Flat connections in three-manifolds and classical Chern-Simons invariant

A general method for the construction of smooth flat connections on 3-manifolds is introduced. The procedure is strictly connected with the deduction of the fundamental group of a manifold M by means of a Heegaard splitting presentation of M. For any given matrix representation of the fundamental group of M, a corresponding flat connection A on M is specified. It is shown that the associated classical Chern-Simons invariant assumes then a canonical form which is given by the sum of two contributions: the first term is determined by the intersections of the curves in the Heegaard diagram, and the second term is the volume of a region in the representation group which is determined by the representation of pi_1(M) and by the Heegaard gluing homeomorphism. Examples of flat connections in topologically nontrivial manifolds are presented and the computations of the associated classical Chern-Simons invariants are illustrated.


Introduction
Each SU(N)-connection, with N ≥ 2, in a closed and oriented 3-manifold M can be represented by a 1-form A = A µ dx µ which takes values in the Lie algebra of SU(N). The Chern-Simons function S[A], can be understood as the Morse function of an infinite dimensional Morse theory, on which the instanton Floer homology [1] and the gauge theory interpretation [2] of the Casson invariant [3] are based. Under a local gauge transformation where Ω is a map from M into SU(N), the function S[A] transforms as where the integer I Ω ∈ Z, is well defined for the gauge orbits of flat SU(N)-connections on M, and it is well defined [4] for the SU(N) representations of π 1 (M) modulo the action of group conjugation. If the orientation of M is modified, one gets cs[A] → −cs [A].
In the case of the structure group SU(2), methods for the computation of cs [A] have been presented in References [5,6,7,8,9,10], where a few non-unitary gauge groups have also been considered. In all the examples that have been examined, cs[A] turns out to be a rational number. In the case of three dimensional hyperbolic geometry, the associated P SL(2, C) classical invariant [7,11,12,13] combines the real volume and imaginary Chern-Simons parts in a complex geometric invariant. The Baseilhac-Benedetti invariant [14] with group P SL(2, C) represents some kind of corresponding quantum invariant.
Precisely because flat connections represent stationary points of the function (1.1), flat connections and the corresponding value of cs[A] play an important role in the quantum Chern-Simons gauge field theory [15]. For instance, the path-integral solution of the abelian Chern-Simons theory has recently been produced [16,17]. In this case, flat connections dominate the functional integration and the value of the partition function is given by the sum over the gauge orbits of flat connections of the exponential of the classical Chern-Simons invariant. The classical abelian Chern-Simons invariant is strictly related [16,17] with the intersection quadratic form on the torsion group of M, which also enters the abelian Reshetikhin-Turaev [18,19] surgery invariant.
In general, the precise expression of the flat connections is an essential ingredient for the computation of the observables of the quantum Chern-Simons theory by means of the pathintegral method. In this article we shall mainly be interested in nonabelian flat connections. We will show that, given a representation ρ of π 1 (M) and a Heegaard splitting presentation [20] of M (with the related Heegaard diagram), by means of a general construction one can define a corresponding smooth flat connection A on M. The method that we describe is related with the deduction [21] of a presentation of the fundamental group of a manifold M by means of a Heegaard splitting of M. Then the associated invariant cs[A] assumes a canonical form, which can be written as the sum of two contributions. The first term is determined by the intersections of the curves in the Heegaard diagram and can be interpreted as a sort of "coloured intersection form". Whereas the second term is the Wess-Zumino volume of a region in the structure group SU(N) which is determined by the representation of π 1 (M) and by the Heegaard gluing homeomorphism.
The procedure that we present for the determination of the flat connections can find possible applications also in the description of the topological states of matter [22,23]. A discussion on the importance of topological configurations and of the holonomy operators in gauge theories can be found for instance in Ref. [24].
Our article is organised as follows. Section 2 contains a brief description of the main results of the present article. The general construction of flat connections in a generic 3-manifold M by means of a Heegaard splitting presentation of M is discussed in Section 3. The canonical form of the corresponding classical Chern-Simons invariant is derived in Section 4, where a two dimensional formula of the Wess-Zumino group volume is also produced. In the remaining sections, our method is illustrated by a few examples. Flat connections in lens spaces are discussed in Section 5 and a non-abelian representation of the fundamental group of a particular 3manifold is considered in Section 6; computations of the corresponding classical Chern-Simons invariants are presented. The case of the Poincaré sphere is discussed in Section 7. One example of a general formula of the classic Chern-Simons invariant for a particular class of Seifert manifolds is given in Section 8. Finally, Section 9 contains the conclusions.

