j. differential geometry 47 (1997) 560-599

CASSON INVARIANT, SIGNATURE DEFECT OF FRAMED MANIFOLDS AND THE SECONDARY

CHARACTERISTIC CLASSES OF SURFACE BUNDLES

SHIGEYUKI MORITA

1. Introduction

The purpose of the present paper is twofold. First we relate the Casson invariant to the signature defect of Hirzebruch, which is a fine invariant defined for framed manifolds. Secondly we introduce the notion of the secondary characteristic classes of surface bundles. This consists of a series of cohomology classes of a certain subgroup of the mapping class group of orientable surfaces. The definition is given by first examining the above relation between the Casson invariant and the signature defect precisely and then generalizing it to higher dimensional context.

The Casson invariant is an integer valued invariant defined for oriented homology 3-spheres. Roughly speaking it counts the number (with signs) of conjugacy classes of irreducible representations of the fundamental group of an oriented homology 3-sphere into the Lie group SU(2). In the cases where the homology 3-spheres are given as the results of performing 1/n Dehn surgeries along knots in the 3-dimensional sphere S^{3}, there is a formula which expresses their Casson invariants in terms of a classical invariant of knots, namely the Alexander polynomial.

In our previous papers [22], [23], we investigated how the Casson invariant is related to the structure of the mapping class group of oriented surfaces through the correspondence between elements of the mapping class group and 3-manifolds via the Heegaard splittings. Here we briefly recall our results. Let M_{g} be the mapping class group of an oriented closed surface of genus g, which we denote by Σ_{g}, and let K_{g} be the subgroup of M_{g} generated by all the Dehn twists along separating simple closed curves on Σ_{g}. Let us fix a Heegaard splitting of S^{3} of genus g. Namely we express S^{3} = H_{g} ∪_{ι_{g}} −H_{g} where H_{g} denotes the handlebody of genus g, and ι_{g} ∈ M_{g} is a certain element so that if we glue H_{g} with −H_{g} along their boundaries by the map ι_{g}, then we obtain S^{3}. Thus to each element φ ∈ K_{g}, we can associate an oriented homology 3-sphere W_{φ} which is the one obtained by first cutting S^{3} along the fixed Heegaard surface Σ_{g} and then reglueing the resulting two pieces by the map φ. Namely we have W_{φ} = H_{g} ∪_{ι_{g}φ} −H_{g}. It can be shown that any homology 3-sphere can be obtained in this way, for some g and φ ∈ K_{g} (see [22]). Then we can define a mapping

Received March 8, 1995, and, in revised form, July 9, 1997.

λ^{∗} : K_{g} ⟶ Z

by setting λ^{∗}(φ) = λ(W_{φ})(φ ∈ K_{g}) where λ(W_{φ}) is the Casson invariant of the homology sphere W_{φ}. In such a situation, the main results of [22] can be summarized as follows. Let M_{g,1} be the mapping class group of Σ_{g} relative to an embedded disc D^{2} ⊂ Σ_{g}, and let K_{g,1} be the subgroup of M_{g,1} generated by all the Dehn twists along separating simple closed curves on Σ_{g}\D^{2}. Then we can define two homomorphisms d : K_{g,1} → Z and q : K_{g,1} → Q such that the equality

λ^{∗}(φ) = \frac{1}{24}d(φ̃) + q(φ̃)

holds for all φ ∈ K_{g} where φ̃ ∈ K_{g,1} is any lift of φ. In other words, the Casson invariant λ^{∗} can be described as the sum of two invariants d and q which are defined for elements of K_{g,1}. The latter invariant q is rather easy to compute and it should measure certain higher order Massey products of the mapping torus M_{φ} = Σ_{g} × I/ ∼, where we identify (p, 0) with (φ(p), 1) (p ∈ Σ_{g}) and M_{φ} homologically the same as Σ_{g} × S^{1}. On the contrary, d is a much deeper invariant than q, and in fact it plays the role of the very core of the Casson invariant from our point of view. It is true that the value of d on each generator of the group K_{g,1}, namely Dehn twist along any separating simple closed curve on Σ_{g} \ D^{2}, is very simple. However at present we do not know any effective method to compute it for arbitrarily given element of K_{g,1}, because there is no algorithm to express any given element as a product of generators. The situation being like this, we must say that d remains to be a rather mysterious invariant.

Now one of the main purposes of the present paper is to give a geometrical interpretation of this invariant d. Precise statements will be described in §5 after a few preparatory sections (§§2-4). In short, we will prove that the value d(φ) is equal to the Hirzebruch’s signature defect (see [14]) of the mapping torus M_{φ} with respect to a certain canonical framing of its tangent bundle (Theorem 5.1). In this way we found a connection between the Casson invariant and the signature defect of certain framed 3-manifolds.

The invariant d is defined not just for elements of the group K_{g,1} but also for any element of the whole mapping class group M_{g,1} (see [22]). Our geometrical interpretation of the invariant d mentioned above is in fact given in this general setting. It takes the form of the signature defect of the mapping torus M_{φ}(φ ∈ M_{g,1}) with respect to a certain 2-framing (in the sense of Atiyah [4]) of its tangent bundle (Theorem 5.2). Our 2framings of these mapping tori M_{φ} are different from Atiyah’s canonical 2-framings, and here we find an interesting interplay between Atiyah’s argument in [4] and ours. Proofs of these results will be given in §6.

In §7 we generalize some of the above arguments given mainly for the 3-dimensional manifolds M_{φ}, to the cases of higher dimensions. In particular we prove that any Σ_{g}-bundle whose holonomy group is contained in the subgroup K_{g,1} has a partial framing on its tangent bundle. We can generalize the definition of the signature defect so that it will be adapted to this weaker setting (Proposition 7.3). By making use of this together with the bordism theory as developed in [1], [8], we obtain the definition of our secondary characteristic classes of surface bundles

d_{i} ∈ H^{4i−3}(K_{g,1}; Q) (i = 1, 2, ⋯),

(see Theorem 7.7). The first class d_{1} ∈ H^{1}(K_{g,1}; Q) is nothing but the above invariant d : K_{g,1} → Z, up to a non-zero factor, which embodies a deep structure of the mapping class group related to the Casson invariant. We expect that other secondary characteristic classes of higher degrees will also reflect certain unknown structure of the mapping class group.

In this paper, we treat our invariant d only locally, namely we consider its value on each individual element of the mapping class group and give a geometrical meaning to it. In future, we would like to investigate this invariant as well as higher secondary classes more globally, perhaps in relation to the global geometry of the Teichmüller space and also of various universal families over it. Here Atiyah’s work [3], which treats the case of genus one, should be considered as a guide.

2. Cocycles for the first characteristic class of surface bundles

Let Σ_{g} be a closed oriented surface of genus g and let M_{g} be the mapping class group of Σ_{g}. Namely it is the group of all the isotopy classes of orientation preserving diffeomorphisms of Σ_{g}. Harer [12] proved that the second cohomology group H^{2}(M_{g}; Z) of M_{g} is an infinite cyclic group (for g ≥ 3) generated by the first Chern class

c_{1} ∈ H^{2}(M_{g}; Z).

Here c_{1} is defined as follows. Let ρ : M_{g} → Sp(2g; Z) be the classical representation of M_{g} onto the Siegel modular group which is induced from the natural action of M_{g} on the integral homology group H_{1}(Σ_{g}; Z) of Σ_{g}. The Siegel modular group is contained in the symplectic group Sp(2g; R), and the maximal compact subgroup of the latter group is isomorphic to the unitary group U(g). Hence we have the usual first Chern class c_{1} ∈ H^{2}(BSp(2g; R); Z). This cohomology class can be pulled back to define an element in H^{2}(Sp(2g; Z); Z) and then in H^{2}(M_{g}; Z) via the homomorphism ρ. This is our first Chern class c_{1} (we use the same letter). In the case where we are given a family of Riemann surfaces of genus g over a differentiable manifold M, our c_{1} is (up to signs) nothing but the first Chern class of the g-dimensional complex vector bundle over M whose fibre over a point p ∈ M is the space of holomorphic differentials of the Riemann surface over p (cf. [2], [19]).

Now let

e_{1} ∈ H^{2}(M_{g}; Z)

be the first characteristic class of surface bundles defined in [18], [19] (see also [2], [17], [27]). Here we briefly recall its definition. Let π : E → M be a differentiable fibre bundle over M with fibre Σ_{g}, and let ξ be the tangent bundle along the fibres of π, namely it is the subbundle of the tangent bundle of E consisting of all vectors which are tangent to the fibres. We assume that π is oriented in the sense that ξ is an oriented plane bundle over E. Let e ∈ H^{2}(E; Z) be the Euler class of ξ, and set

e_{1}(π) = π_{∗}(e^{2}) ∈ H^{2}(M; Z),

where π_{∗} : H^{4}(E; Z) → H^{2}(M; Z) is the Gysin homomorphism or the integration along the fibres. By an obvious naturality of the above construction, we have a well defined cohomology class e_{1} ∈ H^{2}(M_{g}; Z). This is the definition of our first characteristic class e_{1}. If we apply the Gysin homomorphism to powers of the Euler class e of higher degrees, we obtain higher characteristic classes of surface bundles (see [19], [27] for details).

Now the relation between the two cohomology classes c_{1} and e_{1} is very simple; namely we have

e_{1} = 12c_{1}.

This can be proved by applying the Atiyah-Singer index theorem for families of elliptic operators [5] or the Grothendieck Riemann-Roch theorem (cf. [2], [19], [27]). If we assume Harer’s result mentioned above, then the above equality holds over the integers.

Now there are various explicit group cocycles which represent (certain multiples) of the first Chern class c_{1} ∈ H^{2}(M_{g}; Z) (see [9], [11], [16], [21], [25], [29]). Except for just one cocycle defined in [21], [25], all of these cocycles are defined at the level of the Siegel modular group Sp(2g; Z) or the symplectic group Sp(2g; R), and each of them has its own interesting geometrical or algebraic background, e.g. the volumes of geodesic simplices in the Siegel upper half spaces ([9]), the Maslov indices of Lagrangian subspaces of a symplectic vector space ([6], [11], [29], [30]) and so on.

Among these cocycles, the most important one for our purpose is Meyer’s signature cocycle [16] which is defined as follows. For any two elements φ, ψ ∈ M_{g}, let M_{(φ,ψ)} be the Σ_{g}-bundle over the two sphere with three holes (or a pair of pants) such that its boundary is a disjoint union of M_{φ}, M_{ψ} and −M_{φψ}. If we put

τ(φ, ψ) = sign M_{(φ,ψ)},

where sign M_{(φ,ψ)} denotes the signature of the 4-manifold M_{(φ,ψ)}, then it can be shown that τ is a 2-cocycle of the group M_{g} and represents 4c_{1} ∈ H^{2}(M_{g}; Z) (see [3], [16]). It follows that 3τ represents the first characteristic class e_{1}. Meyer gave an explicit formula for the cocycle τ at the level of the Siegel modular group Sp(2g; Z).

Next we recall the definition of our group cocycle c which also represents the first characteristic class e_{1} (see [21]). At present this is the only known group cocycle that cannot be defined at the level of the Siegel modular group. We may say that it reflects deeper structure of the mapping class group than any other cocycles. For technical reasons, this cocycle is defined not on the usual mapping class group M_{g} but on another type of mapping class group M_{g,1} which is the one relative to an embedded disc D^{2} ⊂ Σ_{g}; namely we consider only those diffeomorphisms and isotopies which are the identity on D^{2}. (However over the rationals we do have a related group cocycle which is defined on M_{g}; see [25].) We proved in [20] that the cohomology group H^{1}(M_{g,1}; H_{1}(Σ_{g}; Z)) is an infinite cyclic group. Let us fix a crossed homomorphism

k : M_{g,1} ⟶ H_{1}(Σ_{g}; Z)

which represents a generator of the above cohomology group. Here a crossed homomorphism means that we have the equality k(φψ) = k(φ) + φ_{∗}(k(ψ)) for all φ, ψ ∈ M_{g,1}. Now in terms of this crossed homomorphism, our group cocycle c is defined as

c(φ, ψ) = k(φ^{−1}) · k(ψ),

where · denotes the intersection number. We mention here that the crossed homomorphism k is well defined up to coboundaries, which can be naturally identified with elements of H_{1}(Σ_{g}; Z).

