Journal of Knot Theory and Its Ramificationsc©World Scientific Publishing Company

SIGNATURES OF COLORED LINKSWITH APPLICATION TO REAL ALGEBRAIC CURVES

V. FLORENS

Universidad de Valladolidvincent florens@yahoo.fr

Received

Revised

ABSTRACTWe contruct the signature of a µ-colored oriented link, as a locally constant in-

teger valued function with domain (S1 − 1)µ. It restricts to the Tristram-Levine’s

signature on the diagonal and the discontinuities can occur only at the zeros of the col-ored Alexander polynomial. Moreover, the signature and the related nullity verify the

Murasugi-Tristram inequality. This gives a new necessary condition for a link to bound

a smoothly and properly embedded surface in B4, with given Betti numbers. As anapplication, we achieve the classification of the complex orientations of maximal plane

non-singular projective algebraic curves of degree 7, up to isotopy.

1. Introduction

The Tristram-Levine [1, 2] signature of a link, constructed in terms of a Seifertform, is a locally constant integer valued function with domain the unit circle. Thediscontinuities can occur only at the roots of the Alexander polynomial, whereasthe related nullity is non zero only at these roots. It plays an important role in knottheory, especially to construct necessary conditions for a link to bound a smoothlyand properly embedded surface with given Betti numbers in the ball B4. Indeed, thesignature is invariant by link concordance, vanishes for slice links in the strong sense[4] and the so-called Murasugi-Tristram inequality [1, 4] states that the existence ofa smooth surface in B4, with given Betti numbers and which bounds a given link,imposes restrictions on the possible values of the signature and the nullity of thelink.

We define the signature and nullity for µ-colored links, as integer valued func-tions in a dense subset of the µ-torus. They restrict to the Tristram-Levine signatureand nullity on the diagonal. For this, we mainly use Casson and Gordon construc-tions [7, 8] related to the twisted homology and intersection, see also [9, 10]. Roughlyspeaking, we apply the Atiyah-Singer α-invariant [11] to finite cyclic coverings ofprime power order of a closed three-manifold related to the link complement. Thisextends the interpretation of Viro [12] of the Tristram-Levine signature in terms ofintersection forms, and follows the spirit of [14]. We show that the signature can be

1

2

extended to a locally constant function on the complement of zeros of the colored(multi-variables) Alexander polynomial, whereas the nullity is zero only on thiscomplement. Moreover, they verify the Murasugi-Tristram inequality. This gives anew necessary condition for a link to bound a smooth non connected surface in B4,with given Betti numbers. We also show that the signature vanishes for links withnon-zero Alexander polynomial which bound a smooth surface with Euler charac-teristic 1. Note that, as an extension of the Fox-Milnor theorem for slice knots [5],the Alexander polynomial of such links is of the form ff , for f ∈ Z[t±1

1 , . . . , t±1µ ],

see [15]. Of particular interest for these constructions is that a formula of Gilmer[17] for the Atiyah-Singer invariant allows an easy computation of the signature, interms of a link diagram or a word in the braid group.

We apply these constructions to the study of algebraic curves. The 16th problemof Hilbert requires, in particular, a classification of the oval arrangements of realnon-singular algebraic curves in RP 2, up to isotopy. It is only known for the degrees≤ 7. Since Arnold’s work, the question of the existence of such an algebraic curvewith prescribed topology can be approached by studying topological properties ofsurfaces in CP 2. In particular, many necessary conditions were constructed by theuse of the intersections of 2-cycles in the two-fold covering of CP 2, branched alongthe curve. See [13, 20] for a detailed history and references. Recently, Orevkov[21] observed that the quasipositivity of a certain braid provides a new necessarycondition, and this condition is equivalent in the case of J-holomorphic curves. Itimplies that the link in S3 obtained as the closure of the braid bounds a surfacewith given Betti numbers, smoothly and properly embedded in the ball B4. Usingthe Murasugi-Tristram inequality and the Fox-Milnor theorem, Orevkov [21, 22, 24]obtained very promising results related to Hilbert’s 16th problem. This gives animportant motivation to develop necessary conditions (or extend previous existent)in terms of link invariants.

We use the Murasugi-Tristram inequality for the signature of colored links toshow that a complex orientation of an oval configuration in RP 2 is not realizable bya maximal non-singular algebraic curve of degree 7, using Orevkov’s method. Thisresult, which achieves the classification of the complex orientations of maximalplane non-singular algebraic curves of degree 7, were not showed by previous knownmethods. Note that, in general, by Orevkov’s method, the considered surfaces inB4 are non-connected, at least for separating curves.

1.1. Historic

The Tristram signature and nullity of an oriented link L, denoted by σL andηL, are defined for all λ in the unit circle, as the signature and the nullity of thehermitian form (1 − λ)θ + (1 − λ)θt, where θ is any Seifert form of L. Note thatthis definition of ηL differs of 1 from Tristram’s.

1.1.1. Link concordance and Tristram-Levine signatures

3

A knot is slice if it bounds a smooth 2-disk in the ball B4, or equivalently if it isconcordant to the trivial knot. Levine [2] showed that a slice knot is algebraicallyslice, i.e. any Seifert form of a slice knot is metabolic. It follows that the Tristram-Levine signature vanishes for slice knots. Levine also showed that the converseholds in high odd dimension, i.e. any algebraically slice knot is slice. This is falsein dimension 3: Casson and Gordon [7, 8] showed that certain two-bridges knotsin S3, which are algebraically slice are not slice. For this purpose, they definedknot invariants constructed by applying the Atiyah-Singer invariant [11] to abeliancoverings of the two-fold covering of the manifold obtained by a zero surgery on theknot.

Cappell and Shaneson [25] introduced a 4-dimensional viewpoint of algebraicallyslice knots. Roughly speaking, a knot is (4-dimensionally) algebraically slice if theinfinite cyclic covering of the zero surgery manifold bounds a spin 4-manifold suchthat the inclusion induces an isomorphism on the first homology group and thetwisted intersection form with coefficients in Z[t±1] has a metabolizer which projectsonto a metabolizer for the ordinary intersection form. They show that this conditionis equivalent to Levine’s condition. Cochran, Teichner and Orr [26] generalized thisapproach and constructed a geometric filtration of the knot concordance group andan infinite sequence of new obstructions that vanish on slice knots.

In the case of links, they are several generalizations of sliceness. Fox [4] definedslice links in the strong sense as links concordant to the trivial link. The Tristram-Levine signature is invariant by concordance, and in particular, it vanishes for slicelinks. Levine [3] also constructed signatures for links such that the linking numberof any two components is zero by applying the Atiyah-Singer invariant to the finitecyclic coverings of the zero surgery manifold on the link. They are concordanceinvariant and vanish for slice links. See also [27].

1.1.2. The Murasugi-Tristram inequality

The following theorem is the so-called Murasugi-Tristram inequality.

Theorem 1.1. [1, 6] If L bounds a surface F smoothly and properly embedded inB4 with µ connected components and rk H1(F ) = b1, then, for any root of unity ofprime power order λ the following inequality holds:

|σL(λ)|+ |ηL(λ)− µ+ 1| ≤ b1.

Murasugi [6] first proved this inequality for λ = −1 without the term |ηL(λ)−µ+1|. Tristram [1] then proved it for λ of the form e2iπ[q/2] where q is a prime power.The methods of both Murasugi and Tristram are purely 3-dimensional. They relatethe considered link to the trivial link by elementary transformations and study theeffect of these transformations to the signatures and nullity. Viro [12] interpretedσL and ηL as the signatures and nullities of intersection forms related to coverings

4

of B4 branched along a surface with boundary L, obtained by pushing a Seifertsurface from S3. Next Kauffman and Taylor [28] gave a proof of the inequality,using coverings, in the case λ = −1. Kauffman [29] gave also a proof of it in thecase of connected surfaces (µ = 1), for all λ of prime power order. Gilmer [17]finally gave a complete proof using branched coverings of S4, obtained by gluingtwo copies of B4.

1.2. Signatures of colored links

Consider a compact closed 3-manifold M with b1 = b1(M) ≥ 1 and τ(M) theMilnor torsion of M , defined as the Reidemeister torsion associated to the universalfree abelian covering of M , see Section 2.

In Section 3, we construct the signature σ(M) and the related nullity η(M) asmaps T b1P −→ Z, where T b1P is the set of primary points of the b1-torus T b1 (seeDefinition 3.14), dense in T b1 . The signature is defined by considering a reformula-tion, due to Casson and Gordon [7, 8], of the Atiyah-Singer α-invariant [11] of finitecyclic coverings of prime power order of M .

These maps have properties closely related to τ(M). First recall that this latercoincides, up to factors of the form ti − 1, with the Alexander polynomial ∆(M),see [30]. We show that the signature can be extended (by continuity) to a locallyconstant map on the complement of τ(M) = 0 in T b1 , and at each primary points,the nullity is zero in this complement. See Section 3.2 and 3.3. Note that in the caseτ(M) 6= 0, this complement has codimension one by the symmetry of the torsion.

In Section 3.4, we show that both signatures and nullities are invariant by ho-mology cobordism. We also show that the signature vanishes for a special classof 3-manifolds, that we call bordant manifolds. This class, which is invariant byhomology cobordism, coincides in the case b1 = 1, with 0.5-solvable (rationally)manifolds in the sense of Cochran, Teichner and Orr [26].

In Section 4, we apply these constructions to the boundary of the complementof a smoothly embedded surface in B4 with boundary a given oriented link L andobtain isotopy invariants σL and ηL for links which admits a decomposition intoµ sublinks with linking number zero. They are maps TµP −→ Z, which restrictto Tristram’s signatures and nullities in the diagonal. Link signatures constructedin terms of finite cyclic covers of the complement of the link in S3 were alreadyconsidered in various contexts by Gilmer [17], Gordon and Litherland [33], Levine[3] and Smolinsky [34]. We show that σL can be extended to a locally constant mapon the complement of ∆L = 0 in Tµ∗ = (S1 − 1)µ, where ∆L(t1, . . . , tµ) is theAlexander polynomial associated to the coloring of L. We note that this signatureseems to be closely related to the signature that could be defined in terms of ageneralized Seifert matrix (for colored links), according to a C-complex [35, 36].For this, one may show that the generalized Seifert matrix evaluated at a primarypoint of Tµ is a matrix for the intersection form with twisted coefficients in thecyclotomic field of the complement of the C-complex pushed in B4. This is similar

5

to Viro’s construction [12]. More generally, the generalized Seifert matrix should beclosely related to the intersection form in Z[t±1

1 , . . . , t±1µ ] of this complement. Note

that Cimasoni and Cooper [35, 36] have already proved that the determinant of thisSeifert matrix is the Alexander polynomial of the colored link.

1.2.1. Colored concordance

In Section 4, the generalized signature and nullities are shown to be coloredconcordance invariants. The fact that the signature vanishes for slice links can berecovered as a particular case of the generalized Murasugi-Tristram inequality (seeSection 1.2.2.).

We now restrict ourselves to colored links with non-zero colored Alexander poly-nomial. In Section 5, we define geometrically bordant colored links as links bound-ing a smooth surface in B4 with Euler characteristic 1. In particular, geometricallybordant knots are slice knots. We show that geometrically bordant links are alge-braically bordant, in the sense that some manifold derived from the link complementis bordant, see previous section. This “algebraic” class of links has vanishing sig-natures. The Alexander polynomial of geometrically bordant links is of the formf · f for f ∈ Z[t1, . . . , tµ], see [15] and we suggest a way to show that it holds alsofor algebraically bordant links. In this way, algebraically bordant links are naturalgeneralization of algebraically slice knots, in the sense of [25].