Outlook
The main steps of our construction can be summarised as follows. For any given SU(N) representation ρ of π 1 (M), one can find a corresponding flat connection A on M whose structure is determined by a Heegaard splitting presentation M = H L ∪ f H R of M. In this presentation, the manifold M is interpreted as the union of two handlebodies H L and H R which are glued by means of the homeomorphism f : ∂H L → ∂H R of their boundaries, as sketched in Figure 1. Let the fundamental group of M be defined with respect to a base point x b which belongs to the boundaries of the two handlebodies. Then the representation ρ of π 1 (M) canonically defines a representation of the fundamental group of each of the two handlebodies H L and H R . As shown in Figure 2, in each handlebody the generators of its fundamental group can be related with a set of corresponding disjoint meridinal discs. To each meridinal disc is associated a matrix which is specified by the representation ρ; this matrix can be interpreted as a "colour" which is attached to each meridinal disc. With the help of these colored meridinal discs, one can construct a smooth flat connection A 0 L in H L -and similarly a smooth flat connection A 0 R in H R -whose holonomies correspond to the elements of the representation ρ in the handlebody H L (or H R ). The precise definition of A 0 L and A 0 R is given in Section 3. In general, A 0 L and A 0 R do not coincide with the restrictions in H L and H R of a single connection A in M, because the images -under f -of the boundaries of the meridinal discs of H L are not the boundaries of meridinal discs of H R . So, in order to define a connection A which is globally defined in M, one needs to combine A 0 L with A 0 R in a suitable way. In facts, the exact matching of the gauge fields A 0 L and A 0 R in M is specified by the homeomorphism f through the Heegaard diagram, which shows precisely how the boundaries of the meridinal discs of H L are pasted onto the surface ∂H R , in which the boundaries of the meridinal discs of H R are also placed. Let us denote by f * A 0 L the image of A 0 L under f . The crucial point now is that, on the surface ∂H R , the connections A 0 because their holonomies define the same representation of π 1 (∂H R ). The value of the map U 0 from the surface ∂H R on the group SU(N) is uniquely determined by equation (2.2) and by the condition U 0 (x b ) = 1. In facts, we will demonstrate that where Φ R and Φ f * L denote the developing maps associated respectively with A 0 R and f * A 0 L from the universal covering of ∂H R into the group SU(N). The definition of the developing map will be briefly recalled in Section 3.3. Then the map U 0 can smoothly be extended to the whole handlebody H R ; this extension will be denoted by U. The values of U : H R → SU(N) inside H R are not constrained and can be chosen without restrictions apart from smoothness. As far as the computation of the classical Chern-Simons invariant is concerned, the particular choice of the extension U of U 0 turns out to be irrelevant. To sum up, the connection A -which is well defined in M and whose holonomies determine the representation ρtakes the form

6)
and The function X [A] is defined on the surface ∂H R , and similarly the value of the Wess-Zumino volume Γ[U] mod Z only depends [25,26,27] on the values of U in ∂H R (i.e., it only depends on U 0 ). A canonical dependence of Γ on U 0 will be produced in Section 4.4. Therefore both terms in expression (2.5) are determined by the data on the two-dimensional surface ∂H R of the Heegaard splitting presentation M = H L ∪ f H R exclusively. This is why the particular choice of the extension of U 0 inside H R is irrelevant. The remaining part of this article contains the proof of Proposition 1 and a detailed description of the construction of the flat connection A. Examples will also be given, which elucidate the general procedure and illustrate the computation of cs[A].

