A Salvetti complex for Toric Arrangements and its fundamental group
aa r X i v : . [ m a t h . C O ] F e b A Salvetti complex for Toric Arrangements and itsfundamental group
Giacomo d’Antonio and Emanuele DelucchiNovember 21, 2018
Abstract
We describe a combinatorial model for the complement of a com-plexified toric arrangement by using nerves of acyclic categories. Thisgeneralizes recent work of Moci and Settepanella [12] on thick toricarrangements.Moreover, we study its fundamental group and compute a presen-tation thereof.
Introduction A toric arrangement is, roughly speaking, a family of subtori of a complextorus ( C ∗ ) n . The study of the topology and the combinatorics of suchobjects is a fairly new, yet thriving topic. As the very first attempt inthis direction we can cite the work of Lehrer [8], where the representationtheory on the cohomology of the configuration space F ( C ∗ , n ) of n pointsin the pointed complex plane is studied. This configuration space is indeedthe complement of a toric arrangement. Its topology is already well known,since F ( C ∗ , n ) ≃ F ( C , n + 1).The foundation of the topic can be traced to the paper [3] by De Conciniand Procesi. There the main objects are defined, the cohomology of thecomplement of a toric arrangement is studied (mainly from the point ofview of algebraic geometry) and some applications of the theory are out-lined. In particular, these authors treat the topic with the explicit goal ofgeneralizing the theory of hyperplane arrangements, and they put all this ina wider context that encompasses applications in topics such as the studyof integer points of Zonotopes and box splines. An extensive account of the1ork of De Concini and Procesi on this new subject can be found in theirforthcoming book [4].Ehrenborg, Readdy and Slone [5] take another point of view, study-ing toric arrangements on the “compact torus” ( S ) n and considering theproblem of enumerating faces of the induced decomposition of the compacttorus.The next step is the work of Moci, in particular his papers [9], [10] and[11], developing the theory with a special focus on combinatorics. In partic-ular, Moci introduces a two-variable polynomial that encodes enumerativeinvariants of many of the different objects populating the landscape out-lined by De Concini and Procesi in [4]. The same author, in joint work withSettepanella [12], studied the homotopy type of the complement of a specialclass of toric arrangements ( thick arrangements, see Section 2 below). Inthis work we will use a similar but more general approach, so that our re-sults hold for a wider class of toric arrangements, which we call complexified because of structural affinity with the case of hyperplane arrangements.Indeed, a rich and lively theory exists for arrangements of hyperplanesin affine complex space. An affine hyperplane is the (translate of the) ker-nel of a linear form. An affine arrangement is called complexified if thedefining linear forms are real linear forms. Equivalently, a complexifiedarrangement induces an arrangement of real (affine) hyperplanes that de-termines it completely. It is from this equivalent formulation that we takeinspiration for our definition of complexified toric arrangements : these arethe arrangements that induce an arrangement in the compact torus andare determined by it. Every ‘thick’ arrangement in the sense of [12] is com-plexified, and there are nonthick complexified arrangements.It is our explicit goal to try to present the theory and the results in away that at once underlines the structural similarities with the theory ofhyperplane arrangements and shows where (and why) the peculiarities ofthe toric theory are.We will try to do so by using a combinatorial tool that aptly generalizesthe idea of a poset and its order complex: acyclic categories and theirnerves.Our first main result shows that the combinatorial structure of a com-plexified toric arrangement can be used to construct an acyclic categorywhose nerve is homotopy equivalent to the complement of the arrange-ment. It is this acyclic category that we suggest to call Salvetti category .2ccordingly, we suggest to call the complex obtained as the nerve of theSalvetti category the
Salvetti complex of the toric arrangement. Our resultspecializes to the construction of [12] for the case of thick arrangements.The second main result is the computation of a (finite) presentation forthe fundamental group of the arrangement’s complement, appearing herefor the first time, to the best of our knowledge.Our paper begins with a review of the relevant background facts abouthyperplane arrangements and acyclic categories: this will be the content ofSection 1. Then, in Section 2 we give a brief account of the theory of toricarrangements, with the special goal to set some notations, terminology andbasic facts that will be relevant for the sequel. With Section 3 we will enterthe core ouf our work, defining our combinatorial model (Definition 14)and proving our first main result (Theorem 1): the nerve of the Salvetticategory models the homotopy type of any complexified toric arrangement.The computation of our presentation for the fundamental group will becarried out in Section 4, and the presentation itself will be given as oursecond main result, Theorem 2.
Acknowledgements
The second author was introduced to this subject by Luca Moci, whom hethanks for the many interesting and insightful conversations.The two authors started their collaboration during the special periodon arrangements and configuration spaces at the Centro De Giorgi of theScuola Normale Superiore. This work was mainly carried out at the de-partment of mathematics of the University of Bremen.Both authors gladly acknowledge conversations with prof. Mario Sal-vetti, thanking him for pointing out a flaw in Section 4 of an early versionof this paper.
Before turning our attention to toric arrangements, let us briefly reviewsome basics about hyperplane arrangements.Let families of linear forms l , . . . l n ∈ Hom( C d , C ) and scalars z , . . . z n
3e given. For every i = 1 , . . . , n we have then an affine hyperplane H i := { z ∈ C d : l i ( z ) = z i } . The (affine) hyperplane arrangement in C d defined by the given linear formsand scalars is the set A = { H , . . . , H n } . The arrangement is called complexified if its defining forms are real, i.e., l i ∈ Hom( R d , R ) for all i .There are several descriptions of the homotopy type of the complementof a set of hyperplanes in complex space. In this paper we will take in-spiration by the work of Salvetti [14], where a regular polytopal complexwhich embeds in the complement of a complexified real arrangement as adeformation retract is constructed: the Salvetti complex . Definition 1.
Let A be a complexified real arrangement in C n . We write D = D ( A ) for the cellular decomposition induced by A on R n and F = F ( A ) for its face poset (ordered by inclusion ). The maximal elements of F are called chambers .Given a face F ∈ F , we can consider the affine subspace | F | it generates,say | F | = y + L for a linear subspace L . The projection map π F : R n → R n /L maps chambers of A on chambers of the arrangement A F = { π F ( H ) : F ⊆ H } . (1)We define the Salvetti poset
Sal( A ) on the element set { [ F, C ] :
F, C ∈ D and F ≤ C in F } by the order relation[ F , C ] ≤ [ F , C ] ⇐⇒ F ≤ F in F and π F ( C ) = π F ( C ) . (2) Definition 2.
Let A be a complexified real arrangement in C n ; the Salvetticomplex of A is the simplicial complex S = S ( A ) := ∆(Sal( A )). Proposition 1.1 (Salvetti [14]) . The complex S ( A ) is a deformation re-tract of the arrangement’s complement, i.e., of the space C d \ S ni =1 H i The reader should be aware that this is in contrast to some of the existing literature. S is the barycentric subdivision of a regularpolytopal complex that we want now to describe.Consider the graph G ( A ) with the set of chambers of A as vertex setand edge set given by E = { e [ F,C ] = ( C, D ) : F ∈ D , codim ( F ) = 1 , F ≤ C, op( C, F ) = D } where op( C, F ) is the opposite chamber of C with respect to F . We can as-sign a direction to an edge e [ F,C ] by thinking it oriented from C to op( C, F ).We say that every edge e [ F,C ] of G ( A ) ‘crosses’ the hyperplane which sup-ports F . A hyperplane H separates two chambers C and D if a straightline segment from any point in the interior of C to any point in the interiorof D intersects H .A path in G ( A ) from a vertex (chamber) C to a vertex (chamber) D is positive minimal if it is directed and if it never crosses any hyperplanemore than once. Definition 3.
The unsubdivided Salvetti complex is the polytopal complex(i) whose 1-skeleton is the realization of the graph G ( A );(ii) whose k -cells corresponds to the pairs [ F, C ] with F ∈ F ( A ) a faceof codimension k and C a chamber with F ≤ C ;(iii) where the 1-skeleton of a k -cell e [ F,C ] is attached along the minimalpositive directed paths in G ( A ) from C to OP ( C, F ).The reader can now easily convince her- or himself that condition (2)states exactly when a cell e [ F ,C ] lies in the boundary of the cell e [ F ,C ] in the unsubdivided Salvetti complex. In other words, the poset Sal( A )is the face poset of the unsubdivided Salvetti complex (and hence S is itsbarycentric subdivision).We close this section by noting that the coarser structure of the unsub-divided complex has been used already in the seminal paper by Salvetti[14] to compute the fundamental group of the complement of a complexi-fied hyperplane arrangement. We will return to this topic and review thetechniques introduced by Salvetti when we will compute our presentationfor the fundamental group of complexified toric arrangements.5 .2 Acyclic categories Let us now introduce the idea of acyclic categories. We can think of acycliccategories as posets in which more than one relation between two elementsis allowed. Our main general reference for this topic is Kozlov’s book [7]and, for specifics about actions of infinite groups, Babson and Kozlov’spaper [1].
Definition 4. An acyclic category is a small category C , such that:(i) the only morphisms that have inverses are the identities;(ii) the only morphism from an object to itself is the identity.We will write O ( C ) for the objects of C and M ( C ) for its morphisms.Acyclic categories occur sometimes in the literature as “loop-free cate-gories” or “scwol”s (small category without loops, cfr. [2]). To an acyclic category we can associate its nerve . This is the generalizationof the order complex of a poset. Meaning that, if the category is indeed aposet (that is, between two arbitrary objects there is at most a morphism),then its nerve is indeed its order complex. In general, however, the nerveof an acyclic category will not be a simplicial complex. Instead it will bea regular trisp . Trisps –also called ∆ -complexes in [6]– are a generalizationof simplicial complexes.To define trisps we start with the notion of a polytopal complex . This is,roughly speaking, a complex obtained gluing polytopal cells. We will followKozlov’s book ([7, Definition 2.39]), except that we don’t require polytopalcomplexes to be regular . More precisely:
Definition 5. A polytopal complex is a topological space X obtained withthe following construction:(i) Start with the 0-skeleton X , a discrete set of points.(ii) At the k -th step we attach all the k -dimensional faces. These areconvex polytopes P ⊆ R k , attached along the maps f : ∂P → X k − .The attaching maps are required to be cellular . Furthermore, theinterior of each face of P has to be attached homeomorphically to theinterior of a face in X k − . The k -skeleton is defined as X k = (cid:16)G P ⊔ X k − (cid:17) / x ∼ f ( x ) X = ∪ k ∈ N X k .A trisp can be described then as a polytopal complex in which everycell is a simplex. For more details about trisps and for the precise definitionwe refer to [7].Having introduced trisps, we can now define the nerve of an acycliccategory. Definition 6.
Let C be an acyclic category; the nerve ∆( C ) is the trisp(i) whose k -dimensional simplexes are k -length chains of composable mor-phisms σ = a m → a m → a m → · · · m k → a k , (ii) where the boundary simplexes of a simplex σ as above are defined as: ∂ σ = a m → a m → · · · m k → a k ∂ j σ = a m → · · · m j − → a j − m j +1 ◦ m j → a j +1 m j +2 → · · · m k → a k ∂ k σ = a m → a m → a m → · · · m k − → a k − Acyclic categories can be used to describe the topology of a polytopal com-plex. For this section we refer to [2, III C .1]. Definition 7.
Let X be a polytopal complex; its face category is the acycliccategory F ( X )(i) whose set of objects O ( F ( X )) corresponds to the set of cells of X ,(ii) where for every cell P of X and for every face F of the polytope P there is a morphism m P,F : Q → P ∈ M ( F ( X )), where Q is the faceof X upon which F is glued,(iii) where if P m P ,F → P m P ,F → P is a composable chain of morphisms in F ( X ), then m P ,F ◦ m P ,F = m P ,F ′ (here F ′ is the face of F which is glued upon F ⊆ P , and henceupon P ). 7 emark . We notice that in point (iii) of definition 7 the face F ′ is uniquelydetermined, since the (restriction of the) gluing map F → P is a cellularhomeomophism. Definition 8.
The barycentric subdivision of a polytopal complex X , isthe regular trisp B ( X ) = ∆( F ( X )): the nerve of the face category.The face category describes the topology of a polytopal complex in thefollowing sense: Proposition 1.2.
Let X be a polytopal complex, then the geometric real-ization of B ( X ) is homeomorphic to X . These concepts have been already used in metric geometry and espe-cially in geometric group theory. There acyclic categories are called scwol s,the nerve of a category is called the geometric realization and the face cat-egory of a polytopal complex is called the barycentric subdivision . Moredetails can be found in [2, III C ]. We will now introduce toric arrangements together with some constructionthat will be needed in the following.The n -dimensional complex torus is the space ( C ∗ ) n ; the n -dimensional compact torus is ( S ) n . A character of a complex torus T is an affinehomomorphism χ : T → C ∗ , i.e., a Laurent polynomial in C [ x ± , . . . x ± n ]that is also a group homorphism with respect to the complex multiplication.One can easily see that, then, χ is a Laurent monomial and for x ∈ T wehave χ ( x ) = x α x α · · · x α n n with α = ( α , . . . , α n ) ∈ Z n . The correspondence between a character χ ∈ Λ and the associated integervector α χ makes the set of characters into a lattice Λ ∼ = Z n with the oper-ation defined by pointwis multiplication of characters.The above, “concrete” definitions suffice for many purposes. It is how-ever convenient for us and common in the literature to give a more abstractdefinition, starting with any (finitely generated) lattice Λ, which will be ourcharacter lattice. We then define the corresponding torus to be T Λ := Hom Z (Λ , C ∗ ) . T Λ ∼ = ( C ∗ )rk Λ whose compo-nents are the evaluation maps on the elements of the basis. Analogously,the compact torus on the lattice Λ is defined as Hom Z (Λ , S ). Definition 9. A complexified toric arrangement is a finite collection A = { ( χ, a ) : χ ∈ Λ , a ∈ S } , where Λ is a finitely generated lattice. We may think of A as the arrange-ment of the hypersurfaces H χ,a = { x ∈ T Λ : χ ( x ) = a } , where ( χ, a ) runsover A .The complement of A is then M ( A ) := ( C ∗ ) n \ [ ( χ,a ) ∈ A H χ,a . Remark . Toric arrangements were first defined in [3] as sets of pairs( χ, a ) with a ∈ C ∗ . Restricting the constants to S allows for the same A to define an arrangement of subtori on the compact torus ( S ) n (sincea Laurent monomial maps ( S ) n on S ). The analogy with the case ofcomplexified hyperplane arrangements motivates our terminology. Definition 10.
Let A be a complexified toric arrangement. With D = D ( A ) we will denote the induced cell-decomposition of the compact torus( S ) n . Remark . On the other hand, [10] and [12] define a toric arrangement asan arrangement of kernels of characters (thus requiring a = 1). This cutsout a whole class of arrangements (e.g. A = { t = − , s = − } in ( C ∗ ) ).Moreover one can have hypersurfaces with many connected components,which are not in general kernels of characters (e.g. t = 1). Definition 11.
A toric arrangement A on a k -dimensional torus T Λ iscalled essential ifrk A := rk (cid:10) χ ∈ Λ : ( χ, a ) ∈ A for some a ∈ S (cid:11) = k. This can be stated equivalently by saying that the layers of maximal codi-mension are points. 9 emark . Consider a (non essential) arrangement A = { ( χ , a ) , . . . , ( χ n , a n ) } with rk A = l < k . Then there exists an essential arrangement A ′ (the essentialisation of A ) such that M ( A ) = M ( A ′ ) × ( C ∗ ) k − l . With the notation of Definition 13, A ′ = A Γ whereΓ = { χ ∈ Λ : ∃ k ∈ Z : χ k ∈ h χ , . . . , χ n i} . In other words, it is not restrictive to consider essential arrangements.
Assumption 1.
Unless otherwise stated, we will always assume our ar-rangement to be complexified and essential . Remark . As is the case in the theory of hyperplane arrangements, one ofthe goals of the study of toric arrangements is to relate topological prop-erties of the complement M ( A ) to the combinatorics of the arrangement A . In the hyperplane case, the combinatorics is expressed by the poset ofintersections L ( A ) of elements of A . In the case of toric arrangements, theresults of [3] suggest that the right combinatorial invariant may be the posetof layers C ( A ), where a layer is a connected component of an intersectionof hypersurfaces H χ,a , and the partial order is given by inclusion.In the case of hyperplane arrangements, L ( A ) does not suffice to de-termine the homotopy type of the complement: indeed, there are explicitexamples of arrangements with isomorphic intersection poset, whose com-plements are not homeomorphic (see [13]). In the case of a complexifiedreal hyperplane arrangement, the homeomorphism type of the complementis determined instead by the face poset of the induced (regular CW) de-composition D ( A ) of R n .In general, the homotopy type of a complexified toric arrangement can-not be described in terms of the face poset of the induced decompositionof the compact torus. Indeed Moci and Settepanella in [12] characterizeexactly the arrangements for which this poset describes the homotopy typeof M ( A ): these are the arrangements A for which D ( A ) is a regular cell-complex or, in the terminology of [12], thick arrangements.In our take at this matter we would like to keep full generality andtherefore suggest to replace the poset of faces with the following moregeneral object. 10 efinition 12. Let A be a complexified toric arrangement. Then F ( A )will denote the face category of the complex D ( A ) (see Definition 10). Remark . Thick arrangements are precisely those arrangement for whichthe face category F ( A ) is a poset. For such arrangements the constructionof the Salvetti complex in the affine case translates almost literally to thetoric case (see [12] for the details).Our construction is more general in the sense that it does not assumethickness and, moreover, in the thick case it specializes to the complexconsidered by Moci and Settepanella. The operation of passing to sub arrangements, while intuitive and elemen-tary in the case of hyperplane arrangements, needs some careful consider-ation in the toric case.Let Γ be a subgroup of the lattice Λ. Then T Γ := Hom Z (Γ , S ) is acompact (rk Γ)-torus and the inclusion i Γ : Γ → Λ induces a surjection π Γ : T Λ → T Γ given by restriction: π Γ ( p ) = p | Γ . Definition 13.
Given a subgroup Γ ⊆ Λ and an arrangement A in T Λ , wedefine the arrangement A Γ = { ( χ, a ) ∈ A : χ ∈ Γ } . Proposition 2.1.
The map π Γ : T Λ → T Γ induces a cellular map π cell Γ : D ( A ) → D ( A Γ ) .Proof. We can choose a basis x , . . . , x n for Λ such that Γ = h x k , . . . , x k l l i .The isomorphism T Λ ≃ C n is given by evaluation on the chosen basis: p ( p ( x ) , . . . p ( x n )). Therefore the projection ( C ∗ ) n → ( C ∗ ) l is given bythe map ( y , . . . , y n ) ( y k , . . . , y k l l ). This map is continuous and mapshypersurfaces (of A Γ ⊆ A in ( C ∗ ) n ) onto hypersurfaces (of A Γ in ( C ∗ ) l ),hence is cellular.The construction of A Γ is to be thought of as the analogue of the quo-tient construction in (1). In particular, given any face F ∈ F ( A ) we canlet Γ be the lattice Λ F := { χ ∈ Λ | χ is constant on F } . A F := A Λ F , π F := π cellΛ F : D ( A ) → D ( A F ) . (3)The fact that π F is cellular implies that π F induces a morphism of acycliccategories π F : F ( A ) → F ( A F ). In order to connect the theory of toric arrangements to that of hyperplanearrangements, we will look at a particular covering space of a toric arranga-ment complement. Again, for our purposes it is convenient to work withabstract tori.Consider the following covering map p : Hom Z (Λ , C ) → Hom Z (Λ , C ∗ ) ϕ exp ◦ ϕ where exp : C → C ∗ is the exponential map, i.e., z e πiz . Notice thatHom Z (Λ , C ) ∼ = C n and, through this isomorphism, p is just the universalcovering map ( t , . . . , t n ) ( e πit , . . . , e πit n )of the torus T Λ . Furthermore, p restricts to a universal covering map R n ∼ = Hom Z (Λ , R ) → Hom Z (Λ , S ) ∼ = ( S ) n of the compact torus, under which the preimage of a toric arrangement A is the (infinite) affine hyperplane arrangement A ↾ = { ( χ, a ′ ) ∈ Λ × R | ( χ, e πia ′ ) ∈ A } , or, in coordinates: A ↾ = {h α, x i = a ′ | ( x α , e πia ′ ) ∈ A } . Here α ∈ Z n and x α is the associated character x α · · · x α n n . With this def-inition p induces a cellular map p : D ( A ↾ ) → D ( A ).The arrangement A ↾ is a locally finite complexified affine hyperplanearrangement and therefore admits a Salvetti complex S ↾ = S ↾ ( A ) := S ( A ↾ ) . A ↾ The character lattice Λ acts cellulary on S ↾ and continously on the coveringspace M ( A ). These two actions are compatible, meaning that the embed-ding S ↾ → M ( A ↾ ) constructed in [14] is Λ-equivariant (more precisely, itcan be so constructed). Example . Figure 1 shows the Salvetti complex for the arrangement A ↾ ,with A = { ( ts, , ( ts − , } . The green cells belong to the same Λ-orbit.With the previous constructions in mind, we can now restate a keyresult of [12]. Proposition 2.2 ([12, Lemma 1.1]) . Let A be an essential toric arrange-ment; the embedding S ↾ → M ( A ↾ ) induces an embedding S ↾ / Λ → M ( A ) of the quotient S ↾ in the complement M ( A ) as a deformation retract.Remark . In the proof of Proposition 2.2 given in [12] the hypotesis ofessentiality is required. Indeed the construction of the homotopy inverse ψ : S ↾ / Λ → M ( A ) does not work for non-essential arrangements. We now head towards the first main theorem of this paper, introducing thenotion of Salvetti complex for general complexified toric arrangements with13 construction that specializes to the complex of [12] in the case of thickarrangements.
Definition 14 (Salvetti category) . Let A be a toric arrangement on ( C ∗ ) n .The Salvetti Category of A is the acyclic category ζ = ζ ( A ) defined asfollows:(i) the objects are the morphisms in F ( A ) between faces and chambers O ( ζ ) = { m : F → C : m ∈ M ( F ( A )) , C chamber } ;(ii) for every morphism n : F → F in F ( A ), and for every pair m : F → C , m : F → C in O ( ζ ) there is a morphism ( n, m , m ) : m → m if and only if π F ( m ) = π F ( m ); (4)where π F is the morphism of face categories induced by the cellularmap in (3);(iii) let m i : F i → C i for i = 1 , , O ( ζ ), suppose the pairs( m , m ) and ( m , m ) satisfy condition (4), then the pair ( m , m )satisfies the same condition and we can define for morphisms n : F → F , n ′ : F → F the composition( n ′ , m , m ) ◦ ( n, m , m ) = ( n ◦ n ′ , m , m ) . Definition 15.
Let A be a toric arrangement; its Salvetti complex is thenerve ∆( ζ ( A )).We can now state the main theorem of this section. Theorem 1.
Let Λ be a lattice and A be a complexified toric arrangementin T Λ . The nerve ∆( ζ ( A )) embeds in M ( A ) as a deformation retract.Remark . Being the nerve of an acyclic category, ∆( ζ ( A )) is a regulartrisp. Remark . In the case of affine arrangements of hyperplanes, the Salvettiposet defined in Section 1.1 is indeed the poset of cells of a regular CW-complex, of which the (simplicial) Salvetti complex is the barycentric sub-division. Earlier we have called this the “unsubdivided” Salvetti complex.14ur goal now is to describe a CW complex of which the nerve ∆( ζ ) is thebarycentric subdivision. This complex will not be regular in general, butthe resulting economy in terms of cells will come in handy in the followingconsiderations.Let then A denote a toric arrangement. Every cell of the unsubdividedSalvetti complex of A ↾ corresponds to the topological closure of the star of avertex [ F, C ] of the subdivided complex. Because the projection Sal( A ↾ ) → ζ is a covering of categories, the interior of the star of any vertex of thenerve ∆(Sal( A ↾ )) is mapped homeomorphically to the interior of the starof its image. This gives a canonical CW-structure on ∆( ζ ). The acycliccategory ζ is precisely the face category of the resulting CW complex.In particular, the explicit determination of the boundary maps of thiscomplex is now reduced to a straightforward computation.Before we can get to the proof, some preparatory considerations are inorder. In order to proceed with the argument we still need to spend a few wordson the quotient construction of (1) and its toric analogue.Let F be a face of D ( A ) and let Λ F be the sublattice of characters inΛ that are constant on F . Every ϕ ∈ Λ F is then constant on the affinesubspace spanned by F , which we write y + L for y ∈ R n and L a linearsubspace of R n : therefore ϕ vanishes on L . Then we have an isomorphism ρ : R n /L → Hom Z (Λ F , R ) . (5)Recall from (3) the arrangement A F = { ( χ, a ) ∈ A : χ ∈ Λ F } ⊆ A in Hom Z (Λ F , R ). The isomorphism ρ from (5) does not map the arrange-ment ( A ↾ ) F onto ( A F ) ↾ . Indeed ( A F ) ↾ contains all the translates of thehyperplanes in ( A ↾ ) F . That is( A ↾ ) F ⊆ A F ↾ = { ( χ, a + k ) | ( χ, a ) ∈ ( A ↾ ) F , k ∈ Z } and therefore we have a natural cellular support map s : D ( A F ↾ ) → D ( A ↾ F )15 π F ↾ π ↾ F Figure 2: Restriction vs. CoveringThe map π F of (3) lifts (via p ) to a map R rk Λ → R rk Λ F which in-duces a cellular map π F ↾ : D ( A ↾ ) → D (( A F ) ↾ ) and the following diagramcommutes D ( A ↾ ) π F ↾ / / p (cid:15) (cid:15) D (( A F ) ↾ ) p (cid:15) (cid:15) D ( A ) π F / / D ( A F ) (6)On the other hand, in Hom(Λ , R ) we have the projection from (2),which we call π ↾ F and in terms of which the Salvetti complex of A ↾ isdefined, which is π ↾ F : D ( A ↾ ) → D (( A ↾ ) F )and is related to π F ↾ via π ↾ F = s ◦ π F ↾ . Figure 2 shows an example of projections π ↾ F and π F ↾ . Lemma 3.1.
Let F , F , C , C ∈ F ( A ↾ ) with C , C chambers, F ≤ C and F ≤ F ≤ C . Then π F ↾ ( C ) = π F ↾ ( C ) ⇐⇒ π ↾ F ( C ) = π ↾ F ( C ) . roof. The direction ⇒ follows since π ↾ F = s ◦ π F ↾ . For ⇐ : if π ↾ F ( C ) = π ↾ F ( C ), then π F ↾ ( C ) = π F ↾ ( C + λ ), for some λ ∈ Λ F . But since F isa common face of C and C , it has to be λ = 0. Corollary 3.2.
Let [ F , C ] , [ F , C ] denote two elements of Sal A ↾ , theSalvetti poset of A ↾ . Then [ F , C ] ≤ [ F , C ] ⇐⇒ F ≥ F in F ( A ) and π F ↾ ( C ) = π F ↾ ( C ) Our strategy for the proof of Theorem 1 will be to prove that the toricSalvetti complex ∆( ζ ) is the quotient of the action Λ y S ↾ in the categoryof trisps. For this, we need first to take care of some ground work. Lemma 3.3.
Let A be a complexified toric arrangement. Then there is acovering q : F ( A ↾ ) → F ( A ) of acyclic categories with Galois group Λ and F ( A ) = F ( A ↾ ) / Λ as a quotient of acyclic categories.Proof. Let F ∈ D ( A ↾ ) be a face of the affine arrangement A ↾ . In particular F is a polytope and p ( F ) ∈ D ( A ) is a face of A . We can then use F apolytopal model of p ( F ) in Definition 7 and map a morphism F ′ ≤ F tothe corresponding morphism m F ′ ,F .This defines a functor q : F ( A ↾ ) → F ( A ). Furthermore q is a coveringof categories in the sense of [2, Definition A.15] with Λ as automorphismgroup and Λ acts transitively on the fibers of q . It then follows that F ( A ) ∼ = F ( A ↾ ) / Λ.In particular, we note the following consequence.
Corollary 3.4.
The morphisms in F ( A ) correspond to the orbits { Λ( F ≤ F ) | F , F ∈ D ( A ↾ ) } . Now we can prove a key lemma, finally making sense of our definitionof ζ . Lemma 3.5.
The category ζ is the quotient Sal ( A ↾ ) / Λ in the category ofacyclic categories. roof. We first need to construct a projection, i.e., a functor Π : Sal ( A ↾ ) → ζ . Recall that the objects of Sal ( A ↾ ) are of the form [ F, C ] with
F, C ∈F ( A ↾ ), F ≤ C , and C a chamber of A ↾ . Also, from the proof of Lemma3.3 we recall the projection q : F ( A ↾ ) → F ( A ). It is now possible to defineΠ on the objects as follows:Π([ F, C ]) = q ( F ≤ C ) : q ( F ) → q ( C ) . According to Corollary 3.2, relations in F ( A ↾ ) are of the form [ F , C ] ≤ [ F , C ] where F ≤ F and π F ↾ ( C ) = π F ↾ ( C ) . On the other hand, morphisms in ζ ( A ) are given by triples ( n, m , m )where m : F → C , m : F → C are objects of ζ , n : F → F is amorphism in F ( A ) and the following condition holds: π F ( m ) = π F ( m ) . Therefore, in order to able to map a relation [ F , C ] ≤ [ F , C ] tothe morphism ( q ( F ≤ F ) , Π([ F , C ]) , Π([ F , C ])) and for this map to besurjective, we need to verify the following condition: π F ↾ ( C ) = π F ↾ ( C ) ⇐⇒ π q ( F ) (Π([ F , C ])) = π q ( F ) (Π([ F , C ])) . We go back to the diagram (6), and write the corresponding commutativediagram of face categories: F ( A ↾ ) π F ↾ / / q (cid:15) (cid:15) F ( A F ↾ ) q (cid:15) (cid:15) F ( A ) π q ( F / / F ( A q ( F ) )Now π F ↾ is a map of posets and since π F ↾ ( F ) = π F ↾ ( F ) we have π F ↾ ( C ) = π F ↾ ( C ) ⇐⇒ π F ↾ ( F ≤ C ) = π F ↾ ( F ≤ C ) . Furthermore q is a covering of categories, in particular is injective on themorphisms incident on π F ↾ ( F ). It then follows that π F ↾ ( F ≤ C ) = π F ↾ ( F ≤ C ) ⇔ q ◦ π F ↾ ( F ≤ C ) = q ◦ π F ↾ ( F ≤ C ) ⇔ π q ( F ) ( q ( F ≤ C )) = π q ( F ) ( q ( F ≤ C )) . Concluding: the functor Π is well defined and it now follows easilyfrom Lemma 3.3 that it is a Galois covering of acyclic categories with Λ asautomorphism group. 18e want to show that, in our particular case, the nerve constructioncommutes with the quotient. Babson and Kozlov in [1] give a necessaryand sufficient condition for this:
Proposition 3.6 ([1, Theorem 3.4]) . Let C be an acyclic category equippedwith a group action G y C . A canonical isomorphism ∆( C ) /G ∼ = ∆( C /G ) exists if and only if the following condition is satisfied:Let t ≥ and let ( m , . . . , m t − , m a ) , ( m , . . . , m t − , m b ) composablemorphism chains. Let Gm a = Gm b , then ther exists some g ∈ G ,such that g ( m a ) = m b and g ( m i ) = m i , ∀ i ∈ { , . . . , t − } . The next lemma ensures that we can apply the previous proposition toour case.
Lemma 3.7.
Let C be an acylic category and G y C act as the Galoisgroup of a covering map. Then the condition of proposition 3.6 is satisfied.Proof. Consider two composable morphism chains as in the condition ofproposition 3.6. Since t ≥ m a and m b must have the same domain, m a : p → q , m b : p → r . Furthermore there isa g ∈ G , such that m b = gm a .Let ϕ : C → D be a covering map with Galois group G . Then ϕ ( m a ) = ϕ ( m b ) ⇒ m a = m b and the condition is trivially satisfied.We finally get to the proof of Theorem 1, which now follows as anapplication of the previous considerations. Proof of Theorem 1.
According to proposition 2.2 the statement holds forthe complex S ↾ / Λ = ∆(Sal A ↾ ) / Λ. The lattice Λ acts on S ↾ as the auto-morphism group of a covering map, in particular lemma 3.7 holds and wehave: S ↾ / Λ = ∆(Sal A ↾ ) / Λ ∼ = ∆(Sal A ↾ / Λ) ∼ = ∆( ζ ) . As an application of the results of the previous sections, and in a struc-tural tribute to the seminal paper of Salvetti [14], we would like to give apresentation for the fundamental group of a complexified toric arrangement.19 .1 Product structure
First, note that the inclusion M ( A ) → T Λ induces an epimorphism ofgroups ε : π ( M ( A )) → π ( T Λ ) ≃ Z n . Lemma 4.1.
The map ε has a section ξ .Proof. Choose a point y ∈ R n in a chamber of A ↾ . Then for all choices of x ∈ R n we have x + iy ∈ M ( A ↾ ) . Accordingly, for every choice of arguments θ , . . . , θ n ∈ R ,( λ e πiθ , . . . , λ n e πiθ n ) ∈ M ( A )where, for all j = 1 , . . . , n , λ j := e − πy j This defines a map f : T Λ → M ( A ) , z ( λ e πi arg z , . . . , λ n e πi arg z n )that induces a homomorphism ξ : π ( T Λ ) → π ( M ( A )) . Since f is a homotopy (right-) inverse to the inclusion M ( A ) → T Λ , εξ = id and ξ is the required section. Lemma 4.2.
The sequence → p ∗ ( π ( S ↾ )) ι → π ( M ( A )) ε → π ( T Λ ) → is split exact. Therefore π ( M ( A )) ≃ π ( S ↾ ) ⋊ π ( T Λ ) . Proof.
We already showed that the map ε has a section, we then need onlyto prove ι ( p ∗ ( π ( S ↾ ))) = Ker ε . It is clear that ι ( p ∗ ( π ( S ↾ ))) ⊆ Ker ε . Forthe opposite inclusion we consider the sequence0 → p ∗ ( π ( M ( A ↾ ))) → π ( M ( A )) → π ( T Λ ) → γ ] ∈ π ( M ( A )) be an element of Ker ε . Let j be the inclusion of M ( A ) in the ambient torus T Λ . Then j ◦ γ is a null homotopic loop in T Λ and lifts therefore to a closed path γ ′ in the universal cover C n . Let γ ↾ bethe lift of γ to M ( A ↾ ) with base point x , then γ ′ = j ↾ ◦ γ ↾ and γ ↾ is also aclosed path. That is, [ γ ] = p ∗ [ γ ↾ ] ∈ p ∗ ( π ( M ( A ↾ ))) ∼ = p ∗ ( π ( S ↾ )).20 .2 Presentation of π ( M ( A ↾ )) As a stepping stone towards the computation of a presentation for thefundamental group of M ( A ), we establish some notation and recall thepresentation of π ( S ↾ ) given by Salvetti in [14].Choose - and from now fix - a chamber C of A ↾ , and let x be a genericpoint in C - i.e. such that for all i = 1 , . . . , d the straight line segment s i from x to u i x meets only faces of codimension at most 1. Remark . In general, given a set K of cells of a complex, K i will denotethe subset of cells of codimension i .Also, to streamline notation we will from now write F , respectively F ↾ for F ( A ), F ( A ↾ ). F F Figure 3: Generators, an example: β F = l F l F l − F Recall the graph G ↾ := G ( A ↾ ) of Definition 3. Herewe will adopt a useful notational convention inspired by [14]: we will writeedges of G ↾ as indexed by the face of codimension 1 they cross, and inwriting a path we will write l F for a crossing of F ‘along the direction ofthe edge’, l − F for a crossing ‘against the direction’ of the edge. By specifyingthe first vertex of the path then there is no confusion about which edge isused, and in which direction.A positive path then is a path of the form l F l F . . . l F k for F , . . . F k ∈ F ↾ . It is also minimal if the hyperplane supporting F i isdifferent from the hyperplane supporting F j for all i = j .Since any two positive minimal paths with same origin and same endare homotopic, given C, C ′ ∈ F ↾ we will sometimes write ( C → C ′ ) for the(class of) positive minimal paths starting at C and ending at C ′ .21or every F ∈ F ↾ we define a path as follows: β F := ( C → ( C ) F ) l F ( C → ( C ) F ) − , (7)where, here and in the following, for a chamber C and a face F theexpression C F will denote the unique chamber in π − F ( π F ( C )) that contains F in its boundary. Lemma 4.3 (p. 616 of [14]) . The group π ( S ↾ ) is generated by the set { β F | F ∈ F ↾ } . Given a positive path ν = l F , . . . , l F k define loops β νF i := l F · · · l F i − l F i l − F i − · · · l − F . (8)Moreover, let F j , . . . , F j l be the sequence obtained from F , . . . , F k by re-cursively deleting faces F j that are supported on a hyperplane which sup-ports an odd number of elements of F j +1 , . . . , F k (compare [14, p. 614])and define Σ( ν ) := ( F i l , . . . , F i l ) . (9) Lemma 4.4 (Lemma 12 in [14]) . Given a positive path ν = l F , · · · , l F k starting in the chamber C and ending in C ′ . Then there is a homotopy ν ≃ (cid:0) Y G ∈ Σ( ν ) β νG (cid:1) ( C → C ′ ) . From this Lemma another useful result follows.
Lemma 4.5 (Corollary 12 in [14]) . Let F , G be two faces of codimension that are supported on the same hyperplane. Then β F is homotopic to ( h Y i =1 β νj i ) β G ( h Y i =1 β νj i ) − , where ν is a positive minimal path from C to π G ( C ) , and j , . . . , j h arethe indices of the edges in ν that cross a hyperplane that does not separate C from π F ( C ) , in the order in which they appear in ν . .2.2. Relations. For every face G ∈ F ↾ consider a chamber C > G andlet C ′ be its opposite chamber with respect to G . Consider a minimal posi-tive path ω from C to C ′ . Let us then consider the set h ( G ) := { F , . . . F k } of the codimension 1 faces adjacent to G , indexed according to the orderin which the positive minimal path ω ‘crosses’ them. This ordering is welldefined up to cyclic permutation. Let now for i = 1 , . . . k F i + k be the facetopposite to F i with respect to G . Define a path α G ( C ) := l F l F . . . l F k . (10)Salvetti introduces a set of relations associated with G : R G : β F . . . β F k = β F . . . β F k β F = . . . stating the equality of all cyclic permutations of the product. In fact, forevery cyclic permutation σ of { , . . . , k } β F σ (1) · · · β F σ ( k ) ≃ ( C → e C ) α G ( e C )( C → e C ) − (11)where e C := ( C ) G and ≃ means homotopy. One of the results of [14] is that the fundamentalgroup of M ( A ↾ ) can be presented as π ( S ↾ ) = h β F , F ∈ F ↾ | R G , G ∈ F ↾ i . We describe the action of u ∈ Λ on a path γ ∈ G ↾ by writing u.γ for thepath obtained by translation of γ with u . Definition 16.
Choose a basis u , . . . u n of Λ, and for i = 1 , . . . d let ω i = ω (1) i be the positive minimal path of G ↾ from C to u i C obtainedby crossing the faces met by the straight line segment s i (which connectsfrom x to u i x ). Also, for k ≥ ω ( k ) i = ω i ( u i .ω ( k − i ). Similarly, let ω ( − i := ω − i and ω ( − k ) i := ω ( − i ( u − i .ω (1 − k ) i ). Given any u ∈ Λ write u = u q · · · u q n n and define ω u := ω ( q )1 u q .ω ( q )2 · · · (cid:0) r − Y j =1 u q n n (cid:1) .ω ( q n ) r . (12)Let then τ i := p ∗ ( ω i ) , τ u := p ∗ ( ω u ) . j ( t ′ i ) s j ( t ′ i +1 ) s j ( t i ) P i s ′ j ( t ) w ( j, t ) r j ( t ) F i + i ( C ) F i r i, r i, Figure 4: Construction for the proof of Lemma 4.6Notice that a path ω u needs not be minimal, nor positive. In fact, itis positive if and only if u has nonnegative coordinates in Λ. Given i and k , the path ω ( k ) i is positive if and only if k ≥
0, and in this case it is alsominimal.
Lemma 4.6. In π ( M ( A )) , p ( ω ( k ) i ) = τ ki and τ i τ j = τ j τ i for all i, j . The ε ∗ τ i generate π ( T Λ ) .Proof. Let X = f ( T Λ ) be the image of the map f in the proof of Lemma4.1, where we now choose y to be a point of our base chamber C .Let the straight line segment s j be parametrized by s j ( t ) := tx + (1 − t ) u j x , ≤ t ≤ . The Minkowski sum X ′ := s + · · · + s n ⊂ R n is a fundamental regionfor the action of Λ on R n . For Y := X ′ + iy ⊆ M ( A ↾ ) we have p ( Y ) = X .In particular, the segments s j map under ε to a system of generators of π ( T Λ ) - in fact, the one associated with the basis u , . . . , u n of Λ.We will next show that for all j = 1 , . . . , d the path s ′ j ( t ) := s j ( t ) + iy is homotopic to the positive minimal path ω j ∈ ( C → u j C ).Indeed, write ω j = l F . . . l F k and let t , . . . t k be such that s j ( t i ) ∈ F i for all i = 1 , . . . , k . Also, write C i , C i +1 for the source and target chambers24f l F i (note: C k +1 = u j C ) and for i = 1 , . . . , k − t ′ i ∈ ] t i − , t i [, t ′ k := 1, t ′ := 0. Then s ′ j ( t ′ i ) ∈ C i for all i = 1 , . . . , k .Recall now that the subset of M ( A ↾ ) with real part x ∈ F consistsof points with imaginary part belonging to the chambers of A ↾ F . In fact,the edge l F i , directed from C i to C i +1 , is by construction ([14, p. 608])the union of two segments, one from a point in P ′ i ∈ C i + 0 i to a point P i ∈ F + i ( C ) F , the other from P i to a point P ′ i +1 ∈ C i +1 + 0 i . We willparametrize these segments as r i, ( t ), t ′ i ≤ t ≤ t i and r i, ( t ), t i ≤ t ≤ t ′ i +1 .Together, they give a parametrization r j ( t ), 0 ≤ t ≤ ω j .The key observation is now that, having chosen y ∈ C , we have that s j ( t h ) ∈ F + i ( C ) F for all h = 1 , . . . , k. Since chambers of arrangements are convex, for all t ∈ [0 ,
1] there is astraight line segment w ( j, t ) joining s j ( t ) and r j ( t ) in M ( A ↾ ).The (topological) disk W j := S t ∈ [0 , w ( j, t ) defines the desired homo-topy between s j and ω j .Now fix i, j ∈ { , . . . , n } clearly s i u i . ( s j ) is homotopic to s j u j . ( s i ), andin π ( M ( A )) we thus have τ i τ j = p ∗ ([ ω i u i .ω j ]) = p ∗ ([ s i u i .s j ])= p ∗ ([ s j u j .s i ]) = p ∗ ([ ω j u j .ω i ]) = τ j τ i . Definition 17.
Let Q be the set of faces that intersect the fundamentalregion X ′ of the proof of Lemma 4.6. Then Q contains C and x . Let Q i := Q ∩ F ↾ i . In particular, Q contains the set of faces crossed by s i , forall i .Recall the parametrization s i ( t ) of the segments s i , and call B the setof faces of the polyhedron X ′ which intersect the convex hull of { s i ([0 , | i ∈ I } for some I ⊆ { , . . . , d } . Notice that every face of X ′ is a translateof some face in B by an element u m · · · u m n n with m , . . . , m n ∈ { , } . Definition 18.
Let F ↾ := { F ∈ Q | F ∩ B = ∅ for all B
6∈ B} µ Γ F F Figure 5: Construction for the proof of lemma 4.7Then F ↾ is a set of representatives for the orbits of the action of Λ on F ↾ . Definition 19.
For any given F ∈ F ↾ let F be the unique element ofΛ F ∩ F ↾ . Then, call u F the unique element of Λ such that F = u F F .Define Γ F := ω u F ( u F .β F ) ω − u F Remark . (1) For all F ∈ F ↾ and all u ∈ Λ p ∗ (Γ uF ) = τ u p ∗ (Γ F ) τ − u . (2) If F ∈ F ↾ , then Γ F = β F .(3) If F ∈ Q , then u F has nonnegative coordinates with respect to u , . . . , u n . (Recall the discussion before Definition 18.)(4) Since X ′ is convex, Q contains the vertices of a positive minimalpath between any two elements of Q . Definition 20.
For j = 1 , . . . , d letΩ j := { F ∈ F ↾ : F is crossed by ω ( k ) j for some k } , And set Ω := S j Ω j . Lemma 4.7.
For all i = 1 , . . . , n , the subgroup of π ( M ( A ↾ )) generatedby the elements β F with F ∈ Ω i is contained in the subgroup generated bythe Γ F , F ∈ Ω i . roof. Let w.l.o.g. F ∈ Ω , and say that F = u k F . If k ≥
0, by construc-tion we have Γ F = β F .Suppose then k <
0, and in this case C ′ := ( C ) F = ( u k C ) F . Let ν denote the positive minimal path from C ′ to C that follows the segments s . We argue by induction on the length d ( F ) of ν : if d ( F ) = 0 we have infact Γ F = β F .Now let d ( F ) >
0. ThenΓ F ≃ ν − l F ν ; β F = µl F µ − where µ is the positive minimal path from C to C ′ following s . Thus β F = µνν − l F ν ( µν ) − = ( µν )Γ F ( µν ) − where µν is the product of all β F ′ with F ′ crossed by µ - therefore, with F ′ ∈ Ω and d ( F ′ ) < d ( F ). By induction, the claim follows. Lemma 4.8.
The set { Γ F | F ∈ Ω } generates π ( M ( A ↾ )) .Proof. Let F ∈ F ↾ , and let H the affine hyperplane supporting F .By construction, there is i ∈ { , . . . , d } and k ∈ Z such that H is crossedby ω ( k ) i in, say, the face G (‘every hyperplane is cut by the coordinate axes’).By Lemma 4.5, β F is then product of β G and other β ± G ′ with G ′ ∈ Ω.These can be written in terms of the Γ F by Lemma 4.7. We now turn to the study of the relations.
Lemma 4.9.
