Twisted acyclicity of a circle and signatures of a link

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 $\mathbb{C}$, see Appendix.

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 $\mathbb{C}$ 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 B).

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.

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.

Recall that a classical knot is a smooth simple closed curve in the 3-sphere $S^{3}$. 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 $S^{3}$. Classical knots incarnate the idea of knottedness: both the curve and $S^{3}$ are topologically standard, but the position of the curve in $S^{3}$ may be arbitrarily complicated topologically. Therefore a classical knot is rather a pair $(S^{3},K)$, where $K$ is a smooth submanifold of $S^{3}$ diffeomorphic to $S^{1}$.

A classical link is a pair $(S^{3},L)$, where $L$ is a smooth closed one-dimensional submanifold of $S^{3}$. If $L$ is connected, then this is a knot.

An exterior of a classical link $(S^{3},L)$ 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 $S^{1}$. An exterior $X(L)$ is a deformation retract of the complement $S^{3}\smallsetminus L$. It’s a nice replacement of $S^{3}\smallsetminus L$, because $\operatorname{Int}X(L)$ is homeomorphic to $S^{3}\smallsetminus L$, but $X(L)$ is compact manifold and has a nice boundary.

If $L$ consists of $m$ connected components, $L=K_{1}\cup\dots\cup K_{m}$, then by the Alexander duality $H_{0}(X(L))=\mathbb{Z}$, $H_{1}(X(L))=\mathbb{Z}^{m}$ and $H_{i}(X(L))=0$ for $i\neq 0,1$. The group $H_{1}(X(L))$ is dual to $H_{1}(L)$ with respect to the Alexander linking pairing $H_{1}(L)\times H_{1}(X(L))\to\mathbb{Z}$. Hence a basis of $H_{1}(L)$ defines a dual basis in $H_{1}(X(L))$. An orientation of $L$ determines a basis $[K_{1}]$, …, $[K_{m}]$ of $H_{1}(L)$, and the dual basis of $H_{1}(X(L))$, which is realized by meridians $M_{1}$, …, $M_{m}$ positively linked to $K_{1}$, …, $K_{m}$, respectively. (The meridians are fibers of a tubular fibration $\partial X(L)\to L$ over points chosen on the corresponding components.)

Therefore, if $L$ is oriented, then a local coefficient system on $X(L)$ with fiber $\mathbb{C}$ is defined by an $m$-tuple of complex numbers $(\zeta_{1},\dots,\zeta_{m}$, the images under the monodromy homomorphism $H_{1}(X(L))\to\mathbb{C}^{\times}$ of the generators $[M_{1}]$, …, $[M_{m}]$ of $H_{1}(X(L))$.

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

Let $L=K_{1}\cup\dots\cup K_{m}\subset S^{3}$ be a classical link, $\zeta_{i}\in\mathbb{C}$, $|\zeta_{i}|=1$, $\zeta=(\zeta_{1},\dots,\zeta_{m})\in(S^{1})^{m}$ and $\mu:H_{1}(S^{3}\smallsetminus L)\to\mathbb{C}^{\times}$ takes to $\zeta_{i}$ a meridian of $K_{i}$ positively linked with $K_{i}$.

Let $F_{1},\dots F_{m}\subset D^{4}$ be smooth oriented surfaces transversal to each other with $\partial F_{i}=F_{i}\cap\partial D^{4}=K_{i}$. Extend the tubular neighborhood of $L$ involved in the construction of $X(L)$ to a collection of tubular neighborhoods $N_{1}$, …, $N_{m}$ of $F_{1}$, …, $F_{m}$, 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 $N_{i}\cap N_{j}$ would contain only one point of $F_{i}\cap F_{j}$ 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 $D^{2}\times D^{2}$ on $B$.

To achieve this, one has to make the fibers of the tubular fibration $N_{i}\to F_{i}$ 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 $X(F)=D^{4}\smallsetminus\cup_{i=1}^{m}\operatorname{Int}N_{i}$. 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 $i=1,\dots,m$. These pieces are fibered over the corresponding pieces of $F_{i}$ with fiber $S^{1}$.