In an other hand, one may define algebraically slice colored links as links forwhich any generalized Seifert matrix, according to a C-complex [35, 36], is meta-bolic. We conjecture that if L is geometrically bordant, then it is algebraically sliceand ask if, similarly to knots, this 3-dimensional point of view of sliceness is equiv-alent to the 4-dimensional (algebraic) point of view. Indeed, is L algebraically sliceif and only if it is algebraically bordant ?

1.2.2. Generalized Murasugi-Tristram inequality

In Section 5.2, we show that σL and ηL verify the Murasugi-Tristram inequality,see Theorem 1.1 above. This gives a new obstruction to the existence of a non-connected surface F1 t · · · t Fµ smoothly embedded in B4 with a given genus (orfirst Betti number) and boundary L. This result can be viewed as a specializationof a theorem of Gilmer ([17], Theorem 4.1): glue together two copies of B4 alongtheir boundary, gluing the surface F in the first copy to the cone on L on the secondcopy. One obtains the sphere S4 with a configuration of colored surfaces. ApplyingGilmer’s Theorem 4.1 to this pair, one obtains the result. We show that the objectsdiscussed there can be reinterpreted in ways that lead to significant simplificationsof Gilmer’s proof. In Section 6, we give an algorithm to compute σL and ηL and tocheck if the generalized Murasugi-Tristram inequality is satisfied for a given genus.Using this algorithm, we compute σL for a family of links coming from real algebraiccurves (see Section 1.3). For these examples, our result is stronger than the classicalMurasugi-Tristram inequality.

6

In conclusion, we mention recent results of Rudolph [37] and Ackbulut andMatveyev [38]. They also give necessary conditions for links to bound smooth givensurfaces, that use link invariants such as the Thurston-Bennequin invariant and arederived from the adjunction inequality. It happens that they are of independentinterest. For example these results can be used to show that the connected sum oftwo trefoils (2, 3) is not slice while the result of Fox and Milnor does not show it;conversely, the Murasugi-Tristram inequality or the theorem of Fox and Milnor canbe used to show that the trefoil (2,−3) is not slice, while the recent results do notshow it.

1.3. Real plane algebraic curves

In this section, we state our result on algebraic curves. All the references forprecise definitions and results are given in [13, 20].

A real plane non-singular projective algebraic curve A of degree m is an ho-mogeneous real polynomial of degree m, without singularities, in three variables,considered up to constant factors. If P is such a polynomial, then the equationP (x0, x1, x2) = 0, where the xi are real, defines the set RA of the real points ofthe curve. RA is a closed one-dimensional manifold homeomorphic to a family ofdisjoint embedded S1 in RP 2. For a long time, according to the 16th Hilbert’s prob-lem, the main question concerning the topology of real algebraic curves was how anonsingular curve of degree m can be arranged in RP 2, up to an isotopy. It canbe formulated as the determination of which isotopy types of embeddings of familyof disjoint S1 in RP 2 are realized by nonsingular plane real projective algebraiccurves of degree m. The first prohibitions come from the topological consequencesof Bezout’s theorem. In particular the Harnack inequality shows that the numberof components of RA is at most (m−1)(m−2)

2 + 1.At present this problem has been solved only for m ≤ 7. The complete list of

curves of degree 7, up to isotopy, was given by Viro. If we use his notation, theyare of the form 〈J〉, 〈J t1〈1〈1〉〉〉, and 〈J tβ〉t1〈α〉〉 where α+β ≤ 14 and α < 14.

Following Klein, one is also interested in how the isotopy type of a curve isconnected to the way the set RA of its real points is embedded to the set CAof its complex points (i.e. the set of points of the complex projective plane whosehomogeneous coordinates satisfy the equation defining the curve). CA is an orientedsmooth two-dimensional submanifold of the complex projective plane CP 2. It isinvariant under the antiholomorphic involution conj : (z0 : z1 : z2) → (z0 : z1 : z2).If the curve A is maximal (called also an M -curve) i.e. if the number of componentsof RA is exactly (m−1)(m−2)

2 + 1, then RA divides CA into two connected pieces.The natural orientations of these two halves determine two opposite orientationson RA as their common boundary; these orientations of RA are called the complexorientations. The scheme of mutual arrangements of the connected components ofRA enriched by the description of one of these complex orientations is called thecomplex scheme of the curve, following the terminology used by Viro.

7

The classification of complex schemes of maximal curves of degree m is solvedonly for m ≤ 6. In the case m = 7, after the work of Orevkov and Le Touze-Fiedler[39], there remains one case, namely 〈J t7+t3−t1−〈2+t2−〉〉, whose realizabilitywas still unknown.

In Section we show

Theorem 1.2. The complex scheme 〈J t 7+ t 3− t 1−〈2+ t 2−〉〉 is not realizableby a real algebraic maximal curve of degree 7.

Let Bn be the group of n-braids. A braid b in Bn is called quasipositive ifb =

∏mi=1 aiσ1a

−1i for some ai ∈ Bn. Orevkov observed that the quasipositivity of a

certain braid provides a necessary condition for the realizability of a given scheme,see [21]. Applying this approach, it is proved in [22] that Theorem 1.2 would followfrom the fact that none of the following 4-braids is quasipositive:

σ−31 σ−1

2 σ1σ−β12 σ−1

3 σ2σ−β23 σ−1

2 σ3σ−β32 σ−1

3 σ22σ

21σ2σ3σ1σ2σ1σ3σ2σ3σ2σ

21σ2σ1 in B4

with βi ≡ 1 mod 2, β1 + β2 + β3 = 9.

These are pure braids and we denote successively the components of their closureby L1, L2(1), L2(2), L2(3). Let L2 = L2(1) ∪ L2(2) ∪ L2(3). The quasipositivityof these braids implies (see [21]) the existence of a planar surface (i.e. of genus0) smoothly and properly embedded in B4 with two connected components andboundary L1 t L2. In order to prove that L does not bound such a surface inB4, Orevkov used the Murasugi-Tristram inequality (see Theorem 1.1 above) andexcluded all the possibilities for (β1, β2, β3) except:

(7, 1, 1), (3, 5, 1), (1, 5, 3), (1, 3, 5), (1, 1, 7).

By computing the generalized signatures and nullities for these five links, we showthe negative answer by Theorem 5.19. This proves Theorem 1.2 and achieves theclassification of complex scheme realizable by real algebraic maximal non singularcurves of degree 7. Note that the negative answer for three of these five links wereproved with unitary representations of braid group, see [22].

2. Milnor torsion and twisted homology

This section is devoted to classical constructions and results related to the Milnortorsion of a closed 3-manifold. For more details, see [30].

Let C∞ be a free infinite cyclic (multiplicative) group, with fixed generatort. For any positive integer r, the group ring Z[Cr∞] is identified with the Laurentpolynomial ring Λ = Z[t±1

1 , . . . , t±1r ]. Moreover, for the rest of this paper, polynomial

will mean Laurent polynomial, and the equality up to a unit is denoted .=.Let QΛ be the quotient field of Λ. By the inclusion, it is a Λ-module.Since Λ is commutative, left or right Λ-modules are not distinguished and this

allows the tensor product of Λ-modules to be modules as well.

8

2.1. Torsion for acyclic complexes

Let us consider the chain complex

C∗ = (Cm∂m−1−→ Cm−1

∂m−2−→ · · · ∂0−→ C0),

where each Ci is a finite dimensional vector space over QΛ and ∂i is a linear mapsatisfying ∂i ∂i+1 = 0 for all i = 0, . . . ,m − 2. Suppose also that C∗ is based, i.e.each Ci is equipped with a preferred basis, which will be denoted by ci.

Suppose that C∗ is acyclic, i.e. Hi(C∗) = 0 for all i. Let Bi = Im∂i. Since C∗ isacyclic Bi = Ker∂i−1. It follows that Ci/Bi ' Im∂i−1 = Bi−1. This is equivalentto the statement that the following sequence is exact:

0 −→ Bi −→ Ci −→ Bi−1 −→ 0.

For each i = 1, . . . ,m, choose a basis bi of Bi. Let bi−1 ⊂ Ci be a lift of bi−1 to Ci.From the above exact sequence it is clear that bi∪ bi−1 forms a basis for Ci. Denotethis basis by bibi−1 and the determinant of the transition matrix for the change ofbasis ci to bibi−1 by [bibi−1/ci]. One shows that the element [bibi−1/ci] is non-zeroin QΛ and does not depend on the choice of the lift bi−1 of bi−1.

Definition 2.1. The torsion of the acyclic based complex C∗ over Λ is defined as

τ(C∗) =m∏i=0

[bibi−1/ci](−1)i+1∈ QΛ∗.

The torsion τ(C∗) is independent of the choice of the bases bi.

2.2. Homological computation of the torsion

Let H be a finitely generated Λ-module. A presentation of H is an exact se-quence

Λm −→ Λn −→ H −→ 0,

where n is a positive integer and m is 0,∞ or a positive integer. The bases vectors inΛn determine a system of generators in H and each bases vector in Λm correspondsto a relation between these generators. A (m × n) matrix of the map Λm −→ Λn

with respect to the standard bases in Λm and Λn is called a presentation matrix ofH. The elementary ideal of H is the ideal of Λ generated by the (n × n)-minorsof a presentation matrix of H. Note that it does not depend on the choice of thepresentation matrix. The order of H is a generator of the smallest principal idealof Λ containing the elementary ideal. It is denoted ord H.

Let us now consider a non-acyclic chain complex C∗ over Λ, where each Ciis free and finitely generated Λ-module. Assume that C∗ is based and that rankH∗(C) = 0. Since H∗(C ⊗Λ QΛ) ' H∗(C) ⊗Λ QΛ, this hypothesis is equivalent todimQΛH∗(C)⊗Λ QΛ = 0 or to the fact that the complex C∗ ⊗Λ QΛ is acyclic overQΛ. Note that Λ is a Noetherian unique factorization domain and the followingformula for the torsion of C∗ ⊗Λ QΛ holds.

9

Lemma 2.1. [31] Theorem 4.7. If C∗ ⊗Λ QΛ is acyclic, then

τ(C∗ ⊗Λ QΛ) =m∏i=0

(ord Hi(C)

)(−1)i+1

.

2.3. Twisted chain complexes and homology

Let X be a finite CW-complex with b1(X) ≥ 1 and p : X −→ X be its universalcovering. Orient all open cells of X and the cells of X in such a way that the re-striction of p to each cell is orientation preserving. Let π = π1(X,x) for x ∈ X. Theaction of π on X by covering transformations induces an action of π on the cellularchain groups Ck(X). Extending this action by linearity to an action of Z[π], Ck(X)can be considered as a Z[π]-module. Note that the boundary homomorphisms arelinear over Z[π]. Choose a lift in X of each cell of X, where the cells of X areordered in an arbitrary way, to form a basis of C∗(X; Z). It follows that C∗(X; Z)is a free based chain complex over Z[π], where the Ci(X; Z) are finitely generated.

Let α : Z[π] −→ Λ be a ring homomorphism. The twisted homology of (X,α)with coefficient in Λ is the homology of the complex

Cα(X; Λ) = Λ⊗α C(X; Z).

