ) are smoothmanifolds. Proposition 3.4.
We have M ( X ) = M s , ( X ) . Furthermore, for any ( u, v ) ∈ P , R u,v is a diffeomorphism from M ( X, u, v ) to M (Σ , u ) .
7t is possible that the various M ( X, u, v ) have different dimensions.Actually, dim M (Σ , u ) = 2( g −
1) dim G + n X i =1 dim u i . Proof.
Let u ∈ C ( G ) n and consider the set M u of ( a, b, c ) ∈ ( G g + n ) satis-fying the relations[ a , b ] . . . [ a g , b g ] c . . . c n = id , c i ∈ u i , ∀ i. Here we used the same notation ( G g + n ) as in Appendix A. It is knownthat M u is a smooth submanifold of G g + n .Choose a standard set of generators ( x, y, z ) of π (Σ) and let t ∈ π ( X )be isotopic to S . The map π (Σ × S ) → π ( X ) being onto, ( x, y, z, t ) isa set of generators of π ( X ). Through these generators, R ( π ( X )) getsidentified with a subset A of G g + n +1 as explained in Appendix A. Bythe decomposition (4), A is the union of the M u × { v } where ( u, v ) runsover P . P being finite, A is a submanifold of G g + n +1 , which shows that R ( π ( X )) = R s , ( π ( X )) in the notation of Appendix A and consequentlythat M s , ( X ) = M ( X ). For any ( u, v ) ∈ P , the projection M u ×{ v } → M u being a diffeomorphism, we conclude that R u,v is a diffeomorphism.Consider again a compact connected manifold and a representation ρ of π ( Y ) in G . Composing ρ with the adjoint representation, the Lie al-gebra g becomes a π ( Y )-module. Denote by H • ( π ( Y ) , Ad ρ ) the groupcohomology with coefficient in g . Alternatively, we may consider the flatvector bundle Ad ρ → Y associated to ρ via the adjoint representation. Let H • ( Y, Ad ρ ) be the cohomology of Y with local coefficient. Then for j = 0or 1, H j ( π ( Y ) , Ad ρ ) ≃ H j ( Y, Ad ρ ). Lemma 3.5.
For any irreducible representation ρ of π ( X ) , we have anatural identification between H ( X, Ad ρ ) and T [ ρ ] M ( X ) .Proof. By appendix A, T [ ρ ] M ( X ) is naturally identified with a subspaceof H ( X, Ad ρ ). Similarly, it is known that T [ ρ ] M (Σ , u ) gets identified tothe kernel of the morphism H (Σ , Ad ρ ) → H ( ∂ Σ , Ad ρ ). Furthermore,we easily see that the tangent linear map to R u,v is the restriction of themorphism H ( X, Ad ρ ) → H (Σ , Ad ρ ). As we will see in Theorem 4.2, thefollowing sequence is exact0 → H ( X, Ad ρ ) → H (Σ , Ad ρ ) → H ( ∂ Σ , Ad ρ ) → . This implies that H ( X, Ad ρ ) = T [ ρ ] M ( X ).8 The homology groups H ( X, Ad ρ ) and H (Σ , Ad ρ ) As in the previous section, for any compact connected topological space Y and representation ρ : π ( Y ) → G , we consider the flat vector bundleAd ρ → Y . We are interested in corresponding homology groups H • ( Y, Ad ρ )for Y = X or Σ. As a first remark, if ρ is irreducible, then by AppendixA, H ( Y, Ad ρ ) = H ( π ( Y ) , Ad ρ ) = 0 because the center of G is finite. Byduality, H ( Y, Ad ρ ) = 0.Consider the surface Σ and an irreducible representation ρ : π (Σ) → G .For any boundary component C i , choose a base point x i ∈ C i and let V i =ker(hol i − id) where hol i : Ad ρ | x i → Ad ρ | x i is the holonomy of C i in Ad ρ .We have two isomorphisms H ( C i , Ad ρ ) ≃ V i , H ( C i , Ad ρ ) ≃ V i . sending u ∈ V i into [ x i ] ⊗ u and [ C i ] ⊗ u respectively. Lemma 4.1.