By the Alexander duality, the orientation of $F_{i}$ defines a homomorphism $H_{1}(X(F))\to\mathbb{Z}$, the linking number with $F_{i}$. These homomorphisms altogether determine a homomorphism $H_{1}(X(F))\to\mathbb{Z}^{m}$. For any $\zeta=(\zeta_{1},\dots,\zeta_{m})$, the composition of this homomorphism with the homomorphism

is a homomorphism $H_{1}(X(F))\to(\mathbb{C}^{\times})^{m}$ extending $\mu$. If each $F_{i}$ has no closed connected components, then this extension is unique. Let us denote it by $\overline{\mu}$.

According to D.6, in $H_{2}(X(F);\mathbb{C}_{\overline{\mu}})$ there is a Hermitian intersection form. Denote its signature by $\sigma_{\zeta}(L)$.

In the definition of signature $\sigma_{\zeta}(L)$ above one needs to numerate the components $K_{i}$ of $L$ to associate to each of them the corresponding component $\zeta_{i}$ of $\zeta$, 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 $S^{3}$ together with a map (called coloring) assigning to each connected component of $L$ a color in $\{1,\dots,m\}$. The sublink $L_{i}$ is constituted by the components of $L$ with color $i$ for $i=1,\dots,m$.

For an $m$-colored link $L=L_{1}\cup\dots\cup L_{m}$ and $\zeta=(\zeta_{1},\dots,\zeta_{m})\in(S^{1})^{m}$, the signature $\sigma_{\zeta}(L)$ is defined as above, but each component $K_{j}$ colored with color $i$ is associated to $\zeta_{i}$.

If $\zeta_{i}=-1$ for all $i=1,\dots,m$, then the signature $\sigma_{\zeta}(L)$ coincides with the Murasugi signature $\xi(L)$ introduced in [11]. If all $\zeta_{i}$ are roots of unity of a degree, which is a power of a prime number and all linking numbers $\operatorname{lk}(L_{i},L_{j})$ vanish, then $\sigma_{\zeta}(L)$ coincides with the signature defined by Florens [2].

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

There is a spectrum of objects considered as generalizations of classical knots and links. The closest generalization of classical knots are pairs $(S^{n},K)$, 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 $S^{n}$ 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 to 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 $\mathbb{C}^{2}$. Recall that for an algebraic curve $C\subset\mathbb{C}^{2}$ and a point $p\in C$ the boundary of a sufficiently small ball $B$ centered at $p$, the link $(\partial B,\partial B\cap C)$ 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 $C\subset\mathbb{C}^{n}$ 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.

By an $m$-colored link of dimension $n$ we shall mean a collection of $m$ oriented smooth closed $n$-dimensional submanifolds $L_{1}$, …, $L_{m}$ of the sphere $S^{n+2}$ such that any sub-collection has transversal intersection. The latter means that for any $x\in L_{i_{1}}\cap\dots\cap L_{i_{k}}$ the tangent spaces $T_{x}L_{i_{1}}$, …, $T_{x}L_{i_{k}}$ are transverse, that is, $\dim(T_{x}L_{i_{1}}\cap\dots\cap T_{x}L_{i_{k}})=n+2-2k$.

More generally, an $m$-colored configuration of transversal submanifolds in a smooth manifold $M$ is a family of $m$ smooth submanifolds $L_{1}$, …, $L_{m}$ 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 $\partial M\cap L_{i}=\partial L_{i}$ for any $i=1,\dots,m$.

As above, in Section 2.3, for any $m$-colored configuration $L$ of transversal submanifolds $L_{1}$, …, $L_{m}$ in $M$ one can find a collection of their tubular neighborhoods $N_{1}$, …, $N_{m}$ which agree with each other in the sense that for any sub-collection $L_{i_{1}}$, …, $L_{i_{\nu}}$ the intersection of the corresponding neighborhoods $N_{i_{1}}\cap\dots\cap N_{i_{\nu}}$ is neighborhood of the intersection $L_{i_{1}}\cap\dots\cap L_{i_{\nu}}$ fibered over this intersection with the corresponding poly-disk fiber.