Henceforth let us call this cocycle c, which represents the first characteristic class e_{1} ∈ H^{2}(M_{g,1}; Z), the intersection cocycle.

3. A central extension of the mapping class group and framings of 3-manifolds

In this section, first we recall the notion of 2-framings of 3-manifolds and the definition of a certain central extension of the mapping class group M_{g} from Atiyah’s paper [4], in which they are fully developed. According to [4], these ideas are all due to Witten. In the same paper, Atiyah clarified the relationship between Meyer’s signature cocycle and the above ideas by introducing the notion of canonical 2-framing to every 3-manifold. In the latter half of this section, we will consider how our intersection cocycle is related to these ideas.

Let M be a closed oriented 3-manifold. Then it is easy to see that the twice of the tangent bundle of M, denoted by 2T M = T M ⊕ T M, has a natural spin structure. Moreover it is a trivial bundle because T M is trivial. A homotopy class of trivializations of 2T M as a Spin(6)-bundle is called a 2-framing of M. The set of 2-framings of any 3-manifold can be naturally identified with the set of integers Z as follows. Let α be a 2-framing of M. Choose a compact oriented 4-manifold Z such that ∂Z = M, and set δ(α) = \frac{1}{6}p_{1}(2T Z; α) − sign Z where the relative Pontrjagin number p_{1} is defined by trivializing 2T Z on ∂Z = M by the 2-framing α. It is easy to see that δ(α) is well defined independent of the choice of Z. In fact it is the signature defect of Hirzebruch adapted to the context of 2-framings. Now the identification of the set of 2framings with Z mentioned above is given by associating 3δ(α) to each 2-framing α. In these terminologies, Atiyah’s canonical 2-framing of a 3-manifold is defined to be the unique 2-framing α such that δ(α) = 0.

Next we recall the definition of a central extension

0 ⟶ Z ⟶ M̂_{g} ⟶ M_{g} ⟶ 1.

As a set, M̂_{g} is the one consisting of isomorphism classes of oriented surface bundles M → S^{1} over the circle with fibre Σ_{g} together with a choice of 2-framing α on the total space M as well as an identification of the fibre over the base point ∗ ∈ S^{1} with Σ_{g} (the definition given in [4] is insufficient. One must fix a trivialization of each surface bundle over the base point as above). Thus any element of M̂_{g} can be represented by a pair (φ, α) where φ ∈ M_{g} and α is a 2-framing on the mapping torus M_{φ}. The group law is defined as (φ, α)(ψ, β) = (φψ, γ) where γ is the 2-framing on M_{φψ} such that the relative p_{1} of M_{(φ,ψ)} with respect to the 2-framing α + β − γ on ∂M_{(φ,ψ)} = M_{φ} + M_{ψ} − M_{φψ} vanishes.

Now let s_{0} : M_{g} → M̂_{g} be the canonical section to the central extension Z → M̂_{g} → M_{g} given by s_{0}(φ) = (φ, α_{φ}) (φ ∈ M_{g}), where α_{φ} is Atiyah’s canonical 2-framing on M_{φ}. Then it turns out that the associated 2-cocycle of the above extension

M_{g} ∋ φ, ψ ↦ s_{0}(φ)s_{0}(ψ){s_{0}(φψ)}^{−1} ∈ Z

is equal to thrice Meyer’s signature cocycle τ. It follows that the extension class of the above central extension is 12c_{1} and hence is nothing but the first characteristic class e_{1} ∈ H^{2}(M_{g}; Z).

Now we would like to investigate the influence of our intersection cocycle c, which also represents e_{1}, on the above central extension. But before doing so, we make a short digression to consider a general fact about central extensions which should be well known.

Geometrically a central extension corresponds to a principal SO(2)- bundle or equivalently an oriented S^{1}-bundle. To mesure how such a bundle is twisted globally, we put a connection on it which is a 1form on the total space satisfying certain conditions. We then have the associated curvature form which is a closed 2-form on the base space. The de Rham cohomology class of this closed 2-form is nothing but the Euler class of the given bundle (up to the factor 2π). Another method to compute the Euler class is to try to make a cross section to the associated oriented plane bundle whose image is transverse to the zero-section. Then the intersection of the image of the section with the zero-section represents the Poincaré dual of the Euler class. Now let us consider how these geometrical ideas can be converted into the algebraic context of central extensions of groups.

Let

0 ⟶ Z ⟶ Γ̂ ⟶ Γ ⟶ 1

be a central extension and let χ ∈ H^{2}(Γ; Z) be its Euler class. A section of this central extension is a set map s : Γ → Γ̂ which is a right inverse to the projection π : Γ̂ → Γ. A retraction is a mapping r : Γ̂ → Z such that the equality r(f̂+ m) = r(f̂) + m holds for any f̂ ∈ Γ̂ and m ∈ Z where we write the multiplication by m ∈ Z additively. A cocycle of this central extension is a group 2-cocycle c ∈ Z^{2}(Γ; Z) which represents the Euler class χ. In these terminologies, we have

Proposition 3.1. Let 0 → Z → Γ̂ → Γ → 1 be a central extension. Then there is a natural bijection between the following two sets:

(i) the set of all sections,

(ii) the set of all retractions.

Moreover if we assume that H^{1}(Γ; Z) = 0, then there is also a canonical bijection between any of the above two sets and

(iii) the set of all cocycles.

The normalizing condition (i) s(id) = id for elements of the first set corresponds to the conditions (ii) r(id) = 0 and (iii) c(f, id) = c(id, f) = 0 for any f ∈ Γ, for elements of the other two sets respectively, under the above bijections.

Proof. The natural one-to-one correspondence between the two sets (i) and (ii) is given as follows. To each section s : Γ → Γ̂ we associate a retraction r : Γ̂ → Z such that r(f̂) = f̂s(f)^{−1} where f̂ ∈ Γ̂ is any element and f = π(f̂). Conversely, if we are given a retraction r : Γ̂ → Z, then we can construct a section s : Γ → Γ̂ by setting s(f)(f ∈ Γ) to be the unique element f̂ ∈ Γ̂ such that π(f̂) = f and r(f̂) = 0. It is easy to see that the above two correspondences are mutually inverses.

Next we consider the relation between the two sets (i) and (iii). Any section s : Γ → Γ̂ gives rise to a cocycle c which is defined by

c(f, g) = s(f)s(g){s(fg)}^{−1}(f, g ∈ Γ).

For later use, let us fix a section s_{0} and let c_{0} be the associated cocycle. Now let c be any cocycle. Both of the cocycles c_{0} and c represent the same cohomology class, namely the Euler class of the given central extension. Hence there exists a 1-cochain d : Γ → Z such that c = c_{0} + δd. If we define a section s : Γ → Γ̂ by s = s_{0} + d, then it is easy to see that the associated cocycle is nothing but the given one c. Now we consider the assumption that H^{1}(Γ; Z) = 0. This condition implies that the 1-cochain d above is uniquely defined so that we can associate a unique section to each cocycle. It is then a routine matter to verify that the above correspondences define a bijection between the set of all sections and that of all cocycles. If we write r_{0} for the retraction associated to the fixed section s_{0} (or equivalently to the fixed cocycle c_{0}), then the retraction r corresponding to a general section s (or a general cocycle c) is given by r(f̂) = r_{0}(f̂) − d(f) where f̂ ∈ Γ̂ and f = π(f̂).

Finally the last statement of our proposition can be easily checked by a straightforward inspection. This completes the proof. q.e.d.

Now we finish our digression and go back to our specific central extension 0 → Z → M̂_{g} → M_{g} → 1 or rather to a similar one

0 ⟶ Z ⟶ M̂_{g,1} ⟶ M_{g,1} ⟶ 1

which corresponds to the mapping class group M_{g,1}. It is well known that the mapping class groups M_{g} and M_{g,1} satisfy the assumption of Proposition 3.1 (see [12]). Hence we can apply it to the above central extension. Let s_{0} : M_{g,1} → M̂_{g,1} be the section defined by Atiyah’s canonical 2-framing. Then as was mentioned before, Atiyah [4] proved that the associated cocycle is nothing but 3τ, where τ is Meyer’s signature cocycle. The retraction M̂_{g,1} → Z associated to the canonical section s_{0} is of course given by thrice the signature defect δ.

Now recall from the previous section that we have our intersection cocycle c which also represents the Euler class of the above central extension. Hence, by Proposition 3.1, we have the associated section

s : M_{g,1} ⟶ M̂_{g,1}.

If we write c = 3τ + δd where d : M_{g,1} → Z is a certain 1-cochain which is uniquely defined, then s = s_{0} + d. One of the main purposes of the present paper is to give a geometrical meaning to this section s or equivalently to the associated 1-cochain d.

4. Vector fields on surfaces and the mapping class group

Our main concern in this paper is the intersection cocycle c ∈ Z^{2}(M_{g,1}; Z) which represents the first characteristic class e_{1}. This cocycle is defined once we fix a crossed homomorphism

k : M_{g,1} → H_{1}(Σ_{g}; Z)

which represents a generator of H^{1}(M_{g,1}; H_{1}(Σ_{g}; Z)) ≅ Z. More precisely we have

c(φ, ψ) = k(φ^{−1}) · k(ψ).

In our previous papers [20], [21], [24], [26] we showed various ways of constructing crossed homomorphisms M_{g,1} → H_{1}(Σ_{g}; Z) which represent a generator of H^{1}(M_{g,1}; H_{1}(Σ_{g}; Z)). The purpose of this section is to give another crossed homomorphism M_{g,1} → H_{1}(Σ_{g}; Z). This crossed homomorphism was proposed by Furuta [10] and it turned out that it also represents a generator of H^{1}(M_{g,1}; H_{1}(Σ_{g}; Z)), as will be shown below. It is defined in terms of vector fields on surfaces . At present it is the most geometrical one and is fit best to our purpose. Later Trapp [28] defined yet another crossed homomorphism which is given in terms of the concept of winding numbers ([7]) and is closely related to Furuta’s one.

Now let T_{1}Σ_{g} be the unit tangent bundle of Σ_{g}. Then π_{1}(T_{1}Σ_{g}) can be naturally considered as a subgroup of M_{g,1} (see [15]). It was actually proved in [20] that a crossed homomorphism k : M_{g,1} → H_{1}(Σ_{g}; Z) represents a generator of H^{1}(M_{g,1}; H_{1}(Σ_{g}; Z)) if and only if k(γ) = ±(2 − 2g)[γ] for any (and hence all) element γ ∈ π_{1}(T_{1}Σ_{g}) ⊂ M_{g,1} with [γ] ≠ 0 where [γ] ∈ H_{1}(Σ_{g}; Z) is the homology class of γ projected to the base space Σ_{g}. Now we define Furuta’s crossed homomorphism

k : M_{g,1} ⟶ H_{1}(Σ_{g}; Z).

First choose a non-singular vector field X on Σ_{g}^{0} = Σ_{g} \Int D^{2}, where D^{2} is the fixed embedded disc in Σ_{g}. Also choose a Riemannian metric g_{0} on Σ_{g}. For each element φ ∈ M_{g,1}, select a representative φ ∈ Diff_{+} Σ_{g} (we use the same letter) which is the identity on the disc D^{2}, and consider the vector field φ_{∗}X which is also non-singular. We define a mapping

f_{φ} : Σ_{g}^{0} → S^{1}

as follows, where S^{1} stands for the set of all angles mod 2π. For each point p ∈ Σ_{g}^{0}, we have two tangent vectors X_{p} and (φ_{∗}X)_{p} of Σ_{g}^{0} at p ∈ Σ_{g}^{0}. We set f_{φ}(p) to be ∠(X_{p}, (φ_{∗}X)_{p}) where the symbol ∠(X_{p}, (φ_{∗}X)_{p}) denotes the angle from X_{p} to (φ_{∗}X)_{p} which is measured with respect to the fixed orientation and the metric on Σ_{g}^{0} and is counted mod 2π. Let [f_{φ}] ∈ H^{1}(Σ_{g}^{0}; Z) = H^{1}(Σ_{g}; Z) be the cohomology class determined by the map f_{φ}; namely [f_{φ}] = f_{φ}^{∗}(ι) where ι ∈ H^{1}(S^{1}; Z) is the generator. Let us write k(φ) ∈ H_{1}(Σ_{g}; Z) for the Poincaré dual of [f_{φ}]. Then we obtain a mapping k : M_{g,1} → H_{1}(Σ_{g}; Z). It is easy to see that the homotopy class of the mapping f_{φ} does not depend on the choice of the representative of φ in Diff_{+} Σ_{g} nor the choice of the metric g_{0} on Σ_{g}^{0} so that the mapping k is well defined depending only on the choice of the non-singular vector field X.