We have H (Σ , Ad ρ ) = H (Σ , Ad ρ ) = 0 . Furthermore thenatural map f : H ( ∂ Σ , Ad ρ ) → H (Σ , Ad ρ ) is injective.Proof. Σ being connected with a non empty boundary, H (Σ , ∂ Σ , Ad ρ ) = 0,so by Poincar´e duality, H (Σ , Ad ρ ) = 0. Since ρ is irreducible, H (Σ , Ad ρ ) =0 and by Poincar´e duality, H (Σ , ∂ Σ , Ad ρ ) = 0. Writing the long exact se-quence associated to the pair (Σ , ∂ Σ), we deduce that f is one-to-one.Consider now the Seifert manifold X and an irreducible representation ρ : π ( X ) → G . Since Σ is a submanifold of X , we have a natural morphism g : H (Σ , Ad ρ ) → H ( X, Ad ρ )By lemma 3.1, the restriction of ρ to S is central. So the restriction of thebundle Ad ρ to Σ × S is isomorphic to Ad ρ | X ⊠ R S . Here we denote by R S the trivial vector bundle over S with fiber R . This allows to define asecond application h : H (Σ , Ad ρ ) → H ( X, Ad ρ )which sends α ∈ H (Σ , Ad ρ ) into the image of α ⊠ [ S ] ∈ H (Σ × S , Ad ρ )by the natural morphism H (Σ × S , Ad ρ ) → H ( X, Ad ρ ). If E → B and E ′ → B ′ are two vector bundles, we denote by E ⊠ E ′ the vector bundle( π ∗ E ) ⊗ (( π ′ ) ∗ E ′ ) where π and π ′ are the projection from B × B ′ onto B and B ′ . heorem 4.2. We have H ( X, Ad ρ ) = H ( X, Ad ρ ) = 0 . Furthermore thefollowing sequences are exact: → H ( ∂ Σ , Ad ρ ) f −→ H (Σ , Ad ρ ) g −→ H ( X, Ad ρ ) → , → H ( ∂ Σ , Ad ρ ) f −→ H (Σ , Ad ρ ) h −→ H ( X, Ad ρ ) → . Proof.
Since ρ is irreducible, H ( X, F ) = 0. By Poincar´e duality, H ( X, F ) =0. To prove that the sequences are exact, we will consider the Mayer-Vietorislong exact sequence associated to the decomposition (3) of X .Since the restriction of Ad ρ to Σ × S is isomorphic to Ad ρ | Σ ⊠ R S , wecan compute by applying the K¨unneth theorem to the maps H j ( ∂ Σ × S , Ad ρ ) → H j (Σ × S , Ad ρ ) , j = 3 , , , . (6)We have that H j ( S , R ) = R for j = 0 , H j (Σ , Ad ρ ) = 0for j = 0 ,
2. We deduce that H (Σ × S , Ad ρ ) = H (Σ × S , Ad ρ ) = 0and H ( ∂ Σ × S , Ad ρ ) ≃ H ( ∂ Σ , Ad ρ ), which determines (6) for j = 0 and3. For j = 2, the map (6) identifies with the map f : H ( ∂ Σ , Ad ρ ) → H (Σ , Ad ρ ) and for j = 1 with f ⊕ H ( ∂ Σ , Ad ρ ) ⊕ H ( ∂ Σ , Ad ρ ) → H (Σ , Ad ρ ) , because H (Σ , Ad ρ ) = 0. Applying again the K¨unneth theorem, the maps H j (Σ × S , Ad ρ ) → H j ( X, Ad ρ ) identify with g and h for j = 1 and 2respectively.It remains to compute the maps H j ( C i × S , Ad ρ ) → H j ( D × S , ˜ ϕ ∗ i Ad ρ ).Here we denote by ˜ ϕ i the embedding of D × S into X extending ϕ i . Since D is contractible, H j ( D × S , ˜ ϕ ∗ i Ad ρ ) = 0 for j = 2 ,
3. Let us determine theholonomy of S in the bundle ˜ ϕ ∗ i Ad ρ → D × S . It is equal to the holonomyof ϕ i ( S ) in Ad ρ → C i × S . For any loop γ of C i × S based at ( x i , γ : Ad ρ | x i → Ad ρ | x i the holonomy of γ in Ad ρ → C i × S .Since D is contractible, hol ϕ i ( ∂D ) is trivial, so thatker(hol ϕ i ( S ) − id) = ker(hol ϕ i ( ∂D ) − id) ∩ ker(hol ϕ i ( S ) − id)= ker(hol C i − id) ∩ ker(hol S − id)= ker(hol C i − id)= V i ϕ i is a diffeomorphism and second that hol S is trivial. We deduce that H j ( D × S , ˜ ϕ ∗ i Ad ρ ) ≃ V i for j = 0 or 1. As above, let us identify H ( C i × S , Ad ρ ) with H ( C i , Ad ρ ) ⊕ H ( C i , Ad ρ ) = V i ⊕ V i . Then by Equation (2), the map H ( C i × S , Ad ρ ) → H ( D × S , ˜ ϕ ∗ i Ad ρ ) corresponds to V i ⊕ V i → V i , ( u, v ) → q i u + p i v. Putting everything together and setting V = L V i , we obtain the followinglong exact sequence0 → V f −→ H (Σ , Ad ρ ) h −→ H ( X, Ad ρ ) → V ⊕ V ˜ f −→ H (Σ , Ad ρ ) ⊕ V [ g, ˜ g ] −−→ H ( X, Ad ρ ) → V id −→ V → g : V → H ( X, Ad ρ ) is unknown and ˜ f is the map (cid:18) f q p (cid:19) with q, p : V → V the maps whose restriction to V i are the multiplications by q i , p i respectively.We recover the fact that f is injective. Since f is injective and the p i don’t vanish, ˜ f is injective too. Furthermore the identity of V is certainlyinjective. So the Mayer-Vietoris long exact sequences breaks into three exactsequences: 0 → V f −→ H (Σ , Ad ρ ) h −→ H ( X, Ad ρ ) → → V ⊕ V ˜ f −→ H (Σ , Ad ρ ) ⊕ V [ g, ˜ g ] −−→ H ( X, Ad ρ ) → → V id −→ V → X and Σ Choose an invariant scalar product on the Lie algebra of G . For anytopological space Y and representation ρ of π ( Y ) in G , the flat vectorbundle Ad ρ inherits a flat metric. This allows to define a cup product11 k ( Y, Z, Ad ρ ) × H ℓ ( Y, Z, Ad ρ ) → H k + ℓ ( Y, Z, R ) for any closed subspace Z of Y . We will us these products for X and Σ.Consider an irreducible representation ρ of π (Σ) in G . We have a bilin-ear map H (Σ , ∂ Σ , Ad ρ ) × H (Σ , Ad ρ ) → R , ( α, β ) → α · β (10)sending ( α, β ) to the evaluation of the cup product α ∪ β on the fundamentalclass of (Σ, ∂ Σ). Consider the following portion of the long exact sequenceassociated to the pair (Σ , ∂ Σ) ... → H (Σ , ∂ Σ , Ad ρ ) π −→ H (Σ , Ad ρ ) f ∗ −→ H ( ∂ Σ , Ad ρ ) → ... and introduce the space K := ker f ∗ = Im π ⊂ H (Σ , Ad ρ ). For any α , β ∈ K , we set Ω( α, β ) := ˜ α · β (11)where ˜ α is any element of H (Σ , ∂ Σ , Ad ρ ) such that π ( ˜ α ) = α . Lemma 5.1.