In fact one may use the universal free abelian covering of X, instead of the universalcovering. Let G = H1(X; Z)/TorsH1(X; Z), where TorsH1(X; Z) is the subgroupof H1(X; Z) consisting of elements of finite order. Let h : Z[π] −→ Z[G] be theprojection and p : X −→ X be the covering induced by h. As above, endow X withthe CW-decomposition induced by X and the action of G on X induces an actionof Z[G] on C∗(X; Z).

Consider now a ring homomorphism κ : Z[G] −→ Λ and set α = κ h. There isa Λ-isomorphism

Cα(X; Λ) = Λ⊗α C(X; Z) = Λ⊗κ C(X; Z) = C(X; Z)⊗κ Λ.This equality allows us to compute the twisted homology of (X,α) directly from X

and κ. The chain complex Cα(X; Λ) is now denoted Cκ(X; Λ).

Definition 2.2. The twisted homology of (X,κ), denoted as Hκ∗ (X; Λ), is the

homology of the complex Cκ(X; Λ).

The twisted homology depends neither on the CW-structure on X nor on thechoice of the cell orientations.

Remark 2.3. If κ : G −→ Cr∞ is a group homomorphism, it extends uniquely intoa ring homomorphism Z[G] −→ Λ that is also denoted by κ. This convention iscontinued in the rest of the paper.

This construction can easily be extended to the case of pairs. If Y is a subcom-plex of X, then the cellular chain complex C∗(Y ) is clearly a subcomplex of C∗(X)

10

and by definition C∗(X,Y ) = C∗(X)/C∗(Y ). Obviously p−1(Y ) is a subcomplexof X. Since the action of Λ on X preserves p−1(Y ), one obtains an action of Λon C∗(X, p−1(Y )). In this way, the cellular chain complex C∗(X, p−1(Y )) is a freechain complex over Λ with a basis determined by a lift of the open oriented andordered cells of X−Y to X−p−1(Y ). The twisted homology Hκ

∗ (X,Y ; Λ) is definedas the homology of the complex

Cκ∗ (X,Y ; Λ) = C∗(X, p−1(Y ); Z)⊗κ Λ.

One can make similar constructions for the twisted cohomology.

Definition 2.3. The twisted cohomology of (X,κ), denoted as H∗κ(X; Λ), is the

homology of the complex

HomΛ(Cκ(X; Λ); Λ).

2.4. Generalities on pairs over Cr∞

Definition 2.4. A pair (X,κ) over Cr∞ is a compact smooth connected orientedmanifold with b1(X) ≥ 1 and a surjective group homomorphism κ : H1(X) −→ Cr∞.

By a theorem of Whitehead, X has a canonical piecewise-linear structure, uniqueto ambient isotopy. More precisely, it is endowed with a maximal family of pl-triangulations, where any two pl-triangulations have a common linear subdivisionwhich is pl. Endow X with the CW-decomposition induced by one of these pl-triangulations. Since X is compact, this CW-complex is finite.

Definition 2.5. The pair (Y, κ) is the boundary of the pair (X, κ) over Cr∞ if∂X = Y and the following diagram commutes:

H1(Y ) κ //

i1

Cr∞

H1(X)eκ

;;wwwwwwwww

where i1 is the map induced by the inclusion. Note that κ is necessarily surjective.In the rest of the paper, κ is also denoted by κ.

Geometrically, that means that the covering of X induced by κ has boundarythe covering of Y and the group action coincides with those given in the boundary.

Definition 2.6. Two pairs (M1, κ1) and (M2, κ2) are homology Cr∞-cobordant ifthere exist a pair (W,κ) such that

. ∂W = M1 t −M2

. H∗(W,Mi) = 0 for i = 1, 2.

. κ|H1(Mi) = κi for i = 1, 2.

11

2.5. Milnor torsion of a pair (M,κ)

Let (M,κ) be a pair over Cr∞ where M is 3-dimensional. The chain complexCκ∗ (M ;QΛ) over QΛ is based by construction, see previous section.

Definition 2.7. If the chain complex Cκ∗ (M ;QΛ) is acyclic, then the Milnor torsionof (M,κ) is defined as

τκ(M) = τ(Cκ∗ (M ;QΛ)).The torsion τκ(M) is a non-zero element of QΛ, well-defined up to sign and multi-plication by monomials.

One shows that τκ(M) is invariant under subdivision and then is independentof the choice of the triangulation. Following Section , it can be computed withhomological methods. The Alexander module of (M,κ) is defined as the Λ-moduleHκ

1 (M ; Λ), i.e. H1(C∗(M ; Z)), where C∗(M ; Z) is considered as a Λ-module. SinceΛ is a Noetherian ring, it is finitely generated.

Definition 2.8. The Alexander polynomial of (M,κ) is the order of the Λ-moduleHκ

1 (M ; Λ) and is denoted by ∆κ(M).

The Fox differential calculus [4] can be used to compute a presentation matrixof the Alexander module of (M,κ) (and thus computes ∆κ(M)) in terms of κ andof a presentation of the group π1(M).

Remark 2.4. The rational Alexander polynomial of a pair (M,κ) over C∞ is theorder of the Alexander module Hκ

1 (M ; Q[t±1]). Since Q[t±1] is a principal idealdomain, the Alexander module is principal and the polynomial is the determinantof a sub-matrix of the Fox matrix.

Theorem 2.5. [40] Suppose that M is a compact connected 3-manifold whoseboundary is non-empty and consists of tori. If r = 1, let t be a generator of C∞.Then,

τκ(M) =

∆κ(M)(t− 1)−1 if b1(M) = 1∆κ(M) if b1(M) ≥ 2.

Theorem 2.5 is due to Milnor but a detailed proof can also be found in [31],which uses more explicitly Lemma 2.1.

2.6. Determinant of intersection forms over Q[t±1]

Let (M3, κ) be a pair over C∞. In this section we show that the (rational)Alexander polynomial of (M3, κ) can be interpreted as the determinant of the in-tersection form in Hκ

2 (W ; Q[t±1]), for some (W,κ) with boundary (M,κ). It followsfrom the fact that this intersection form provides a matrix presentation of theAlexander module.

12

Note that the twisted homology Q[t±1]-modules Hκ∗ (M ; Q[t±1]) are isomorphic

to H∗(M∞; Q) considered as Q[t±1]-modules, where M∞ −→ M is the infinitecyclic covering induced by κ.

Definition 2.9. The intersection form ψκ of a pair (W 4, κ) over C∞ is the bilinearform, hermitian for the involution ·

ψκ : Hκ2 (W ; Q[t±1])×Hκ

2 (W ; Q[t±1]) −→ Q[t±1]

so that ψκ(x, y) =∑i∈Z

< x, ti · y > t−i,

where <,> is the ordinary intersection pairing.

It is easy to prove that for each given (x ⊗ f, y ⊗ g), the sum is finite and ψκ

takes values in Q[t±1].

Proposition 2.1. Any pair (M3, κ) over C∞ bounds a pair (W 4, κ) such thatHκ

1 (W ; Q[t±1]) = 0.

Proof. Since the bordism group Ω3(B(C∞)) = 0, there exist a pair (W ′, κ′)with boundary (M,κ). In the next paragraph, we will show that one may performa finite number of surgeries of index 2 on (W ′, κ′) to obtain a new pair (W,κ) withthe same boundary and π1(W ) = 1. A surgery of index 2 is the following process:

W ′ − (c×B3) ∪c×S2 B2 × S2

for c an S1 embedded in W ′. It “kills” the corresponding homotopy class of c inW . For details on the surgery on 4-dimensional manifolds see [44].

Consider a differentiable map f : W ′ −→ S1 which induces κ : H1(W ′) −→ Z =H1(S1) and choose a regular value p for which f−1(p) = Σ is a connected threedimensional sub-manifold of W ′, with boundary S a connected surface embeddedin M . Since ambient dimension is greater than three we can make surgeries oncurves disjoint from Σ. We then get a 5-dimensional cobordism U with boundaryW ′ ∐−W such that ∂W ′ = ∂W = M . Moreover, f extends to a mapping F fromU where F−1(p) = Σ× [0; 1] is a regular fiber and W −Σ×1 is simply connected.This implies that π1(W ) is cyclic and κ′ gives an isomorphism of π1(W ) with Z(or C∞). Obviously the infinite cyclic covering W∞ −→ W induced by κ is theuniversal covering of W . Hence π1(W∞) = 1 and Hκ

1 (W ; Q[t±1]) = 0.

Lemma 2.2. Let (W,κ) with boundary (M,κ) be given by Proposition 2.1. ThenHκ

2 (W ; Q[t±1]) is a free Q[t±1]-module and

detψκ .= ∆κ(M).

Proof. We show that a Gramm matrix of ψκ is a presentation matrix of theQ[t±1] Alexander module of (M,κ). Let W ′ be W with the dual cell structure andsubcomplex M ′. Following Milnor [40] (see also [9]), one defines sesqui-linear forms,for all i = 0, . . . , 4:

13

Cκi (W ; Q[t±1])× Cκ4−i(W′,M ′; Q[t±1]) −→ Q[t±1]

(c, c′) 7−→∑i∈Z

< c, ti · c′ > t−i,

They induce Q[t±1]-isomorphisms

Cκ4−i(W′,M ′; Q[t±1]) −→ HomQ[t±1](Cκi (W ; Q[t±1]); Q[t±1]).

These isomorphisms take the differential of Cκ∗ (W ′,M ′; Q[t±1]) to the dual of thedifferential of Cκ∗ (W ; Q[t±1]) and induce in particular a Poincare duality isomor-phism

Hκ2 (W,M ; Q[t±1]) '−→ H2

κ(W ; Q[t±1]).

The universal coefficient theorem applied to the chain complex Cκ∗ (W ′,M ′; Q[t±1])implies that evaluation induces an isomorphism, since Hκ

1 (W ; Q[t±1]) = 0,

H2κ(W ; Q[t±1]) ' HomQ[t±1](Hκ

2 (W ; Q[t±1]),Q[t±1]). (∗)By similar arguments, since Hκ

1 (W,M ; Q[t±1]) = 0, one shows that

Hκ2 (W ; Q[t±1]) ' HomQ[t±1](Hκ

2 (W,M ; Q[t±1]),Q[t±1]).

Since Q[t±1] is a principal ideal domain, it follows that Hκ2 (W ; Q[t±1]) is a free

Q[t±1]-module.Moreover, following (∗), the pairing induces an unimodular pairing

Hκ2 (W ; Q[t±1])×Hκ

2 (W,M ; Q[t±1]) −→ Q[t±1].

To compute the determinant of i2 : Hκ2 (W ; Q[t±1]) −→ Hκ

2 (W,M ; Q[t±1]) in-duced by the inclusion, consider the following diagram with coefficients in Q[t±1]:

Hκ2 (W )×Hκ

2 (W )Id×i2 //

ψκ

''OOOOOOOOOOOOHκ

2 (W )×Hκ2 (W,M)

vvmmmmmmmmmmmmm

vvmmmmmmmmmmmmm

Q[t±1]

The diagonal map on the right is just the unimodular pairing considered above. Itfollows that

detψκ .= det i2.

By the exact sequence on homology with twisted coefficients in Q[t±1] of the pair(W,M), since Hκ

1 (W ; Q[t±1]) = 0 and i0 is an isomorphism, we get

Hκ2 (W ) i2−→ Hκ

2 (W,M) −→ Hκ1 (M) −→ 0.