Proposition 4.1. The mapping k : M_{g,1} → H_{1}(Σ_{g}; Z) defined above is a crossed homomorphism. Moreover its cohomology class [k] is independent of the choice of the non-singular vector field X and is a generator of the infinite cyclic group H^{1}(M_{g,1}; H_{1}(Σ_{g}; Z)).

Proof. We divide the proof into three steps.

Step I. First we prove that the mapping k : M_{g,1} → H_{1}(Σ_{g}; Z) is a crossed homomorphism. To show this, for any two elements φ, ψ ∈ M_{g,1} and a point p ∈ Σ_{g}^{0}, we consider three tangent vectors X_{p}, (φ_{∗}X)_{p} and ((φψ)_{∗}X)_{p} at p ∈ Σ_{g}^{0}. By the definition of the mappings f_{φψ} and f_{φ}, we have f_{φψ}(p) = ∠(X_{p}, ((φψ)_{∗}X)_{p}) and f_{φ}(p) = ∠(X_{p}, (φ_{∗}X)_{p}). Hence we have

f_{φψ}(p) = f_{φ}(p) + ∠((φ_{∗}X)_{p}, ((φψ)_{∗}X)_{p}).

Now ∠((φ_{∗}X)_{p}, ((φψ)_{∗}X)_{p}) = ∠(φ_{∗}(X_{φ^{−1}(p)}), φ_{∗}((ψ_{∗}X)_{φ^{−1}(p)}) so that this angle is equal to \overline{f}_{ψ}(φ^{−1}(p)) where \overline{f}_{ψ} : Σ_{g}^{0} → S^{1} is the mapping defined similarly as f_{ψ} but with respect to the Riemannian metric φ^{∗}(g_{0}) on Σ_{g}^{0}. In particular the two mappings f_{ψ} and \overline{f}_{ψ} are homotopic each other. Now we can write

f_{φψ}(p) = f_{φ}(p) + (\overline{f}_{ψ} ∘ φ^{−1})(p).

If we consider the corresponding cohomology classes, we obtain [f_{φψ}] = [f_{φ}] +(φ^{−1})^{∗}([f_{ψ}]) ∈ H^{1}(Σ_{g}; Z), and taking the Poincaré duals we finally have

k(φψ) = k(φ) + φ_{∗}(k(ψ)).

Hence k is a crossed homomorphism.

Step II. Next we show that the cohomology class [k] is independent of the choice of the non-singular vector field X on Σ_{g}^{0}. Let X′ be another non-singular vector field on Σ_{g}^{0}, and for each element φ ∈ M_{g,1} we denote f′_{φ} : Σ_{g}^{0} → S^{1} for the associated map, and also write

k′ : M_{g,1} → H_{1}(Σ_{g}; Z)

for the crossed homomorphism corresponding to X′. Let us write w : Σ_{g}^{0} → S^{1} for the mapping defined as w(p) = ∠(X_{p}, X′_{p}) (p ∈ Σ_{g}^{0}), and let [w] ∈ H^{1}(Σ_{g}; Z) be the cohomology class determined by w. Also let \overline{w} : Σ_{g}^{0} → S^{1} be the mapping defined similarly as w but with respect to the Riemannian metric φ^{∗}(g_{0}) on Σ_{g}^{0}. Thus \overline{w}(p) = ∠(φ_{∗}(X_{p}), φ_{∗}(X′_{p})). It is easy to see that the two maps w and \overline{w} are homotopic each other so that [w] = [\overline{w}]. Now we compute

f′_{φ}(p) =∠(X′_{p}, (φ_{∗}X′)_{p})
=∠(X_{p}, (φ_{∗}X)_{p}) + ∠((φ_{∗}X)_{p}, (φ_{∗}X′)_{p}) − ∠(X_{p}, X′_{p})
=f_{φ}(p) + \overline{w}(φ^{−1}(p)) − w(p).

Hence we have

[f′_{φ}] = [f_{φ}] + (φ^{−1})^{∗}([w]) − [w] ∈ H^{1}(Σ_{g}; Z).

If we take the Poincaré duals, we obtain

k′(φ) = k(φ) + φ_{∗}([w]^{∗}) − [w]^{∗}

where [w]^{∗} ∈ H_{1}(Σ_{g}; Z) is the Poincaré dual of [w]. We can now conclude that the two crossed homomorphisms k, k′ : M_{g,1} → H_{1}(Σ_{g}; Z) are cohomologous to each other, because their difference is a coboundary corresponding to [w]^{∗}.

Step III. Finally we prove that the cohomology class [k] is a generator of the infinite cyclic group H^{1}(M_{g,1}; H_{1}(Σ_{g}; Z)). To show this, we may assume that g = 2 because we know from [20] that the natural inclusion i : M_{g,1} → M_{g+1,1} induces an isomorphism

i^{∗} : H^{1}(M_{g+1,1}; H_{1}(Σ_{g+1}; Z)) ≅ H^{1}(M_{g,1}; H_{1}(Σ_{g}; Z))

for all g ≥ 2.

Figure 1a

Figure 1b

Now choose a free generator α_{1}, α_{2}, β_{1}, β_{2} of π_{1}(Σ_{2}^{0}) as depicted in Figure 1a and let x_{1}, x_{2}, y_{1}, y_{2} be the corresponding homology classes in H_{1}(Σ_{2}; Z). Also let m and m′ be two simple closed curves on Σ_{2}^{0} as shown in Figure 1b. Set γ = μ(μ′)^{−1} ∈ M_{2,1} where μ and μ′ are the Dehn twists along m and m′ respectively. Then it is easy to see that γ belongs to the subgroup π_{1}(T_{1}Σ_{2}) ⊂ M_{2,1} and also that the homology class of γ projected to the base space Σ_{2} is equal to y_{2}.

Figure 2a

Figure 2b

Now choose a vector field X on Σ_{2} as shown in Figure 2a, where we deform the embedded disc D^{2}, by an isotopy, so that it will contain the two hyperbolic singular points of X in its interior. Thus we obtain a non-singular vector field X on Σ_{2}^{0}. Let

k_{X} : M_{2,1} ⟶ H_{1}(Σ_{2}; Z)

be the crossed homomorphism associated to X. We claim that the equality k_{X}(γ) = 2y_{2} holds. By the above arguments, this will then imply that the crossed homomorphism k_{X} represents a generator of the infinite cyclic group H^{1}(M_{g,1}; H_{1}(Σ_{g}; Z)) for all g ≥ 2 and X. More precisely, we have k_{X}(γ) = (2g − 2)[γ] for all γ ∈ π_{1}(T_{1}Σ_{g}) ⊂ M_{g,1} (notice that the value of 2g − 2 for g = 2 is equal to 2).

Now we prove k_{X}(γ) = 2y_{2}. We know from the outset that k_{X}(γ) is a multiple of y_{2}. By the definition of the crossed homomorphism k_{X}, k_{X}(γ) is the Poincaré dual of the cohomology class [f_{γ}] = f_{γ}^{∗}(ι) where

f_{γ} : Σ_{2}^{0} ⟶ S^{1}

is the mapping defined by f_{γ}(p) = ∠(X_{p}, (γ_{∗}X)_{p}). Now we know that [f_{γ}] is a multiple of the Poincaré dual of y_{2} which is x_{2}^{∗}. Here x_{1}^{∗}, x_{2}^{∗}, y_{1}^{∗}, y_{2}^{∗} are the dual basis of x_{1}, x_{2}, y_{1}, y_{2}. Hence we have only to count the number of times the point f_{γ}(p) goes around the circle S^{1} according as the point p goes around the oriented closed curve ℓ, which is shown in Figure 1b and represents the homology class x_{2}, once. Since the mapping γ does not act on the homology of Σ_{g}^{0}, to compute the above number we may replace the mapping f_{γ} by f_{γ} ∘ γ. Then (f_{γ} ∘ γ)(p) = ∠(X_{γ(p)}, γ_{∗}(X_{p})). Now the vector field X along the oriented closed curve γ(ℓ) looks like the picture shown in Figure 2b. By a direct inspection, we can now conclude that the above number is equal to 2. This completes the proof. q.e.d.

Since we know that any crossed homomorphism M_{g,1} → H_{1}(Σ_{g}; Z) is trivial on the subgroup K_{g,1} (see [21], [22]), we have the following immediate corollary to Proposition 4.1.

Corollary 4.2. Let X be any non-singular vector field on Σ_{g}^{0}. Then for any element φ ∈ K_{g,1}, the vector field φ_{∗}X is homotopic to X in the sense that there exists a one-parameter family X_{t}(t ∈ R) of non-singular vector fields on Σ_{g}^{0} such that X_{0} = X and X_{1} = φ_{∗}X.

5. The Casson invariant as the signature defect of certain framed 3-manifolds

As was explained in the Introduction (see also [22] for details), the influence of the Casson invariant on the structure of the mapping class group is focussed upon an invariant d which defines a homomorphism

d : K_{g,1} ⟶ Z.

The value of this invariant d on each generator of K_{g,1} is very simple. More precisely, let ω be a separating simple closed curve on Σ_{g} \ D^{2} and let h be the genus of the subsurface of Σ_{g} which is bounded by ω and does not contain the embedded disc D^{2}. Then d(D_{ω}) = 4h(h − 1) where D_{ω} is the Dehn twist along ω.

Now one of our main results is to give a geometrical meaning to this invariant d. To formulate this, for each element φ ∈ K_{g,1} we consider the mapping torus M_{φ} which is defined as Σ_{g} × I/ ∼ where we identify (p, 0) with (φ(p), 1) (p ∈ Σ_{g}). We will see below that the 3manifold M_{φ} has a canonical framing t_{φ} on its tangent bundle T M_{φ}. We can therefore consider the signature defect of the framed 3-manifold (M_{φ}, t_{φ}) which we write δ(M_{φ}, t_{φ}). Namely if Z is a compact oriented 4manifold with ∂Z = M_{φ}, then we have δ(M_{φ}, t_{φ}) = \frac{1}{3}p_{1}(T Z; t_{φ})−sign Z, where p_{1}(T Z; t_{φ}) denotes the relative p_{1} of the tangent bundle of Z with respect to the framing t_{φ} on its boundary.

Theorem 5.1. Let K_{g,1} be the subgroup of the mapping class group M_{g,1} generated by all the Dehn twists along separating simple closed curves on Σ_{g} \D^{2}. Then for each element φ ∈ K_{g,1}, there is a canonical framing t_{φ} on the mapping torus M_{φ} such that the equality

d(φ) = 3δ(M_{φ}, t_{φ})

holds, where the right-hand side denotes the signature defect of the framed 3-manifold (M_{φ}, t_{φ}).