Denote the complement $M\smallsetminus\cup_{i=1}^{m}\operatorname{Int}N_{i}$ 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 $L_{1}$, …, $L_{m}$. However, for the purposes of this paper it is easier to stay with the choices.

Let $L=L_{1}\cup\dots\cup L_{m}$ be an $m$-colored link of dimension $2n-1$ in $S^{2n+1}$.

As well known (see, e.g., [7]), for each oriented closed codimension 2 submanifold $K$ of $S^{2n+1}$ there exists an oriented smooth compact submanifold $F$ of $D^{2n+2}$ such that $\partial F=K$. Choose for each $L_{i}$ such a submanifold of $D^{2n+2}$, 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 $D^{2n+2}$, $F=F_{1}\cup\dots\cup F_{m}$ has an exterior $X(F)$. By the Alexander duality, $H^{1}(X(F);\mathbb{C}^{\times})$ is naturally isomorphic to $H_{2n}(F,L;\mathbb{C}^{\times})$. Let $\zeta=(\zeta_{1},\dots,\zeta_{m})\in(S^{1})^{m}$. Take $\sum_{i=1}^{m}\zeta_{i}[F_{i}]\in H_{2n}(F,L;\mathbb{C}^{\times})$ and denote by $\mu$ the Alexander dual cohomology class considered as a homomorphism $H_{1}(X(F))\to\mathbb{C}^{\times}$. Denote by $\mathbb{C}_{\mu}$ the local coefficient system on $X(F)$ corresponding to $\mu$.

According to D.6, in $H_{n+1}(X(F);\mathbb{C}_{\mu})$ there is an intersection form, which is Hermitian, if $n$ is odd, and skew-Hermitian, if $n$ is even. Denote its signature by $\sigma_{\zeta}(L)$.

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].

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

Recall that $\sigma_{\zeta}(L)$ can be obtained from $F$: for an appropriate local coefficient system $\mathbb{C}_{\mu}$ on $X(F)$, this is the signature of a Hermitian intersection form defined in $H_{n+1}(X(F);\mathbb{C}_{\mu})$. The signature of an Hermitian form cannot be greater than the dimension of the underlying space. In particular,

This can be considered as a restriction on a homological characteristic of $F$ in terms of invariants of $L$. However, $\dim_{\mathbb{C}}H_{n+1}(X(F);\mathbb{C}_{\mu})$ is not a convenient characteristic of $F$. It can be estimated in terms of more convenient ones.

Let $\zeta=(\zeta_{1},\dots,\zeta_{m})\in(S^{1})^{m}$. Let $p_{1},\dots,p_{k}\in\mathbb{Z}[t_{1},t_{1}^{-1}\dots,t_{m},t_{m}^{-1}]$ be generators of the deal of relations satisfied by complex numbers $\zeta_{i}$. Let $d$ be the greatest common divisor of the integers $p_{1}(1,\dots,1)$, …, $p_{k}(1,\dots,1)$, if at least one of these integers does not vanish, and zero otherwise. Cf. C.6. Let

By C.3.1,

The advantage of passing to homology with non-twisted coefficients is that we can use the Alexander duality:

Hence,

The inequality (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 (1) may be replaced by a smaller quantity, the rank of the form. The rank is known to be the rank of the homomorphism $H_{n+1}(X(F);\mathbb{C}_{\mu})\to H_{n+1}(X(F),\partial X(F);\mathbb{C}_{\mu})$. Let us estimate this rank.

The sum in the left hand side of the inequalities (4) is an invariant of the link $L$. Its special case for classical links with $r=0$ is know as $\zeta$-nullity and appeared in the Murasugi-Tristram inequalities and their generalizations.