It follows that any matrix of i2 is a presentation of the Alexander Q[t±1]-module of(M,κ) and

det i2.= ord Hκ

1 (M) .= ∆κ(M).

14

3. Signature and nullity invariants

3.1. Casson-Gordon σ-invariant

In this section, we review the construction of a Casson and Gordon [7, 8] invari-ant for a pair (M, ζ) over the finite cyclic group Cq. It is a reformulation of theAtiyah-Singer α-invariant. We also recall Gilmer’s construction [17] to compute itin terms of a surgery presentation of (M, ζ).

LetM be an oriented compact three manifold and ζ : H1(M) −→ C∗ be a charac-ter of finite order. Let q, a positive integer, be such that the image of ζ is a cyclic sub-group of order q, generated by α = e2iπ/q. Since Hom(H1(M),Cq) = [M,B(Cq)],ζ induces q-fold covering Mq −→ M , with a choosen deck transformation denotedalso by α. As ζ maps onto Cq, the choosen deck transformation sends x to the otherendpoint of the arc that begins at x and covers a loop representing an element of(ζ)−1(α).

The bordism group Ω3(B(Cq)) = Cq, and for some integer n, n disjoint copiesof M bound a compact 4-manifold W over B(Cq). Note n can be taken to be q.Let W q be the induced covering with the deck transformation, denoted also by α,that restricts to α on the boundary. This induces a Z[Cq]- module structure onC∗(W q; Q), where the multiplication by α ∈ Z[Cq] corresponds to the action of αon W q. The cyclotomic field Q(α) is a natural Z[Cq]-module.

Definition 3.10. Let Hζ∗ (W ; Q(α)) be the twisted homology of (W, ζ), i.e the

homology of the complexC∗(W q; Z)⊗Z[Cq ] Q(α).

We define similarly the twisted homology of (M, ζ).

Note that we have an isomorphism

Hζ∗ (W ; Q(α)) ' H∗(W q; Q)⊗Q[Cq ] Q(α).

Let a→ a denotes the involution on Q(α) induced by complex conjugation.

Definition 3.11. The twisted intersection form of (W, ζ) is defined as

φζ : Hζ2 (W ; Q(α))×Hζ

2 (W ; Q(α)) −→ Q(α)

so that, for all a, b in Q(α) and x, y in H2(W q; Z),

φζ(x⊗ a, y ⊗ b) = ab

q∑i=1

< x,αiy > αi.

Definition 3.12. The Casson-Gordon σ-invariant of (M, ζ) and the related nullityare

σ(M, ζ) =1n

(Sign (φζ)− Sign (W )

)

15

η(M, ζ) = dim Hζ1 (M ; Q(α)).

Let U be a closed 4-manifold, ζ : H1(U) −→ Cq and φζ be defined as above.Modulo torsion the bordism group Ω4(B(Cq)) is generated by the constant map fromCP (2) to B(Cq). If ζ is trivial, then Sign (φζ) = Sign (U). Since both signaturesare invariant under cobordism, one has Sign (φζ) = Sign (U). The independence ofσ(M, ζ) from the choice of W and n follows from this and Novikov additivity. Onemay see directly that these invariants do not depend on the choice of q. In this wayCasson and Gordon argued that σ(M, ζ) is an invariant. Alternatively one may usethe Atiyah-Singer G-Signature [11] theorem and Novikov additivity.

The following Lemma 3.3 will be very useful in the following section. It is anatural generalization of Theorem 3.4 [28]. Let βζi denotes the ith Betti numberwith twisted coefficient in Q(α) and Kζ be Hζ

2 (W ; Q(α))/(Radical φζ).

Lemma 3.3. Suppose that (W, ζ) is a pair over Cq with b boundary components(Mi, ζi), where Mi is connected and ζi is non-zero for all i. Then the followingholds

dim Kζ = −βζ3(W ) + βζ1(W ) + βζ2(W )−b∑i=1

η(Mi, ζi).

In particular, if (W, ζ) has connected boundary (M, ζ) and satisfies βζ3(W ) = 0,then

η(M, ζ) = null(φζ) + βζ1(W ),and η(M, ζ) = 0 implies that φζ is non-degenerate.

The proof of Lemma 3.3 follows from the exact sequence in twisted homologyof the pair (W,ti=1,...,bMi).

We now describe a way to compute σ(M, ζ) in terms of a surgery presentationof (M, ζ).

Definition 3.13. Let K be an oriented knot in S3. Let A be an embedded annulussuch that ∂A = K tK ′ with lk(K,K ′) = f . A p-cable on K with twist f is definedas the union of oriented parallel copies of K lying in A such that the number ofcopies with the same orientation minus the number with opposite orientation isequal to p.

Suppose that M is obtained by surgery on a link L1 ∪ · · · ∪ Lν with framingsf1, . . . , fν .

Remark 3.6. The linking matrix Λ with framings in the diagonal is a presentationmatrix of H1(M), where the generators are the meridians of the components. Forthis, consider the 4-manifold W with one 0-handle and only 2-handles attachedalong the components of L, with respect to the framing. We have then ∂W =M and H1(W ) = 0. It follows that the intersection form with matrix Λ is apresentation matrix of H1(M). Note that Lemma 2.2 is an analogous result, withtwisted coefficients.

16

Let ω = e2iπr/q with (r, q) = 1 and suppose that ζ(mi) = ωpi for a given~p = (p1, . . . , pν) in Zν with gcd(p1, . . . , pν) = 1. The following proposition is ageneralization of a formula given in [8] Lemma 3.1, where all pi are assumed to be1, that is given in [17] Theorem 3.6.

Proposition 3.2. Let L′ be the link with ν′ components obtained from L by re-placing each component by a non-empty algebraic pi-cable with twist fi along thiscomponent. One has

σ(M, ζ) = σL′(ω)− Sign (Λ) + 2r(q − r)q2

~pTΛ~p,

η(M, ζ) = ηL′(ω)− ν′ + ν.

3.2. Signature of a pair (M,κ)

Let (M3, κ) be a pair over Cr∞ and ∆κ(M) be the Alexander polynomial of(M3, κ). Let T r = S1 × · · · × S1 ⊂ Cr. In this section we construct the signatureof (M3, κ) and show that it can be extended to a locally constant map in thecomplement of ∆κ(M) = 0 in T r.

Definition 3.14. The set T rP of primary points of T r is the set of elements ofthe form ω~p = (ωp1 , . . . , ωpr ) where ω is a primitive root of prime power order q,~p = (p1, . . . , pr) ∈ Zr with gcd(p1, . . . , pr) = 1 and gcd(pi, q) = 1. In the case r = 1,we simply denote T 1

P by S1P .

Note that T rP is dense in T r.

Proposition 3.3. If ∆κ(M) is not identically zero, its zero set in T r is a realalgebraic hypersurface of codimension 1.

The proof follows since ∆κ(M) has real coefficients and symmetry, i.e. ∆κ(M) =∆κ(M), see [30].

For any primary point ω~p in T rP , we denote the projection

sω~p : Cr∞ −→ C∗

tk 7−→ ωpk .

and κω~p = sω~p κ, for a pair (M,κ) over Cr∞.See Definition 3.12 for the construction of σ(M,κω~p).

Theorem 3.7. Let (M,κ) be a pair over Cr∞. The map

σκ(M) : T rP −→ Z

ω~p 7−→ σ(M,κω~p)

can be extended to a locally constant map on the complement of ∆κ(M) = 0 inT r .

17

The rest of the section is devoted to the proof of Theorem 3.7. In Paragraph ,we prove it in the case of pairs over C∞ (i.e. r = 1). In Paragraph , we prove it inthe general case.

3.2.1. Proof of Theorem 3.7 when r = 1.

We suppose in this paragraph that κ : H1(M) −→ C∞. For any ω ∈ S1P , let

κω = sω κ, where sω sends t to ω.For any λ in S1, Q(λ) is a Q[t±1]-module by the map t 7−→ λ.

Definition 3.15. For all λ ∈ S1, let Hκ∗ (W ; Q(λ)) be the homology of the complex

Cκ∗ (W ; Q(λ))def= C∗(W ; Q)⊗κλ

Q(λ) ' C∗(W∞)⊗sλQ(λ),

where W∞ −→W is the covering induced by κ.

Propositions 3.4 and 3.5 below were suggested by the proof of Theorem 3 in [7].For any ω ∈ S1

P of order q, we denote the homology of (W,κω) by Hω2 (W ; Q(α)),

where α = e2iπ/q, see Definition 3.10.

Proposition 3.4. If Hκ1 (W ; Q[t±1]) = 0, then for any ω of order q in S1

P there isan isomorphism of Q(α)-vector spaces

Hκ2 (W ; Q(α)) ' Hω

2 (W ; Q(α)).

Proof. Let W −→ W be the universal free abelian covering of W , see Section. We have the following isomorphism of Q[t±1]-modules:

Cκ∗ (W ; Q[t±1]) def= C∗(W ; Q)⊗κ Q[t±1]) ' C∗(W∞; Q).

Following Milnor, there is an exact sequence of Q[t±1]-modules associated to theinfinite cyclic covering W∞ −→W q:

Hκ2 (W ; Q[t±1]) t

q−1−→ Hκ2 (W ; Q[t±1]) −→ H2(W q; Q) −→ Hκ

1 (W ; Q[t±1]) = 0.

Hence, we have an isomorphism of Q[t±1]-modules

H2(W q; Q) ' coker (tq − 1) ' Hκ2 (W ; Q[t±1])⊗sω Q[Cq].

We obtain

Hω2 (W ; Q(α)) ' H2(W q; Q)⊗Q[Cq ] Q(α) ' Hκ

2 (W ; Q[t±1])⊗sωQ(α).

Let ψκ be the intersection form on Hκ2 (W ; Q[t±1]). Recall that, by the universal

coefficient theorem, if Hκ1 (W ; Q[t±1]) = 0, then Hκλ

2 (W ; Q(λ)) ' Hκ2 (W ; Q[t±1])⊗

Q(λ).

18

Definition 3.16. Suppose that Hκ1 (W ; Q[t±1]) = 0. For all λ ∈ S1, let ψλ be the

intersection form

ψλ : Hκ2 (W ; Q(λ))×Hκ

2 (W ; Q(λ)) −→ Q(λ)

constructed from ψκ by tensoring with Q(λ).

Let I(t) be a Gramm matrix of ψκ with respect to some basis in Hκ2 (W ; Q[t±1]).

By Lemma 2.2,Hκ2 (W ; Q[t±1]) is a free Q[t±1]-module and dim Q(λ)H

κ2 (W ; Q[t±1])⊗

Q(λ) = dim Q[t±1]Hκ2 (W ; Q[t±1]). Hence I(λ) is a Gramm matrix of ψλ with re-

spect to the image of the chosen basis in Hκ2 (W ; Q[t±1]).

For any ω ∈ S1P , we denote the intersection form of (W,κω) by ψω, see Definition

3.11.

Proposition 3.5. For any ω in S1P , the following equality holds

ψω = φω.

In particular, I(ω) is a Gramm matrix for φω.