We now define the framing t_{φ} on M_{φ}. For that purpose we first choose a non-singular vector field X on Σ_{g}^{0} = Σ_{g} \ Int D^{2}. Since the mapping φ is the identity on D^{2}, M_{φ} contains D^{2} × S^{1} as a natural submanifold. Let us write M′_{φ} for M_{φ} \ Int D^{2} × S^{1}. Since we are assuming that the mapping class φ belongs to K_{g,1}, the vector field φ_{∗}X is homotopic to X, as was proved in the previous section (Corollary 4.2). Hence we can define a non-zero section of ξ|M′_{φ} by making use of a homotopy between X and φ_{∗}X where ξ has been used to denote the tangent bundle along the fibres of the surface bundle M_{φ}. This in turn defines a framing of ξ|M′_{φ} by choosing the second vector of the framing to be the one which is orthogonal to the non-zero section and has the right direction with respect to the orientation on ξ. This induces a framing of T M_{φ} over M′_{φ} because T M_{φ} ≅ ξ ⊕ ε where ε is the trivial line bundle. Now observe that the restriction of the above framing of ξ to ∂(D^{2} × S^{1}) is independent of φ. In fact it rotates (2 − 2g)-times around the boundary of D^{2} and is constant along the direction of S^{1}. Hence if we glue the tangent bundle of D^{2} × S^{1} along its boundary according to the above framing and then pull it back to (D^{2}/∂D^{2})×S^{1}, then we obtain (η_{2−2g} ˆ⊗ε)⊕ε = (η_{2−2g} ⊕ε) ˆ⊗ε. Here η_{2−2g} is an oriented plane bundle over D^{2}/∂D^{2} = S^{2} with the Euler number equal to 2−2g, ε is the trivial line bundle and the symbol ˆ⊗ denotes the exterior tensor product. Now the bundle η_{2−2g} ⊕ ε over S^{2} is trivial and has a unique trivialization up tp homotopy because π_{2}(SO(3)) = 0. Therefore the framing of T M_{φ} over M′_{φ} obtained above can be naturally extended to the interior of D^{2} × S^{1}. We thus obtain a global framing t_{φ} of the 3-manifold M_{φ}. Unfortunately this framing is not uniquely defined because of the non-vanishing of H_{1}(M_{φ}, Z). However it does hold that the induced 2-framing 2t_{φ} is uniquely defined. Moreover it turns out that it does not depend even on the choice of the non-singular vector field X. Hence we can say that the 2-framing 2t_{φ} is a canonical one for the 3-manifold M_{φ}.

Our original proof of Theorem 5.1, which was obtained some years ago, was given by a direct computation of the signature defects. However we think that the best way to prove and also to understand the above results would be to generalize everything from the context of the subgroup K_{g,1} to that of the whole mapping class group M_{g,1} as follows. This will allow us to obtain a more conceptual proof of Theorem 5.1 than before. Recall that we have fixed a non-singular vector field X on Σ_{g}^{0}. This induces a crossed homomorphism k_{X} : M_{g,1} ⟶ H_{1}(Σ_{g}; Z) (see §4) and therefore we have the associated intersection cocycle c_{X} defined by

c_{X}(φ, ψ) = k_{X}(φ^{−1}) · k_{X}(ψ)(φ, ψ ∈ M_{g,1})

(see §2). According to Proposition 3.1, we have a section

s_{X} : M_{g,1} → M̂_{g,1}

which corresponds to c_{X}. As in §3, let us write

c_{X} = 3τ + δd_{X},

where d_{X} : M_{g,1} → Z is a certain 1-cochain which is uniquely defined. Then s_{X} = s_{0} + d_{X} where s_{0} is Atiyah’s canonical section. We have the following result which is a generalization of Theorem 5.1 to the context of the whole mapping class group M_{g,1}.

Theorem 5.2. Let M_{g,1} be the mapping class group of Σ_{g} relative to an embedded disc D^{2}. Choose a non-singular vector field X on

Σ_{g}^{0} = Σ_{g} \ Int D^{2}.

Then for each element φ ∈ M_{g,1}, there is a canonical 2-framing t_{X}(φ) on the mapping torus M_{φ}, which is uniquely defined, such that the equality

d_{X}(φ) = 3δ(M_{φ}, t_{X}(φ))

holds. Here the right-hand side denotes the signature defect of the 2framed 3-manifold (M_{φ}, t_{X}(φ)). Equivalently, if we define a section t_{X} : M_{g,1} → M̂_{g,1} by

t_{X}(φ) = (M_{φ}, t_{X}(φ))(φ ∈ M_{g,1}),

then it corresponds to the intersection cocycle c_{X}:

c_{X}(φ, ψ) = t_{X}(φ)t_{X}(ψ){t_{X}(φψ)}^{−1}.

Here we would like to emphasize that our canonical 2-framing mentioned in the above theorem is different from Atiyah’s canonical 2framing which is defined for general 3-manifolds. Now we define our canonical 2-framing t_{X}(φ) on M_{φ}.

We have the mapping f_{φ} : Σ_{g}^{0} → S^{1} defined by

f_{φ}(p) = ∠(X_{p}, (φ_{∗}X)_{p}) (p ∈ Σ_{g}^{0}),

(see §4). We may assume that π ∈ S^{1} is a regular value of f_{φ} by perturbing, if necessary, the vector field φ_{∗}X slightly (cf. the argument in the proof of Proposition 6.1). Similarly we may assume that the length of the vector fields X and φ_{∗}X (after a necessary perturbation) are constantly equal to 1, because it is only the direction of the tangent vectors that is essential in the following arguments and the length is irrelevant. Let us write K_{φ} ⊂ Σ_{g}^{0} for the submanifold f_{φ}^{−1}(π). We put an orientation on K_{φ} so that its homology class [K_{φ}] is equal to k_{X}(φ). We also consider K_{φ} as a submanifold of M_{φ} by embedding it into Σ_{g}^{0} × \frac{1}{2} ⊂ M_{φ}. Now we define a section σ of the oriented plane bundle ξ over M′_{φ} = M_{φ} \ Int D^{2} × S^{1} by

σ(p, t) = (1 − t)X_{p} + t(φ_{∗}X)_{p} (p ∈ Σ_{g}^{0}, t ∈ [0, 1]).

Then it is easy to see that the image Im σ of this section is transverse to the zero-section Z_{0} of ξ and also that Im σ ∩ Z_{0} = K_{φ}. Moreover if we put an orientation on the normal bundle of K_{φ} which is induced from those on M_{φ} and K_{φ}, then it can be checked that the section σ restricted to the normal direction of any point p on K_{φ}, followed by the projection to the fibre of ξ at p with respect to a trivialization near p, is orientation preserving. We summarize the above argument as the following proposition for later use (compare with Proposition 4.1 of [21]).

Proposition 5.3. Let φ ∈ M_{g,1} and let M_{φ} → S^{1} be the corresponding oriented Σ_{g}-bundle. Then the Poincaré dual of the Euler class χ(ξ) ∈ H^{2}(M_{φ}; Z) of the tangent bundle ξ along the fibres of this surface bundle can be represented by the 1-dimensional submanifold

K_{φ} ∪ (2 − 2g)S^{1}

of M_{φ} where S^{1} denotes any “horizontal circle” q × S^{1}(q ∈ D^{2}).

Now we go on to define our 2-framing t_{X}(φ) on M_{φ}. The section σ defines a framing of M′_{φ} away from the submanifold K_{φ}. This framing restricted to ∂(D^{2} × S^{1}) is independent of φ and it has a canonical extension to the interior of D^{2} ×S^{1} as before. Thus we obtain a framing of the tangent bundle of M_{φ} outside of K_{φ} and this in turn induces a 2-framing there. Now let us look into the situation near K_{φ}. Let K be any connected component of K_{φ}; it is an oriented circle embedded in Σ_{g}^{0} × \frac{1}{2} ⊂ M_{φ}. Let N(K) be a small tubular neighborhood of K in M_{φ}. Then we can define a bundle trivialization

ξ|_{N(K)} ≅ N(K) × R^{2}

as follows. Recall that X is a non-singular vector field on Σ_{g}^{0}. Hence it defines a non-singular vector field over Σ_{g}^{0} × [\frac{1}{2} − ϵ, \frac{1}{2} + ϵ] ⊂ M_{φ} which is constant along the second factor. Thus we can consider X also as a non-zero section of ξ there. In particular X defines a non-zero section of ξ|_{N(K)}. This induces the desired bundle trivialization by sending the vector X_{q}(q ∈ N(K)) to the first unit vector of R^{2}. Next let us define an identification N(K) ≅ K × D^{2} as follows, where D^{2} is the unit disc in R^{2}. For each point p ∈ K, let D_{p} ⊂ N(K) denote the normal disc at p so that N(K) = ∪_{p∈K}D_{p}, and let q_{p} ∈ ∂D_{p} be the point such that its second coordinate as a point in Σ_{g}^{0} × I attains the minimum among all the points in D_{p}. Then the required identification is characterized (up to homotopy) by the condition that each point q_{p} goes to (p, 1) ∈ K × ∂D^{2}. Now if we combine the above two trivializations, we obtain a bundle isomorphism

ξ|_{N(K)} ≅ (K × D^{2}) × R^{2}.

By the definition of the section σ, the vector σ(q_{p}) is equal to X_{p} (up to a positive constant) which is sent to the first unit vector of R^{2} under the above isomorphism. Moreover if the point q goes around ∂D_{p} once in the positive direction starting from the point q_{p}, then the corresponding vector σ(q), projected to the fibre of ξ at p, goes around the origin once in the positive direction starting from the vector X_{p}. Hence, by perturbing if necessary the section σ appropriately, we can conclude that the section σ̄ of the trivial bundle (K × D^{2}) × R^{2}, which is induced from σ via the above bundle isomorphism, is the “identity”. Namely we have σ̄(p, v) = v (p ∈ K, v ∈ D^{2} ⊂ R^{2}). Hence if we glue the bundle ξ over N(K) = K × D^{2} along its boundary according to the framing there induced from the section σ and pull it back to K × (D^{2}/∂D^{2}), then we obtain ε ˆ⊗η_{1}. Here ε is the trivial line bundle over K, and η_{1} denotes an oriented plane bundle over D^{2}/∂D^{2} = S^{2} with the Euler number exactly equal to 1. Since π_{2}(Spin(4)) = 0, 2η_{1} has a unique trivialization as a Spin(4)-bundle over S^{2}. Hence we can conclude that the above 2-framing of M_{φ}, which was defined outside of K_{φ}, can be uniquely extended to the whole of M_{φ} as a global 2-framing. This is the definition of our canonical 2-framing t_{X}(φ). We will see later that this 2-framing is well defined independent of the various choices made (see Proposition 6.1).

6. Proofs of Theorem 5.1 and Theorem 5.2

In this section we prove our main results (Theorem 5.1 and Theorem 5.2). First we show

Proof of Theorem 5.1 assuming Theorem 5.2. Let X be any nonsingular vector field on Σ_{g}^{0}. Then we know that for any element φ ∈ K_{g,1}, the vector field φ_{∗}X is homotopic to X or equivalently we have k_{X}(φ) = 0 (see Corollary 4.2). Hence, by the definitions, the 2-framing t_{X}(φ) of the mapping torus M_{φ} is equal to twice the usual framing t_{φ}. It follows that δ(M_{φ}, t_{φ}) = δ(M_{φ}, t_{X}(φ)). On the other hand, we know also that the value d_{X}(φ) is independent of the choice of X and is equal to d(φ) (see [22]). It is now clear that Theorem 5.1 follows from Theorem 5.2. In particular we have proved that the 2-framing 2t_{φ} does not depend on the choice of the non-singular vector field X. q.e.d.

Next we prove Theorem 5.2. For that, we first generalize the definition of the 2-framing t_{X}(φ) given after Proposition 5.3 as follows. Thereby we also prove that this 2-framing is uniquely defined.

Proposition 6.1. Let φ ∈ M_{g,1} and let M_{φ} be the corresponding mapping torus. Let K_{φ} be any oriented 1-dimensional submanifold of M_{φ} embedded in the locus Σ_{g}^{0} × \frac{1}{2} ⊂ M_{φ} such that its homology class in Σ_{g}^{0} is equal to k_{X}(φ). Then there exists a section σ of the oriented plane bundle ξ over M′_{φ} such that:

(i) its value at any point p × 0 ∈ Σ_{g}^{0} × 0 is equal to X_{p},

(ii) it is transverse to the zero-section Z_{0} of ξ, and Im σ ∩ Z_{0} = K_{φ}.