Denote $\sum_{s=0}^{2r}(-1)^{s}\dim H_{n-s}(S^{2n-1}\smallsetminus L;\mathbb{C}_{\mu})$ by $n^{r}_{\zeta}(L)$ and call it $r$th $\zeta$-nullity of $L$.

By the twisted Poincaré duality (see Appendix D.3), $H_{n-s}(S^{2n-1}\smallsetminus L;\mathbb{C}_{\mu})$ is isomorphic to $H^{n+1+s}(S^{2n-1}\smallsetminus L;\mathbb{C}_{\mu})$. The latter complex vector space is dual to $H_{n+1+s}(S^{2n+1}\smallsetminus L;\mathbb{C}_{\mu^{-1}})$ and anti-isomorphic to $H_{n+1+s}(S^{2n+1}\smallsetminus L;\mathbb{C}_{\mu})$, see Appendix D.5. Therefore,

and $n^{r}_{\zeta}(L)=n^{r}_{\overline{\zeta}}(L)$. This sum is a part of the left hand side of (5).

Now we can rewrite Theorem 4.C as follows:

If $F_{i}$ are pairwise disjoint, than the right hand sides of (15) and (16) are equal due to Poincaré-Lefschetz duality for $F$, but we do not assume that $F=\cup F_{i}$ is a manifold, and therefore the inequalities (15) and (16) are not equivalent and we have to keep both of them.

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, $\zeta=(\zeta_{1},\dots,\zeta_{m})\in(S^{1})^{m}$, $p_{1},\dots,p_{k}\in\mathbb{Z}[t_{1},t_{1}^{-1}\dots,t_{m},t_{m}^{-1}]$ are generators of the ideal of relations satisfied by complex numbers $\zeta_{i}$. Integer $d$ is the greatest common divisor of the integers $p_{1}(1,\dots,1)$, …, $p_{k}(1,\dots,1)$, if at least one of them does not vanish, and $d=0$ otherwise. Cf. 4.2 and C.6. Finally,

The local coefficient system $\mathbb{C}_{\mu}$ on $S^{2n+1}\smallsetminus L$ defined by the monodromy $\mu:H_{1}(S^{2n+1}\smallsetminus L)\to\mathbb{C}^{\times}$ which maps the meridian of $L_{i}$ to $\zeta_{i}$ extends to $S^{2n+2}\smallsetminus\Lambda$. We will denote the extension by the same symbol $\mathbb{C}_{\mu}$.

The sphere $S^{2n+1}$ bounds in $S^{2n+2}$ two balls, hemi-spheres $S^{2n+2}_{+}$ and $S^{2n+2}_{-}$ such that $\partial S^{2n+2}_{+}=S^{2n+1}$ and $\partial S^{2n+2}_{-}=-S^{2n+1}$ with the orientations inherited from the standard orientation of $S^{2n+2}$. In $H_{n+1}(S^{2n+2}\smallsetminus\Lambda;\mathbb{C}_{\mu})$ there is a (Hermitian or skew-Hermitian) intersection form. Its signature is zero by Theorem D.7.2, because $\Lambda$ bounds a configuration of pairwise transversal submanifolds $\Delta=\Delta_{1}\cup\dots\cup\Delta_{m}$ in $D^{2n+3}$ and $\mathbb{C}_{\mu}$ extends over $D^{2n+3}\smallsetminus\Delta$.

Proof of Theorem 5.B. Sum up the inequalities of the last three Lemmas.$\square$

Let $X$ be a topological space, and $\xi$ be a $\mathbb{C}$-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 fibre $\mathbb{C}_{x}$ over $x$ to the fibre $\mathbb{C}_{y}$ over $y$, such that the parallel transport along product of paths equals the composition of parallel transports along the factors. In formula: $T_{uv}=T_{v}\circ T_{u}$. 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 $\xi$ over a simply connected $X$ gives a trivialization of $\xi$.