The bilinear map Ω is well-defined, antisymmetric and nondegenerate. So ( K, Ω) is a symplectic vector space.
Proof.
For any ˜ α , ˜ β ∈ H (Σ , ∂ Σ , Ad ρ ), ˜ α · π ( ˜ β ) + ˜ β · π ( ˜ α ) = 0. Assumingthat π ( ˜ α ) = α and π ( ˜ β ) = β , we get thatΩ( α, β ) = ˜ α · β = − ˜ β · α, which proves that Ω( α, β ) does not depend on the choice of ˜ α and thatΩ( α, β ) = − Ω( β, α ). By Poincar´e duality, the pairing (10) is non degenerate,so the same holds for Ω.Consider now an irreducible representation ρ of π ( X ) in G . By Poincar´eduality, we have a nondegenerate pairing H ( X, Ad ρ ) × H ( X, Ad ρ ) → R (12)sending ( α, β ) to the evaluation of α ∪ β ∈ H ( X ) on the fundamental class.By Theorem 4.2, the maps g ∗ and h ∗ induce isomorphisms from H ( X, Ad ρ )and H ( X, Ad ρ ) to K . 12 heorem 5.2. For any α ∈ H ( X, Ad ρ ) and β ∈ H ( X, Ad ρ ) , we have α · β = Ω( g ∗ α, h ∗ β ) . Proof.
We will use de Rham cohomology. First let us prove that any ele-ment in H ( X, Ad ρ ) has a representative β ∈ Ω ( X, Ad ρ ) which vanishesidentically on a neighborhood of Z = ˜ ϕ ( D × S ) ∪ . . . ∪ ˜ ϕ n ( D × S ) andsuch that β = p ∗ Σ ˜ β ∧ p ∗ S τ + dγ on Σ × S (13)where p Σ and p S are the projection from Σ × S onto Σ and S respectively,˜ β ∈ Ω (Σ , Ad ρ ) is closed, τ ∈ Ω ( S ) satisfies R S τ = 1 and γ belongs toΩ (Σ × S , Ad ρ ).To check that, let us start with any representative β ∈ Ω ( X, Ad ρ ).Since H ( Z, Ad ρ ) = 0, we have β = dµ on Z . We can even assume that thisholds on a neighborhood U of Z . Let ϕ ∈ C ∞ ( X ) with support contained in U and identically equal to 1 on Z . Replacing β with β − d ( ϕµ ), we have that β ≡ Z . By K¨unneth theorem, H (Σ × S , Ad ρ ) = H (Σ , Ad ρ ) ⊗ H ( S , R ), which implies that β has the form (13) on Σ × S .Let us prove that any element in H ( X, Ad ρ ) has a representative α ∈ Ω ( X, Ad ρ ) such that α = p ∗ Σ ˜ α on Σ × S (14)where ˜ α ∈ Ω (Σ , Ad ρ ) is closed and vanishes identically on ∂ Σ.To check that, we start with any representative α ∈ Ω ( X, Ad ρ ). ByK¨unneth theorem, H (Σ × S , Ad ρ ) = H (Σ , Ad ρ ) so that we have onΣ × S the equality α = p ∗ Σ ˜ α + dγ with ˜ α ∈ Ω (Σ , Ad ρ ) closed and γ ∈ Ω (Σ × S , Ad ρ ). Observe that [ ˜ α ] = g ∗ [ α ], so by Theorem 4.2, f ∗ [ ˜ α ] = 0.Thus adding to ˜ α an exact form (which modifies γ ), the restriction of ˜ α to ∂ Σ vanishes. Finally, extending γ to X , and replacing α by α − dγ , weobtain Equation (14).Now consider α and β as above. Then[ α ] · [ β ] = Z X α ∧ β = Z Σ × S α ∧ β because β vanishes identically on a neighborhood of Z . To evaluate thislast integral, we replace α and β by their expressions (14), (13). By Stokes’theorem, Z Σ × S p ∗ Σ ˜ α ∧ dγ = Z ∂ Σ × S p ∗ Σ ˜ α ∧ γ = 013ecause ˜ α vanishes identically on ∂ Σ. By Fubini theorem and because R S τ = 1, Z Σ × S p ∗ Σ ˜ α ∧ p ∗ Σ ˜ β ∧ p ∗ S τ = Z Σ ˜ α ∧ ˜ β Since ˜ α vanishes on ∂ Σ, it is the representative of a class in H (Σ , ∂ Σ , Ad ρ ).So this last integral is equal to Ω([ ˜ α ] , [ ˜ β ]). Furthermore [ ˜ α ] = g ∗ [ α ] and[ ˜ β ] = h ∗ [ β ], which concludes the proof. X Let ρ be an irreducible representation ρ of π ( X ) in G . Since H ( X, Ad ρ ) = H ( X, Ad ρ ) = 0, the torsion of the flat euclidean vector bundle Ad ρ is anon vanishing vector of the linedet H • ( X, Ad ρ ) ≃ (cid:0) det( H ( X, Ad ρ )) (cid:1) − ⊗ det( H ( X, Ad ρ ))well-defined up to sign. In the appendix B, we recall its definition andthe properties we will need to compute it. By Theorem 4.2, we have anisomorphism ψ : H ( X, Ad ρ ) → H ( X, Ad ρ )sending g ( β ) into h ( β ) for any β ∈ H (Σ , Ad ρ ). The determinant of ψ belongs to det H • ( X, Ad ρ ).Let ∆ : C ( G ) → R be the function given by∆( u ) = (cid:12)(cid:12) det H g (Ad g − id) (cid:12)(cid:12) / where g is any element in the conjugacy class u and H g is the orthocomple-ment of ker(Ad g − id). Theorem 6.1.