Flat connections
Given a matrix representation ρ of π 1 (M), we would like to determine a corresponding flat connection A on M whose holonomies agree with ρ; then we shall compute S[A].
In order to present a canonical construction which is not necessarily related with the properties of the representation space, we shall use a Heegaard splitting presentation M = H L ∪ f H R of M. The construction of A is made of two steps. First, in each of the two handlebodies H L and H R we define a flat connection, A 0 L and A 0 R respectively, whose holonomies coincide with the elements of the matrix representation of the fundamental group of the handlebody which is induced by ρ. Second, the components A 0 L and A 0 R are combined according to the Heegaard diagram to define A on M.

Heegaard splitting
Let us recall [4,20] that the fundamental group of a three-dimensional oriented handlebody H of genus g is a free group with g generators {γ 1 , γ 2 , ..., γ g }. A disc D in H is called a meridinal disc if the boundary of D belongs to the boundary of H, ∂D ⊂ ∂H, and ∂D is homotopically trivial in H. Let {D 1 , D 2 , ..., D g } be a set of disjoint meridinal discs in H such that H − {D 1 , D 2 , ..., D g } is homeomorphic with a 3-ball with 2g removed disjoint discs in its boundary. These meridinal discs {D 1 , D 2 , ..., D g } can be put in a one-to-one correspondence with the g handles of the handlebody H or, equivalently, with the generators of π 1 (H), and can be oriented in such a way that the intersection of γ j with D k is δ jk . For instance, in the case of a handlebody of genus 2, a possible choice of the generators {γ 1 , γ 2 } and of the discs {D 1 , D 2 } is illustrated in Figure 2, where the base point x b is also shown.

By means of a Heegaard presentation
one can find a presentation of the fundamental group π 1 (M). Suppose that the two handlebodies Thus each Heegaard splitting can be described by a diagram which shows the set of the characteristic curves {C ′ j } on the surface ∂H R . One example of Heegaard diagram is shown in Figure 3. Figure 3. Example of a genus 2 Heegaard diagram.
Let {γ 1 , γ 2 , ..., γ g } be a complete set of generators for π 1 (H R ) which are associated to a complete set of meridinal discs of H R . The fundamental group of M is specified by adding to the generators {γ 1 , γ 2 , ..., γ g } the constraints which implement the homotopy triviality condition of the curves {C ′ j }. Indeed, since each curve C j is homotopically trivial in M, the fundamental group of M admits [20,21] the presentation where [C ′ j ] denotes the π 1 (H R ) homotopy class of C ′ j expressed in terms of the generators are determined by the intersections of the boundaries of the meridinal discs of H L and H R , which can be inferred from the Heegaard diagram.

Flat connection in a handlebody
Let us consider the handlebody H L of the Heegard splitting M = H L ∪ f H R of genus g and a corresponding set {D 1 , D 2 , ..., D g } of disjoint meridinal discs in H L . For each j = 1, 2, ..., g, consider a collared neighbourhood N j of D j in H L . As shown in Figure 4, where the hermitian traceless matrix b j belongs to the Lie algebra of SU(N). Let θ(t) be a C ∞ real function, with θ ′ (t) = dθ(t)/dt > 0, satisfying θ(0) = 0 and θ(ǫ) = 1. Then the value of A 0 L in the region N j is given by A 0 The orientation of the parameterization (or the sign in equation (3.4)) is fixed so that the holonomy of the connection (3.4) coincides with expression (3.3). As a consequence of equation (3.4) one has dA 0 L = 0 and also, since N j ∩ N k = ∅ for j = k, one finds A 0 L ∧ A 0 L = 0. By construction, the smooth 1-form A 0 L represents a flat connection on H L whose holonomies coincide with the matrices that represent the elements of the fundamental group of H L . The restriction of A 0 L on the boundary ∂H L has support on g ribbons and its values are determined by equation (3.4); the j-th ribbon represents a collared neighbourhood of the curve C j = ∂D j in ∂H L . The same construction can be applied to define a flat connection A 0 R on H R .