Another name for $\xi$ is a local coefficient system with fibre $\mathbb{C}$.

Recall that if $X$ is a path-connected locally contractible space (and in more general situations, which would not be of interest here) a local coefficient system with fibre $\mathbb{C}$ on $X$ is defined by the monodromy reprensentation $\pi_{1}(X,x_{0})\to\mathbb{C}^{\times}$, where $\mathbb{C}^{\times}=\mathbb{C}\smallsetminus 0$ is the multiplicative group of $\mathbb{C}$. The monodromy representation assigns to $\sigma\in\pi_{1}(X,x_{0})$ a complex number $\zeta$ such that the parallel transport isomorphism along a loop representing $\sigma$ is multiplication by $\zeta$

Since $\mathbb{C}^{\times}$ is commutative, a homomorphism $\pi_{1}(X,x_{0})\to\mathbb{C}^{\times}$ factors through the abelianization $\pi_{1}(X,x_{0})\to H_{1}(X)$. Thus a local coefficient system with fibre $\mathbb{C}$ is defined also by a homology version $\mu:H_{1}(X)\to\mathbb{C}^{\times}$ of the monodromy representation, which can be considered also as a cohomology class belonging to $H^{1}(X;\mathbb{C}^{\times})$.

The local coefficient system defined by a monodromy representation $\mu:H_{1}(X)\to\mathbb{C}^{\times}$ is denoted by $\mathbb{C}^{\mu}$. Sometimes instead of $\mu$ we will write data which defines $\mu$, for example the images under $\mu$ of generators of $H_{1}(X)$ selected in a special way.

Homology groups $H_{n}(X;\xi)$ of $X$ with coefficients in $\xi$ is a classical invariant studied in algebraic topology. It is an immediate generalization of $H_{n}(X;\mathbb{C})$. 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 $\mathbb{C}$ is a formal finite linear combination of singular simplices $f_{i}:T^{p}\to X$ with complex coefficients.

A singular chain of $X$ with coefficients in $\xi$ is also a formal finite linear combination of singular simplices, but each singular simplex $f_{i}:T^{p}\to X$ appears in it with a coefficient taken from the fiber $\mathbb{C}_{f_{i}(c)}$ of $\xi$ over $f_{i}(c)$, where $c$ is the baricenter of $T^{p}$. Of course, all the fibres of $\xi$ are isomorphic to $\mathbb{C}$. So, a chain with coefficients in $\xi$ can be identified with a chain with coefficients in $\mathbb{C}$, provided the isomorphisms $\mathbb{C}_{f_{i}(c)}\to\mathbb{C}$ are selected. But they are not.

All singular $p$-chains of $X$ with coefficients in $\xi$ form a complex vector space $C_{p}(X;\xi)$.

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 $i$th 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 $\xi$ 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 $\xi$ and denoted by $H_{p}(X;\xi)$.

Homology with coefficients in the local coefficient system corresponding to the trivial monodromy representation $1:H_{1}(X)\to\mathbb{C}^{\times}$ coincides with homology with coefficients in $\mathbb{C}$.

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 $\xi$ 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 $(z\sigma_{x}:\tau)_{y}\in\mathbb{C}_{y}$ where $\sigma$ is a $p$-cell, $\tau$ is a $(p-1)$-cell, $z\in\mathbb{C}_{x}$, $x\in\sigma$, $y\in\tau$. The boundary operator is defined by the incidence numbers: $\partial(z\sigma)=\sum_{\tau}(z\sigma_{x}:\tau)_{y}\tau$.

Let $f:D^{p}\to X$ be a characteristic map for $\sigma$. Assume that a point $y$ in $(p-1)$-cell $\tau$ is a regular value for $f$. This means that $y$ has a neighborhood $U$ in $\tau$ such that $f^{-1}(U)\subset S^{p-1}\subset D^{p}$ is the union of finitely many balls mapped by $f$ homeomorphically onto $U$. Connect $f^{-1}(x)\in D^{p}$ with all the points of $f^{-1}(y)$ by straight paths. Compositions of these paths with $f$ are paths $s_{1}$,…$s_{N}$ connecting $x$ with $y$. Then put

where $\varepsilon_{i}=+1$ or $-1$ according to whether $f$ preserves or reverses the orientation on the $i$th ball out of $N$ balls constituting $f^{-1}(U)$.

