A functorial extension of the abelian Reidemeister torsions of three-manifolds
Let be a field and let be a multiplicative subgroup. We consider the category of -dimensional cobordisms equipped with a representation of their fundamental group in , and the category of -linear maps defined up to multiplication by an element of . Using the elementary theory of Reidemeister torsions, we construct a “Reidemeister functor” from to . In particular, when the group is free abelian and is the field of fractions of the group ring , we obtain a functorial formulation of an Alexander-type invariant introduced by Lescop for -manifolds with boundary; when is trivial, the Reidemeister functor specializes to the TQFT developed by Frohman and Nicas to enclose the Alexander polynomial of knots. The study of the Reidemeister functor is carried out for any multiplicative subgroup . We obtain a duality result and we show that the resulting projective representation of the monoid of homology cobordisms is equivalent to the Magnus representation combined with the relative Reidemeister torsion.
Let be the category of -dimensional cobordisms introduced by Crane and Yetter [CY99], and whose definition we briefly recall. The objects of are integers , and correspond to compact connected oriented surfaces of genus with one boundary component. Indeed, we fix for every a model surface whose boundary is identified with , and we also fix a base point on . The morphisms in the category are the equivalence classes of cobordisms between the surfaces and . To be more specific, a cobordism from to is a pair consisting of a compact connected oriented -manifold and an orientation-preserving homeomorphism where
two such pairs and are equivalent if there exists a homeomorphism such that . We shall denote a pair simply by the upper-case letter , with the convention that the boundary-parametrization is always denoted by the lower-case letter ; besides, we denote by the restriction of composed with the inclusion of into . Thus the cobordism “runs” from the bottom surface to the top surface . The degree of the cobordism is the integer .
The composition of two cobordisms in is defined by identifying to and, for any integer , the identity of the object is the cylinder with the obvious boundary-parametrization. Our model surfaces also come with an identification of the boundary-connected sum with the surface for any . Thus the category is enriched with a monoidal structure : the tensor product of two integers is the sum , and the tensor product of two cobordisms is their boundary-connected sum .
Let now be an abelian group. The category can be refined to the category of cobordisms equipped with a representation of the first integral homology group in . To be more specific, an object of is a pair consisting of an integer and a group homomorphism . A morphism in the category is a pair where and is a group homomorphism such that . The composition of two morphisms and , such that , is defined by
where is the composition in and is defined from and by using the Mayer–Vietoris theorem. The monoidal structure of also extends to the category : the tensor product of objects is
where is identified with , and the tensor product of morphisms is
where is identified with .
Consider now a commutative ring and fix a subgroup of its group of units. Let be the category whose objects are -graded -modules and whose morphisms are graded -linear maps of arbitrary degree, up to multiplication by an element of . The usual tensor product of graded -modules defines a monoidal structure on the category : here the tensor product of two graded -linear maps and is defined with Koszul’s rule, i.e we set for any homogeneous elements . In this paper, we construct and study two functors from to for some specific rings and specific subgroups .
Our first functor is based on the “Alexander function” introduced by Lescop [Les98].
For any compact orientable -manifold with boundary, this function is defined on an exterior power of the Alexander module of relative to a boundary point,
and it takes values in a ring of Laurent polynomials. Lescop’s definition proceeds in a rather elementary way using a presentation of the Alexander module.
Let be a finitely generated free abelian group, and let be its group ring. Then there is a degree-preserving monoidal functor
which, at the level of objects, assigns to any
the exterior algebra of the -twisted relative homology group of the pair .
The -linear map associated to a morphism of is defined in a very simple way from the Alexander function of using the decomposition of into two parts, and . The fact that the Alexander function gives rise to a functor on the category of cobordisms is somehow implicit in [Les98], where Lescop studies the behaviour of her invariant under some specific gluing operations. As it contains the Alexander polynomial of knots in a natural way, we call the Alexander functor.
Since the works of Milnor [Mil62] and Turaev [Tur75], it is known that the Alexander polynomial of knots
and -manifolds can be interpreted as a special kind of abelian Reidemeister torsion.
We follow this direction to define our second functor, which we call the Reidemeister functor.
In the sequel, the category associated to a field and a subgroup of is denoted by .
Let be a field and let be a subgroup of . Then there is a degree-preserving monoidal functor
which, at the level of objects, assigns to any
the exterior algebra of the -twisted relative homology group of the pair .
The construction of the functor uses the elementary theory of Reidemeister torsions, but note that we need to consider cell chain complexes which are not necessarily acyclic. When is a finitely generated free abelian group and is the field of fractions of , we recover the functor by extension of scalars. Thus it suffices to study the functor and this is done using basic properties of combinatorial torsions. For instance, we compute its restriction to the monoid of homology cobordisms (which includes the mapping class group of a surface): we find that the representation induced by is equivalent to the Magnus representation combined with the Reidemeister torsion of cobordisms relative to the top surface. We also give a formula for in terms of Heegaard splittings and we show that satisfies some duality properties, which generalize the symmetry properties of the Alexander polynomial of knots and -manifolds.
It is expected that Turaev’s refinements of the Reidemeister torsion [Tur86, Tur89] can be adapted to refine to a kind of “monoidal” degree-preserving functor from to the category of graded -vector spaces: the sign ambiguity would presumably be fixed using homological orientations on the manifolds, while the ambiguity in would be fixed by adding Euler structures. (Observe however that, since we use Koszul’s rule and we allow morphisms in to have non-zero degree, this category is not monoidal in the usual sense of the word.)
We now explain how our constructions are related to prior works. Soon after the emergence of quantum invariants of -manifolds in the late eighties, there have been several works which showed how to interpret the classical Alexander polynomial in this new framework. A more general problem was then to extend the Alexander polynomial to a functor from a category of cobordisms to a category of vector spaces following, as close as possible, the axioms of a TQFT [Ati88]. This problem has been solved by Frohman and Nicas who used elementary intersection theory in -representation varieties of surfaces [FN91]. (See also [FN94] for a much more general construction using -representations.) Later, Kerler showed that the Frohman–Nicas functor is in fact equivalent to a TQFT based on a certain quasitriangular Hopf algebra [Ker03a]. The Alexander polynomial of a knot in an integral homology -sphere is recovered from this functor by taking the “graded” trace of the endomorphism associated to the cobordism that one obtains by “cutting” along a Seifert surface of . It turns out that, in the case , the Alexander functor is equivalent to the Frohman–Nicas functor. Note that the way how their functor determines the Alexander polynomial is somehow extrinsic, in that it goes through the choice of a Seifert surface. On the contrary, the functor for intrinsically contains the Alexander polynomial of oriented knots in oriented integral homology -spheres by considering any knot of this type as a “bottom knot” in the style of [Hab06], i.e by regarding its exterior as a morphism in . Since this functorial extension of the Alexander polynomial applies to cobordisms equipped with an element of , it should be regarded as a kind of HQFT with target – see [Tur10] – rather than a TQFT.
Our constructions are also related to the work of Bigelow, Cattabriga and the first author [BCF12], which provides a functorial extension of the Alexander polynomial to the category of tangles instead of the category of cobordisms. To describe this relation, let be the monoidal category whose objects are pairs of non-negative integers – corresponding to surfaces with punctures – and whose morphisms are cobordisms with tangles inside. Clearly the category contains the category of [CY99] as well as the usual category of (unoriented) tangles in the standard ball; for any abelian group , there is an obvious refinement of the category . When is the infinite cyclic group generated by , the usual category of oriented tangles in the standard ball can be regarded as a subcategory of by only considering those representations of tangle exteriors that send any oriented meridian to the generator . The functors and constructed in this paper could be extended to the category using similar methods, but with more technicality. When is infinite cyclic, the restriction of the resulting functor to would coincide with the “Alexander representation of tangles” constructed in [BCF12]. We also mention Archibald’s extension of the Alexander polynomial [Arc10], which is based on diagrammatic presentations of tangles: her invariant seems to be very close to the invariant constructed in [BCF12] and it is stronger since it is defined without ambiguity in .
Finally, our approach is related to the work of Cimasoni and Turaev on “Lagrangian representations of tangles” [CT05, CT06]. These representations are functors from the category to the category of “Lagrangian relations” (which generalizes the category of -modules equipped with non-degenerate skew-hermitian forms) and, for string links, they are equivalent to the (reduced) Burau representation [LD92, KLW01]. The constructions of [CT05, CT06] could be adapted to the case of cobordisms in order to obtain a functor from to the category of “Lagrangian relations” over the ring . In the case of homology cobordisms, the resulting functor would be equivalent to the (reduced) Magnus representation but it would miss the relative Reidemeister torsion: so it would be weaker than the functor .
The paper is organized as follows. A first part deals exclusively with the Alexander functor:
§2 gives the construction of the functor (Theorem I)
and §3 explains how the classical Alexander polynomial of knots is contained in .
Next, the Reidemeister functor is constructed in §4 (Theorem II) and it is proved to be a generalization of in §5.
(Thus, we provide two different proofs of the functoriality of .)
Starting from there, we focus on the study of and indicate the resulting properties for .
The abelian Reidemeister torsions of knot exteriors and closed -manifolds are shown to be determined by in §6.
The functor restricts to a projective representation of the monoid of homology cobordisms, which we fully compute in §7.
We also explain in §8 how to calculate using Heegaard splittings of cobordisms,
and we prove in §9 a duality result for which involves the twisted intersection form of surfaces.
Finally, the paper ends with a short appendix recalling the definition and basic properties of the torsion of chain complexes.
Notation and conventions. Let be a commutative ring. The exterior algebra of an -module is denoted by
the multivector defined by a finite family of elements of is still denoted by . If is free of rank , a volume form on is an isomorphism of -modules .
Let be a topological space with base point . The maximal abelian cover of based at is denoted by , and the preferred lift of is denoted by . (Here we assume the appropriate assumptions on to have a universal cover.) For any oriented loop in based at , the unique lift of to starting at is denoted by .
Unless otherwise specified, (co)homology groups are taken with coefficients in the ring of integers ; (co)homology classes are denoted with square brackets . For any subspace such that and any ring homomorphism , we denote by the -twisted homology of the pair , namely
If is another pair of spaces and is a continuous map,
the corresponding homomorphism is still denoted by .
If a base point is given and ,
the -linear map induced by is also denoted by .
Acknowledgements. This work was partially supported by the French ANR research project “Interlow” (ANR-09-JCJC-0097-01). The authors would like to thank the referee for some useful comments.
2. The Alexander functor
We firstly review the Alexander function of a -manifold with boundary following [Les98]. (Note that the terminology “Alexander function” has a very different meaning in [Tur86].) Next, we construct the Alexander functor . In this section, we fix a finitely generated free abelian group ; the extension of a group homomorphism to a ring homomorphism is still denoted by .
2.1. The Alexander function
Let be a compact connected orientable -manifold with connected boundary. We fix a base point and a group homomorphism . The genus of is the integer , i.e the genus of the surface .
There exists a presentation of the -module whose deficiency is .
We consider a decomposition of with a single -handle, -handles and -handles. Since the boundary of has genus , we have . This handle decomposition defines a -dimensional complex onto which deformation retracts. The complex has a single -cell (which we assume to be ), -cells and -cells. Thus we obtain a presentation of the -module with generators and relations. ∎
We now simplify our notation by setting and .
Definition 2.2 (Lescop [Les98]).
Consider a presentation of the -module with deficiency :
Let be the -module freely generated by the symbols , and regard as elements of . Then the Alexander function of with coefficients is the -linear map defined by
for any , which we lift to some in an arbitrary way.
The map can be concretely computed as follows: if one considers the matrix defined by the presentation (2.1) of , and if one adjoins to this matrix some row vectors giving in the generators , then is the determinant of the resulting matrix. It is shown in [Les98, §3.1] that, up to multiplication by a unit of (i.e., an element of ), the map does not depend on the choice of the presentation (2.1).
Let be the field of fractions of . The following lemma, which is implicit in [Les98], shows that either the Alexander function is trivial or it induces by extension of scalars a volume form on .
We have , and if and only if .
Let be the matrix with entries in corresponding to the presentation (2.1) of the -module . The multiplication defines a linear map whose cokernel is . Therefore
Clearly, we have so that .
Assume that and let be a matrix obtained by adding arbitrary rows to . Then so that all the minors of of order vanish. By expanding the determinant of successively along the last rows, we see that and deduce that .
Assume that . Then so that has a non-zero minor of order . Let be the indices of the columns of not pertaining to . Then . ∎
2.2. Definition of
In order to define a functor , we associate to any object of the exterior algebra
of the -module , which is free of rank . Next, we associate to any morphism a -linear map
of degree as follows. We denote by the interval , which connects the base point of the bottom surface to that of the top surface . We set , and . Then, for any integer , the image of any is defined by the following property:
Here is an arbitrary volume form on . Due to the choices of and of the presentation of , the map is only defined up to multiplication by an element of . Besides, observe that is trivial on for any and any .
The next two lemmas show that the above paragraph defines a monoidal functor from to , which proves Theorem I of the Introduction. The first lemma is related to Property 6 of the Alexander function in [Les98], while the second lemma seems to be new.
For any morphisms and , we have
We set , , , and
In the statement of the lemma and in the proof below, we identify
in the obvious way with
Since the intersection of and in is a -disk which retracts onto , the Mayer–Vietoris theorem gives an isomorphism . If , then by Lemma 2.3 so that ; the same lemma applied to shows that
so that and (2.2) trivially holds true. Therefore, we can assume in the sequel that and .
Let : we aim at showing that is equal to
(Recall that we are using Koszul’s rule in the definition of the tensor product of morphisms in the category .) It is enough to prove that, for any integers such that and any , the identity
holds true up to multiplication by an element of independent of (and, in particular, independent of ). In the sequel, we fix some volume forms and on and respectively, and we assume that the volume form on is defined by
for any and . By definition of , we have
If , then by our assumptions, so that is torsion; we deduce that ; on the other hand, the degree of is so that as well; thus (2.3) trivially holds true if . If , then and the same conclusion applies. Therefore, we can assume in the sequel that and .
To proceed, we consider a presentation and a presentation . By the above-mentioned isomorphism between and , we obtain a presentation
Note that, with these choices of presentations, the matrix corresponding to is the direct sum of the matrices corresponding to and . Therefore, we get
For any morphisms and such that , we have
The next subsection is devoted to the proof of Lemma 2.5.
2.3. Proof of the functoriality of
We use the notations of Lemma 2.5 and we set
Let be a basis of : we set and for all . We consider presentations of the following form:
Applying the Mayer–Vietoris theorem to , we obtain that the -module is generated by
subject to the relations
In the sequel, we set and . Let and : we wish to compute
using the previous presentation of . For this, we do some computations in where and denotes the free -module generated by the symbols listed at (2.6). Set , . Then, we have
Here the sums are taken over all parts , denotes the complement of , is the wedge of the for , is the wedge of the for and is the signature of the permutation (where the elements of in increasing order are followed by the elements of in increasing order). A sign is missing in the second sum but, since the presentation of is arbitrary of deficiency , we can assume that its number of relations is even.
In the sequel, we omit the “tilde” notation to distinguish elements of from their lifts to . Note that, in the above sums, the multivector has degree which is greater than as soon as ; similarly, the multivector has degree which is greater than as soon as ; since and are respectively the numbers of generators of and in the above presentations, the summand corresponding to vanishes for and for . Therefore the above sums are actually indexed by the subsets having cardinality , and we get