For any irreducible representation ρ of π ( X ) in G , thetorsion of Ad ρ → X is given by τ (Ad ρ ) = n Y i =1 p dim V i i ∆ ( ρ ( C i ) r i ) det ψ (15) where r i is any inverse of q i modulo p i and V i = ker(Ad ρ ( C i ) − id) . Let us make a few remark on the left hand side of (15).1. It follows from the relation (2) and the fact that ρ ( S ) is central byLemma 3.1, that (Ad ρ ( C i ) ) p i is the identity. So the right hand side of(15) does not depend on the choice of r i .14. V i is the Lie algebra of the centralizer of ρ ( C i ) in G . So the dimensionof V i is equal to dim G − dim u i where u i is the conjugacy class of ρ ( C i ). Proof.
By the proof of Theorem 4.2, the Mayer-Vietoris long exact se-quence breaks into three short exact sequences: (7), (8) and (9). Choose α ∈ det V and β ∈ V dim H (Σ , Ad ρ ) − dim V H (Σ , Ad ρ ) such that f ( α ) ∧ β ∈ det H (Σ , Ad ρ ) does not vanish. By (7), we have an isomorphism R ≃ det V ⊗ (cid:0) det H (Σ , Ad ρ ) (cid:1) − ⊗ det H ( X, Ad ρ ) (16)sending 1 into α ⊗ (cid:0) f ( α ) ∧ β (cid:1) − ⊗ h ( β ). By (8), we have an isomorphism R ≃ (det V ) − ⊗ (cid:0) det H (Σ , Ad ρ ) ⊗ det V (cid:1) ⊗ (cid:0) det H ( X, Ad ρ ) (cid:1) − (17)sending 1 into α − ⊗ (cid:0) ( f ( α ) ∧ β ) ⊗ (det p ) α (cid:1) ⊗ g ( β ) − where p is the mapintroduced in the proof of Theorem 4.2. We easily compute that:det p = n Y i =1 p dim V i i By (9), we have an isomorphism R ≃ det V ⊗ (cid:0) det V (cid:1) − (18)sending 1 into α ⊗ α − . Taking the tensor product of (16), (17) and (18), weget the isomorphism associated to the Mayer-Vietoris long exact sequence: R ≃ (cid:0) det H ( X, Ad ρ ) (cid:1) − ⊗ det H ( X, Ad ρ ) . (19)It sends 1 into (cid:0) det p (cid:1) h ( β ) /g ( β ) = (cid:0) det p (cid:1)(cid:0) det ψ (cid:1) .Let us compute the torsion of the restrictions of Ad ρ to C i × S , Σ × S and ˜ ϕ i ( D × S ) respectively. We will use the identifications made previouslyfor the various cohomology groups. First, the torsion of Ad ρ → C i × S is1 ∈ R ≃ det V i ⊗ (cid:0) det V i (cid:1) − ⊗ det V i ⊗ (cid:0) det V i (cid:1) − . Indeed, the bundle Ad ρ | C i × S is isomorphic to Ad ρ | C i ⊠ R S . Furthermore, χ ( C i ) = χ ( S ) = 0. By property 2 of the appendix B, this implies that τ (Ad ρ, C i × S ) = 1.Second the torsion of Ad ρ → Σ × S is1 ∈ R ≃ det H (Σ , Ad ρ ) ⊗ (cid:0) det H (Σ , Ad ρ ) (cid:1) − ρ | Σ × S is isomorphic to Ad ρ | Σ ⊠ R S . Since χ ( S ) =0, we deduce from properties 2 and 4 of the appendix B that τ (Ad ρ, Σ × S ) = τ ( R S ) χ (Ad ρ | Σ ) = 1.Third the torsion of ˜ ϕ ∗ i Ad ρ → D × S belongs to R ≃ det V i ⊗ (cid:0) det V i (cid:1) − .Since ϕ i is a diffeomorphism from ∂D × S to C i × S reversing the orientationand satisfying (2), we have the following relation in H ( C i × S ) ϕ i ([ S ]) = r i [ C i ] + s i [ S ]where r i , s i are such that such that p i s i + q i r i = 1. Since ρ ( S ) is central,Ad ρ ( S ) is the identity, so Ad ρ ( ϕ i ( S )) = Ad r i ρ ( C i ) . By Property 4 of the appendix B, we conclude that the torsion of ˜ ϕ ∗ i Ad ρ is equal to the square of ∆( ρ ( C i ) r i ).By Property 3 of the appendix B, we deduce from the previous compu-tations that (cid:0) det p (cid:1)(cid:0) det ψ (cid:1) = τ ( X, Ad ρ ) n Y i =1 ∆ ( ρ ( C i ) r i )which concludes the proof.Using the duality between homology and cohomology and Poincar´e du-ality, we havedet H • ( X, Ad ρ ) ≃ det( H ( X, Ad ρ )) ⊗ (cid:0) det( H ( X, Ad ρ )) (cid:1) − ≃ (cid:0) det( H ( X, Ad ρ )) (cid:1) (20)So (det ψ ) − may be viewed as the square of a volume element of H ( X, Ad ρ ).Recall that g ∗ induces an isomorphism from H ( X, Ad ρ ) to a symplectic vec-tor space ( K, Ω). The following lemma is an easy consequence of Theorem5.2.