According to one of the most fundamental properties of homology, the dimension of $H_{0}(X;\mathbb{C})$ is equal to the number of path-connected components of $X$. In particular, $H_{0}(X;\mathbb{C})$ does not vanish, unless $X$ is empty.

This is not the case for twisted homology. A crucial example is the circle $S^{1}$. Let $\mu:H_{1}(S^{1})\to\mathbb{C}^{\times}$ maps the generator $1\in\mathbb{Z}=H_{1}(S^{1})$ to $\zeta\in\mathbb{C}^{\times}$.

Probably, the simplest application of Lemma C.2.1 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.

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

Let $H_{1}(X)$ be generated by $g$ and $\zeta=\mu(g)$ be an algebraic number. Assume that $p$ is the minimal integer polynomial with relatively prime coefficients which annihilates $\zeta$. Assume also that $g(1)$ is divisible by a prime number $p$. Then for $R$ we can take $\mathbb{Q}[\zeta]\subset\mathbb{C}$, for $P$ the field $\mathbb{Z}/p\mathbb{Z}$, and for $h$ the ring homomorphism $\mathbb{Q}[\zeta]\to\mathbb{Z}/p\mathbb{Z}$ mapping $\zeta\mapsto 1$.

Here is a more general situation: Let $H_{1}(X)$ be generated by $g_{1}$,…$g_{k}$, and $\zeta_{i}=\mu(g_{i})$ be an algebraic number for each $i$. Assume that $p_{i}$ is the minimal integer polynomial with relatively prime coefficients which annihilates $\zeta_{i}$. Assume also that the greatest common divisor of $g_{1}(1)$,…, $g_{k}(1)$ is divisible by a prime number $p$. Then for $R$ we can take $\mathbb{Q}[\zeta_{1},\dots,\zeta_{k}]\subset\mathbb{C}$, for $P$ the field $\mathbb{Z}/p\mathbb{Z}$, and for $h$ the ring homomorphism $\mathbb{Q}[\zeta_{1},\dots,\zeta_{k}]\to\mathbb{Z}/p\mathbb{Z}$ mapping $\zeta_{i}\mapsto 1$ for all $i$.

Let $H_{1}(X)$ is generated by $g$ and $\zeta=\mu(g)$ is transcendent. Then for $R$ we can take the ring $\mathbb{Z}[\zeta,\zeta^{-1}]$, for $Q$ the field $\mathbb{Q}(\zeta)$, for $P$ the field $\mathbb{Q}$, and for $h$ the ring homomorphism $\mathbb{Z}[\zeta]\to\mathbb{Q}$ which maps $\zeta$ to 1.

Let $H_{1}(X)$ is generated by $g_{1}$,…$g_{k}$, and the ideal of relations over the ring of integers satisfied by complex numbers $\zeta_{i}=\mu(g_{i})$ is generated by Laurent polynomials $p_{1},\dots,p_{k}\in\mathbb{Z}[t_{1},t_{1}^{-1}\dots,t_{m},t_{m}^{-1}]$. Let $d$ be the greatest common divisor of the integers $p_{1}(1,\dots,1)$, …, $p_{k}(1,\dots,1)$, 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 $\mathbb{Z}[\zeta_{1},\zeta_{1}^{-1},\dots,\zeta_{k},\zeta_{k}^{-1}]$. For $Q$ we can take the quotient field of $R$, but since both $Q$ and its quotient field are contained in $\mathbb{C}$, let us take $Q=\mathbb{C}$.