Moreover the corresponding 2-framing, which is defined in a similar way as t_{X}(φ), is homotopic to t_{X}(φ). It follows in particular that the homotopy class of this 2-framing depends only on X and φ and not on the choice of the submanifold K_{φ}.

Proof. As before, we consider the mapping f_{φ} : Σ_{g}^{0} → S^{1} defined by f_{φ}(p) = ∠(X_{p}, (φ_{∗}X)_{p}) (p ∈ Σ_{g}^{0}), and assume that π ∈ S^{1} is a regular value of f_{φ}. We put K_{φ}^{0} = f_{φ}^{−1}(π), and consider it as an oriented submanifold of M_{φ} embedded in Σ_{g}^{0} × \frac{1}{2} ⊂ M_{φ}. Our 2-framing t_{X}(φ) was defined in terms of this picture. Now we compare it with the more general situation described in Proposition 6.1. It is easy to see that we can construct a smooth mapping f : Σ_{g}^{0} → S^{1} such that (i) f sends any point on ∂Σ_{g}^{0} to 0 ∈ S^{1} and (ii) π ∈ S^{1} is a regular value of f and f^{−1}(π) coincides with the given 1-dimensional submanifold K_{φ} ⊂ Σ_{g}^{0} × \frac{1}{2}. Since the homology class of K_{φ} is equal to k_{X}(φ), by the assumption, the mapping f is homotopic to f_{φ} relative to ∂Σ_{g}^{0}. Now let X′ be a nonsingular vector field on Σ_{g}^{0} defined by the conditions ∠(X_{p}, X′_{p}) = f(p) and ∥X′_{p}∥ = ∥X_{p}∥(p ∈ Σ_{g}^{0}). Since f is homotopic to f_{φ}, it is easy to see that X′ is homotopic to φ_{∗}X. Now we define a section σ of ξ over M′_{φ} by

σ(p, t) = \begin{cases}X_{p} & t ∈ [0, \frac{1}{4}], \\ (\frac{3}{2} − 2t)X_{p} + (2t − \frac{1}{2})X′_{p} & t ∈ [\frac{1}{4}, \frac{3}{4}], \\ X_{p}^{t} & t ∈ [\frac{3}{4}, 1], \end{cases}

where p ∈ Σ_{g}^{0}, and X^{t}(t ∈ [\frac{3}{4}, 1]) is a homotopy of non-singular vector fields such that X^{\frac{3}{4}} = X′ and X^{1} = φ_{∗}X. Then it is easy to see that σ is transverse to the zero-section Z_{0} of ξ and Im σ ∩ Z_{0} = K_{φ}. This proves (i).

Next we prove that the 2-framing associated to the above picture is homotopic to the original t_{X}(φ). Since the two mappings f_{φ}, f : Σ_{g}^{0} → S^{1} are homotopic to each other, there is a homotopy F : Σ_{g}^{0} × I → S^{1} such that F (p, 0) = f_{φ}(p) and F (p, 1) = f(p). We may assume that π ∈ S^{1} is a regular value of F, and put K̃ = F^{−1}(π). Then K̃ is a submanifold of Σ_{g}^{0} × I of codimension one such that it meets the boundary transversely, and K̃ ∩ Σ_{g}^{0} × 0 = K_{φ}^{0}, K̃ ∩ Σ_{g}^{0} × 1 = K_{φ}. We consider K̃ as a submanifold of M_{φ} × I by embedding it into the locus Σ_{g}^{0} × \frac{1}{2} ×I ⊂ M_{φ}×I, and let N(K̃) denote a small tubular neighborhood of K̃ in M_{φ} × I. Now we consider the oriented plane bundle ξ̃ = ξ ˆ⊗ε over M_{φ} × I where ε denotes the trivial line bundle over I. We define a bundle trivialization

ξ̃|K̃ ≅ K̃ × R^{2}

as follows. The non-singular vector field X on Σ_{g}^{0} defines the one on a neighborhood of Σ_{g}^{0} × \frac{1}{2} × I, which is constant along the second as well as the third factor. In particular we can consider X as a non-zero section of ξ̃|_{N(K̃)}. Then the required trivialization is defined by sending

each vector Xq̃(q̃ ∈ N(K̃)) to the first unit vector of R^{2}.

Now it can be easily shown that the section of ξ̃ given over M′_{φ} ×{0, 1} can be extended to a section σ̃ which is defined over M′_{φ} ×I such that it is transverse to the zero-section Z̃_{0} of ξ̃ and Im σ̃ ∩ Z̃_{0} = K̃. This section σ̃ induces a trivialization N(K̃) = K̃ × D^{2} as follows. For each point p̃ ∈ K̃, let Dp̃ be the normal disc at p̃ so that we have N(K̃) = ∪_{p̃∈K̃}Dp̃. By the condition that σ̃ is transverse to the zerosection, we see that there is (essentially) a unique point q̃p̃ ∈ ∂Dp̃ such that the vector σ̃(q̃p̃) is equal to Xq̃p̃ up to a positive scalar. Then the required trivialization is defined by sending the point q̃p̃ to (p̃, 1). It is easy to see that the restriction of the above trivialization to the boundary coincides with those of N(K_{φ}^{0}) and N(K_{φ}). Thus we obtain a bundle isomorphism

ξ̃|K̃ ≅ (K̃ × D^{2}) × R^{2},

which coincides with the previously given one on the boundary. Then by exactly the same reason as before, the section σ̄ of the trivial bundle (K̃ × D^{2}) × R^{2} which is induced from σ̃ via the above isomorphism is given by the “identity”, namely,

σ̄(p̃, v) = v(p̃ ∈ K̃, v ∈ D^{2} ⊂ R^{2}).

If we glue the bundle ξ̃ over N(K̃) along its boundary according to the framing there induced from the non-zero section σ̃ and pull it back to K̃ × (D^{2}/∂D^{2}), then we obtain ε ˆ⊗η_{1} where ε is the trivial line bundle over K̃. Hence the 2-framing on M′_{φ} × {0, 1}, which is a usual framing of the Spin(6)-bundle 2T M_{φ} = 2(ξ ⊕ ε), can be extended to a framing of 2T M_{φ} ˆ⊗ε over M′_{φ} × I. On the other hand, the framings of ξ ⊕ ε over each embedded solid torus D^{2}×S^{1} in M_{φ}×0 and M_{φ}×1 are the same so that they can be extended constantly over D^{2} × S^{1} × I ⊂ M_{φ} × I. Thus we have shown that the 2-framing on M_{φ} × {0, 1} can be extended to a global framing of 2T M_{φ} ˆ⊗ε over the whole of M_{φ} × I. This completes the proof of (ii) and hence that of Proposition. q.e.d.

Now we begin the proof of Theorem 5.2.

Proof of Theorem 5.2. The situation is as follows. We fix a nonsingular vector field X on Σ_{g}^{0}. We then have the associated intersection cocycle c_{X} given by

c_{X}(φ, ψ) = k_{X}(φ^{−1}) · k_{X}(ψ)(φ, ψ ∈ M_{g,1}),

where k_{X} : M_{g,1} → H_{1}(Σ_{g}; Z) is the crossed homomorphism corresponding to X. Let us define a section t_{X} : M_{g,1} → M̂_{g,1} by

t_{X}(φ) = (M_{φ}, t_{X}(φ))(φ ∈ M_{g,1}),

where t_{X}(φ) is the 2-framing on M_{φ} defined in §5. Then our task is to prove the identity

c_{X}(φ, ψ) = t_{X}(φ)t_{X}(ψ){t_{X}(φψ)}^{−1}

for all φ, ψ ∈ M_{g,1}. Now we consider the Σ_{g}-bundle

M_{(φ,ψ)} ⟶ P,

where P is a pair of pants (or the 2-sphere with three holes) such that its boundary is a disjoint union of M_{φ}, M_{ψ} and −M_{φψ}. Now the value t_{X}(φ)t_{X}(ψ){t_{X}(φψ)}^{−1} is equal to the “difference degree” of the two 2-framings t_{X}(φ)t_{X}(ψ) and t_{X}(φψ) on M_{φψ}. In other words, it is equal to

−\frac{1}{2}p_{1}(2T (M_{φψ} × I); −t_{X}(φ)t_{X}(ψ) + t_{X}(φψ)),

which is minus half of the relative p_{1} of twice the tangent bundle of the 4-manifold M_{φψ} × I with respect to the 2-framing −t_{X}(φ)t_{X}(ψ) on −M_{φψ} ×0 and t_{X}(φψ) on M_{φψ} ×1. On the other hand, by the definition of the multiplication in the group M̂_{g,1}, the relative p_{1} of the 4-manifold M_{(φ,ψ)} with respect to the 2-framing t_{X}(φ) + t_{X}(ψ) − t_{X}(φ)t_{X}(ψ) on its boundary vanishes. Hence we can conclude that

t_{X}(φ)t_{X}(ψ)\{t_{X}(φψ)\}^{−1} = \frac{1}{2}p_{1}(2T M_{(φ,ψ)}; t_{X}(φ) + t_{X}(ψ) − t_{X}(φψ)).

Thus we have only to prove the following equality

\frac{1}{2}p_{1}(2T M_{(φ,ψ)}; t_{X}(φ) + t_{X}(ψ) − t_{X}(φψ)) = k_{X}(φ^{−1}) · k_{X}(ψ).

Namely our task is to prove that the half of the relative p_{1} of the 4manifold M_{(φ,ψ)} with respect to our canonical 2-framings on each connected component of its boundary is given by the intersection number k_{X}(φ^{−1}) · k_{X}(ψ) = −k_{X}(φ) · φ_{∗}(k_{X}(ψ)).

Now instead of the Σ_{g}-bundle M_{(φ,ψ)} over P, we consider the following model for it. Namely we take the product bundle Σ_{g} × △ over the standard 2-simplex △ spanned by the three vertices {0, 1, 2}, and then identify the point (p, 0) with (φ(p), 1) and (φψ(p), 2) for all p ∈ Σ_{g}. The resulting Σ_{g}-bundle is essentially the same as M_{(φ,ψ)}. We have the non-singular vector field X on Σ_{g}^{0} × 0 and it corresponds to φ_{∗}X (resp. (φψ)_{∗}X) on Σ_{g}^{0} × 1 (resp. Σ_{g}^{0} × 2) under the above identification. Now choose any oriented 1-dimensional submanifold K_{φ}, K_{ψ} and K_{φψ} of Int Σ_{g}^{0} whose homology classes are k_{X}(φ), k_{X}(ψ) and k_{X}(φψ) respectively. We embed the three manifolds K_{φ}, φ(K_{ψ}), and K_{φψ} into the locus Σ_{g}^{0} × m_{01}, Σ_{g}^{0} × m_{12} and Σ_{g}^{0} × m_{02} respectively, where m_{ij} ∈ △ is the midpoint of the 1-simplex ℓ_{ij} spanned by the two vertices {i, j}. We have the identity k_{X}(φψ) = k_{X}(φ) + φ_{∗}(k_{X}(ψ)) so that there exists an oriented properly embedded submanifold W of Σ_{g} × △ of codimension two such that ∂W = K_{φ} ∪ φ(K_{ψ}) ∪ −K_{φψ} (see Figure 3).

Figure 3

Now let ξ be the tangent bundle along the fibre of the trivial Σ_{g}- bundle Σ_{g} × △. The boundary of this trivial bundle can be written as

∂(Σ_{g} × △) = Σ_{g} × ℓ_{01} ∪ Σ_{g} × ℓ_{12} ∪ −Σ_{g} × ℓ_{02}

and each of the product bundle Σ_{g} × ℓ_{ij} on the right-hand side will serve as a model for the Σ_{g}-bundles M_{φ}, φ(M_{ψ}) and M_{φψ} respectively. Here the meaning of the second bundle φ(M_{ψ}) is the following. Namely we transfer everything from the bundle M_{ψ} to Σ_{g} × ℓ_{12} by the map φ (see Figure 3). By virtue of Proposition 6.1, We have the section σ of ξ over Σ_{g}^{0} × ∂△, which is transverse to the zero-section Z_{0} such that Im σ∩Z_{0} = K_{φ} ∪φ(K_{ψ})∪−K_{φψ}. We also have the associated 2-framing t_{X}(φ) + φ(t_{X}(ψ)) − t_{X}(φψ) on the boundary. Clearly

p_{1}(2T M_{(φ,ψ)}; t_{X}(φ) + t_{X}(ψ) − t_{X}(φψ))

= p_{1}(2T (Σ_{g} × △); t_{X}(φ) + φ(t_{X}(ψ)) − t_{X}(φψ)).