Proof. Let us consider two elements of Hω2 (W ; Q(α)). Up to the isomorphism

of Proposition 3.4 they are sums of terms on the form (x ⊗ ωj , y ⊗ ωk) wherex, y ∈ Hκ

2 (W ; Q[t, t−1]). Let p : W∞ −→W q be the projection. We have

ψω(x⊗ ωj , y ⊗ ωk)

= ωk−jψω(x, y

)= ωk−j

∑i∈Z

< x, tiy > ω−i

and we get the result by:∑i∈Z

< x, tiy > ω−i =∑

i=0...q−1

< p(x), p(tiy) > ω−i

=∑

i=0...q−1

< p(x), ωip(y) > ω−i.

The general case is easy to deduce from this.We now give the proof of Theorem 3.7 in the case r = 1.

Let (W,κ) with boundary (M,κ) and Hκ1 (W ; Q[t±1]) = 0 be given by Proposition

2.1. The map σκ(M) is defined as

S1P −→ Z

ω 7−→ Sign (φω)− Sign (W ),

where φω is the intersection form of (W,κω), see Definition 3.12. By Proposi-tion 3.5, for all ω ∈ S1

P , one has ψω = φω, where ψω is the intersection form of(W,κ) evaluated at ω, see Definition 3.16. Hence, λ 7−→ Sign (ψλ) − Sign (W )extends σκ(M) to S1. Moreover, by Lemma 2.2, det(ψκ) .= ∆κ(M). It follows thatdet(I(λ)) = ∆κ(M)(λ) for all λ ∈ S1, where I(λ) is a Gramm matrix for ψλ. Since

19

the entries of I(λ) are continuous (as polynomial maps) in λ, for λ 6= 1, the resultfollows.

3.2.2. Proof of Theorem 3.7 when r > 1.

Let ~p = (p1, . . . , pr) be in Zr with gcd(p1, . . . , pr) = 1. Consider the map

s~p : Cr∞ −→ C∞

tk −→ tpk ,

and define κ~p = s~p κ. Since the pi are co-prime, κ~p is surjective.Let Ω be a connected component of the complement of ∆κ(M) = 0 in T r and

ΩP = T rP ∩Ω be the primary points of Ω. Consider the curve (embedded circle) S~pin T r, defined as

S~p = (λp1 , . . . , λpr );λ ∈ S1 ⊂ T r.Following Theorem 2.5 above and [31] Theorem 13.3 and Remark 13.6, the Alexan-der polynomial of (M,κ~p) is given by ∆κ(M)(tp1 , . . . , tpµ). It follows that there isa bijective correspondence between ∆κ~p

(M) = 0 and ∆κ(M) = 0 ∩ S~p. More-over, the restriction σκ(M)|S~p

coincides with σκ~p(M) at each corresponding pair of

primary points. By Theorem 3.7 in the case r = 1, it follows that for any connectedcomponent C of S~p ∩ Ω, the restriction σκ(M)|C∩ΩP is constant. The conclusionfollows from the following purely topological lemma.

Lemma 3.4. Let Ω be an open connected subset of T r and ΩP = T rP ∩ Ω be theprimary points of Ω. Let σ be a map ΩP −→ Z.

If for any ~p ∈ Zr with gcd(p1, . . . , pr) = 1 and for any connected component Cof S~p ∩ Ω, the restriction σ|C∩ΩP is constant, then σ is constant on ΩP .

Proof. The proof is done the case r = 2, since it can be easily generalized.Let us consider the torus T 2

∗ as R2/Z2. The curves S~p are the projections ofprimary lines in R2. Note that each primary line contains a point with integercoordinates.

Consider now the problem in R2 and Ω as an open connected set of R2. Let x, ybe two primary points in Ω. Since Ω is connected there exists a path c : [0; 1] −→ Ωsuch that c(0) = x and c(1) = y. Let Bii be a covering by balls of c([0; 1]) in Ω(note that a ball is convex). By compactness, there exist a covering of c([0; 1]) bya finite subset B1, . . . , Bn of Bi Let us prove that σ is locally constant on theset of primary points of each ball of Ω. This will implies that σ(x) = σ(y) and σ isconstant on ΩP .

Let p be a point of Ω and r ∈ R∗+ such that B(p, r) ⊂ Ω. We prove that σ islocally constant around each primary point a ∈ B(p, r). For this, we prove thatfor any primary point m ’close’ to a, there exists a path joining a to m in B(p, r)constituted by two segments of primary lines. Then, by hypothesis, σ(m) = σ(a)and σ is constant on a neighborhood of a.

20

Let D(0,0),a be the line which passes through (0, 0) and a. Let D(0,1),m be theline which pass through (0, 1) and m. Let us consider the following map F :

F : B(p, r) −→ R2

m 7−→ D(0,0),a

⋂D(0,1),m

It is continuous as a rational function on the coefficients of m. Note that F (a) = a.The two lines D(0,0),a and D(0,1),m are primary lines. We prove that for any m closeto a, F (m) is in B(p, r). Let s = r − ‖a− p‖. By continuity of F , there exist ε ≥ 0such that ‖m − a‖ ≤ ε =⇒ ‖F (m) − a‖ ≤ s. Then, for all m in the ball B(a, ε),‖F (m)−p‖ ≤ ‖F (m)−a‖+‖a−p‖ ≤ s+‖a−p‖ = r. Then, [m;F (m)]∪ [F (m); a]is a polygonal path from a to m in B(p, r) whose segments lie on primary lines.

3.3. Nullity of a pair (M,κ)

Let (M,κ) be a pair over Cr∞ with Alexander polynomial ∆κ(M), as in theprevious section.

Definition 3.17. The nullity of (M,κ) is the map defined as

ηκ(M) : T rP −→ Z

ω~p 7−→ η(M,κω~p).

Theorem 3.8. For all ω~p in T rP ,

∆κ(M)(ω~p) = 0 if and only if ηκ(M)(ω~p) 6= 0

Proof. Let ω~p be a fixed point of T rP , such that ω has (prime power) order q.By Propositions 2.1 and 2.2, the pair (M,κ~p) bounds a pair (W,κ~p) over C∞ suchthat

detψκ~p = ∆κ(M)(tp1 , . . . , tpµ),

where ψκ~p is the intersection form on Hκ~p

2 (W ; Q[t±1]). Let φω~p

be the twisted in-tersection form of (W,κω~p), see Definition 3.11. By Definition 3.16 and Proposition3.5, if I(t) is a Gramm matrix for ψκ~p , then I(ω) is a Gramm matrix for φω

~p

. Inparticular, we have

detφω~p

= ∆κ(M)(ω~p). (∗)Since H1(W ; Z) = Z and gcd(p1, . . . , pµ) = 1 (the covering is connected), by [17]Proposition 1.5, one shows

Hω~p

1 (W ; Q(α)) = 0.

Lemma 3.3 implies that null (φω~p

) = η(M,κω~p). Hence,

detφω~p

= 0 if and only if η(M,κω~p) > 0

and the result follows from (∗).

21

3.4. Homology Cr∞-cobordisms

Theorem 3.9. If two pairs (M1, κ1) and (M2, κ2) are Cr∞-homology cobordant,then for all ω~p ∈ T rP ,

σκ1(M1)(ω~p) = σκ2(M2)(ω~p),

and ηκ1(M1)(ω~p) = ηκ2(M2)(ω~p).

The first part of Theorem 3.9 follows from [45] Theorem 9.Proof. For the reader convenience, we give a complete proof, as a natural gener-

alization of the proof of Theorem 3.8 of [28]. Let (W,κ) with boundary components(Mi, κi) be the homology cobordism. Note that χ(W ) = 0. Let ω~p be in T rP andφω

~p

be the intersection form of (W,κω~p). Since Sign (W ) = 0, we have

σ(M1, κ1ω~p

)− σ(M2, κ2ω~p

) = Sign φω~p

.

Let βω~p

i = dim Hω~p

i (W ; Q(α)). Since χ(W ) = χκω~p (W ) = 0, we have

βω~p

1 = βω~p

2 − βω~p

3 .

By Lemma 3.3, we obtain dim Kω~p

= −η(M1, κ1ω~p

)−η(M2, κ2ω~p

)+2βω~p

1 . Moreover,

by Proposition 1.5 of [17], we have Hω~p

1 (W,Mi; Q(α))) = 0 for i = 1, 2. Hence,from the exact sequence in homology with twisted coefficients in Q(α) of the pairs(W,Mi), one obtains

βω~p

1 ≤ η(Mi, κω~p) for i = 1, 2.

This gives dim Kω~p

= 0 and Sign φω~p

= 0. Furthermore,

βω~p

1 = η(M1, κ1ω~p

) = η(M2, κ2ω~p

).

3.5. Pairs (M,κ) with vanishing signature

3.5.1. The case of pairs over C∞

A metabolizer is a totally isotropic module direct summand of half rank.

Definition 3.18. A pair (M,κ) over C∞ is C∞-bordant if:• there exists a pair (W,κ) where W is spin, with boundary (M,κ).• the intersection form on Hκ

2 (W ; Q[t±1]) has a metabolizer whose image is ametabolizer for the form in H2(W ).

Definition 3.18 is a rational version of 0.5-solvability, see [26] Theorem 8.4. Theproperty of being C∞-bordant is invariant by homology C∞-cobordism, see [26]remark 8.6.

22

Theorem 3.10. If (M,κ) with τκ(M) 6= 0 is C∞-bordant, then the following holds:• There exists f ∈ Q[t±1] such that τκ(M) .= f · f .• σκ(M) = 0.

Proof. Let (W,κ) with boundary (M,κ) be given in Definition 3.18. Let (W ′, κ)with boundary (M,κ) be obtained by surgeries on (W,κ) such thatHκ

1 (W,Q[t±1]) =0, see Proposition 2.1. Let ψκ be the intersection form on Hκ

2 (W ′; Q[t±1]). Byhypothesis, ψκ is metabolic.

Following Proposition 2.2, detψκ .= τκ(M). Hence there exists f ∈ Q[t±1] suchthat τκ(M) .= f · f . Let I(t) be a Gramm matrix for ψκ . From Proposition 3.5,for any ω ∈ S1

P , σκ(M)(ω) = Sign I(ω) − Sign (W ). It follows that the signaturevanishes.

3.5.2. The case of pairs over Cr∞

Definition 3.19. A pair (M,κ) over Cr∞ is Cr∞-bordant if for all ~p in Zr withgcd(p1, . . . , pr) = 1 and τκ(M)(tp1 , . . . , tpµ) 6= 0, the pair (M,κ~p) is C∞-bordant.

Note that by the hypothesis gcd(p1, . . . , pr) = 1, the corresponding infinite cycliccoverings are connected.

The property of being Cr∞-bordant is invariant by homology Cr∞-cobordism.

Theorem 3.11. If (M,κ) with τκ(M) 6= 0 is Cr∞-bordant, then σκ(M) = 0.

Proof. By Theorem 3.11, the restriction of σκ(M) to all the curves S~p vanishes.Hence, σM (κ) vanishes.

Remark 3.12. If the pair (M,κ) over Cr∞, with τκ(M) 6= 0 satisfies the hypothesisof Theorem 3.11, then by Theorem 3.10, for all ~p in Zr with gcd(p1, . . . , pr) = 1,there exists f ∈ Q[t±1] such that

τκ(M)(tp1 , . . . , tpµ) .= f · f.

That suggests that τκ(M) .= g · g for some g ∈ Q[t±11 , . . . , t±1

r ], but we have notproved this.

4. Link invariants

4.1. Colored links and concordance

A µ-colored link is an oriented link L with ν components in S3 together with asurjective map assigning to each component a color in 1, . . . , µ.

For k = 1, . . . , µ, let Lk be the sublinks of L corresponding to all the componentsof L with the same color k.