Flat connection in a 3-manifold
Let us now construct a flat connection A in M = H L ∪ f H R which is associated with the representation ρ of π 1 (M). As far as the value of A on H L is concerned, one can put 2), in which U 0 must assume the unit value at the base point x b . Then the map U 0 can smoothly be extended in H R , let U denote this extension. The value of A on H R is taken to be The value of U 0 on the surface ∂H R represents a fundamental ingredient of our construction, so we now describe how it can be determined. To this end, we need to introduce the concept of developing map. Let us recall that any flat SU(N)-connection A defined in a space X can be locally trivialized because, inside a simply connected neighbourhood of any given point of X, A can be written as A = −iΦ −1 dΦ. The value of Φ coincides with the holonomy of A. When the representation of π 1 (X) determined by A is not trivial, Φ cannot be extended to the whole space X. A global trivialisation of A can be found in the universal covering X of X; in this case, the map Φ : X → SU(N) represents the developing map. For any element γ of π 1 (X) acting on X by covering transformations, the developing map satisfies in agreement with equations (1.6). Now, on the surface ∂H R we have the two flat connections f * A 0 L and A 0 R which are related by a gauge transformation, equation (2.2). Thus, for each oriented path γ ⊂ ∂H R connecting the starting point x 0 with the final point x, the corresponding holonomies are related according to equation (1.6) which takes the form When the starting point x 0 coincides with the base point x b of the fundamental group, one has U(x b ) = 1, and then This equation is equivalent to the relation (2.3). Indeed, because of the transformation property (3.7), the combination Φ −1 R Φ f * L is invariant under covering translations acting on the universal covering of ∂H R (and then Φ −1 R Φ f * L is really a map from ∂H R into SU(N)), and locally coincides with the product  Since dA 0 Moreover, a direct computation shows that As before, the first term on the r.h.s of equation (4.3) is vanishing By using equation (2.2), the second term can be written as the surface integral By combining equations (4.1)-(4.5) one finally gets When the representation ρ is abelian, Γ[U] vanishes and the classical Chern-Simons invariant is completely specified by X [A] which assumes the simplified form

Group volume
The term Γ[U] can be interpreted as the 3-volume of the region of the structure group which is bounded by the image of the surface ∂H R under the map Φ −1 R Φ f * L : ∂H R → SU(N). In this case also, the combination Φ −1 R Φ f * L of the two developing maps, which are associated with f * A 0 L and A 0 R , is characterized by the homeomorphism f : ∂H L → ∂H R which topologically identifies M.
In general, the direct computation of Γ[U] is not trivial, and the following properties of Γ[U] turns out to be useful. When U(x) can be written as where W (x) ∈ SU(N) and Z(x) ∈ SU(N), one obtains By means of equation (4.9) one can easily derive the relation Therefore relation (4.10) becomes where it is understood that one possibly needs to decompose the integral into a sum of integrals computed in different regions of ∂H R where V (x) and H(x) are well defined [30]. Expression (4.13) explicitly shows that the value of Γ[U] (modulo integers) is completely specified by the value of U on the surface ∂H R . In the case of the structure group SU(2) ∼ S 3 , the computation of Γ[U] can be reduced to the computation of the volume of a given polyhedron in a space of constant curvature. Discussions on this last problem can be found, for instance, in the articles [31,32,33,34,35,36,37,38].

Canonical extension
The reduction of the Wess-Zumino volume Γ[U] into a surface integral on ∂H R can be done in several inequivalent ways, which also depend on the choice of the extension of U 0 from the surface ∂H R in H R . Let us now describe the result which is obtained by means of a canonical extension of U 0 . We shall concentrate on the structure group SU(2), the generalisation to a generic group SU(N) is quite simple.
Suppose that the value of U 0 on the surface ∂H R can be written as U 0 (x, y) = e in(x,y)σ = e i 3 a=1 n a (x,y) σ a = cos n(x, y) + i n(x, y)σ sin n(x, y) , (4.14) where (x, y) designate coordinates of ∂H R , n = 3 b=1 n b n b 1/2 , the components of the unit vector n are given by n a = n a /n, and {σ a } (with a = 1, 2, 3) denote the Pauli sigma matrices. The canonical extension of U 0 is defined by U(τ, x, y) = e i τ n(x,y)σ , (4.15) where the homotopy parameter τ takes values in the range 0 ≤ τ ≤ 1. When τ = 1 one recovers the expression (4.14), whereas in the τ → 0 limit one finds U = 1. A direct computation gives in which Σ(x, y) = 3 a=1 n a (x, y)σ a . Therefore, by using the identity  This equation will be used in Section 6, Section 7 and Section 8.