Lemma 6.2. g ∗ sends (det ψ ) − to the square of the Liouville form Ω N /N ! ,where N = dim K . Application to moduli spaces
Let us apply the previous results to the character manifold M ( X ). Recallthat a density of a n dimensional manifold M is a section of the line bundle,whose fiber at x is the space of applications f : ( T x M ) n → R satisfying f ( Ax , . . . , Ax n ) = | det A | f ( x , . . . , x n ) for any endomorphism A of T x M .Here, we have natural densities on M ( X ) and M (Σ , u ) defined as follows: • Since the tangent space T [ ρ ] M ( X ) is H ( X, Ad ρ ), by the isomor-phism (20), the torsion τ (Ad ρ ) is the inverse of the square of a densityof T [ ρ ] M ( X ). This defines a density µ X of M ( X ) whose value at [ ρ ]is τ (Ad ρ ) − / . • For any u ∈ C ( G ) n , M (Σ , u ) is a symplectic manifold, so it hasa canonical density µ u . The symplectic structure of T ρ M (Σ , u ) isthe form Ω considered in Lemma 5.1 and if N = dim M (Σ , u ), µ u ([ ρ ]) = | Ω ∧ N | /N !.For any ( u, v ) ∈ P , we defined a diffeomorphism R u.v from M ( X, u, v ) to M (Σ , u ). By the proof of Lemma 3.5, the linear tangent map of R u,v at[ ρ ] is the map g ∗ : H ( X, Ad ρ ) → K . We deduce from Theorem 6.1 andTheorem 5.2 via Lemma 6.2 our main result. Theorem 7.1.
For any ( u, v ) ∈ P , we have on M ( X, u, v ) µ X = n Y i =1 ∆( u r i i ) p dim V i / i ! R ∗ u,v µ u with r i any inverse of q i modulo p i . In this section, we adapt the previous result to the constant coefficient case.We consider the same Seifert manifold X as above and we assume that theEuler number χ = − n X i =1 q i p i does not vanish.For Y = X, Σ , ∂ Σ, we let H j ( Y ) := H j ( Y, R ). In contrast to the previouscase, the groups H ( X ) and H ( X ) do not vanish. Introduce the three maps f : H ( ∂ Σ) → H (Σ) , g : H (Σ) → H ( X ) , h : H (Σ) → H ( X )17efined as follows. f and g are the morphisms corresponding to the inclusions ∂ Σ ⊂ Σ and Σ ⊂ X respectively. h sends γ ∈ H (Σ) to the image of γ ⊠ [ S ] ∈ H (Σ × S ) in H ( X ). Proposition 8.1. g and h are surjective and their kernel is the image of f .Proof. The proof is similar to the one of Theorem 4.2, with the additionaldifficulty that H ( X ) ≃ R , H (Σ) ≃ R and H ( X ) ≃ R . The Mayer-Vietorislong exact sequence splits into three exact sequences:0 → R → R n f −→ H (Σ) h −→ H ( X ) → → R n B −→ R n ⊕ H (Σ) ⊕ R A −→ H ( X ) → → R n C −→ R n ⊕ R → R → C and B are given by C ( x ) = ( x, x + . . . + x n ) and B ( x, y ) =( z, f ( x ) , y + . . . + y n ) with z ∈ R n given by z i = q i x i + p i y i . By the firstsequence in (21), h is surjective and its kernel is the image of f . One checksthat Im B and 0 ⊕ H (Σ) ⊕ ⊕ Im f ⊕
0. Using that for any γ ∈ H (Σ), A (0 , γ,
0) = g ( γ ), onededuces from the second sequence of (21) that g is surjective with kernel theimage of f .Let Σ be the closed surface obtained by gluing a disc to each boundarycomponent of Σ. The inclusion Σ ⊂ Σ induces an isomorphism H (Σ) ≃ H (Σ) / Im f . So by Proposition 8.1, we have two isomorphisms˜ g : H (Σ) → H ( X ) , ˜ h : H (Σ) → H ( X ) . Proposition 8.2.