Definition 4.20. A µ-colored link L is algebraically split if lk(Lk, Lk′) = 0 for allk 6= k′.

23

Definition 4.21. A µ-colored link L bounds a surface F = F1 t · · · t Fµ with µ

connected components if ∂Fk = Lk for all k = 1, . . . , µ.

Note that if a µ-colored link L bounds a surface F smoothly embedded in B4,then it is algebraically split, since the intersection number Fk · Fk′ vanishes inB4. Conversely, any algebraically split µ-colored link bounds a smoothly embeddedsurface in B4 (with arbitrarily high genus).

Definition 4.22. Two µ-colored links L and L′ with ν components are µ-concordantif there exist a smooth, oriented submanifold T = T1 t · · · t Tν of S3 × I, home-omorphic to a disjoint union of ν copies of S1 × I such that for all i = 1, . . . , ν,Ti ∩S3 × 0 and Ti ∩S3 × 1 are respectively a component of L and L′ with the samecolor.

The µ-concordance is an equivalence relation for µ-colored links.

4.2. Exteriors of surfaces in B4

Let L be a µ-colored algebraically split link with ν components. Let F with µ

connected components be a smoothly and properly embedded surface in B4 withboundary L, see previous section.

Let WF = B4−T (F ) be the exterior of an open tubular neighborhood of F . Bythe Thom isomorphism, excision, and the long exact sequence of the pair (B4,WF ),the homology of WF depends only on those of F ; in particular, H1(WF ) = Zµ isgenerated by the meridians mk for k = 1, . . . , µ of the connected components of F .Let us consider the pair (WF , κ) where κ is the isomorphism:

κ : H1(WF ) −→ Cµ∞

mk 7−→ tk.

Let (MF , κ) be the boundary of (WF , κ). If there are no ambiguity about the surfaceF , we simply denote the pairs as (W,κ) and (M,κ).

Remark 4.13. Let E = S3−T (L) be the exterior of an open tubular neighborhood ofL in S3. Consider a Seifert surface for each sublink of L, and let γi for i = 1, . . . , νbe the curves where it intersects the boundary of E. The manifold MF can bedescribed as the gluing of E with F ×S1 along their boundary, where F ×1 is gluedalong the curves γi. In particular, MF does not depend on the embbeding of Fin B4. By a similar construction to [16] Proposition 3.13, one may use a surgerypresentation of MF to describe (MF , κ) without using WF .

4.3. Alexander polynomial of links

In this section, we recall the definition of the Alexander polynomial of a µ-coloredalgebraically split link (with ν components). We show that it coincides with thetorsion of the boundary of the complement of a surface in B4, see previous section.

24

The group H1(E) is free abelian of rank ν, generated by the meridians of thecomponents of L. Let us consider the homomorphism κ : H1(E) −→ Cµ∞ whichsends the meridians of the components of the same sublink Lk to the generator tkin Cµ∞.

Definition 4.23. The Alexander polynomial of the µ-colored link L is the elementof Λ:

∆L(t1, . . . , tµ) = ∆κ(E).

Up to the multiplication by a unit in Λ, it is invariant under an isotopy of L.

By Theorem 2.5, ∆L is also the torsion of (E, κ), up to a factor t− 1 in the caseof a knot.

Proposition 4.6. The torsion of the pair (MF , κ) is given by the formula

τκ(M) =

(t− 1)−2 ·∆L(t) if ν = µ = 1∏µk=1(tk − 1)−χ(Fk) ·∆L(t1, . . . , tµ).

Proof. By identifying properly F × S1 in MF , κ induces ρk : H1(Fk × S1) −→Cµ∞ and ε : H1(E) −→ Cµ∞. We first compute separately the torsions of (Fk×S1, ρk)and (E, ε).

Obviously ε sends all the meridians of the same sublink Lk to tk. By Theorem2.5, the torsion of (E, ε) is related to the Alexander polynomial as follows:

If µ = 1, we set t = t1.

τε(E) =

∆L(t− 1)−1 if ν = µ = 1∆L if ν ≥ 2.

We now compute τρk(Fk × S1). Note that H1(Fk × S1) = H1(Fk)⊕ < mk > and

ρk sends mk to tk. The Alexander polynomial ∆ρk(Fk × S1) can be computed by

using the Fox differential calculus. If Fk is a disk, then π1(Fk × S1) = C∞ and∆ρk

(Fk×S1) = 1. If Fk is not a disk, then the group π1(Fk×S1) can be presentedby b1(Fk) + 1 generators a1, . . . , ab1(Fk), mk with the relations aimk = mkai for alli = 1, . . . , b1(Fk). The corresponding presentation matrix of the Alexander moduleHρk

1 (Fk×S1; Λ) has b1(Fk)+1 columns and b1(Fk) of them have only one non-zeroentry tk − 1. Thus,

∆ρk(Fk × S1) = (tk − 1)b1(Fk)−1 = (tk − 1)−χ(Fk) for all k = 1, . . . , µ.

By the multiplicativity of the torsion [32], τκ(M) is the product of the torsions of thepairs (E, ε) and (Fk × S1, ρk), and the result follows. Let i1 : H1(E) −→ H1(MF )and i2 : H1(F ×S1) ' ⊕µk=1H1(Fk×S1) −→ H1(MF ) be induced by the inclusions.Thus, κ i1 is defined on H1(E) and sends all the meridians of the same sublink Lkto tk in Cµ∞. By Theorem 2.5, the torsion of (E, κ i1) is ∆L(t1, . . . , tµ) if ν > 1and (t− 1)∆L(t) if ν = µ = 1, i.e if L is a knot.

25

Consider the morphism

κ i2 : ⊕µk=1H1(Fk × S1) = ⊕µk=1H1(Fk)⊕(⊕µk=1 < mk >

)−→ Cµ∞

mk 7−→ tk.

Note that ψ i2 is the direct sum of the characters H1(Fk × S1) −→ C∞ withmk 7−→ tk. The result follows from the multiplicativity applied to the pairs (E, ε)and (Fk × S1, ρk).

4.4. Link signatures and nullities

Let (MF , κ) be defined as in Section .

Definition 4.24. The signature and nullity of the µ-colored algebraically split linkL are defined as

σL = σκ(MF ),

ηL = ηκ(MF ).

They are integer valued functions with domain T µP .

For short the twisted homology of (., κω~p) is denoted by Hω~p

∗ (.; Q(α)).Proof. We show that σL and ηL do not depend on the surface F . By Remark

4.13, they do not depend on the embedding of F . Hence, we have to show that theydo not depend on the choice of the abstract surface.

First consider the case of the signatures. Let F0 be a planar abstract surfacewith µ connected components and ν boundary components. The boundary of F0

is a collection of ν abstract colored circles, where the coloring is induced by theconnected components of F0. Suppose that F0 is chosen in such a way that thiscoloring agrees with those of L. Let MF0 be constructed by gluing F0 × S1 to E,in a similar way to Remark 4.13.

The following construction is similar to those of [16, 18]. Let ω~p be a fixedelement of T µP . If we identify properly F × S1 in MF , κω~p induces a characteron H1(F × S1) which maps H1(Fi) to 1 in Cq. Choose inductively a collection ofg disjoint curves in the kernel of κω~p that form a metabolizer for the intersectionform on H1(F )/H1(∂F ). By taking a tubular neighborhood of these curves in F ,we obtain a collection of (S1 × I) embedded in F . Using these embeddings, attachround 2-handles (B2× I)×S1 along (S1× I)×S1 to the trivial cobordism MF × Iand obtain a cobordism Ω between MF and MF0 .

Let U = WF ∪MFΩ with boundary MF0 . The Cq-covering of WF extends

uniquely to U . Note that Ω may also be viewed as the result of attaching round1-handles to MF0 × I. Since the intersection form on Ω vanishes, Sign (U) =Sign (WF ) = 0. If Mq

F is the Cq-covering induced by κω~p , the Cq-covering of Ω,restricted to each round 2-handles is B2 × I × c attached to the trivial cobordismMqF × I along S1× I × c. Since the Cq- action on these handles is given by rotation

on c, by a Mayer-Veitoris argument, the inclusion induces an isomorphism whichpreserves the intersection form:

26

Hω~p

2 (U ; Q(α)) ' Hω~p

2 (WF ; Q(α)).

and the corresponding twisted signatures coincide.Consider now the case of the nullity. Since the deck transformation of the

corresponding coverings acts by identity, one has Hω~p

1 (L×S1) = Hω~p

1 (F ×S1) = 0.Thus, by the exact sequence with twisted coefficients of the pair (E,F × S1), thereis an isomorphism

Hω~p

1 (MF ) ' Hω~p

1 (E)

and ηL(ω~p) depends only on L.Let us denote Tµ∗ =

(S1 − 1

)µ.Theorem 4.14. If ∆L(t1, . . . , tµ) 6= 0, then• σL can be extended to a locally constant map in Tµ∗ − ∆L = 0.• For all primary point ω~p of Tµ∗ , ηL(ω~p) = 0 if and only if ∆L(ω~p) 6= 0.

Proof. Following Theorem 4.6, for (M,κ) defined in Section , the zeros of∆κ(M) are those of ∆L and the curves of the form 1×. . . 1×S1×1×. . . 1. InSection these curves are denoted S~p for ~p of the form (0, . . . , 0, pi, 0, . . . , 0). SinceσL is the signature of (M,κ), the result follows from Theorem 3.7. The secondstatement is also a consequence of Theorem 3.8 and Proposition 4.6.

Theorem 4.15. The signature and nullity are invariant by a µ-concordance.

Proof. Let L and L′ be two algebraically split µ-colored links and T be a µ-concordance, see Definition 4.22. Let E (resp. E′) be the exterior in S3 of an opentubular neighborhood of L (resp. L′). Let ω~p be a fixed element in T µ.

Since L and L′ are µ-concordant, they bound the same surfaces in B4. Let Fwith boundary L and F ′ with boundary L′ be (the images of) two embeddings ofthe same abstract surface in B4. Let (MF , κ) (resp. (MF ′ , κ)) be the boundary ofthe exterior of F (resp. F ′) in B4. Note that MF (resp. MF ′) is obtained by gluingF × S1 to E (resp. F ′ × S1 to E′).

Let W be the exterior of an open tubular neighborhood of T in S3× I. We have

H∗(W,E) = H∗(W,E′) = 0.

Moreover, κ induces characters on H1(E) and H1(E′) which extend naturally toH1(W ). By a Mayer-Veitoris argument, one deduces that MF and M ′

F are Cµ∞-homology cobordant. Following Theorem 3.9, one has σL = σL′ and ηL = ηL′ .

5. Links and smooth surfaces in B4

In all the section, L is a µ-colored algebraically split link.

5.1. Bordant links

27

Let L be a µ-colored link. Let F be an abstract surface with µ connectedcomponents and ν boundary components, according to the coloring of L. FollowingSection , we consider the pair (MF , κ) where MF is obtained by gluing F × S1 tothe exterior of L. See in particular Remark 4.13.

Definition 5.25. A µ-colored algebraically split link is algebraically bordant if(MF , κ) is Cµ∞-bordant.

Lemma 5.5. The property of being bordant is independent of the chosen surface Fto construct (MF , κ).