If $d>1$, then we can take for $P$ the field $\mathbb{Z}/p\mathbb{Z}$ with any prime $p$ which divides $d$. If $d=0$, then let $P=\mathbb{Q}$. 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.32).

Cochain groups $C^{p}(X;\xi)$ (which are vector spaces over $\mathbb{C}$) and cohomology $H^{p}(X;\xi)$ are defined similarly: $p$-cochain with coefficients in $\xi$ is a function assigning to a singular simplex $f:T^{p}\to X$ an element of $\mathbb{C}_{f(c)}$, the fibre of $\xi$ over $f(c)$.

This can be interpreted as the chain complex of the local coefficient system $\operatorname{Hom}(\mathbb{C},\xi)$ whose fibre over $x\in X$ is $\operatorname{Hom}_{\mathbb{C}}(\mathbb{C},\mathbb{C}_{x})$. More generally, for any local coefficient systems $\xi$ and $\eta$ on $X$ with fibre $\mathbb{C}$ there is a local coefficient system $\operatorname{Hom}(\xi,\eta)$ defined fibre-wise with parallel transport defined naturally in terms of parallel transports of $\xi$ and $\eta$. If the monodromy representations of $\xi$ and $\eta$ are $\mu$ and $\nu$, respectively, then the monodromy representation of $\operatorname{Hom}(\xi,\eta)$ is $\mu^{-1}\nu:H_{1}(X)\to\mathbb{C}^{\times}:x\mapsto\mu^{-1}(x)\nu(x)$.

Similarly, for any local coefficient systems $\xi$ and $\eta$ on $X$ with fibre $\mathbb{C}$ there is a local coefficient system $\xi\otimes\eta$. If $\mu,\nu:H^{1}(X)\to\mathbb{C}^{\times}$ are homomorphisms, then $\mathbb{C}^{\mu}\otimes\mathbb{C}^{\nu}$ is the local coefficient system $\mathbb{C}^{\mu\nu}$ corresponding to the homomorphism-product $\mu\nu:H^{1}(X)\to\mathbb{C}^{\times}:x\mapsto\mu(x)\nu(x)$.

If $\nu=\mu^{-1}$ (that is $\mu(x)\nu(x)=1$ for any $x\in H^{1}(X)$), then $\mathbb{C}^{\mu}\otimes\mathbb{C}^{\nu}$ is the non-twisted coefficient system with fibre $\mathbb{C}$.

In contradistinction to non-twisted case, there is no way to calculate $H_{n}(X;\xi\otimes\eta)$ in terms of $H_{*}(X;\xi)$ and $H_{*}(X;\eta)$. Indeed, both $H_{*}(S^{1};\mathbb{C}^{\mu})$ and $H_{*}(S^{1};\mathbb{C}^{\mu^{-1}})$ vanish, unless $\mu:H_{1}(S^{1})\to\mathbb{C}^{\times}$ is trivial, but $H_{0}(S^{1};\mathbb{C}^{\mu}\otimes\mathbb{C}^{\mu^{-1}})=H_{0}(S^{1};\mathbb{C})=\mathbb{C}$.

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 $\xi$, $\eta$ and $\zeta$ with fibre $\mathbb{C}$ on $X$, a pairing $\xi\oplus\eta\to\zeta$ is a fibre-wise map which is bilinear over each point of $X$. Given such a pairing, there are pairings

etc.

A pairing $\xi\oplus\eta\to\zeta$ of local coefficients systems can be factored through the universal pairing $\xi\oplus\eta\to\xi\otimes\eta$.

Since $\mathbb{C}^{\mu}\otimes\mathbb{C}^{\mu^{-1}}$ is a non-twisted coefficient system with fibre $\mathbb{C}$, this gives rise to a non-singular pairing

which induces a non-singular pairing

Thus, the vector spaces $H_{p}(X;\mathbb{C}^{\mu^{-1}})$ and $H^{p}(X;\mathbb{C}^{\mu})$ are dual.