For any α ∈ H ( X ) and β ∈ H ( X ) , we have α · X β = (˜ g ∗ α ) · Σ (˜ h ∗ β ) where · X and · Σ denote the Poincar´e pairings of X and Σ respectively.Proof. The proof is similar to the one of Theorem 5.2. First, since the imageof B in (21) contains 0 ⊕ ⊕ H (Σ), the image of H ( X ) → H (Σ × S ) ≃ H (Σ) ⊕ H (Σ) is contained in H (Σ). Consequently any class α of H ( X )has a representative a ∈ Ω ( X ) such that a = p ∗ ˜ a on Σ × S p : Σ × S → Σ the projection and ˜ a ∈ Ω (Σ) a representative of ˜ g ∗ α .Second, any class β ∈ H ( X ) has a representative b ∈ Ω ( X ) whose supportis contained in an open subset of Σ × S and such that b = p ∗ ˜ b ∧ q ∗ τ + dγ where ˜ b ∈ Ω (Σ) is a representative of ˜ h ∗ β , supported in an open subset ofΣ, q is the projection Σ × S → S , τ ∈ Ω ( S ) is such that R S τ = 1 and γ ∈ Ω (Σ × S ). Finally, one checks that Z X a ∧ b = Z Σ ˜ a ∧ ˜ b. using Stokes’ formula.We can also compute the torsion of X as in Theorem 6.1. Since H ( X )and H ( X ) have rank one, the torsion belongs to H ( X ) ⊗ (cid:0) det H ( X ) (cid:1) − ⊗ det H ( X ) ⊗ (cid:0) H ( X )) − . Proposition 8.3.
The Reidemeister torsion of X is given by τ ( X ) = χ n Y i =1 p i [ x ] ⊗ det ψ ⊗ [ X ] − where χ = − P q i /p i is the Euler number of X , x ∈ X and [ x ] ∈ H ( X ) is the corresponding class, ψ is the map ˜ h ◦ ˜ g − : H ( X ) → H ( X ) and [ X ] ∈ H ( X ) is the fundamental class.Proof. We adapt the proof of Theorem 6.1. Let e = 1 ∈ R , ( e i ) be thecanonical basis of R n , δ = e ∧ . . . ∧ e n , ρ = f ( e ) ∧ . . . ∧ f ( e n − ) and σ ∈∧ g H (Σ) such that ρ ∧ σ is a generator of ∧ g + n − H (Σ). Then one checksthat the isomorphisms corresponding to the three exact sequences in (21)send 1 into e ⊗ δ − ⊗ ( ρ ∧ σ ) ⊗ h ( σ ) − , χ − ( Q p i ) − e ⊗ ( δ ⊗ ( ρ ∧ σ ) ⊗ e ) − ⊗ h ( σ ) − and δ − ⊗ ( δ ⊗ e ) ⊗ e − respectively. The factor χ ( Q p i ) appears because B ( δ ⊗ δ ) = χ ( Q p i ) δ ⊗ ρ ⊗ e . The torsions of ∂ Σ × S , ∂ Σ × D and Σ × S are respectively δ ⊗ ( δ ⊗ δ ) − ⊗ δ , δ ⊗ δ − and ( ρ ∧ σ ) ⊗ ( δ ⊗ ( ρ ∧ σ ) ⊗ e ) − ⊗ e .We conclude with Property 3 of appendix B.Trivialising H ( X ) and H ( X ) by sending [ x ] and [ X ] to 1, and iden-tifying H ( X ) with the dual of H ( X ) by Poincar´e duality, the inverse ofsquare root of the torsion gets identified with an element of det H ( X ). Bypropositions 8.2 and 8.3, the torsion satisfies˜ g ∗ ( τ ( X )) − / = (cid:12)(cid:12)(cid:12) χ n Y i =1 p i (cid:12)(cid:12)(cid:12) − / µ (22)19here µ ∈ det H (Σ) is the Liouville density of H (Σ).This may be applied to the space J ( X ) consisting of representation of π ( X ) in U(1) as follows. First, for any connected compact manifold Y , J ( Y ) is an abelian Lie group, the product being the pointwise multiplica-tion. The Lie algebra of J ( Y ) is the space of morphisms from π ( Y ) to R , which identifies with H ( Y ). In particular for the Seifert manifold X ,the Lie algebra of J ( X ) being H ( X ), ( τ ( X )) − / determines an invariantdensity of J ( X ). Furthermore, the Lie algebra of J (Σ) being H (Σ), J (Σ)has an invariant symplectic structure and a corresponding Liouville density.For any ( u, v ) ∈ U(1) n +1 , let J ( X, u, v ) be the subset of J ( X ) consistingof the representations ρ such that ρ ( C i ) = u i for any i and ρ ( S ) = v . Then J ( X ) = [ ( u,v ) ∈Q J ( X, u, v ) (23)where Q is the set of ( u, v ) ∈ U(1) n +1 such that u . . . u n = 1 and forany i , u p i i = v q i . Since the Euler number χ does not vanish, Q is finite.Furthermore, for any ( u, v ) ∈ Q , J ( X, u, v ) is connected. So (23) is thedecomposition of J ( X ) into connected components.Let = (1 , . . . , ∈ U(1). J ( X, ,