Proof. Suppose that for some surface F , (MF , κ) is Cµ∞-bordant. We want toprove that for any other abstract surface F ′ with boundary L, the correspondingpair (MF ′ , κ) is bordant. Let us fix ~p in Zr, with gcd(p1, . . . , pµ) = 1 and κ~p :H1(M) → C∞ be obtained by composing ti 7→ tpi with κ. Let (W,κ~p) be a pairwith boundary (MF , κ~p), such that the intersection form on Hκ~p

2 (W ; Q[t±1]) has ametabolizer whose image is a metabolizer for the form on H2(W ). Note that Wdepends on ~p.

We mainly use constructions made in the proof of Definition 4.24, and keep thenotations of this proof. Recall that F0 is a planar abstract surface with boundaryL. Consider the cobordism Ω with boundary components MF and MF0 . Let Ube obtained by gluing W and Ω along MF . We denote also by κ~p the extensionof κ~p to H1(U) and its restriction to H1(MF0). A Mayer-Veitoris argument showsthat the inclusion induces an isomorphism which preserves the intersection formHκ~p

2 (U ; Q[t±1]) ' Hκ~p

2 (W ; Q[t±1]). In particular, the form on Hκ~p

2 (U ; Q[t±1]) ismetabolic. Using the proof of Proposition 2.1, by surgeries on (U, κ~p) we obtain acobordant pair (U ′, κ~p) with boundary (MF0 , κ~p), such that the covering of U ′ isthe universal covering. Since the form on Hκ~p

2 (U ′; Q[t±1]) has a metabolizer and itprojects onto a metabolizer for the form on H2(U ′), the pair (MF0 , κ~p) is bordant.By similar arguments, it follows that (MF ′ , κ~p)) is C∞-bordant for all ~p.

Remark 5.16. If L is a knot, then L is bordant (with integral coefficients) if andonly if it is algebraically slice. See [25, 26].

Theorem 5.17. Let L be a µ-colored algebraically split link with ∆L 6= 0. If L isalgebraically bordant, then σL vanishes.

The proof of Theorem 5.17 follows from Theorem 3.11.

Definition 5.26. L is geometrically bordant if it bounds a smoothly embeddedsurface in B4 with Euler characteristic 1.

Theorem 5.18. Let L be a µ-colored link with ∆L 6= 0. If L is geometricallybordant, then L is algebraically bordant.

Proof. Let WF be the exterior of an open tubular neighborhood of F in B4

with χ(F ) = 1. Letκ : H1(WF ) −→ Cµ∞

28

mi 7−→ ti.

Note that σL = σ(MF , κ) where (MF , κ) = (∂WF , κ). We prove that (MF , κ) isCµ∞-bordant, see Definition 3.19.

Since F has µ connected components, then H1(F ) = Zµ−1. By the Thom iso-morphism, excision and the long exact sequence in homology of the pair (B4,WF ),the integral homology of WF is:

H0(WF ) = Z;H1(WF ) = Zµ;H2(WF ) = Zµ−1;H3(WF ) = H4(WF ) = 0.

In particular, χ(WF ) = 0.Let ~p be in Zµ with gcd(p1, . . . , pµ) = 1 and such that ∆L(tp1 , . . . , tpµ) 6= 0. By

Theorem 2.5, the complex Cκ~p∗ (M ; Q(t)) is acyclic, i.e Hκ~p

∗ (M ; Q(t)) = 0. Obvi-ously, Hκ~p

0 (W ; Q(t)) = Hκ~p

4 (W ; Q(t)) = 0. By the Milnor exact sequence [41] forthe infinite cyclic covering W∞

F −→WF induced by κ~p, the Z[t±1]-module H3(W∞F )

is torsion, since H3(WF ) = 0. Thus, Hκ~p

3 (WF ; Q(t)) = 0. Since Hκ~p

2 (M ; Q(t)) = 0,by the exact sequence in twisted homology of the pair (WF ,MF ), this impliesdim H

κ~p

3 (W,M ; Q(t)) = 0. Thus, Hκ~p

1 (W ; Q(t)) = 0. Since the Euler characteristicwith twisted coefficients coincide with the ordinary one, one gets Hκ~p

2 (W ; Q(t)) = 0.Using the proof of Proposition 2.1, make surgeries on (W,κ~p) and obtain a

cobordant pair (W ′, κ~p) with boundary (MF , κ) and such that Hκ~p

1 (W ; Q[t±1]) = 0.Thus the form on H

κ~p

2 (W ′; Q[t±1]) is metabolic. Moreover, since κ~p induces theuniversal covering of W ′, the metabolizer of ψκ~p maps onto a metabolizer for theordinary form on H2(W ).

5.2. Generalized Murasugi-Tristram inequality

Theorem 5.19. Suppose that the µ-colored link L bounds a surface F, smoothlyand properly embedded in B4, with µ connected components. Let b1 the first Bettinumber of F . Then, for all ω~p in T µP

|σL(ω~p)|+ |ηL(ω~p)− µ+ 1| ≤ b1

Remark 5.20. Consider the knot K obtained as the connected sum of the trefoiland its mirror image. Since K is ribbon, it is slice; but the Tristram nullity ηK(e

2iπ6 )

is non zero and the inequality of Theorem 5.19 does not hold. This illustrates thatthe hypothesis that ω~p is a primary point, in particular that ω has prime powerorder, is necessary in Theorem 5.19.

Theorem 5.19 provides also an obstruction for a link to be slice in the strongsense, see [4].

Corollary 5.1. If L is a slice link, then σL = 0 and ηL = µ− 1.

The proof follows from b1 = 0. This result was shown in [3] in the followingsense. An elementary necessary condition for L to be slice is that the linking matrix

29

of L is zero. If we take the maximal coloring for the link (i.e. each component hasa different color), then our signatures coincides with Levine’s signature.

Remark 5.21. One may show as a direct consequence of Theorem 5.19 that linkswith non-zero Alexander polynomial bounding smooth surfaces with Euler charac-teristic 1 in B4 have vanishing re-defined signatures. For this, write the inequalityat each ω~p in T µP with ∆L(ω~p) 6= 0, i.e. ηL(ω~p) = 0.

Proof. Let WF be the exterior of a tubular neighborhood of F in B4. By theThom isomorphism, excision, and the long exact sequence of the pair (B4,WF ), theintegral homology of WF depends only on those of F :

H0(WF ) = Z ; H1(WF ) = Zµ ; H2(WF ) = Zb1 ; H3(WF ) = H4(WF ) = 0.

In particular, the Euler characteristic χ(WF ) = 1 − µ + b1. Let ω~p be in T µ andβω

~p

i = dim Hω~p

i (W ; Q(α)). By Lemma 3.3 one has the following estimate

|σL(ω~p)| ≤ dim Kω~p

≤ βω~p

1 − βω~p

3 + βω~p

2 − ηL(ω~p) (∗)

Obviously, βω~p

0 = 0. Since H3(WF ) = 0, by [17] Proposition 1.4, βω~p

3 = 0. Since thetwisted homology is constructed with the cells of WF , the Euler characteristic withtwisted coefficients coincides with the ordinary one. Thus, χω

~p

= 1− µ+ b1 and

βω~p

2 = βω~p

1 + b1 − µ+ 1.

Thus, (∗) gives|σL(ω~p)|+ ηL(ω~p) ≤ b1 − µ+ 1 + 2βω

~p

1 .

There are two different estimation of βω~p

1 .. From the exact sequence Hω~p

1 (M) → Hω~p

1 (W ) → Hω~p

1 (W,M) → 0, one has

βω~p

1 ≤ ηL(ω~p).

One obtains

|σL(ω~p)| − ηL(ω~p) + µ− 1 ≤ b1. (1)

. Since H1(W ) = Zµ, by [17], one has

βω~p

1 ≤ µ− 1.

One obtains|σL(ω~p)|+ ηL(ω~p)− µ+ 1 ≤ b1. (2)

The result follows from (1) and (2).

6. Computation

The inequality of Theorem 5.19 can be used to prove that L does not bounds asmooth surface in B4 with given Betti numbers. If there exists ω~p in TµP such thatthe inequality does not hold, then the embedded surface does not exists.

30

Suppose that L is a µ-colored link with components, such that each sublinkLk = Lk(1)∪ · · · ∪Lk(νµ) corresponds to the components with the same color k fork = 1, . . . , µ. Note that

∑νk = ν.

From now on, fix a point ω~p in T µP , such that ω = e2iπr/q and give an ex-plicit algorithm to compute σL(ω~p) and ηL(ω~p). This algorithm arises as a directconsequence of Gilmer’s formula, see Theorem 3.7 [17], and Proposition 3.2. Let

~pν = (p1, . . . , p1︸ ︷︷ ︸ν1

, . . . , pk, . . . , pk︸ ︷︷ ︸νk

, . . . , pµ, . . . , pµ︸ ︷︷ ︸νµ

),

and introduce the notation :

~f =(f1(1), . . . , f1(ν1), . . . , fk(1), . . . , fk(νk), . . . , fµ(1), . . . , fµ(νµ)

)Theorem 6.22. Suppose that ~f is a solution of the following system of simultaneouscongruences and equalities, where Λ the linking matrix of L with the vector ~f in thediagonal:

Λ ~pν ≡ 0 mod q (i)∑νk

i=1 fk(i) = −∑i,j;i 6=j lk(Lk(i), Lk(j)), for all k = 1 . . . µ (ii)

Let L′ be a non-empty link where the component Lk(i) of L is replaced by a pk-cable on Lk(i) with twist fk(i), see Definition 3.13. If ν′ denotes the number ofcomponents of L′, then

σL(ω~p) = σL′(ω)

ηL(ω~p) = ηL′(ω)− ν′ + ν

Proof. Let WF with boundary MF be the exterior of an open tubular neighbor-hood of a surface F in B4 with µ connected components and boundary L. Consider

ζ : H1(WF ) −→ C∗

mk 7−→ ωpk .

The character ζ induces a character on H1(M)F and a one on H1(E) which sendsall the meridians of the same sublink Lk to ωpk . The same notation ζ is used forall of them. By Definition 4.24,

σL(ω~p) = σ(MF , ζ) and ηL(ω~p) = η(MF , ζ).

We now show that there exists ~f such that ζ induces a character on H1(ML),where ML is obtained by surgery on L with framing vector ~f . Moreover, we showthat ~f can be chosen such that (ii) holds. The formula then follows from Theorem3.7 of [17].

Recall that if ML is obtained by surgery on L with vector framings ~f , then thematrix Λ with entries linking numbers of L and ~f in the diagonal is a presentationmatrix of H1(ML), see Remark 3.6. If ~f is the solution of Λ ~pν ≡ 0 mod q, then

31

the character denoted ζ, which sends all the meridians of the same sublink Lk toωpk , is well-defined on H1(ML).

The congruence Λ ~pν ≡ 0 mod q is a linear system in the variables fk(i) wherethe congruences are of the form pk · fk(i) ≡ ak(i) mod (q) for ak(i) in Z. Since(pk, q) = 1, pk is invertible in (Z/qZ)∗ and each equation has a solution. It followsthat there exist many ~f which verify (i). We now show that we can choose one ofthem such that (ii) holds also. Let i be in 1, . . . , µ. For short, we denote here byΓj,k for j, k = 1 . . . νi, the linking matrix of the sublink Li of L.