Rationality
As it has already been mentioned, in all the considered examples the value of the SU(N) classical Chern-Simons invariant is given by a rational number. Let us now present a proof of this property for a particular class of 3-manifolds. Suppose that the universal covering M of the three-manifold M is homeomorphic with S 3 , so that M can be identified with the orbit space [39] which is obtained by means of covering translations (acting on S 3 ) which correspond to the elements of the fundamental group π 1 (M). Given a flat connection A on M, let us denote by A the flat connection on M ∼ S 3 which is the upstairs preimage of A. By construction, one has S[A] where |π 1 (M)| denotes the order of π 1 (M). On the other hand, since S 3 is simply connected, one can find a map Ω : S 3 → SU(N) such that where n is an integer. Equations

First example
In order to illustrate how to compute X [A], let us consider the lens spaces L(p, q), where the coprime integers p and q verify p > 1 and 1 ≤ q < p. The manifolds L(p, q) admit [4, 20] a genus 1 Heegaard splitting presentation, L(p, q) = H L ∪ f H R where H L and H R are solid tori. The fundamental group of L(p, q) is the abelian group π 1 (L(p, q)) = Z p .

The representation
Let us concentrate, for example, on L(5, 2) whose Heegaard diagram is shown in Figure 5, where the image C ′ of a meridian C of the solid torus H L is displayed on the surface ∂H R . The torus ∂H R is represented by the surface of a 2-sphere with two removed discs +F and −F . The boundaries of +F and −F must be identified (the points with the same label coincide). A possible choice of the base point x b of the fundamental group is also depicted. In the solid torus H L , let the meridian C be the boundary of the meridinal disc D L ⊂ H L , which is oriented so that the intersection of D L with the generator γ L ⊂ H L of π 1 (H L ) is +1. Suppose that the representation ρ : π 1 (L(5, 2)) = Z 5 → SU(4) is specified by where Y is given by The restriction of A 0 L on the boundary ∂H L is nonvanishing inside a strip which is a collared neighbourhood of C. Therefore the image f * A 0 Let us now consider H R . The meridinal disc D R ⊂ H R can be chosen in such a way that the boundary of D R coincides with the boundaries of +F (and −F ) of Figure 5. The image on ∂H R of the corresponding generator γ R of π 1 (H R ) is associated to +F , and it can be represented by an arrow intersecting the boundary of the disc +F and oriented in the outward direction. As in the previous case, we introduce a collared neighbourhood where Y represents an element of the Lie algebra of SU(N). The restriction of A 0 R on the boundary ∂H R is nonvanishing inside a collared neighbourhood of ∂(+F ). The value taken by A 0 R must be consistent with the given representation ρ : π 1 (L(5, 2)) → SU(4) which is specified by equation (5.1). In order to determine A 0 R , one can consider a closed path γ ⊂ ∂H R with base point x b . One needs to impose that the holonomy of A 0 R along γ must coincide with the holonomy of f * A 0 L along γ. One then finds Y = (4π/5)Y , and consequently As shown in the Heegaard diagram of Figure 5, the collar neighbourhood of C ′ and the collar neighbourhood of ∂(+F ) -where the connections f * A 0 L and A 0 R are nonvanishingintersect in five (rectangular) regions of ∂H R . Only inside these rectangular regions is the 2- As far as the computation of the Chern-Simons invariant is concerned, these five regions are equivalent and give the same contribution to X [A]. The values of the connections inside one of the five rectangular intersection regions are shown in Figure 6. In the intersection region shown in Figure 6, one then finds one region Therefore the value of the classical Chern-Simons invariant which, in this abelian case, takes the form is given by