Let $X$ be an oriented connected compact manifold of dimension $n$. Then $H_{n}(X,\partial X)$ is isomorphic to $\mathbb{Z}$ and the orientation is a choice of the isomorphism, or, equivalently, the choice of a generator of $H_{n}(X,\partial X)$. We denote the generator by $[X]$.

Let $\mu:H_{1}(X)\to\mathbb{C}^{\times}$ 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

However, the local coefficient systems of the homology or cohomology groups involved in a pairing are different, unless $\operatorname{Im}\mu\subset\{\pm 1\}$.

Recall that for vector spaces $V$ and $W$ over $\mathbb{C}$ a map $f:V\to W$ is called semi-linear if $f(a+b)=f(a)+f(b)$ for any $a,b\in V$ and $f(za)=\overline{z}f(a)$ for $z\in\mathbb{C}$ and $a\in V$. This notion extends obviously to fibre-wise maps of complex vector bundles. If $\xi$ and $\eta$ local coefficient systems of the type that we consider, then fibre-wise semi-linear bijection $\xi\to\eta$ commuting with all the transport maps is called a semi-linear equivalence between $\xi$ and $\eta$.

For any local coefficient system $\xi$ with fiber $\mathbb{C}$ on $X$ there exists a unique local coefficient system on $X$ with a semi-linear equivalent to $\xi$. It is denoted by $\overline{\xi}$ and called conjugate to $\xi$. If $\xi=\mathbb{C}^{\mu}$, then $\overline{\xi}$ is $\mathbb{C}^{\overline{\mu}}$, where $\overline{\mu}(x)=\overline{\mu(x)}$ for any $x\in H_{1}(X)$.

A homomorphism $\mu:H_{1}(X)\to\mathbb{C}^{\times}$ is called unitary if $\operatorname{Im}\mu\subset S^{1}=U(1)=\{z\in\mathbb{C}\mid|z|=1\}$. In $S^{1}$ the inversion $z\mapsto z^{-1}$ coincides with the complex conjugation: if $|z|=1$, then $z^{-1}=\overline{z}$. Therefore if $\mu:H_{1}(X)\to\mathbb{C}^{\times}$ is unitary, then $\overline{\mathbb{C}^{\mu}}=\mathbb{C}^{\mu^{-1}}$ and there exists a semi-linear equivalence $\mathbb{C}^{\mu}\to\mathbb{C}^{\mu^{-1}}$.

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.33) we get a sesqui-linear pairing

(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.33) is non-singular, and (D.34) differs from it by a semi-linear equivalence on the second variable.

Let $X$ be an oriented connected compact smooth manifold of even dimension $n=2k$ and $\mu:H_{1}(X)\to\mathbb{C}^{\times}$ be a unitary homomorphism. Combining the relativisation homomorphism

with the pairing (D.34) for $p=k$ define sesqui-linear form

It is called the intersection form of $X$.

If $k$ is even, this form is Hermitian, that is $\alpha\circ\beta=\overline{\beta\circ\alpha}$. If $k$ is odd, it is skew-Hermitian, that is $\alpha\circ\beta=-\overline{\beta\circ\alpha}$.

The difference between Hermitian and skew-Hermitian forms is not as deep as the difference between symmetric and skew-symmetric bilinear forms. Multiplication by $i=\sqrt{-1}$ 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.35) may be singular. Its radical, that is the orthogonal complement of the whole $H_{k}(X;\mathbb{C}^{\mu})$, is the kernel of the relativisation homomorphism $H_{k}(X;\mathbb{C}^{\mu})\to H_{k}(X,\partial X;\mathbb{C}^{\mu})$. It can be described also as the image of the inclusion homomorphism

where $\operatorname{in}_{*}$ is the inclusion homomorphism $H_{1}(\partial X)\to H_{1}(X)$.

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 $\mu:H_{1}(X)\to\mathbb{C}$ the signature and nullity of the intersection form

are denoted by $\sigma_{\mu}(X)$ and $n_{\mu}(X)$, 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: