Homology of the circle with non-trivial local coefficients is trivial. From this well-known fact we deduce geometric corollaries involving codimension-two links. In particular, the Murasugi-Tristram signatures are extended to invariants of links formed of arbitrary oriented closed codimension two submanifolds of an odd-dimensional sphere. The novelty is that the submanifolds are not assumed to be disjoint, but are transversal to each other, and the signatures are parametrized by points of the whole torus. Murasugi-Tristram inequalities and their generalizations are also extended to this setup.

1. Introduction

The goal of this paper is to simplify and generalize a part of classical link theory based on various signatures of links (defined by Trotter [19] Murasugi [10],[11], Tristram [18], Levine [7] [8], Smolinsky [17], Florens [2] and Cimasoni and Florens [1]). This part is known for its relations to topology of 4-dimensional manifolds, see [18], [20], [21] [4], [6] and applications in topology of real algebraic curves [12], [13] and [2].

Similarity of the signatures to the new invariants [15], [14], which were defined in the new frameworks of link homology theories and had spectacular applications [15], [9], [16] to problems on classical link cobordisms, gives a new reason to revisit the old theory.

There are two ways to introduce the signatures: the original 3-dimensional, via Seifert surface and Seifert form, and 4-dimensional, via the intersection form of the cyclic coverings of 4-ball branched over surfaces. I believe, this paper clearly demonstrates advantages of the latter, 4-dimensional approach, which provides more conceptual definitions, easily working in the situations hardly available for the Seifert form approach.

In the generalization considered here the classical links are replaced by collections of transversal to each other oriented submanifolds of codimension two.

Technically the work is based on a systematic use of twisted homology and the intersection forms in the twisted homology. Only the simplest kinds of twisted homology is used, the one with coefficients in ℂ, see Appendix.

1.1. Twisted acyclicity of a circle

A key property of twisted homology, which makes the whole story possible, is the following well-known fact, which I call twisted acyclicity of a circle:

Twisted homology of a circle with coefficients in ℂ and non-trivial monodromy vanishes.

This implies that the twisted homology of this kind completely ignores parts of the space formed by circles along which the monodromy of the coefficient system is non-trivial (for precise and detailed formulation see Section Appendix B).

1.2. How the acyclicity works

In particular, twisted acyclicity of a circle implies that the complement of a tubular neighborhood of a link looks like a closed manifold, because the boundary, being fibered to circles, is invisible for the twisted homology.

Moreover, the same holds true for a collection of pairwise transversal generically immersed closed manifolds of codimension 2 in arbitrary closed manifold, provided the monodromy around each manifold is non-trivial. The twisted homology does not feel the intersection of the submanifolds as a singularity.

The complement of a cobordism between such immersed links looks (again, from the point of view of twisted homology) like a compact cobordism between closed manifolds.

This, together with classical results about signatures of manifolds and relations between twisted homology and homology with constant coefficients, allows us to deal with a link of codimension two as if it was a single closed manifold.

1.3. Organization of the paper

I cannot assume the twisted homology well-known to the reader, and review the material related to it. Of course, the material on non-twisted homology is not reviewed. The review is limited to a very special twisted homology, the one with complex coefficients. More general twisted homology is not needed here.

The review is postponed to appendices. The reader somehow familiar with twisted homology may visit this section when needed. The experts are invited to look through appendices, too.

We begin in Section 2 with a detailed exposition restricted to the classical links. Section 3 is devoted to higher dimensional generalization, including motivation for our choice of the objects. Section 4 is devoted to span inequalities, that is, restrictions on homology of submanifolds of the ball, which span a given link contained in the boundary of the ball. Section 5 is devoted to slice inequalities, which are restrictions on homology of a link with given transversal intersection with a sphere of codimension one.

2. In the classical dimension

2.1. Classical knots and links.

Recall that a classical knot is a smooth simple closed curve in the 3-sphere S³. This is how one usually defines classical knots. However it is not the curve per se that is really considered in the classical knot theory, but rather its placement in S³. Classical knots incarnate the idea of knottedness: both the curve and S³ are topologically standard, but the position of the curve in S³ may be arbitrarily complicated topologically. Therefore a classical knot is rather a pair (S³,K), where K is a smooth submanifold of S³ diffeomorphic to S¹.

A classical link is a pair (S³,L), where L is a smooth closed one-dimensional submanifold of S³. If L is connected, then this is a knot.

2.2. Twisted homology of a classical link exterior

An exterior of a classical link (S³,L) is the complement of an open tubular neighborhood of L. This is a compact 3-manifold with boundary. The boundary is the boundary of the tubular neighborhood of L. Hence, this is the total space of a locally trivial fibration over L with fiber S¹. An exterior X(L) is a deformation retract of the complement S³ \ L. It’s a nice replacement of S³ \ L, because IntX(L) is homeomorphic to S³ \ L, but X(L) is compact manifold and has a nice boundary.

If L consists of m connected components, L = K₁ ∪∪ K_m, then by the Alexander duality H₀(X(L)) = ℤ, H₁(X(L)) = ℤ^m and H_i(X(L)) = 0 for i≠0,1. The group H₁(X(L)) is dual to H₁(L) with respect to the Alexander linking pairing H₁(L) × H₁(X(L)) → ℤ. Hence a basis of H₁(L) defines a dual basis in H₁(X(L)). An orientation of L determines a basis [K₁], …, [K_m] of H₁(L), and the dual basis of H₁(X(L)), which is realized by meridians M₁, …, M_m positively linked to K₁, …, K_m, respectively. (The meridians are fibers of a tubular fibration ∂X(L) → L over points chosen on the corresponding components.)

Therefore, if L is oriented, then a local coefficient system on X(L) with fiber ℂ is defined by an m-tuple of complex numbers (ζ₁,…,ζ_m), the images under the monodromy homomorphism H₁(X(L)) → ℂ^× of the generators [M₁], …, [M_m] of H₁(X(L)).

Thus for an oriented classical knot L consisting of m connected components, local coefficient systems on X(L) with fiber ℂ are parametrized by (ℂ^×)^m.

2.3. Link signatures

Let L = K₁ ∪∪K_m ⊂ S³ be a classical link, ζ_i ℂ, |ζ_i| = 1, ζ = (ζ₁,…,ζ_m) (S¹)^m and μ : H₁(S³ \ L) → ℂ^× takes to ζ_i a meridian of K_i positively linked with K_i.

Let F₁,…F_m ⊂ D⁴ be smooth oriented surfaces transversal to each other with ∂F_i = F_i ∩ ∂D⁴ = K_i. Extend the tubular neighborhood of L involved in the construction of X(L) to a collection of tubular neighborhoods N₁, …, N_m of F₁, …, F_m, respectively.

Without loss of generality we may choose N_i in such a way that they would intersect each other in the simplest way. Namely, each connected component B of N_i ∩N_j would contain only one point of F_i ∩F_j and no point of others F_k and would consist of entire fibers of N_i and N_j, so that the fibers define a structure of bi-disk D² × D² on B.

To achieve this, one has to make the fibers of the tubular fibration N_i → F_i at each intersection point of F_i and F_j coinciding with a disk in F_j and then diminish all N_i appropriately.

Now let us extend X(L) to X(F) = D⁴ \ ∪_i=1^m IntN_i. This is a compact 4-manifold. Its boundary contains X(L), the rest of it is a union of pieces of boundaries of N_i with i = 1,…,m. These pieces are fibered over the corresponding pieces of F_i with fiber S¹.

By the Alexander duality, the orientation of F_i defines a homomorphism H₁(X(F)) → ℤ, the linking number with F_i. These homomorphisms altogether determine a homomorphism H₁(X(F)) → ℤ^m. For any ζ = (ζ₁,…,ζ_m), the composition of this homomorphism with the homomorphism

According to Appendix D.6, in H₂(X(F); ℂ_μ) there is a Hermitian intersection form. Denote its signature by σ_ζ(L).

Proof. Any F′_i with ∂F′_i = F′_i ∩ ∂D⁴ = K_i is cobordant to F_i. The cobordisms W_i ⊂ D⁴ × I can be made pairwise transversal. They define a cobordism D⁴ × I \ ∪_i IntN(W_i) between X(F) and X(F′). By Theorem App.D.B,

2.4. Colored links

In the definition of signature σ_ζ(L) above one needs to numerate the components K_i of L to associate to each of them the corresponding component ζ_i of ζ, but there is no need to require connectedness of each K_i. This leads to a notion of colored link.

An m-colored link L is an oriented link in S³ together with a map (called coloring) assigning to each connected component of L a color in {1,…,m}. The sublink L_i is constituted by the components of L with color i for i = 1,…,m.

For an m-colored link L = L₁ ∪∪ L_m and ζ = (ζ₁,…,ζ_m) (S¹)^m, the signature σ_ζ(L) is defined as above, but each component K_j colored with color i is associated to ζ_i.

2.5. Relations to other link signatures

If ζ_i = -1 for all i = 1,…,m, then the signature σ_ζ(L) coincides with the Murasugi signature ξ(L) introduced in [11]. If all ζ_i are roots of unity of a degree, which is a power of a prime number and all linking numbers lk(L_i,L_j) vanish, then σ_ζ(L) coincides with the signature defined by Florens [2].

In the most general case, σ_ζ(L) coincides with the signature defined for arbitrary ζ by Cimasoni and Florens [1] using a 3-dimensional approach, with a version of Seifert surface, C-complex.

3. In higher dimensions

3.1. Apology for the generalization of higher dimensional links

There is a spectrum of objects considered as generalizations of classical knots and links. The closest generalization of classical knots are pairs (Sⁿ,K), where K is a smooth submanifold diffeomorphic to S^n-2. Then the requirements on K are weakened. Say, one may require K to be only homeomorphic to S^n-2, not diffeomorphic. Or just a homology sphere of dimension n - 2. The codimension is important in order to keep any resemblance to classical knots.

In the same spirit, for the closest higher-dimensional counter-part of classical links one takes a pair consisting of Sⁿ and a collection of its disjoint smooth submanifolds diffeomorphic to S^n-2. One allows to weaken the restrictions on the submanifolds. Up to arbitrary closed submanifolds.

I suggest to allow transversal intersections of the submanifolds.

Of course, the main excuse for this is that some results can extended to this setup. Here is a couple of other reasons.

First, in the classical dimension it is easy for submanifolds to be disjoint. Generically curves in 3-sphere are disjoint. If they intersect, it is a miracle or, rather, has a special cause.

Generic submanifolds of codimension two in a manifold of dimension > 3 intersect. If they do not intersect, this is a miracle, or consequence of a special cause.

Second, classical links emerge naturally as links of singular points of complex algebraic curves in ℂ². Recall that for an algebraic curve C ⊂ ℂ² and a point p C the boundary of a sufficiently small ball B centered at p, the link (∂B,∂B ∩ C) is well-defined up to diffeomorphism, and it is called the link of C at p.

An obvious generalization of this definition to an algebraic hypersurface C ⊂ ℂⁿ gives rise to a pair (S^2n-1,K) with connected K. It cannot be a union of disjoint submanifolds of S^2n-1.

It would not be difficult to extend the results of this paper to a more general setup. For example, one can replace the ambient sphere with a homology sphere, or even more general manifold. However, one should stop somewhere. The author prefers this early point, because the level of generality accepted here suffices for demonstrating the new opportunities open by a systematic usage of twisted homology. On the other hand, further generalizations can make formulations more cumbersome.

3.2. Colored links

By an m-colored link of dimension n we shall mean a collection of m oriented smooth closed n-dimensional submanifolds L₁, …, L_m of the sphere Sⁿ⁺² such that any sub-collection has transversal intersection. The latter means that for any x L_i1 ∩∩ L_ik the tangent spaces T_xL_i1, …, T_xL_ik are transverse, that is, dim(T_xL_i1 ∩∩ T_xL_ik) = n + 2 - 2k.

3.3. Generic configurations of submanifolds

More generally, an m-colored configuration of transversal submanifolds in a smooth manifold M is a family of m smooth submanifolds L₁, …, L_m of M such that any sub-collection has transversal intersection. If M has a boundary, the submanifolds are assumed to be transversal to the boundary, as well as the intersection of any sub-collection. Furthermore, assume that ∂M ∩ L_i = ∂L_i for any i = 1,…,m.

As above, in Section 2.3, for any m-colored configuration L of transversal submanifolds L₁, …, L_m in M one can find a collection of their tubular neighborhoods N₁, …, N_m which agree with each other in the sense that for any sub-collection L_i1, …, L_iν the intersection of the corresponding neighborhoods N_i1 ∩∩N_iν is neighborhood of the intersection L_i1 ∩∩ L_iν fibered over this intersection with the corresponding poly-disk fiber.

Denote the complement M \ ∪_i=1^m IntN_i by X(L) and call it an exterior of L. This is a smooth manifold with a system of corners on the boundary. The differential type of the exterior does not depend on the choice of neighborhoods. Moreover, one can eliminate the choice of neighborhoods and deleting of them from the definition. Instead, one can make a sort of real blowing up of M along L₁, …, L_m. However, for the purposes of this paper it is easier to stay with the choices.

3.4. Link signatures

Let L = L₁ ∪∪ L_m be an m-colored link of dimension 2n - 1 in S²ⁿ⁺¹.

As well known (see, e.g., [7]), for each oriented closed codimension 2 submanifold K of S²ⁿ⁺¹ there exists an oriented smooth compact submanifold F of D²ⁿ⁺² such that ∂F = K. Choose for each L_i such a submanifold of D²ⁿ⁺², denote it by F_i, and make all the F_i transversal to each other by small perturbations.

As a union of m-colored transversal submanifolds of D²ⁿ⁺², F = F₁ ∪∪ F_m has an exterior X(F). By the Alexander duality, H¹(X(F); ℂ^×) is naturally isomorphic to H_2n(F,L; ℂ^×). Let ζ = (ζ₁,…,ζ_m) (S¹)^m. Take ∑ _i=1^mζ_i[F_i] H_2n(F,L; ℂ^×) and denote by μ the Alexander dual cohomology class considered as a homomorphism H₁(X(F)) → ℂ^×. Denote by ℂ_μ the local coefficient system on X(F) corresponding to μ.

According to Appendix D.6, in H_n+1(X(F); ℂ_μ) there is an intersection form, which is Hermitian, if n is odd, and skew-Hermitian, if n is even. Denote its signature by σ_ζ(L).

Proof. Any F′_i with ∂F′_i = F′_i ∩ ∂D²ⁿ⁺² = L_i is cobordant to F_i. The cobordisms W_i ⊂ D²ⁿ⁺² ×I can be made pairwise transversal to form m-colored configuration W of transversal submanifolds of D²ⁿ⁺² × I. They define a cobordism X(W) between X(F) and X(F′). By Theorem App.D.B,

4. Span inequalities

Let L = L₁ ∪…,∪L_m be an m-colored link of dimension 2n - 1 in S²ⁿ⁺¹. Let F = F₁ ∪∪ F_m be an m-colored configuration of transversal oriented compact 2n-dimensional submanifolds of D²ⁿ⁺² with ∂F_i = F_i ∩∂D²ⁿ⁺² = L_i. In this section we consider restrictions on homological characteristics of F in terms of invariants of L.

4.1. History

The first restrictions of this sort were found by Murasugi [10] and Tristram [18] for classical (1-colored) links. To m-colored classical links and pairwise disjoint surfaces F_i the Murasugi-Tristram inequalities were generalized by Florens [2]. A further generalization to m-colored classical links and intersecting F_i was found by Cimasoni and Florens [1]. Higher dimensional generalizations for 1-colored links were found by the author [21], [22].

4.2. No-nullity span inequalities

The most general results in this direction are quite cumbersome. Therefore, let me start with weak but simple ones.

Recall that σ_ζ(L) can be obtained from F: for an appropriate local coefficient system ℂ_μ on X(F), this is the signature of a Hermitian intersection form defined in H_n+1(X(F); ℂ_μ). The signature of an Hermitian form cannot be greater than the dimension of the underlying space. In particular,

(4.1)

This can be considered as a restriction on a homological characteristic of F in terms of invariants of L. However, dim_ℂH_n+1(X(F); ℂ_μ) is not a convenient characteristic of F. It can be estimated in terms of more convenient ones.

Let ζ = (ζ₁,…,ζ_m) (S¹)^m. Let p₁,…,p_k ℤ[t₁,t₁^-1…,t_m,t_m^-1] be generators of the ideal of relations satisfied by complex numbers ζ_i. Let d be the greatest common divisor of the integers p₁(1,…,1), …, p_k(1,…,1), if at least one of these integers does not vanish, and zero otherwise. Cf. Appendix C.6. Let

By App.C.C,

4.3. General span inequalities

The inequality (4.1) can be improved. Indeed, the manifold X(F) has a non-empty boundary. Therefore, its intersection form may be degenerate and the right hand side of (4.1) may be replaced by a smaller quantity, the rank of the form. The rank is known to be the rank of the homomorphism H_n+1(X(F); ℂ_μ) → H_n+1(X(F),∂X(F); ℂ_μ). Let us estimate this rank.

Lemma 4.A For any exact sequence … C_k C _k-1… of vector spaces and any integers n and r

(4.2)

Proof. The Euler characteristic of the exact sequence

is the difference between the left and right hand sides of (4.2). On the other hand, it vanishes, as the Euler characteristic of an exact sequence.

Lemma 4.B Let X be a topological space, A its subspace, ξ a local coefficient system on X with fiber ℂ. Then for any natural n and r ≤

(4.3)

where b_k(*) = dim_ℂH_k(*;ξ)

Proof. Apply Lemma 4.A to the homology sequence of pair (X,A) with coefficients in ξ.

Theorem 4.C For any integer r with 0 ≤ r ≤

(4.4)

(4.5)

where ζ and P are is in Section 4.2

Proof. As mentioned above,

(4.6)

By Lemma 4.B,

(4.7)

Summing up these inequalities and moving one of the sums from the right hand side to the left, we obtain:

(4.8)

The left hand sum of (4.8) coincides with the left hand side of (4.4). The right hand side can be estimated using Theorem App.C.C:

(4.9)

Further, By the Alexander duality, By exactness of the pair sequence, H^n+1+s(D²ⁿ⁺²,F;P) = H^n+s(F;P).

Similarly,

The last equality in this sequence holds true if n + 1 + s < 2n + 1, that is, s < n.

Since P is a field,

dimPHn+s(F;P) =	dim PHn+s(F;P),	(4.10)
dimPHn+1+s(F,L;P) =	dim PHn+1+s(F,L;P).	(4.11)

Combining formulas (4.10), (4.11) with the calculations above and equalities (4.9) and (4.8), we obtain the first desired inequalities (4.4).

The inequalities (4.5) are proved similarly. Namely, by Lemma 4.B

(4.12)

Summing up inequalities (4.6) and (4.12) and moving one of the sums from the right hand side to the left, we obtain:

(4.13)

After this the same estimates and transformations as in the proof of (4.4) gives rise to (4.5).

4.4. Nullities

The sum in the left hand side of the inequalities (4.4) is an invariant of the link L. Its special case for classical links with r = 0 is known as ζ-nullity and appeared in the Murasugi-Tristram inequalities and their generalizations.

Denote ∑ _s=0^2r(-1)^s dimH_n-s(S²ⁿ⁺¹ \ L; ℂ_μ) by n_ζ^r(L) and call it rth ζ-nullity of L.

By the Poincaré duality (see Appendix D.3), H_n-s(S²ⁿ⁺¹ \ L; ℂ_μ) is isomorphic to H^n+1+s(S²ⁿ⁺¹ \ L; ℂ_μ). The latter complex vector space is dual to H_n+1+s(S²ⁿ⁺¹ \ L; ℂ_μ^-1) and anti-isomorphic to H_n+1+s(S²ⁿ⁺¹ \ L; ℂ_μ), see Appendix D.5. Therefore,

(4.14)

and n_ζ^r(L) = n_ζ^r(L). This sum is a part of the left hand side of (4.5).

Now we can rewrite Theorem 4.C as follows:

Theorem 4.D For any integer r with 0 ≤ 2r ≤ n

(4.15)

(4.16)

If F_i are pairwise disjoint, than the right hand sides of (4.15) and (4.16) are equal due to Poincaré-Lefschetz duality for F, but we do not assume that F = ∪F_i is a manifold, and therefore the inequalities (4.15) and (4.16) are not equivalent and we have to keep both of them.

5. Slice inequalities

Again, as in the preceding section, let L₁,…,L_m ⊂ S²ⁿ⁺¹ be smooth oriented transversal to each other submanifolds constituting an m-colored link L = L₁ ∪∪L_m of dimension 2n - 1.

Let Λ_i ⊂ S²ⁿ⁺² be oriented closed smooth submanifolds transversal to each other and to S²ⁿ⁺¹, with ∂Λ_i ∩S²ⁿ⁺¹ = L_i. In this section we consider restrictions on homological characteristics of Λ = ∪_i=1^mΛ_i in terms of invariants of link L. Of course, some results of this kind can be deduced from the results of the preceding section, but an independent consideration gives better results.

5.1. No-nullity slice inequalities

The most general results in this direction are quite cumbersome. Therefore, let me start with weak but simple ones.

We will use the same algebraic objects as in the preceding section. In particular, ζ = (ζ₁,…,ζ_m) (S¹)^m, p₁,…,p_k ℤ[t₁,t₁^-1…,t_m,t_m^-1] are generators of the ideal of relations satisfied by complex numbers ζ_i. Integer d is the greatest common divisor of the integers p₁(1,…,1), …, p_k(1,…,1), if at least one of them does not vanish, and d = 0 otherwise. Cf. 4.2 and Appendix C.6. Finally,

Let μ : H₁(S²ⁿ⁺¹ \ L) → ℂ^× be the homomorphism which maps the meridian of L_i to ζ_i. The local coefficient system ℂ_μ on S²ⁿ⁺¹ \ L defined by μ extends to S²ⁿ⁺² \ Λ. We will denote the extension by the same symbol ℂ_μ.

The sphere S²ⁿ⁺¹ bounds in S²ⁿ⁺² two balls, hemi-spheres S₊²ⁿ⁺² and S_-²ⁿ⁺² such that ∂S₊²ⁿ⁺² = S²ⁿ⁺¹ and ∂S_-²ⁿ⁺² = -S²ⁿ⁺¹ with the orientations inherited from the standard orientation of S²ⁿ⁺². In H_n+1(S²ⁿ⁺² \ Λ; ℂ_μ) there is a (Hermitian or skew-Hermitian) intersection form. Its signature is zero by Theorem App.D.B, because Λ bounds a configuration of pairwise transversal submanifolds Δ = Δ₁ ∪∪ Δ_m in D²ⁿ⁺³ and ℂ_μ extends over D²ⁿ⁺³ \ Δ.

Theorem 5.A Under the assumption above,

(5.1)

Proof. The intersection form on H_n+1(S²ⁿ⁺² \ Λ; ℂ_μ) restricted to the images of H_n+1(S₊²ⁿ⁺² \ Λ; ℂ_μ) and H_n+1(S_-²ⁿ⁺² \ Λ; ℂ_μ) has signatures σ_ζ(L) and -σ_ζ(L), respectively. Therefore the dimension of each of the images is at least |σ_ζ(L)|.

The images are obviously orthogonal to each other with respect to the intersection form, because their elements can be realized by cycles lying in disjoin open hemi-spheres. Hence

On the other hand, by Theorem App.C.C, Summing up these two inequalities, we obtain the desired one.

5.1.1. General slice inequalities

Theorem 5.B Under assumptions above

(5.2)

Lemma 5.C Let j be the inclusion S²ⁿ⁺¹ \ L → S²ⁿ⁺² \ Λ. Then

(5.3)

Proof. Denote by i^± the inclusion S_±²ⁿ⁺² \ Λ → S²ⁿ⁺² \ Λ. Observe that the space H_n+1(S²ⁿ⁺² \ Λ; ℂ_μ) has a natural filtration:

(5.4)

The inclusion homomorphisms and the boundary homomorphism of the Mayer-Vietoris sequence of the triad (S²ⁿ⁺² \ Λ;S₊²ⁿ⁺² \ Λ,S_-²ⁿ⁺² \ Λ) are dual to each other with respect to the intersection forms: Since the intersection forms are non-singular, it follows that rkj_* = rk∂.

By exactness of the Mayer-Vietoris sequence, the rank of ∂ is the dimensions of the top quotient of the filtration (5.4), while the rank of j_* is the dimension of the smallest term j_*H_n+1(S²ⁿ⁺¹ \ L; ℂ_μ) of this filtration.

The middle term of the filtration contains the subspaces i_*⁺H_n+1(S₊²ⁿ⁺² \ Λ; ℂ_μ) and i_*^-H_n+1(S_-²ⁿ⁺² \ Λ; ℂ_μ). Their intersection is the smallest term, which is orthogonal to both of the subspaces. Therefore the dimension of the quotient of the middle term of the filtration by the smallest term is at least 2|σ_ζ(L)|

The dimension of the whole space H_n+1(S²ⁿ⁺² \ Λ; ℂ_μ) is the sum of the dimensions of the factors. We showed above that the top and lowest factor have the same dimensions equal to rkj_* and that the dimension of the middle factor is at least 2|σ_ζ(L)|.

Lemma 5.D For any exact sequence … C_k C _k-1… of vector spaces and any integers n and t

(5.5)

Proof. The Euler characteristic of the exact sequence

is rk(ρ_n) -∑ _s=0^2t-1(-1)^s dimC_n+s - rk(ρ_n + 2t), that is the difference between the left and right hand sides of (5.5). On the other hand, it vanishes, as the Euler characteristic of an exact sequence.

Lemma 5.E Let X be a topological space, A its subspace, ξ a local coefficient system on X with fiber ℂ. Then for any natural n and integer r

(5.6)

where b_k(*) = dim_ℂH_k(*;ξ).

Proof. Apply Lemma 5.D to the homology sequence of pair (X,A) with coefficients in ξ.

Lemma 5.F For any integer r with 0 ≤ r ≤

(5.7)

Proof. By Lemma 5.E applied to the pair (S²ⁿ⁺² \ Λ,S²ⁿ⁺¹ \ L), we obtain

(5.8)

From this inequality and inequality (5.3) we obtain

(5.9)

From this and the Poincaré duality (which states that H_n+1+s(S²ⁿ⁺² \ Λ; ℂ_μ) is isomorphic to H_n+1-s(S²ⁿ⁺² \ Λ; ℂ_μ)) the desired inequality follows.

Lemma 5.G

(5.10)

Proof. By Theorem App.C.C

(5.11)

By Poincaré duality, H_n+1+s(S²ⁿ⁺² \ Λ,S²ⁿ⁺¹ \ L;P) is isomorphic to H^n+1-s(S²ⁿ⁺² \ S²ⁿ⁺¹,Λ \ L;P). The latter is isomorphic to H^n-s(Λ \ L;P). By the universal coefficients formula, H^n-s(Λ \ L;P) is isomorphic to H_n-s(Λ \ L;P).

Lemma 5.H

(5.12)

Proof. By Theorem App.C.C

(5.13)

By Poincaré duality, H_n+1+s(S²ⁿ⁺² \ Λ;P) is isomorphic to H^n+1-s(S²ⁿ⁺²,Λ;P). From the sequence of pair (S²ⁿ⁺²,Λ) it follows that H^n+1-s(S²ⁿ⁺²,Λ;P) is isomorphic to H^n-s(Λ;P). By the universal coefficient formula, H^n-s(Λ;P) is isomorphic to H_n-s(Λ;P).

Proof of Theorem 5.B. Sum up the inequalities of the last three Lemmas. □

Appendix. Twisted homology

Appendix A. Twisted coefficients and chains

Appendix A.1. Local coefficient system

Let X be a topological space, and ξ be a ℂ-bundle over X with a fixed flat connection.

Here by a connection we mean operations of parallel transport: for any path s in X connecting points x and y the parallel transport T_s is an isomorphism from the fiber ℂ_x over x to the fiber ℂ_y over y, such that the parallel transport along product of paths equals the composition of parallel transports along the factors. In formula: T_uv = T_v ∘ T_u. A connection is flat, if the parallel transport isomorphism does not change when the path is replaced by a homotopic path.

A flat connection in a bundle ξ over a simply connected X gives a trivialization of ξ.

Another name for ξ is a local coefficient system with fiber ℂ.

Appendix A.2. Monodromy representation

Recall that for a path-connected locally contractible X (and in more general situations, which would not be of interest here) it is defined by the monodromy reprensentation π₁(X,x₀) → ℂ^×, where ℂ^× = ℂ \ 0 is the multiplicative group of ℂ. The monodromy representation assigns to σ π₁(X,x₀) a complex number ζ such that the parallel transport isomorphism along a loop which represents σ is multiplication by ζ.

Since ℂ^× is commutative, a homomorphism π₁(X,x₀) → ℂ^× factors through the abelianization π₁(X,x₀) → H₁(X). Thus a local coefficient system with fiber ℂ is defined also by a homology version μ : H₁(X) → ℂ^× of the monodromy representation, which can be considered also as a cohomology class belonging to H¹(X; ℂ^×).

The local coefficient system defined by a monodromy representation μ : H₁(X) → ℂ^× is denoted by ℂ^μ. Sometimes instead of μ we will write data which defines μ, for example the images under μ of generators of H₁(X) selected in a special way.

Appendix A.3. Twisted singular chains

Homology groups H_n(X;ξ) of X with coefficients in ξ is a classical invariant studied in algebraic topology. It is an immediate generalization of H_n(X; ℂ). Hence it is quite often ignored in textbooks on homology theory, I recall the singular version of the definition.

Recall that a singular p-dimensional chain of X with coefficients in ℂ is a formal finite linear combination of singular simplices f_i : T^p → X with complex coefficients.

A singular chain of X with coefficients in ξ is also a formal finite linear combination of singular simplices, but each singular simplex f_i : T^p → X appears in it with a coefficient taken from the fiber ℂ_fi(c) of ξ over f_i(c), where c is the baricenter of T^p. Of course, all the fibers of ξ are isomorphic to ℂ. So, a chain with coefficients in ξ can be identified with a chain with coefficients in ℂ, provided the isomorphisms ℂ_fi(c) → ℂ are selected. But they are not.

All singular p-chains of X with coefficients in ξ form a complex vector space C_p(X;ξ).

The boundary of such a chain is defined by the usual formula, but one needs to bring the coefficient from the fiber over f_i(c) to the fibers over f_i(c_i), where c_i is the baricenter of the ith face of T^p. For this, one may use translation along the composition with f_i of any path connecting c to c_i in T^p: since T^p is simply connected and the connection of ξ is flat, the result does not depend on the path.

These chains and boundary operators form a complex. Its homology is called homology with coefficients in ξ and denoted by H_p(X;ξ).

Homology with coefficients in the local coefficient system corresponding to the trivial monodromy representation 1 : H₁(X) → ℂ^× coincides with homology with coefficients in ℂ.

Appendix A.4. Twisted cellular chains

It is possible to calculate the homology with coefficients in a local coefficient system using cellular decomposition. Namely, a p-dimensional cellular chain of a cw-complex X with coefficients in a local coefficient system ξ is a formal finite linear combination of p-dimensional cells in which a coefficient at a cell belongs to the fiber over a point of the cell. It does not matter which point is this, because fibers over different points in a cell are identified via parallel transport along paths in the cell: any two points in a cell can be connected in the cell by a path unique up to homotopy.

In order to describe the boundary operator, let me define the incidence number (zσ_x : τ)_y ℂ_y where σ is a p-cell, τ is a (p- 1)-cell, z ℂ_x, x σ, y τ. The boundary operator is then defined by the incidence numbers:

Let f : D^p → X be a characteristic map for σ. Assume that a point y in (p- 1)-cell τ is a regular value for f. This means that y has a neighborhood U in τ such that f^-1(U) ⊂ S^p-1 ⊂ D^p is the union of finitely many balls mapped by f homeomorphically onto U. Connect f^-1(x) D^p with all the points of f^-1(y) by straight paths. Compositions of these paths with f are paths s₁,…s_N connecting x with y. Then put

where T_si is a parallel transport operator and ε_i = +1 or -1 according to whether f preserves or reverses the orientation on the ith ball out of N balls constituting f^-1(U).

Appendix B. Twisted acyclicity

Appendix B.1. Acyclicity of circle

According to one of the most fundamental properties of homology, the dimension of H₀(X; ℂ) is equal to the number of path-connected components of X. In particular, H₀(X; ℂ) does not vanish, unless X is empty.

This is not the case for twisted homology. A crucial example is the circle S¹. Let μ : H₁(S¹) → ℂ^× maps the generator 1 ℤ = H₁(S¹) to ζ ℂ^×.

Theorem App.B.A Twisted acyclicity of circle. H_*(S¹; ℂ^μ) = 0, iff  ζ≠1.

Proof. The simplest cellular decomposition of S¹ consists of two cells, one-dimensional σ₁ and zero-dimensional σ₀. One can easily see that ∂σ₁ = (ζ - 1)σ₀. Hence ∂ : C₁(S¹; ℂ^μ) → C₀(S¹; ℂ^μ) is an isomorphism, iff ζ≠0.

Appendix B.2. Vanishing of twisted homology

Corollary App.B.B Let X be a path connected space and μ : H₁(S¹×X) → ℂ^× be a homomorphism. Denote by ζ the image under μ of the homology class realized by a fiber S¹ × point. Then H_*(S¹ × X; ℂ^μ) = 0, if  ζ≠0.

Proof. Since H₁(S¹ × X) = H₁(S¹) × H₁(X), the homomorphism μ can be presented as product of homomorphisms μ₁ : H₁(S¹) → ℂ^× and μ₂ : H₁(X) → ℂ^× which can be obtained as compositions of μ with the inclusion homomorphisms. Thus ℂ^μ = ℂ^μ₁ ⊗ ℂ^μ₂, and we can apply Künneth formula

and refer to Theorem App.B.A.

Corollary App.B.C Let B be a path connected space, p : X → B a locally trivial fibration with fiber S¹. Let μ : H₁(X) → ℂ^× be a homomorphism. Denote by ζ the image under μ of homology class realized by a fiber of p. Then H_*(X; ℂ^μ) = 0, if  ζ≠0.

Proof. It follows from Theorem App.B.A via the spectral sequence of fibration p.

Appendix C. Estimates of twisted homology

Appendix C.1. Equalities underlying the Morse inequalities

Lemma App.C.A For a complex C : → C_i C _i-1 → of finite dimensional vector spaces over a field F

(C.1)

Proof. First, prove inequality (C.1) for n = 0. Since H_s(C) = Ker∂_s∕Im∂_s+1, we have dim_FH_s(C) = dimKer∂_s - dim_F Im∂_s+1. Further, dim_F Im∂_s+1 = rk∂_s+1, and dim_F Ker∂_s = dim_FC_s - rk∂_s. It follows

(C.2)

This is a special case of (C.1) with n = 0, r = s.

The general case follows from it: make alternating summation of (C.2) for s = r,…,2n + s.

Appendix C.2. Algebraic Morse type inequalities

Lemma App.C.B Let P and Q be fields, R be a subring of Q and let h : R → P be a ring homomorphism. Let C : → C_p → C_p-1 → → C₁ → C₀ be a complex of free finitely generated R-modules. Then for any n and r

Thus, the greater ranks of differentials, the smaller

Proof. Choose free bases in modules C_i. Let M_i be the matrix representing ∂_i : C_i → C_i-1 in these bases. The same matrix represents the differential ∂_i^Q of C ⊗_RQ. The matrix obtained from M_i by replacement the entries with their images under h represents the differential ∂_i^P of C ⊗_hP. The minors of the latter matrix are the images of the former one under h. Consequently, the rk∂_i^Q ≥ rk∂_i^P.

By Lemma App.C.A

and

Compare the right hand sides of these equalities. Since dim_PC_s ⊗_hP = dim_QC_s ⊗_RQ and, as it was shown above, rk∂_i^Q ≥ rk∂_i^P, the right hand side of the first of them is smaller than the right hands side of the second one.

Probably, the simplest application of Lemma App.C.B gives well-known upper estimation of the Betti numbers with rational coefficients by the Betti numbers with coefficients in a finite field. It follows from the universal coefficients formula.

Appendix C.3. Application to twisted homology

Theorem App.C.C Let X be a finite cw-complex, and μ : H₁(X) → ℂ^× be a homomorphism. If Imμ ⊂ ℂ^× generates a subring R of ℂ and there is a ring homomorphism h : R → Q, where Q is a field, such that hμ(H₁(X)) = 1, then we can apply Lemma App.C.B and get an upper estimation for dimensions of twisted homology groups in terms of dimensions of non-twisted ones.

(C.3)

Here are several situations in which the assumptions of this theorem are fulfilled.

Appendix C.4. Estimates by untwisted ℤ∕pℤ Betti numbers

Let H₁(X) be generated by g and ζ = μ(g) be an algebraic number. Assume that p is the minimal integer polynomial with relatively prime coefficients which annihilates ζ. Assume also that g(1) is divisible by a prime number p. Then for R we can take ℚ[ζ] ⊂ ℂ, for P the field ℤ∕pℤ, and for h the ring homomorphism ℚ[ζ] → ℤ∕pℤ mapping ζ1.

Here is a more general situation: Let H₁(X) be generated by g₁,…g_k, and ζ_i = μ(g_i) be an algebraic number for each i. Assume that p_i is the minimal integer polynomial with relatively prime coefficients which annihilates ζ_i. Assume also that the greatest common divisor of g₁(1),…, g_k(1) is divisible by a prime number p. Then for R we can take ℚ[ζ₁,…,ζ_k] ⊂ ℂ, for P the field ℤ∕pℤ, and for h the ring homomorphism ℚ[ζ₁,…,ζ_k] → ℤ∕pℤ mapping ζ_i1 for all i.

Appendix C.5. Estimates by rational Betti numbers

Let H₁(X) be generated by g and ζ = μ(g) be transcendent. Then for R we can take the ring ℤ[ζ,ζ^-1], for Q the field ℚ(ζ), for P the field ℚ, and for h the ring homomorphism ℤ[ζ] → ℚ which maps ζ to 1.

Appendix C.6. The most general estimates

Let H₁(X) be generated by g₁,…g_k, and the ideal of relations over the ring of integers satisfied by complex numbers ζ_i = μ(g_i) be generated by Laurent polynomials p₁,…,p_k ℤ[t₁,t₁^-1…,t_m,t_m^-1]. Let d be the greatest common divisor of the integers p₁(1,…,1), …, p_k(1,…,1), if at least one of them is not 0. Otherwise, let d = 0

In other words, consider the specialization homomorphism

Let K be the kernel of S, and let d be the generator of the ideal which is the image of K under the homomorphism

Then for R we can take the ring ℤ[ζ₁,ζ₁^-1,…,ζ_k,ζ_k^-1]. For Q we can take the quotient field of R, but since both Q and its quotient field are contained in ℂ, let us take Q = ℂ.

If d > 1, then we can take for P the field ℤ∕pℤ with any prime p which divides d. If d = 0, then let P = ℚ. The case d = 1 is the most misfortunate: then our technique does not give any non-trivial estimate. For d > 1 or d = 0 we have the inequality (C.3).

Appendix D. Twisted duality

Appendix D.1. Cochains, cohomology and tensor products

Cochain groups C^p(X;ξ) (which are vector spaces over ℂ) and cohomology H^p(X;ξ) are defined similarly: p-cochain with coefficients in ξ is a function assigning to a singular simplex f : T^p → X an element of ℂ_f(c), the fiber of ξ over f(c).

This can be interpreted as the chain complex of the local coefficient system Hom(ℂ,ξ) whose fiber over x X is Hom_ℂ(ℂ, ℂ_x). More generally, for any local coefficient systems ξ and η on X with fiber ℂ there is a local coefficient system Hom(ξ,η) constructed fiber-wise with the parallel transport defined naturally in terms of the parallel transports of ξ and η. If the monodromy representations of ξ and η are μ and ν, respectively, then the monodromy representation of Hom(ξ,η) is μ^-1ν : H₁(X) → ℂ^× : xμ^-1(x)ν(x).

Similarly, for any local coefficient systems ξ and η on X with fiber ℂ there is a local coefficient system ξ ⊗ η. If μ,ν : H¹(X) → ℂ^× are homomorphisms, then ℂ^μ ⊗ ℂ^ν is the local coefficient system ℂ^μν corresponding to the homomorphism-product μν : H¹(X) → ℂ^× : xμ(x)ν(x).

If ν = μ^-1 (that is μ(x)ν(x) = 1 for any x H¹(X)), then ℂ^μ ⊗ ℂ^ν is the non-twisted coefficient system with fiber ℂ.

In contradistinction to non-twisted case, there is no way to calculate H_n(X;ξ ⊗η) in terms of H_*(X;ξ) and H_*(X;η). Indeed, both H_*(S¹; ℂ^μ) and H_*(S¹; ℂ^μ-1 ) vanish, unless μ : H₁(S¹) → ℂ^× is trivial, but H₀(S¹; ℂ^μ ⊗ ℂ^μ-1 ) = H₀(S¹; ℂ) = ℂ.

Appendix D.2. Multiplications

Usual definitions of various cohomological and homological multiplications are easily generalized to twisted homology. For this one needs a bilinear pairing of the coefficient systems. (Recall that in the case of non-twisted coefficient system a pairing of coefficient groups also is needed.) For local coefficient systems ξ, η and ζ with fiber ℂ on X, a pairing ξ ⊕η → ζ is a fiber-wise map which is bilinear over each point of X. Given such a pairing, there are pairings

etc.

A pairing ξ ⊕η → ζ of local coefficients systems can be factored through the universal pairing ξ ⊕ η → ξ ⊗ η.

Since ℂ^μ ⊗ ℂ^μ-1 is a non-twisted coefficient system with fiber ℂ, this gives rise to a non-singular pairing

which induces a non-singular pairing Thus, the vector spaces H_p(X; ℂ^μ-1 ) and H^p(X; ℂ^μ) are dual.

Appendix D.3. Poincaré duality

Let X be an oriented connected compact manifold of dimension n. Then H_n(X,∂X) is isomorphic to ℤ and the orientation is a choice of the isomorphism, or, equivalently, the choice of a generator of H_n(X,∂X). We denote the generator by [X].

Let μ : H₁(X) → ℂ^× be a homomorphism. There are the Poincaré-Lefschetz duality isomorphisms

Similarly to the case of non-twisted coefficients, there are non-singular pairings: the cup-product pairing

and intersection pairing

(D.4)

However, the local coefficient systems of the homology or cohomology groups involved in a pairing are different, unless Imμ ⊂{±1}.

Appendix D.4. Conjugate local coefficient systems

Recall that for vector spaces V and W over ℂ a map f : V → W is called semi-linear if f(a + b) = f(a) + f(b) for any a,b V and f(za) = zf(a) for z ℂ and a V . This notion extends obviously to fiber-wise maps of complex vector bundles. If ξ and η local coefficient systems of the type that we consider, then fiber-wise semi-linear bijection ξ → η commuting with all the transport maps is called a semi-linear equivalence between ξ and η.

For any local coefficient system ξ with fiber ℂ on X there exists a unique local coefficient system on X which is semi-linearly equivalent to ξ. It is denoted by ξ and called conjugate to ξ. If ξ = ℂ^μ, then ξ is ℂ^μ, where μ(x) = μ(x) for any x H₁(X).

Appendix D.5. Unitary local coefficient systems

A homomorphism μ : H₁(X) → ℂ^× is called unitary if Imμ ⊂ S¹ = U(1) = {z ℂ∣|z| = 1}. In S¹ the inversion zz^-1 coincides with the complex conjugation: if |z| = 1, then z^-1 = z. Therefore if μ : H₁(X) → ℂ^× is unitary, then ℂ^μ = ℂ^μ-1 and there exists a semi-linear equivalence ℂ^μ → ℂ^μ-1 .

This semi-linear equivalence induces semi-linear equivalence

and similar semi-linear equivalences in cohomology and relative homology and cohomology.

Combining a semi-linear isomorphism

of this kind with the intersection pairing (D.4) we get a sesqui-linear pairing

(D.5)

(Sesqui-linear means that it is linear on the first variable, and semi-linear on the second one.) This pairing is non-singular, because the bilinear pairing (D.4) is non-singular, and (D.5) differs from it by a semi-linear equivalence on the second variable.

Appendix D.6. Intersection forms

Let X be an oriented connected compact smooth manifold of even dimension n = 2k and μ : H₁(X) → ℂ^× be a unitary homomorphism. Combining the relativisation homomorphism

with the pairing (D.5) for p = k define sesqui-linear form

(D.6)

It is called the intersection form of X.

If k is even, this form is Hermitian, that is α ∘ β = β ∘ α. If k is odd, it is skew-Hermitian, that is α ∘ β = -β ∘ α.

The difference between Hermitian and skew-Hermitian forms is not as deep as the difference between symmetric and skew-symmetric bilinear forms. Multiplication by i = turns a skew-Hermitian form into a Hermitian one, and the original form can be recovered. In order to recover, just multiply the Hermitian form by -i.

The intersection form (D.6) may be singular. Its radical, that is the orthogonal complement of the whole H_k(X; ℂ^μ), is the kernel of the relativisation homomorphism H_k(X; ℂ^μ) → H_k(X,∂X; ℂ^μ). It can be described also as the image of the inclusion homomorphism

where in_* is the inclusion homomorphism H₁(∂X) → H₁(X).

Appendix D.7. Twisted signatures and nullities

As well-known for any Hermitian form on a finite-dimensional space V there exists an orthogonal basis in which the form is represented by a diagonal matrix. The diagonal entries of the matrix are real. The number of zero diagonal entries is called the nullity, and the difference between the number of positive and negative entries is called the signature of the form. These numbers do not depend on the basis.

For a skew-Hermitian form by nullity and signature one means the nullity and signature of the Hermitian form obtained by multiplication of the skew-Hermitian form by i.

For a compact oriented 2k-manifold X and a homomorphism μ : H₁(X) → ℂ the signature and nullity of the intersection form

are denoted by σ_μ(X) and n_μ(X), respectively, and called the twisted signature and nullity of X.

The classical theorems about the signatures of the symmetric intersection forms of oriented compact 4k-manifolds are easily generalized to twisted signatures:

Theorem App.D.A Additivity of Signature. Let X be an oriented compact manifold of even dimension. If A and B are its compact submanifolds of the same dimension such that A ∪ B = X, IntA ∩ IntB = ∅ and ∂(A ∩ B) = ∅, then for any μ : H₁(X) → ℂ^×

where in denotes an appropriate inclusion.

Theorem App.D.B Signature of Boundary. Let X be an oriented compact manifold of odd dimension. Then σ_{μ in *}(∂X) = 0 for any μ : H₁(X) → ℂ^×.

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