In what follows, we will compute the right-hand side of the above equality. Now it is easy to see that the section σ of ξ defined on Σ_{g}^{0} × ∂△ can be extended to a global section σ̃ over the whole of Σ_{g}^{0} × △ such that it is transverse to the zero-section Z̃_{0} and Im σ̃ ∩ Z̃_{0} = W. Let N(W) be a small tubular neighborhood of W. Then, as before, we can define a trivialization

ξ|_{N(W)} ≅ N(W) × R^{2}

as follows. We consider X as a non-singular vector field over a neighborhood of W in Σ_{g} × △, which is constant along the factor △. In particular X defines a non-zero section of ξ|_{N(W)}. Then the required trivialization is given by sending each vector X_{q}(q ∈ N(W)) to the first unit vector of R^{2}. Next for each point p ∈ W let D_{p} be the normal disc at p so that N(W) = ∪_{p∈W} D_{p}. Since the section σ̃ is transverse to the zero-section, there exists a unique point q_{p} ∈ D_{p} such that the vector σ̃(q_{p}) is equal X_{q_{p}} up to a positive constant. Then we can define a trivialization N(W) ≅ W × D^{2} by sending each point q_{p} to (p, 1). Thus we have a bundle isomorphism

ξ|_{N(W)} ≅ (W × D^{2}) × R^{2},

and the section σ̄ of the trivial bundle (W ×D^{2})×R^{2} induced from σ̃ via the above isomorphism is given by the “identity”, namely σ̄(p, v) = v p ∈ W, v ∈ D^{2} ⊂ R^{2}. Now the section σ̃ defines a framing of ξ over Σ_{g}^{0} × △ \ Int N(W). If we glue the bundle ξ|_{N(W)} along its boundary according to the above framing and pull it back to W × D^{2}/∂D^{2}, then we obtain ε ˆ⊗η_{1}. By trivializing 2η_{1}, we get a framing of 2ξ over Σ_{g}^{0} ×△. Clearly the induced framing of 2ξ on Σ_{g}^{0} × ℓ_{01} and Σ_{g}^{0} × ℓ_{02} extends to the 2-framings t_{X}(φ) and t_{X}(φψ) respectively. However the induced 2-framing on Σ_{g}^{0} × ℓ_{12}, which we denote t′_{X}(ψ), is different from the original 2-framing φ(t_{X}(ψ)) in general. Now it is easy to see that

p_{1}(2T (Σ_{g} × △); t_{X}(φ) + t′_{X}(ψ) − t_{X}(φψ)) = 0,

because the above framing of 2ξ is defined over the whole of Σ_{g} × △. Hence we can conclude that

p_{1}(2T (Σ_{g} × △); t_{X}(φ) + φ(t_{X}(ψ)) − t_{X}(φψ)) = 2δ(φ(t_{X}(ψ)), t′_{X}(ψ)),

where the right-hand side denotes twice the “difference degree” (= half of the relative p_{1}) of the two 2-framings φ(t_{X}(ψ)), t′_{X}(ψ) of Σ_{g} × ℓ_{12}.

Claim. We have the equality

δ(φ(t_{X}(ψ)), t′_{X}(ψ)) = −k_{X}(φ) · φ_{∗}(k_{X}(ψ)).

The above claim will surely complete the proof of Theorem 5.2. We first remark that it is enough to prove that the difference degree is proportional to the intersection number k_{X}(φ) · φ_{∗}(k_{X}(ψ)). For if it is the case, then it follows that the 2-cocycle

M_{g,1} ∋ φ, ψ ↦ t_{X}(φ)t_{X}(ψ){t_{X}(φψ)}^{−1}

is proportional to the intersection cocycle c_{X}(φ, ψ). But we know that the above two cocycles are cohomologous to each other and both of them represent the same class e_{1} ∈ H^{2}(M_{g,1}; Z) which is nontrivial. Hence they must be equal.

Now we prove the above weakened form of our Claim. First observe that the two 2-framings φ(t_{X}(ψ)), t′_{X}(ψ) are the same outside of a neighborhood of φ(K_{ψ}) so that their difference is concentrated on N(φ(K_{ψ})). To analyse this difference, let us write down explicit formulae for the framings of 2ξ over N(φ(K_{ψ})) which are induced from the above two 2-framings. First we consider the one induced from the 2-framing t′_{X}(ψ). Recall that this framing was defined in terms of the bundle isomorphism ξ|_{N(W)} ≅ (W × D^{2}) × R^{2}, which is induced from the non-singular vector field X near W. We have the associated bundle projection ξ|N(W) → D^{2} × R^{2} followed by the mapping

D^{2} × R^{2} ⟶  E(η_{1})
↓  ↓
D^{2} ⟶ D^{2}/∂D^{2} = S^{2}

which is induced from the framing on ∂D^{2} given by the non-zero section ∂D^{2} ∋ v ↦ v ∈ R^{2} (v ∈ ∂D^{2} ⊂ R^{2}) there. Now let R_{θ} ∈ SO(2) be the rotation matrix corresponding to the angle θ ∈ S^{1}. Then the framing on ∂D^{2} is given by ∂D^{2} × R^{2} ∋ (v, ζ) ↦ R_{θ}^{−1}ζ ∈ R^{2} where v = e^{iθ} ∈ ∂D^{2}. Now let T_{θ} = R_{θ} ⊕ R_{θ} ∈ SO(2) × SO(2) ⊂ SO(4). It is easy to see that the mapping S^{1} ∋ θ ↦ T_{θ} ∈ SO(4) is null homotopic. In fact, if we set

T_{θ,r} = (\begin{matrix}r cos θ & −r sin θ & \sqrt{1 − r^{2}} & 0 \\ r sin θ & r cos θ & 0 & −\sqrt{1 − r^{2}} \\ −\sqrt{1 − r^{2}} & 0 & r cos θ & −r sin θ \\ 0 & \sqrt{1 − r^{2}} & r sin θ & r cos θ \end{matrix}) ∈ SO(4),

then we have T_{θ,1} = T_{θ} and the correspondence

D^{2} ∋ v = re^{iθ} ↦ T_{θ,r} ∈ SO(4)

defines an extension of the above mapping to D^{2}. In terms of these matrices, our framing of 2η_{1} is given, at the level of the trivial bundle D^{2} × (R^{2} ⊕ R^{2}), by

D^{2} × (R^{2} ⊕ R^{2}) ∋ (v = re^{iθ}, ζ) ↦ T_{θ,r}^{−1}ζ ∈ R^{4}.

Now let K be any connected component of φ(K_{ψ}). Then, by the above argument, the framing of 2ξ over N(K) corresponding to the 2-framing t′_{X}(ψ) is given by

2ξ|_{N(K)} ≅ (K × D^{2}) × (R^{2} ⊕ R^{2}) ∋ (p, v, ζ) ↦ T_{θ,r}^{−1}ζ ∈ R^{4},

where p ∈ K, v = re^{iθ} ∈ D^{2} and ζ ∈ R^{2} ⊕ R^{2} = R^{4} as before.

Next we consider the framing of 2ξ over N(K) corresponding to the 2-framing φ(t_{X}(ψ)). The difference of this framing from the former one comes entirely from that of the choice of the bundle isomorphism ξ|_{N(K)} ≅ (K × D^{2}) × R^{2}. This time, instead of X we have to use the non-singular vector field φ_{∗}X, which is now considered as a vector field defined near N(K). For each point p ∈ K, let D_{p} ⊂ N(K) be the normal disc at p. We have the identification ξ|_{D_{p}} ≅ D^{2} × R^{2} which is defined by the vector field X. Then the vector field φ_{∗}X on D_{p} is represented by a section σ_{p} : D^{2} → R^{2} of the trivial bundle D^{2} × R^{2} via the above identification. Now by perturbing the vector field φ_{∗}X appropriately, we may assume that the above section is a constant function for all p. Namely if we set u(p) = ∠(X_{p}, (φ_{∗}X)_{p}) ∈ S^{1}, then we have σ_{p}(v) = u(p). Here we understand u(p) as a vector in R^{2} of length 1. It follows that the bundle trivialization of ξ|_{D_{p}} with respect to the vector field φ_{∗}X is given by the composition

ξ|_{D_{p}} ≅ D^{2} × R^{2} → D^{2} × R^{2},

where the first isomorphism is the one with respect to the vector field X, and the second bundle map is given by

D^{2} × R^{2} ∋ (v, ζ) ↦ (R_{u(p)}^{−1}v, R_{u(p)}^{−1}ζ) ∈ D^{2} × R^{2}.

Hence the bundle trivialization of ξ|_{N(K)} with respect to the vector field φ_{∗}X is given by the composition

ξ|_{N(K)} ≅ (K × D^{2}) × R^{2} → (K × D^{2}) × R^{2},

where the first isomorphism is the one corresponding to X, and the second mapping is given by

(K × D^{2}) × R^{2} ∋ (p, v, ζ) ↦ (p, R_{u(p)}^{−1}v, R_{u(p)}^{−1}ζ) ∈ (K × D^{2}) × R^{2}.

Observe here that if we write v = re^{iθ}, then R_{u(p)}^{−1}v = re^{i(θ−u(p))}. We can now obtain the following formula for the framing of 2ξ over N(K) corresponding to the 2-framing φ(t_{X}(ψ)):

2ξ|_{N(K)} ≅ (K × D^{2}) × (R^{2} ⊕ R^{2}) ∋ (p, v, ζ) ↦ T_{θ−u(p),r}^{−1}T_{u(p)}^{−1}ζ ∈ R^{4},

where p ∈ K, v = re^{iθ} ∈ D^{2}, and ζ ∈ R^{4}.

If we compare the above explicit formulae for the two framings of 2ξ|_{N(K)} corresponding to the 2-framings φ(t_{X}(ψ)) and t′_{X}(ψ), we can conclude that their difference can be described by the mapping

K × D^{2} ∋ (p, v) ↦ T_{θ,r}^{−1}T_{u(p)}T_{θ−u(p),r} ∈ SO(4),

which factors through a mapping

h : K × S^{2} → SO(4).

More precisely, if K_{i} runs through all connected components of φ(K_{ψ}), and h_{i} : K_{i} × S^{2} → SO(4) denotes the corresponding mapping, then

δ(φ(t_{X}(ψ)), t′_{X}(ψ)) = \frac{1}{2} ∑_{i} < [K_{i} × S^{2}], h_{i}^{∗}(CS) > .

Here [K_{i} × S^{2}] is the fundamental homology class of K_{i} × S^{2}, and CS ∈ H^{3}(SO(4); Z) denotes the Chern-Simons class corresponding to the first Pontrjagin class p_{1}. Now let us define a mapping

h_{0} : S^{1} × S^{2} → SO(4)

by the correspondence

S^{1} × D^{2} ∋ (u, v) ↦ T_{θ,r}^{−1}T_{u}T_{θ−u,r} ∈ SO(4),

and set c_{0} =< [S^{1} × S^{2}], h_{0}^{∗}(CS) >. If we define h′_{i} : K × S^{2} → S^{1} × S^{2} by h′_{i}(p, v) = (u(p), v), then clearly we have h_{i} = h_{0} ∘ h′_{i}. Hence if we write deg h′_{i} for the degree of the mapping h′_{i}, then

δ(φ(t_{X}(ψ)), t′_{X}(ψ)) = \frac{1}{2}c_{0} ∑_{i} deg h′_{i}.

Subclaim. We have the equality

∑_{i} deg h′_{i} = −k_{X}(φ) · φ_{∗}(k_{X}(ψ)).

Proof of Subclaim. Clearly we have deg h′_{i} = deg u_{i} where, recall that, for each connected component K_{i} of φ(K_{ψ}), the mapping u_{i} : K_{i} → S^{1} was defined by u_{i}(p) = ∠(X_{p}, (φ_{∗}X)_{p}). On the other hand, we have the mapping f_{φ} : Σ_{g}^{0} → S^{1} defined by f_{φ}(p) = ∠(X_{p}, (φ_{∗}X)_{p}) (p ∈ Σ_{g}^{0}) so that u_{i} is nothing but the restriction of f_{φ} to K_{i} ⊂ Σ_{g}^{0}. Since f_{φ}^{∗}(ι) ∈ H^{2}(Σ_{g}; Z) is the Poincaré dual of k_{X}(φ), we have the equality < u, f_{φ}^{∗}(ι) > = u · k_{X}(φ) which holds for any homology class u ∈ H_{1}(Σ_{g}; Z). Now we compute

∑_{i} deg h′_{i} = ∑_{i} deg u_{i}
= < [φ(K_{ψ})], f_{φ}^{∗}(ι) >
= − k_{X}(φ) · φ_{∗}(k_{X}(ψ)).

This completes the proof of Subclaim.

Thus we have the equality

δ(φ(t_{X}(ψ)), t′_{X}(ψ)) = −\frac{1}{2}c_{0}k_{X}(φ) · φ_{∗}(k_{X}(ψ)),

and this in turn completes the proof of Claim and hence that of Theorem 5.2. q.e.d.

Remark 6.1. As an immediate consequence of the above proof, we can conclude that the value of c_{0} is equal to 2. It is possible to determine this value by an explicit computation. But our conceptual proof as above enables us to avoid this rather troublesome work.

7. The secondary characteristic classes of surface bundles

Up to the previous section, we have given an interpretation of the Casson invariant in terms of Hirzebruch’s signature defect of certain framed 3-manifolds. More precisely, we identified the value of the homomorphism

d : K_{g,1} ⟶ Z

which serves as the core of the Casson invariant from our point of view, with the signature defect of the corresponding mapping torus with respect to some canonical framing of its tangent bundle (Theorem 5.1). In this section we generalize the above procedure but, so to speak, in the opposite direction we define secondary characteristic classes of surface bundles. Namely we make use of the concept of the signature defect to obtain a series of certain cohomology classes

d_{i} ∈ H^{4i−3}(K_{g,1}; Q)^{M_{g,1}} (i = 1, 2, ⋯)

of which the first one d_{1} is nothing but the above invariant d : K_{g,1} → Z (up to the factor \frac{1}{3}) which is known to be M_{g,1}-invariant (see [22], [23]).

To define our secondary classes, we use the generalized form of Thom’s bordism theory given in [1], [8]. Thus for any CW complex K, let Ω_{∗}(K) = ∑_{i} Ω_{i}(K) denote the oriented bordism group of K. It has a natural structure of a graded module over the Thom bordism ring Ω_{∗}. Over the rationals, there exists a natural isomorphism

Ω_{i}(K) ⊗ Q ≅ ∑_{p+q=i} H_{p}(K; Q) ⊗ Ω_{q}

(see [8]). The following is an immediate corollary to this statement.

Proposition 7.1. Let K be a CW complex and let

f : Ω_{i}(K) ⟶ Q

be a homomorphism. Assume that f vanishes on the submodule

∑_{p+q=i,q>0} H_{p}(K; Q) ⊗ Ω_{q} ⊂ Ω_{i}(K) ⊗ Q.

Then f can be considered as an element of H^{i}(K; Q).

Let us write Ω_{i}(K_{g,1}) for the bordism group of BK_{g,1} which is the classifying space for the group K_{g,1}. Hereafter we are going to define a homomorphism

δ_{i} : Ω_{4i−3}(K_{g,1}) ⟶ Q.

Any element of Ω_{i}(K_{g,1}), represented by an oriented Σ_{g}-bundle π : E → M over a closed oriented i-dimensional manifold M, satisfies the following three conditions:

(i) There is given an identification of the fibre E_{q_{0}} over a base point q_{0} ∈ M with Σ_{g}.

(ii) There is given a cross section s : M → E together with a trivialization N(s(M)) ≅ s(M)×D^{2} of the normal disc bundle N(s(M)) of the image of the section s.

(iii) The image of the holonomy homomorphism π_{1}(M) → M_{g,1} is contained in K_{g,1}.

Henceforth let us call such a Σ_{g}-bundle a K_{g,1}-bundle. Two K_{g,1}-bundles π_{k} : E_{k} → M(k = 0, 1) over the same base space M are said to be isomorphic if there exists an isomorphism E_{0} ≅ E_{1}, over the identity of M, as Σ_{g}-bundles such that (i) the isotopy class of the induced diffeomorphism on E_{q_{0}} = Σ_{g} is contained in K_{g,1} and (ii) it preserves the cross sections as well as the trivializations of their normal disc bundles. Then, as usual, there is a canonical bijection between the set of K_{g,1}- isomorphism classes of K_{g,1}-bundles over M and the set [M, BK_{g,1}] of homotopy classes of mappings from M to the classifying space BK_{g,1}.

Now let π : E → M be a K_{g,1}-bundle and let ξ be its tangent bundle along the fibres. Clearly we have an isomorphism

T E ≅ ξ ⊕ π^{∗}(T M).

Since the image of the holonomy of π is contained in K_{g,1} by the assumption, the total space E contains D^{2} × M as a natural submanifold. We write E′ = E \ Int D^{2} × M.

Proposition 7.2. Let π : E → M be a K_{g,1}-bundle. Then for each non-singular vector field X on Σ_{g}^{0}, there is a canonical framing t_{X} of ξ over E′. Moreover if we glue the bundle ξ along E′ according to the above framing and pull it back to (D^{2}/∂D^{2}) × M via the natural map (D^{2}/∂D^{2}) × M → E/E′, then we obtain η_{2−2g} ˆ⊗ε where η_{2−2g} is an oriented plane bundle over D^{2}/∂D^{2} = S^{2} with the Euler number equal to 2 − 2g, and ε denotes the trivial line bundle over M.

Proof. First we make a cell decomposition of the base space M with a single 0-cell at the base point q_{0}. The non-singular vector field X defines a non-zero section of ξ|_{E′_{q_{0}}} via the identification E′_{q_{0}} = Σ_{g}^{0} where E′_{q_{0}} = E_{q_{0}} ∩E′. We extend this non-zero section to the whole of E′ by an induction on the dimension of the cells. Let τ^{1} be any 1-cell of M. Since the holonomy along this 1-cell is contained in K_{g,1} by the assumption, the non-zero section extends to π^{−1}(τ^{1}) ∩ E′ by Corollary 4.2. Here we can take the extension to be constant along π^{−1}(τ^{1}) ∩ ∂E′. Next let τ^{2} be any 2-cell and let Σ_{g}^{0} × D^{2} → π^{−1}(τ^{2}) ∩ E′ be a trivialization over τ^{2}. By pulling back the non-zero section of ξ over π^{−1}(∂τ^{2}) ∩ E′ via the above trivialization, we obtain a non-zero section σ^{1} of ξ_{0} (= the tangent bundle along the fibres of the trivial Σ_{g}^{0}-bundle Σ_{g}^{0} × D^{2}) over Σ_{g}^{0} × ∂D^{2}. This induces a mapping

h^{1} : Σ_{g}^{0} × ∂D^{2} ⟶ S^{1}

by the formula h^{1}(p, q) = ∠(X_{p}, σ^{1}(p, q)) (p ∈ Σ_{g}^{0}, q ∈ ∂D^{2}). Here we consider X as a non-zero section of ξ_{0} which is constant along the second factor of Σ_{g}^{0} × D^{2}. Observe that the above mapping h^{1} is null homotopic. This is because the restriction of h^{1} to Σ_{g}^{0} × q_{0} (q_{0} is the base point of ∂D^{2} which is identified to the base point of M) and also to ∂Σ_{g}^{0} × ∂D^{2} is the constant map to 0 ∈ S^{1}. Hence there is an extension h̃^{1} : Σ_{g}^{0} × D^{2} → S^{1} of h^{1}. By making use of this extension, it is now easy to extend the non-zero section of ξ over each 2-cell τ^{2}.

Next let τ^{3} be any 3-cell. Choose a trivialization

Σ_{g}^{0} × D^{3} → π^{−1}(τ^{3}) ∩ E′.

Then, as before, the pull back of the non-zero section obtained so far to the trivial bundle Σ_{g}^{0} × D^{3} induces a mapping

h^{2} : Σ_{g}^{0} × ∂D^{3} ⟶ S^{1},

which is easily shown to be null homotopic. Hence there is an extension h̃^{2} : Σ_{g}^{0} × D^{3} → S^{1} of h^{2}, and by making use of this we can extend the non-zero section over each 3-cell τ^{3}. Clearly we can continue the above argument over any k-cell (k ≥ 4) so that we eventually obtain the required non-zero section of ξ|_{E′}. This induces our trivialization t_{X} of ξ over E′. Since the above non-zero section of ξ is constant along the second factor of ∂E′ = ∂D^{2}×M, the latter statement of the Proposition is obvious. This completes the proof. q.e.d.

Let π : E → M be a K_{g,1}-bundle and let t_{X} be the framing of ξ over E′ as above. If the base space M is a closed oriented manifold of dimension 4i − 3 (so that dim E = 4i − 1), then we can define a rational number δ(π) ∈ Q as follows. The definition is given by generalizing that of the Hirzebruch’s signature defect to the context of partially framed (4i − 1)-manifold (E, t_{X}).

First observe that all the Stiefel-Whitney numbers of the manifold E is zero so that we can choose a compact oriented 4i-manifold W such that ∂W = E. Clearly we have T W |_{E} = T E ⊕ ε = π^{∗}(T M) ⊕ ξ ⊕ ε. In view of Proposition 7.2, the framing t_{X} of ξ over E′ together with the unique trivialization of η_{2−2g} ⊕ ε (cf. the argument after Theorem 5.1) induces a canonical framing of the bundle ξ ⊕ ε, which we denote also by t_{X}. Now choose any connection on T M. Then the pull back of this connection to π^{∗}(T M) together with the flat connection on ξ ⊕ ε corresponding to the framing t_{X} above defines a connection on T W |_{E}. Extend this connection to the whole of W. Now let L_{i} = L_{i}(p_{1}, ⋯, p_{i}) be Hirzebruch’s i-th L-polynomial ([13]). Then we can define the relative L_{i}-number L_{i}(T W; t_{X}) with respect to the framing t_{X} on the boundary by

L_{i}(T W; t_{X}) = ∫_{W} L_{i}(T W).

Here L_{i}(T W) denotes the top form on W corresponding to the characteristic class L_{i} with respect to the connection on T W defined above. Now we set

δ(π) = L_{i}(T W; t_{X}) − sign W.

and call it the signature defect of the K_{g,1}-bundle π : E → M.

Proposition 7.3. The above definition of the signature defect of a K_{g,1}-bundle π : E → M is well defined. Namely it is independent of the choices of connections on T M and also of the manifold W such that ∂W = E.

Proof. The proof can be given almost in the same way as the case of usual signature defect. It is enough to mention only the fact that any two connections on T M can be connected by a path of connections and that the possible error term, which is an integral over E ×I corresponding to the difference of two connections on T M, vanishes because of the trivial dimension reasons. This completes the proof. q.e.d.

Next we show that our generalized signature defect of K_{g,1}-bundles is a bordism invariant. More precisely we have

Proposition 7.4. If two K_{g,1}-bundles π_{k} : E_{k} → M_{k} (k = 0, 1) over closed oriented (4i − 3)-dimensional manifolds M_{k} are bordant, namely if they define the same element of Ω_{4i−3}(K_{g,1}), then their generalized signature defects coincide: δ(π_{0}) = δ(π_{1}). It follows that the signature defect defines a homomorphism

δ_{i} : Ω_{4i−3}(K_{g,1}) ⟶ Q.