Let fi(1), . . . , fi(νi − 1) be given as solutions of the congruences above.Thus, by construction there exists some integers n1, . . . , nνi−1 such that for all

j = 1 . . . νi − 1, we have

Xj + fi(j).pi +∑k;k 6=j

Γj,k.pi = nj .q

with Xj =∑k;k 6=i lk(Li(j), Lk).pk where lk(Li(j), Lk) is the sum of the linking

number of Li(j) with the components of Lk. Note that we have∑νi

j=1Xj =∑k;k 6=i lk(Li, Lk).pk = 0. Let us consider

fi(νi) := −∑

j,k;j 6=k

Γj,k − fi(1)− · · · − fi(νi − 1)

We now prove that it is solution of Λ.~p ≡ 0 mod (q). For this, let us compute

Xνi+

(−

∑j,k;j 6=k

Γj,k − fi(1)− · · · − fi(νi − 1)).pi +

∑k 6=νi

Γνi,k.pi (∗)

Note that we have:

−∑

j,k;j 6=k

Γj,k = −∑j;j 6=1

Γ1,j · · · −∑j;j 6=νi

Γνi,j

Thus,

(∗) = Xνi+

(−

∑j;j 6=1

Γ1,j − fi(1)).pi + · · ·+

(−

∑j;j 6=νi−1

Γνi−1,j − fi(νi − 1)).pi

= Xνi+ (X1 − n1.q) + . . . (Xνi−1 − nνi−1.q)

= −(n1 + · · ·+ nνi−1).q = 0 mod q

7. Examples, Proof of theorem 1.2

Consider the examples of links given in Section . In all the five cases, L = L1tL2

is a 2-colored link with L2 = L2(1)∪L2(2)∪L2(3) and L1 has one component. Weshow in this section that L cannot bound a smooth planar surface in B4 with twoconnected components. Note that such a surface has a first Betti number b1 = 2.

Tristram-Levine signaturesBy the Murasugi-Tristram theorem (see Theorem 1.1), if L bound a planar sur-

face, then the Levine-Tristram signatures and nullities verify the following inequality:

32

For all ω in S1P , |σL(ω)|+ |ηL(ω)− 1| ≤ 2

The one-variable Alexander polynomial AL(t) has no roots on S1 ⊂ C. It followsthat ηL ≡ 0 and σL is constant on S1. The computation gives σL ≡ σL(−1) = −1in the five cases. Therefore, by the inequality above, one obtains :

| − 1|+ |0− 2 + 1| ≤ 2

The inequality holds and one cannot conclude.Generalized signaturesThe Tristram-Levine signatures are the restriction on the diagonal in T 2 of

the generalized signatures of L. Let ∆L(t, s) be the Alexander polynomial of L,associated to the coloring. We have to look for connected components of Tµ∗ −∆ =0 which do not intersect the diagonal, i.e. where σL has chances to take differentvalues than the Tristram signatures. To compute ∆L(t, s), we used a programwritten by Orevkov [23], and the program Maple as a guide to picking ∆L = 0.

Remark 7.23. The homography

h : C2 −→ C2

(t, s) 7−→ (u, v) :=(it+ 1t− 1

, is+ 1s− 1

)identifies T 2

∗ with R2 in C2. Under this identification, the conjugation has the form(u, v) 7−→ (−u,−v). It follows that for (u, v) in R2, the zero set of ∆′

L(u, v) isinvariant by the involution (u, v) 7−→ (−u,−v). Hence the study of (u, v);u ≥ 0determines the full picture by symmetry with respect to the origin.

The following table gives the values of the signatures with ω = e2iπr/q and~p = (p1, p2), for the five links considered.

Table 1. Values of the signature

q 29 29 29 29 29

(p1, p2) (3,2) (3,2) (3,2) (3,2) (3,2)r 6 6 11 11 5σ -3 -3 -3 -3 -3

By Theorem 3.4, since ∆L is non zero at these points, ηL = 0. We give here thedetails of the computation of the framings in the case of the last link, see Theorem1.2. The linking matrix of L is:

Λ =

f1(1) −5 2 3−5 f2(1) 2 −12 2 f2(2) 13 −1 1 f2(3)

33

The vector of framings f =(f1, f2(1), f2(2), f2(3)

), must verify the system of con-

gruences and equalities: f1(1) ≡ 0 mod (49)2f2(1) ≡ 13 mod (49)2f2(2) ≡ −12 mod (49)2f2(3) ≡ −9 mod (49)

and f1(1) = 0f2(1) + f2(2) + f2(3) + 4 = 0

The vector f = (0,−18,−6, 20) is a solution of these equations. Note that theproof of Theorem 6.22 contains a method to find such solutions. Let L′ be the linkobtained by taking a 3-cable with framing 0 on L1 union a 2-cable with framingsrespectively −18,−6 and 20 on each component of L2 (see Definition 3.13). ByTheorem 5.19, we obtain :

| − 3|+ |0− 2 + 1| ≤ 2

This inequality is obviously false and the 2-colored links do not bound a planarsurface in B4. This proves Theorem 1.2.

Acknowledgements

The author thanks S.Orevkov for much guidance and encouragement, P.M.Gilmerand D.Lines for a very precise help and helpful suggestions. He also thanks C.Blanchetand O.Viro for helpful comments.

References

[1] Tristram A. G., Some cobordism invariants for links, Proc, Camb. Philos. Soc. 66 (1969)p251-264.

[2] Levine J.P., Knot cobordism groups in codimension two, Comment. Math. Helv. 44(1969) p229-244.

[3] Levine J.P., Link invariants via the eta invariant, Comm. Math. Helv. 69 (1984), p82-119.

[4] Fox R. H., A quick trip through knot theory, Topology of 3-Manifolds and RelatedTopics (Proc. The Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, NewJersey (1962) p120-167.

[5] Fox R.H., Milnor J.W., Singularities of 2-spheres in 4-space and cobordism of knots,Osaka J. Math. 3 (1966), p257-267.

[6] Murasugi K., Knot Theory and its Applications, Birkhauser Verlag, 1993.

[7] Casson A. J., Gordon C. Mc A., Cobordism of classical knots in S3, A la recherche dela topologie perdue Birkhauser Boston, Boston (1986) p201-244.

[8] Casson A. J., Gordon C. Mc A., On slice knots in dimension three, Proc. Symp. inPure Math. XXX 2 (1978) p39-53.

[9] Blanchfield, Intersection theory of manifolds with operators with applications to knottheory, Ann. of Math. 65 (1957) p340-356.

34

[10] Gordon C. Mc A., Some aspect of classical knot theory, Proc. Plans-sus-bex Switherland(1977) Lec. Notes in Math. 685 Berlin Heidelberg New-York Springer-Verlag (1978),p1-60.

[11] Atiyah M. F., Singer I. M., The index of elliptic operators. III, Ann. of Math. 2 (1968)p546-604.

[12] Viro O. Ja, Branched coverings of manifolds with boundary and link invariants, I. Math.USSR Izvestia 7 (1973) p1239-1256.

[13] Viro O. Ja, Survey Progress of the last six years in topology of real algebraic varieties,Uspekhi Mat. Nauk 41:3 (1986) p45-67 (Russian), English translation in Russian Math.Surveys 41:3 (1986), p55-82.

[14] Rokhlin D., Two-dimensional submanifolds of four-dimensional manifolds, Funkt-sional’nyi Analiz i Ego Prilozheniya 5 no1 (1971), p48-60.

[15] Florens V., On the Fox-Milnor theorem for the Alexander polynomial of links, Int. Math.Res. Notices. no2 (2004) p55-67.

[16] Florens V., Gilmer P. M., On the slice genus of links, Alg. and Geom. Topology no3(2003) p905-920.

[17] Gilmer P. M., Configurations of Surfaces in 4-manifolds, Trans. Amer. Math. Soc. 264(1981) p353-380.

[18] Gilmer P. M., On the slice genus of knots, Invent. Math. 66 (1982), p191-197.[19] Gilmer P. M., Livingston C., Dicriminant of Casson-Gordon invariants, Math. Proc.

Cambridge Philos. Soc. 112 (1992) no1 p127-139.[20] Degtyarev A.I., Kharlamov V.M., Topological properties of real algebraic varieties:

Rokhlin’s way, Russian Math. Surveys 55 no4 (2000) p735-814.[21] Orevkov S. Yu., Link theory and oval arrangements of real algebraic curves, Topology

38 (1999) p779-810.[22] Orevkov S. Yu., Quasipositivity test via unitary representations of braid groups and its

applications to real algebraic curves, Journal of Knot th. and Ram. 10 (2001) p1005-1023.

[23] Orevkov S. Yu., web page: http://picard.ups-tlse.fr/ orevkov.[24] Orevkov S. Yu., Classification of flexible M-curves of degree 8 up to isotopy, GAFA -

Geometric and Functional Analysis 12-4 (2002), p723-755.[25] Cappell S., Shaneson J.L., The codimension two placement problem and homology equiv-

alent manifolds, Annals of Math. 99 (1974), p227-348.[26] Cochran T., Teichner P., Orr K., Knot concordance, Withney towers and von Neumann

signatures, Annals of Math. 157 no2 (2003) p433-519.[27] Friedl S., Link concordance, boundary link concordance and eta invariant, Math Arxiv

GT/0306149.[28] Kauffman L., Taylor L., Signature of links, Trans. Amer. Math. Soc. 216 (1976) p351-

365.[29] Kauffman L., Signatures of branched fibrations, Knot theory, Proc. Plans-Sur-Bex

Switzerland, (1977) p203-217.[30] Turaev V.G , Reidemeister torsion in knot theory, Russian Math. surveys 41 (1986).[31] Turaev V.G , Introduction to combinatorial torsions, Notes taken by F.Schlenk, Lec-

tures in Meth. ETH Zurich. Birkhauser, Basel (2001).[32] Turaev V.G, Torsion of 3-dimensional manifolds, Birkhauser Verlag Basel-Boston-

Berlin, (2002).[33] Gordon C. McA., Litherland R.A., On the signature of a link, Invent. Math. 47 no1

(1978) p53-69.[34] Smolinsky L., A generalization of the Levine-Tristram link invariant, Trans. of the

A.M.S. 315 (1989) p205-217.

35

[35] Cooper, D., The universal abelian cover of a link, Low-dimensional topology, Bangor,London Math. Soc. Lecture Note Ser. 48 (1979), p51-66.

[36] Cimasoni, D., A geometric construction of the Conway potential function, To appearin Comment. Math. Helv..

[37] Rudolph L., The slice genus and the Thurston-Bennequin invariant of a knot, Proc. ofthe AMS 125 (1997) p3049-3050.

[38] Ackbulut S., Matveyev R., Exotic structures and adjunction inequality, Tr. J. of Math.21 (1997) p47-53.

[39] Le Touze-Fiedler S., Orientations complexes des courbes algebriques reelles, PreprintLaboratoire E.Picard, Universite Paul Sabatier, Toulouse (2000).

[40] Milnor J. W., A duality theorem for Reidemesiter torsion, Ann. of Math 26 (1962)p137-147.

[41] Milnor J. W., Infinite cyclic coverings, Conference on the Topology of Manifolds (Michi-gan State Univ., E. Lansing, Mich., 1967), Prindle, Weber & Schmidt, Boston, Mass.(1968) p115-133.

[42] Kirk P., Livingston C., Twisted Alexander invariants, Reidemeister torsion and Casson-Gordon invariants, Topology 38 no3 (1999) p635-661.

[44] Kirby C.,The topology of 4-manifolds, Springer Verlag Lecture Notes,vol 1374 (1991).[45] Kawauchi, A survey of knot theory, Birkhauser Verlag (1996).