Let F ∈ Q . Then there is a sequence F , . . . , F k of elementsof Q such that β F is homotopic to ( k Y i =1 Γ F i ) − Γ F ( k Y i =1 Γ F i ) . Moreover, ( F , . . . , F k ) = Σ( ω u F ( u F C → ( u F C ) F )) as in Equation 9.In particular, the F i are translates of elements of Ω ∩ F ↾ .Proof. By definition Γ F = ω u F u F .β F ω − u F . Writing µ for a positive minimalpath ( u F C → ( u F C ) F ) we decompose this intoΓ F = ω u F µ ( l F ) ( ω u F µ ) − . ω u F µ is a positive path, and withLemma 4.4 we write it as a product Q j β ω uF µG j ( C → ( C ) F ) where since µ is positive miminal, the G j are crossed by ω u F and thus are translates offaces intersecting the segments s i .Now, by construction β ω uF µG j = Γ G j . Then, set ∆ F := Y j Γ G j . Therefore if ( C ) F = ( u F C ) F we are done withΓ F ≃ ∆ F β F ∆ − F , and thus β F ≃ ∆ − F Γ F ∆ F . If ( C ) F = ( u F C ) F , then we may choose a representant of ( C → ( u F C ) F ) that ends with l F , so its inverse begins with l − F and we have thesame relation as above.Keeping the notations of the Lemma we define, for every F ∈ Q ,∆ F := Y G ∈ Σ( ω uF ( u F C → ( u F C ) F )) Γ G ; Γ ∆ F := ∆ − F Γ F ∆ F (13)Recall from 4.2.II that to every face G ∈ F ↾ we have an ordered set h ( G ) = ( F , . . . , F k ) of incident codimension 1 faces, one for every hyper-plane containing G . The relations associated with G assert the equalityof β F σ (1) . . . β F σ ( k ) (14)where σ is a cyclic permutation, and we write β i for β F i . Lemma 4.10.
Given G ∈ F ↾ there is ∆ G such that, for all cyclic permu-tations σ , we have a homotopy of paths β F σ (1) . . . β F σ ( k ) ≃ ∆ G ω u G u G . (Γ ∆ u − G F σ (1) . . . Γ ∆ u − G F σ ( k ) ) ω − u G ∆ − G . Proof.
Let us fix some notation and let C ′ := ( C ) G , C ′′ := ( u G .C ) G , µ := ( u G C → C ′′ ), ν := ( C ′′ → C ′ ). By equation (11) we have thehomotopy β σ (1) . . . β σ ( k ) ≃ ( C → C ′ ) α G ( C ′ )( C → C ′ ) − α G ( C ′ ) ≃ ν − α G ( C ′′ ) ν ≃ ν − µ − ω − u G ω u G µα G ( C ′′ ) µ − ω − u G ω u G µν expanding µα G ( C ′′ ) µ − according to Equation (11) and defining ∆ G :=( C → C ′ ) ν − µ − ω − u G we have the homotopy β σ (1) . . . β σ ( k ) ≃ ∆ G ω u G ( u G .β u − G F σ (1) ) . . . ( u G .β u − G F σ ( k ) ) ω − u G ∆ − G (15)From which the claim follows by use of Lemma 4.9. Definition 21.
For F ∈ F ↾ let γ F := p (Γ F ) . Moreover, for F ∈ Q let δ F := p (∆ F ); γ δF := δ − F γ F δ F Given G ∈ F ↾ with h ( G ) = ( F , . . . , F k ), let R ⇂ G define the relationstating the equality of all words γ δF σ (1) · · · γ δF σ ( k ) where σ ranges over all cyclic permutations. Lemma 4.11. If G ∈ F ↾ is a face of codimension , then R ⇂ G is equivalentto R ⇂ G Proof.
Let G ∈ F ↾ . With Lemma 4.10 (and the notation thereof) we knowthat every relation R ⇂ G states the equality of all p ∗ (∆ G ) p ∗ (Γ ∆ F σ (1) . . . Γ ∆ F σ ( k ) ) p ∗ (∆ G ) − , where σ runs over all cyclic permutations. The middle term by Equation(15) is represented by the path ω u G ( u G .β u − G F σ (1) ) . . . ( u G .β u − G F σ ( k ) ) ω − u G and thus its image under p ∗ is represented by the same path as p ∗ ( ω u G ) p ∗ ( β u − G F σ (1) . . . β u − G F σ ( k ) ) p ∗ ( ω u G ) − Where u − G F σ ( i ) ∈ Q for all i . Now we apply Lemma 4.9. The element µ := p ∗ ( ω u G ) ∈ π ( T Λ ) is such that, for every cyclic permutation σ , p ∗ (Γ ∆ F σ (1) . . . Γ ∆ F σ ( k ) ) = µ p ∗ (Γ ∆ F σ (1) . . . Γ ∆ F σ ( k ) ) µ − and therefore relation R ⇂ G is equivalent to relation R ⇂ G .29 .5 Presentation In this closing section we discuss presentations for π ( M ( A )). Lemma 4.12.
For all F ∈ Q let ( F , . . . F k ) = Σ( ω u F ( u F C → ( u F C ) F )) .We have δ F = k Y i =1 τ u Fi γ F i τ − u Fi and, in particular, γ δF can be written as a word in the τ , . . . , τ n and γ F with F ∈ F ↾ .Proof. This is an easy computation using Remark 11.(1).In Particular, the relations R ⇂ can be written in terms of the τ i and the γ F with F ∈ F ↾ . We have immediately Theorem 2.
The group π ( M ( A )) is presented as h τ , . . . , τ n ; γ F , F ∈ F | τ i τ j = τ j τ i for i, j = 1 , . . . , n ; R ⇂ G , G ∈ F i , where we identify F with F ↾ and F with F ↾ . This presentation, while not very economical in terms of generators,has the advantage that the relations can be described with an acceptableamount of complexity.Using Lemma 4.8 and Remark 11.(1) we can let, for all G ∈ F ↾ , e R ⇂ G denote the relations obtained from R ⇂ G by substituting every γ F with thecorresponding expression in terms of the generators τ , . . . , τ d and γ F ′ with F ′ ∈ F ↾ ∩ Ω. Under the identification of F with F ↾ , these are the faces onthe compact torus that are crossed by some fixed chosen reppresentants ofthe generators τ , . . . , τ d . Theorem 3.
The group π ( M ( A )) is presented as h τ , . . . , τ n ; γ F , F ∈ p (Ω) ∩F | τ i τ j = τ j τ i for i, j = 1 , . . . , n ; e R ⇂ G , G ∈ F i . Remark . The number of generators (and relations) can in principle bereduced further, by adequate choice of the coordinates of T Λ . The compu-tations, however, become quite more involved and untransparent. We thusomit them here, leaving the question open for a presentation with genera-tors and relations corresponding to layers instead of faces (which exists inthe case of complexified hyperplane arrangements, as shown by Salvetti in[14] by simplifying the presentation given above in 4.2.3).30 eferences [1] E. Babson and D. N. Kozlov. Group actions on posets. Journal ofAlgebra , 285(2):439 – 450, 2005.[2] M.R. Bridson and A. Haefliger.
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