Proof. By the assumption that the two K_{g,1}-bundles π_{0} and π_{1} are bordant, there exists a K_{g,1}-bundle π̃ : Ẽ → M̃ such that (i) ∂M̃ = M_{1} − M_{0} and π̃|_{∂Ẽ} = π_{1} − π_{0} and (ii) the identification of the fibre Ẽq̃_{0} of π̃ over the base point q̃_{0} ∈ M̃ induces those on the boundary. Let ξ̃ be the tangent bundle along the fibres of π̃. It is easy to see that the framing t_{X} given on the boundary can be extended to a framing t̃_{X} of ξ̃ over Ẽ′. Also any connections on T M_{k} can be extended to one on T M̃. These induce a connection on T Ẽ. But it is easy to see that the corresponding L_{i}-form on Ẽ vanishes identically. We know also that sign Ẽ = 0 because the holonomy of this Σ_{g}-bundle is contained in K_{g,1}. The required assertion follows from these facts easily. q.e.d.

Next we show that the signature defect of K_{g,1}-bundles is invariant under the natural action of the mapping class group. Namely we have

Proposition 7.5. The value of the signature defect of K_{g,1}-bundles is independent of the choice of the non-singular vector field X on Σ_{g}^{0}. Hence the homomorphism δ_{i} : Ω_{4i−3}(K_{g,1}) → Q is M_{g,1}-invariant where the mapping class group M_{g,1} acts on Ω_{∗}(K_{g,1}) naturally.

Proof. Let π : E → M be a K_{g,1}-bundle over a closed oriented (4i−3)-dimensional manifold M. We have the partial framing t_{X} of T E corresponding to the non-singular vector field X. Namely it is a framing of the bundle ξ⊕ε. Now let Y be another non-singular vector field on Σ_{g}^{0} and let t_{Y} be the associated partial framing. Now consider the product bundle π × I : E × I → M × I. Choose any connection on T M. Since there is a canonical isomorphism T (E × I) = π^{∗}(T M) ⊕ ε ⊕ (ξ ˆ⊗ε), we can consider the relative L_{i}-number L_{i}(T (E × I); t_{Y} − t_{X}). We claim that this number vanishes. To show this, it is enough to prove that with respect to the framings t_{X} and t_{Y} of ε⊕ξ on the boundary of E ×I, the relative p_{1} of the bundle ε ⊕ (ξ ˆ⊗ε) is zero. Let us denote this relative class by p_{1}(ε⊕(ξ ˆ⊗ε); t_{Y} −t_{X}) ∈ H^{4}(E×I, E×∂I; Q). Now we have also the relative Euler class χ ∈ H^{2}(E ×I, (E ×∂I)∪(D^{2} ×M ×I); Z) of the bundle ξ ˆ⊗ε with respect to the framings t_{X} and t_{Y} of ξ|_{E′}. Let us use the same symbol χ for the pull back of the above class to H^{2}(E ×I, E × ∂I; Z) which is isomorphic to H^{2}(E; Z) ⊕ (H^{1}(E; Z) ⊗ H^{1}(I, ∂I; Z)). But clearly the restriction of χ to E is zero so that we can consider χ ∈ H^{1}(E; Z) ⊗ H^{1}(I, ∂I; Z). Hence p_{1}(ε ⊕ (ξ ˆ⊗ε); t_{Y} − t_{X}) = χ^{2} = 0 as required. Thus we have proved that the relative L_{i}-number of π × I vanishes. We also have the trivial fact that sign E×I = 0. It is now easy to deduce the required result from these two facts. The latter statement of the proposition is clear because the action of M_{g,1} on a K_{g,1}-bundle is simply given by a change of the identification of the fibre over the base point with Σ_{g} so that its effect on the signature defect reduces to the change of the non-singular vector field. This completes the proof.

q.e.d.

As another general property of the homomorphisms δ_{i}, we prove that they are stable with respect to the genus. More precisely we have

Proposition 7.6. Let i : K_{g,1} → K_{g+1,1} be the homomorphism induced from the natural embedding Σ_{g}^{0} ⊂ Σ_{g+1}^{0}. Then the following diagram is commutative

Ω_{4i−3}(K_{g,1}) ⟶^{δ_{i}} Q
i_{∗}↓  ∥
Ω_{4i−3}(K_{g+1,1}) ⟶^{δ_{i}} Q.

Proof. Let π : E → M be a K_{g,1}-bundle over a closed oriented (4i − 3)-dimensional manifold M and let u ∈ Ω_{4i−3}(K_{g,1}) be its bordism class. As before we set E′ = E \Int D^{2} ×M where D^{2} ×M is considered as a natural submanifold of E. Let T_{0} = T \ Int D^{2} be the 2-torus with a hole. If we define Ê to be the manifold obtained by glueing E′ and T_{0} × M along their boundaries, then the natural projection π̂ : Ê → M admits a structure of a K_{g+1,1}-bundle, and its bordism class in Ω_{4i−3}(K_{g+1,1}) is nothing but the element i_{∗}(u). Now let X be a nonsingular vector field on Σ_{g}^{0} and extend it to obtain a non-singular vector field X̂ on Σ_{g+1}^{0}. Then we have partial framings t_{X} and tX̂ of the tangent bundles along the fibres of E and Ê respectively. Choose a compact oriented 4i-dimensional manifold W such that ∂W = E. Let T̂ be the solid torus and let D^{2} ⊂ ∂T̂ be an embedded disc so that ∂T̂ \ Int D^{2} = T_{0}. If we define Ŵ to be the manifold obtained by attaching T̂ × M to W along the “common” D^{2} × M (⊂ ∂(T̂ × M), E), then clearly we have ∂Ŵ = Ê. Now it is easy to see that sign Ŵ = sign W and L_{i}(T Ŵ; tX̂) = L_{i}(T W; t_{X}). Hence δ_{i}(i_{∗}(u)) = δ_{i}(u). This completes the proof. q.e.d.

Now we consider the value of the homomorphism

δ_{i} : Ω_{4i−3}(K_{g,1}) ⟶ Q

on the “decomposable” part

∑_{p+q=4i−3,q>0} H_{p}(K_{g,1}; Q) ⊗ Ω_{q} ⊂ Ω_{4i−3}(K_{g,1}) ⊗ Q.

First it can be easily checked that the homomorphism

δ_{1} : Ω_{1}(K_{g,1}) = H_{1}(K_{g,1}) → Q

coincides with the invariant \frac{1}{3}d : K_{g,1} → Q. Let us write

d_{1} ∈ H^{1}(K_{g,1}; Q)

for δ_{1} = \frac{1}{3}d. Next consider the case of i = 2. Let π : E → S^{1} be a K_{g,1}-bundle over S^{1} and let N be a closed oriented 4-manifold. We can consider the product π̂ : E × N → S^{1} × N as an element of

H_{1}(K_{g,1}; Q) ⊗ Ω_{4} ⊂ Ω_{5}(K_{g,1}) ⊗ Q,

so that we can consider the value δ_{2}(π̂) ∈ Q. We claim that

δ_{2}(π̂) = d_{1}(π)L_{1}[N],

where L_{1}[N] denotes the L_{1}-number of the 4-manifold N. To prove this, choose a compact oriented 4-manifold W such that ∂W = E.

Then clearly ∂(W × N) = E × N. Now we compute

δ_{2}(π̂) =L_{2}(W × N; t̃_{X}) − sign(W × N)
=L_{1}(W; t_{X})L_{1}[N] − sign W sign N
=(L_{1}(W; t_{X}) − sign W)L_{1}[N]
=d_{1}(π)L_{1}[N],

where t_{X} is the partial framing on T E, t̂_{X} denotes the induced partial framing on T (E × N) and we have used the fact that L_{1}[N] = sign N which is a consequence of the Hirzebruch’s signature theorem. Now we consider the formal product d_{1}L_{1} as a generalized Pontrjagin number for the bordism group of K_{g,1}. Namely for any K_{g,1}-bundle ω : F → M over a closed oriented 5-manifold M, we set d_{1}L_{1}(ω) =< [M], d_{1}(ω)L_{1}(M) > where d_{1}(ω) ∈ H^{1}(M; Q) is the d_{1}-class of the bundle ω. In view of the well known general theory (cf. [8]), this number induces a homomorphism

d_{1}L_{1} : Ω_{5}(K_{g,1}) ⟶ Q.

In terms of this notation, it is easy to see that

d_{1}L_{1}(π̂) = d_{1}(π)L_{1}[N].

Set

d_{2} = δ_{2} − d_{1}L_{1} : Ω_{5}(K_{g,1}) ⟶ Q.

Then the above computation shows that d_{2} is trivial on the decomposable summand

H_{1}(K_{g,1}; Q) ⊗ Ω_{4} ⊂ Ω_{5}(K_{g,1}) ⊗ Q.

Hence we can consider d_{2} as an element of H^{5}(K_{g,1}; Q) by Proposition 7.1.

Now we consider the homomorphism δ_{i} : Ω_{4i−3}(K_{g,1}) → Q for general i. Let π : E → M be a K_{g,1}-bundle over a closed oriented (4j − 3)- dimensional manifold M (j < i) and let N be a closed oriented 4(i − j)- dimensional manifold. Let π̂ : E × N → M × N be the product of π with N. Then almost the same proof as before implies the equality δ_{i}(π̂) = δ_{j}(π)L_{i−j}[N] where we use the Hirzebruch’s signature theorem [13] L_{i−j}[N] = sign N in its full generality. Motivated by this, we set

d_{i} = δ_{i} − d_{i−1}L_{1} − d_{i−2}L_{2} − ⋯ − d_{1}L_{i−1} : Ω_{4i−3}(K_{g,1}) ⟶ Q, where d_{j}L_{k} stands for the generalized Pontrjagin number corresponding to the formal product d_{j}L_{k}. Namely for a general K_{g,1}-bundle ω : F → Z over a closed oriented (4j + 4k − 3)-dimensional manifold Z, we define d_{j}L_{k}(ω) =< [Z], ρ^{∗}(d_{j})L_{k}(Z) > where ρ : π_{1}(Z) → K_{g,1} is the holonomy homomorphism, and we understand ρ^{∗}(d_{j}) as an element of H^{4j−3}(Z; Q). Then an easy inductive argument shows that d_{i} is trivial on the decomposable summand

∑_{p+q=4i−3,q>0} H_{p}(K_{g,1}; Q) ⊗ Ω_{q} ⊂ Ω_{4i−3}(K_{g,1}) ⊗ Q.

Hence we can conclude that d_{i} is an element of H^{4i−3}(K_{g,1}; Q) by Proposition 7.1. In view of Proposition 7.5 and Proposition 7.6, we have established.

Theorem 7.7. The generalized signature defect for K_{g,1}-bundles defines a series of characteristic classes

d_{i} ∈ H^{4i−3}(K_{g,1}; Q)^{M_{g,1}} (i = 1, 2, ⋯),

which are M_{g,1}-invariant and stable with respect to the genus g.

We call these cohomology classes the secondary characteristic classes of surface bundles.

8. Concluding remark

In this paper, we have formulated our results mainly at the level of the mapping class group M_{g,1} of Σ_{g} relative to an embedded disc D^{2} ⊂ Σ_{g} and its subgroup K_{g,1}. However we expect that our results should descend to the level of the usual mapping class groups M_{g}, K_{g}. For example, we already have a formula for an explicit group 2-cocycle of M_{g} which is closely related to the intersection cocycle and represents the class e_{1} ∈ H^{2}(M_{g}) (see [25]). The problem is that it is a rational cocycle rather than an integral one so that the associated mapping d̄: M_{g} → Q, which is an analogue of the invariant d for M_{g,1}, is also rational valued. It would be interesting if one could give a geometrical interpretation to this invariant.

If we restrict d̄to the subgroup K_{g}, then we obtain a homomorphism d̄: K_{g} → Q which is M_{g}-invariant. In fact we know that it is a generator of the group H^{1}(K_{g}; Q)^{M_{g}} which is one dimensional (see [23]). Thus, after an appropriate adjustment, the secondary class d_{1} ∈ H^{1}(K_{g,1}; Q) does descend to the one on the usual mapping class group. It seems to be very likely that higher secondary classes d_{i}(i > 1) also descend to cohomology classes of K_{g}.

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