Lens spaces in general
For a generic lens space L(p, q), the corresponding Heegaard diagram has the same structure of the diagram shown in Figure 5. The curve C ′ on ∂H R and the boundary of the disc (+F ) give rise to p intersection regions. Since the group π 1 (L(p, q)) is abelian, the analogues of equations (5.3) and (5.5) take the form where the matrix M belongs to the Lie algebra of SU(N) and satisfies e i2πM = 1 . Expression (5.12) is in agreement with the results [16,17] obtained in the case of the abelian Chern-Simons theory, where it has been shown that the value of the Chern-Simons action is specified by the quadratic intersection form on the torsion component of the homology group of the manifold.

Second example
Let us consider the 3-manifold Σ 3 which is homeomorphic with the cyclic 3-fold branched covering of S 3 which is branched over the trefoil [20]. Σ 3 admits a Heegaard splitting presentation of genus 2 and the corresponding Heegaard diagram is shown in Figure 7. The surface ∂H R is represented by the surface of a 2-sphere with four removed discs: the boundaries of +F and −F (and similarly the boundaries of +G and −G) must be identified. In Figure 7, the two characteristic curves C ′ 1 and C ′ 2 are represented by the continuous and the dashed curve respectively, and the base point x b is also shown. The two meridinal discs D 1R and D 2R of H R are chosen so that their boundaries coincide with the boundaries of the discs +F and +G respectively. The corresponding generators γ 1 and γ 2 of π 1 (H R ) can be represented by two arrows which are based on the boundaries of +F and +G and oriented in the outward direction. By taking into account the constraints coming from the requirement of homotopy triviality of the curves C ′ 1 and C ′ 2 , one finds a presentation of the fundamental group of Σ 3 , The group π(Σ 3 ) is usually called [20] the quaternionic group; it has eight elements which can be denoted by {±1, ±i, ±j, ±k}, in which ij = k, ki = j and jk = i.
Let the representation ρ : π 1 (Σ 3 ) → SU(2) be given by The corresponding flat connection A 0 R on H R vanishes in H R − {N 1R , N 2R }, where N 1R and N 2R are collared neighbourhoods of the two meridinal discs {D 1R , D 2R } of H R , and With the choice of the base point x b shown in Figure 7, the flat connection A 0 L on H L turns out to be In each region, we shall introduce the variables X and Y according to a correspondence of the type The intersection regions are denoted as {F 1, F 2, F 3, F 4, G1, G2, G3, G4} with the convention that, for instance, the region F 3 (or G3) is a rectangle in which one of the vertices is the point denoted by the number 3 of the boundary of the disk +F (or +G). The values of U 0 in these eight regions are in order; in each of the corresponding pictures, the values of U 0 at the vertices of the rectangle are also reported. Y By using the value of U 0 in the eight intersections regions {F 1, F 2, F 3, F 4, G1, G2, G3, G4}, the contribution X [A], defined in equation (4.5), of the Chern-Simons invariant can easily be determined. One finds X [A] = 1 8π 2 Tr − π 4 σ 1 σ 1 + π 4 σ 1 σ 2 + π 4 σ 1 σ 1 + π 4 σ 1 σ 2 − π 4 σ 2 σ 2 + π 4 σ 2 σ 1 + π 4 σ 2 σ 2 + π 4 σ 2 σ 1 = 0 . As sketched in Figure 9, the set of the images of {F 2, F 4, G2, G4} can be globally parametrised by new variables −1 ≤ X ≤ 1 and −1 ≤ Y ≤ 1 according to the relations Figure 9. Images of the regions {F 2, F 4, G2, G4} parametrised in equation (6.7).
The application U 0 : ∂H R → SU(2) maps the boundaries of the rectangles {F 2, F 4} and {G2, G4} into the eight edges in B 1 shown in Figure 11. Equation (6.7) and the picture of Figure 11 demonstrate that the surface U 0 : ∂H R → SU(2) is symmetric under rotations of π/2 around the σ 3 axis and bounds a region R of SU(2) which is contained in half of the ball B 1 . According to the reasoning of Section 4.4, the volume of this region R must take the value n/8, where n is an integer. This integer n is less than 4 because R is contained inside B 1 and satisfies n ≤ 2 because R is contained inside half of B 1 . Finally, the value n = 2 is excluded because a direct inspection shows that R does not cover the upper half-part of B 1 completely. Therefore one finally obtains In Section 8 it will be shown that equation (6.9) is also in agreement with a direct computation of Γ[U] that we have performed by means of the canonical expression (4.18). Finally, the validity of the result (6.9) has also been verified by means of a numerical evaluation of the integral (4.18). To sum up, in the case of the manifold Σ 3 with the specified representation (6.2) of its fundamental group, the value of the classical Chern-Simons invariant is given by Figure 11. U 0 images in B 1 of the boundaries of the regions {F 2, F 4, G2, G4}.