1) is the component of the identityof J ( X ). We have a natural Lie group isomorphism Φ from J ( X, ,
1) to J (Σ), such that for any ρ ∈ J ( X, , ρ and Φ( ρ ) to Σare the same. The linear tangent map at the identity to Φ is the adjointmap to the map ˜ g : H (Σ) → H ( X ). Thus Equation (22) computes theinvariant density of J ( X, ,
1) in terms of the pull back by Φ of the Liouvilledensity. We recover in this way Theorem 9 of [McL15].
A Representation space
The general theory describing the smooth structure of a representation spaceis rather involved and belongs more to algebraic geometry, [LM85]. In thisappendix, we summarize the basic general facts we need, remaining in thecontext of differential geometry.Let G be a connected Lie group and π be a finitely generated group. Let R ( π ) be the space of representations of π in G . For any set of generators a = ( a , . . . , a N ) of π , the map ξ a : R ( π ) → G N , ρ → ( ρ ( a ) , . . . , ρ ( a N )) , is injective, and allows us to identify R ( π ) with ξ a ( R ( π )). If a = ( a , . . . , a N )and b = ( b , . . . , b M ) are two sets of generators, the bijection ξ b ◦ ξ − a from20 a ( R ( π )) onto ξ b ( R ( π )) is a homeomorphism. Indeed, expressing the a i ’s interms of the b j ’s, we obtain a smooth map ϕ : G N → G M extending ξ b ◦ ξ − a .We endow R ( π ) with the topology such that for any set of generators a of π , ξ a is a homeomorphism onto its image.Let R s ( π ) be the set of representations ρ of π in G admitting an openneighborhood U and a set of generators ( a , . . . , a N ) such that ξ a ( U ) is asmooth submanifold of G N . R s ( π ) has a unique manifold structure suchthat for any such pairs ( U, a ), the map ξ a : U → ξ a ( U ) is a diffeomorphism.Indeed, arguing as above, we see that for any two pairs ( U, a ) and (
V, b ) themap ξ b ◦ ξ − a : ξ a ( U ∩ V ) → ξ b ( U ∩ V ) is a diffeomorphism.For any representation ρ of π in G , composing ρ with the adjoint repre-sentation, the Lie algebra g becomes a left G -module. Consider the corre-sponding cochain complex in degrees 0 and 1: C ( π, Ad ρ ) = g , C ( π, Ad ρ ) =Map( π, g ), the differential in degree 0 is d ρ : g → C ( π, Ad ρ ) , ξ → (cid:0) γ → Ad ρ ( γ ) ξ − ξ (cid:1) and the space of 1-cocycle Z ( π, Ad ρ ) = (cid:8) τ : π → g ; ∀ γ , γ ∈ π, τ ( γ γ ) = τ ( γ ) + Ad ρ ( γ ) τ ( γ ) (cid:9) . For any γ ∈ π , the map e γ : R ( π ) → G sending ρ into ρ ( γ ) is continuous.Its restriction to R s ( π ) is smooth. If ρ ∈ R s ( π ), we have a natural map from T ρ R s ( π ) to Z ( π, Ad ρ ) sending ˙ ρ to the cocycle τ given by τ ( γ ) = R ρ ( γ ) − T ρ e γ ( ˙ ρ ) , ∀ γ ∈ π, where for any g ∈ G , R g − : T g G → g is the linear map tangent to theright multiplication by g − . It is easily seen that this map is well-definedand injective, so we consider the tangent space T ρ R s ( π ) as a subspace of Z ( π, Ad ρ ). G acts on R ( π ) by conjugation. The action preserves R s ( π ). A straight-forward computation shows that the infinitesimal action at ρ ∈ R s ( π ) is thedifferential d ρ introduced above.Assume from now on that G is compact. The subset R ( π ) of R ( π )consisting of irreducible representations is open. Indeed, if ( a , . . . , a n ) isany set of generators, then ξ a ( R ( π )) = ξ a ( R ( π )) ∩ ( G N ) where ( G N ) consists of the N -uplets whose centralizer in G is the center. By the slicetheorem for action of compact Lie group, ( G N ) is open in G N , because itis either empty or the principal stratum for the diagonal action of G on G N by conjugation. 21et R s , ( π ) := R ( π ) ∩ R s ( π ). The quotient space M s , ( π ) := R s , ( π ) /G is a smooth manifold because it is the quotient of a smooth manifold by asmooth action of the compact Lie group G with constant isotropy Z ( G ).Furthermore, for any ρ ∈ R s , ( π ), the infinitesimal action at ρ being d ρ , wehave H ( π, Ad ρ ) = ker d ρ = z ( g )where z ( g ) is the Lie algebra of the center of G . Furthermore T [ ρ ] M s , ( π )identifies with a subspace of H ( π, Ad ρ ) = Z ( π, Ad ρ ) / Im d ρ . B Reidemeister Torsion