Poincaré sphere
The Poincaré sphere P admits a genus 2 Heegaard splitting presentation. The corresponding Heegaard diagram [20] is shown in Figure 12. One of the two characteristic curves, C ′ 1 = f (C 1 ), is described by the continuous line, whereas the second curve C ′ 2 = f (C 2 ) is represented by the dashed path; x b designates the base point for the fundamental group.
The values of A 0 L are determined by equation (7.2) and by the choice of the base point. Indeed, let the generators {λ 1 , λ 2 } of π 1 (H L ) be associated with C 1 and C 2 respectively. Then, from the Heegaard diagram and the position for the base point, one finds ρ(λ 1 ) = g 1 = e ib 1 = exp i π 5 σ , ρ(λ 2 ) = g 2 = e ib 2 = exp i π 3 σ . (7.5) Consequently, the image of A 0 L under the gluing homeomorphism f takes values otherwise .  Figure 13. Values of U 0 in the region where f * A 0 L and A 0 R vanish.
One can now determine the map In the region of the surface ∂H R where both f * A 0 L and A 0 R are vanishing, the values of U 0 are shown in Figure 13. By using the method illustrated in the previous examples, one can compute the classical Chern-Simons invariant. The intersection component is given by The image of the map Φ −1 R Φ f * L : ∂H R → SU(2) is a genus 0 surface in the group SU(2). We skip the details, which anyway can be obtained from the Heegaard diagram and equations (7.2)-(7.6). Numerical computations of the integral (4.18) give the following value of the Wess-Zumino volume (with 10 −10 precision) Γ[A] = 0.0090687883 · · · . (7.8) Therefore, the value of the classical Chern-Simons invariant associated with the representation (7.2) of π 1 (P) turns out to be cs[A] = −0.0083333333 · · · = − 1 120 mod Z , (7.9) where the last identity is a consequence of the fact that |π 1 (P)| = 120. The result (7.9) has also been obtained by means of a complete computation of the integral (4.18); this issue is elaborated in Section 8.

Computations of the Wess-Zumino volume
The computation of Γ[U] by means of the canonical expression (4.18) presents general features that are consequences of our construction of the flat connection A by means of a Heegaard splitting presentation of M. This allows the derivation of universal formulae of the classical Chern-Simons invariant for quite wide classes of manifolds. We present here one example; details will be produced in a forthcoming article. Let us consider the set of Seifert spaces Σ(m, n, −2) of genus zero with three singular fibers which are characterised by the integer surgery coefficients (m, 1), (n, 1) and (2, −1). The manifolds Σ(m, n, −2) admit [4,40] a genus two Heegaard splitting M = H L ∪ f H R and their fundamental group can be presented as π 1 (M) = γ 1 , γ 2 | γ m 1 = γ n 2 = (γ 1 γ 2 ) 2 , (8.1) for nontrivial positive integers m and n. The manifold Σ 3 discussed in Section 6 and the Poincaré manifold P considered in Section 7 are examples belonging to this class of manifolds. Let us introduce the representation of π 1 (M) in the group SU(2) given by where σ and σ are combinations of the sigma matrices satisfying σ 2 = 1 = σ 2 , and g m 1 = g n 2 = (g 1 g 2 ) 2 = −1 .