Let M be a compact manifold possibly with a non empty boundary. Let E → M be a flat real vector bundle equipped with a flat metric. Denote bydet H • ( E ) the linedet H • ( E ) = det H ( E ) ⊗ (det H ( E )) − ⊗ . . . ⊗ (det H n ( E )) ( − n where n is the dimension of M and for any finite dimensional vector space V , det V = V top V . In the acyclic case, det H • ( E ) = R . The Reidemeistertorsion of E is a non-vanishing vector τ ( E ) ∈ det H • ( E ) well-defined up tosign. Let us recall briefly its definition.Let K be the simplicial complex of a smooth triangulation of X . For anycell σ of K , let E σ be the space of flat sections of the restriction of E to σ .Introduce the complex C • ( K, E ) where C k ( K, E ) = L dim σ = k E σ with theusual differential. Then the H k ( E ) are the homology groups of C • ( K, E ).Consequently, we have an isomorphism det C • ( K, E ) ≃ det H • ( E ). Further-more, for any cell σ , E σ is an Euclidean space. So C k ( K, E ) has a naturalscalar product where the E σ are mutually orthogonal, and det H • ( E ) inher-its an Euclidean product by the previous isomorphism. The Reidemeistertorsion τ ( E ) is by definition a unit vector of det H • ( E ). It does not dependon the choice of the triangulation, cf. [Mil66], Section 9.The torsion satisfies the following properties, cf. [KS65] for 1, 2 and[Mil66], Section 3 for 3.1. Let E = E ⊕ E where E and E are two flat Euclidean vector bundleswith base M . Then we have a natural isomorphism H • ( E ) ≃ H • ( E ) ⊕ H • ( E ). The corresponding isomorphism det H • ( E ) ≃ det H • ( E ) ⊗ det H • ( E ) sends τ ( E ) into τ ( E ) ⊗ τ ( E ).2. Let E → M and E → M be two flat Euclidean vector bundles.Assume that M is closed. Set M = M × M and E = E ⊠ E . By22¨unneth theorem, we have H • ( E ) ≃ H • ( E ) ⊗ H • ( E ). The corre-sponding isomorphismdet H • ( E ) ≃ (cid:0) det H • ( E ) (cid:1) χ ( E ) ⊗ (cid:0) det H • ( E ) (cid:1) χ ( E ) sends τ ( E ) into τ ( E ) χ ( E ) ⊗ τ ( E ) χ ( E ) .3. Let E be a flat Euclidean vector bundle whose base M is obtainedby gluing two manifolds M , M along their boundary N . By theMayer-Vietoris exact sequence, we have an isomorphismdet H • ( E ) ⊗ det H • ( E | N ) ≃ det H • ( E | M ) ⊗ det H • ( E | M ) . This isomorphism sends τ ( E ) ⊗ τ ( E | N ) to τ ( E | M ) ⊗ τ ( E | M ).Finally, it is a classical exercise to compute the torsion of a bundle over acircle.4. Let E be a flat Euclidean vector bundle E on an oriented circle C .Let p ∈ C and let ϕ : E p → E p be the holonomy of C . Let H =ker( ϕ − id). We have two isomorphisms H ( E ) ≃ H and H ( E ) ≃ H sending u ∈ H into [ p ] ⊗ u and [ C ] ⊗ u respectively. Thus det H • ( E ) ≃ det H ⊗ (det H ) − ≃ R , so that the torsion may be considered as areal number. With this convention, we τ ( E ) = det − (cid:0) ( ϕ − id) | H ⊥ (cid:1) where H ⊥ is the orthogonal complement of H . References [AB83] M. F. Atiyah and R. Bott. The Yang-Mills equations over Riemannsurfaces.
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Laurent Charles,
Institut de Math´ematiques de Jussieu-Paris Rive Gauche,Sorbonne Universit´es, UPMC Univ Paris 06, F-75005, Paris, France
E-mail address : [email protected] isa Jeffrey, Mathematics Departement, University of Toronto, Toronto,ON, Canada M5S 2E4
E-mail address : [email protected]@math.toronto.edu