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Twisted Lefschetz Numbers of Infra-Solvmanifolds and Algebraic Groups

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Twisted Lefschetz Numbers of Infra-Solvmanifolds and Algebraic Groups TWISTED LEFSCHETZ NUMBERS OF INFRA-SOLVMANIFOLDS AND ALGEBRAIC GROUPS HISASHI KASUYA Abstract. Twisted Lefschetz numbers are extensions of the ordinary Lefschetz numbers for cohomologies with values in flat bundles. As a generalization of lin- earization formula for the ordinary Lefschetz number of a diffeomorphism of a nil- manifold, we show that a twisted Lefschetz number of any diffeomorphism of any infra-solvmanifold is equal to the determinant det(I − A) for some matrix A by using the cohomology of algebraic groups. 1. Introduction Let M be a compact manifold and E be a flat vector bundle over M. We consider the de Rham cohomology H∗(M, E) with values in E. Let ϕ : M → M be a smooth map and a morphism Ξ : ϕ∗E → E of flat bundles where ϕ∗E is the pull-back of E. Consider the induced map Ξ ◦ H∗(ϕ) : H∗(M, E) → H∗(M, E). We define the twisted Lefschetz number i ∗ L(ϕ, E, Ξ) = (−1) trace(Ξ ◦ H (ϕ))|Hi(M,E). Xi As for the ordinary Lefschetz number, we can show the Lefschetz fixed point formula for the twisted Lefschetz number by the Atiyah-Bott theorem ([1]). If ϕ is a diffeomorphism so that the graph of ϕ is transversal to the diagonal in M × M, then we have L(ϕ, E, Ξ) = sign(det(I − dϕx)) · trace(Ξx). ϕ(Xx)=x In general, computations of the twisted Lefschetz numbers are harder than compu- tations of usual Lefschetz numbers. In this paper we give a simple formula for twisted arXiv:1404.7690v3 [math.AT] 11 Nov 2019 Lefschetz numbers of diffeomorphisms of solvmanifolds and infra-solvmanifolds which is a generalization of the ”linearization formula” for ordinary Lefschetz numbers of maps of nilmanifolds. In this paper solvmanifolds (resp. nilmanifolds) are compact homogeneous spaces of connected solvable (resp. nilpotent) Lie groups. We say that a solvmanifold (resp. nilmanifold) M has a discrete presentation (G, Γ) if there exists a simply connected solvable (resp. nilpotent) Lie group G with a cocompact discrete subgroup Γ so that M is diffeomorphic to G/Γ. It is known that every nilmanifold M has a discrete presentation (G, Γ) (see [11]). Consider a nilmanifold G/Γ for a discrete presentation (G, Γ). For a smooth map ϕ : G/Γ → G/Γ, since the fundamental group of G/Γ is Γ, we have the map f : Γ → Γ induced by ϕ. It is known that any homomorphism between cocompact discrete subgroups of simply connected nilpotent Lie groups can be extended to a Lie group homomorphism. Hence we have an extension T : G → G of f. In [14], Nomizu showed 2010 Mathematics Subject Classification. 22E25; 20G20 ; 54H25; 55M20 . Key words and phrases. Topological fixed point theory, cohomology of algebraic group, twisted lefschetz number, infra-solvmanifold. 1 2 HISASHI KASUYA that the de Rham cohomology of a nilmanifold G/Γ for a discrete presentation (G, Γ) is isomorphic to the Lie algebra cohomology of the Lie algebra g of G. Hence, considering the induced homomorphism T∗ : g → g, taking a matrix presentation A of T∗ : g → g, we can get the ”linearization formula” L(ϕ) = det(I − A) where L(ϕ) is the ordinary Lefschetz number. We are interested in getting a ”linearization formula” for solvmanifolds. But there are many difficulties: • In general, a solvmanifold M does not have a discrete presentation (G, Γ). • In general, for a simply connected solvable Lie group with a cocompact dis- crete subgroup Γ, a homomorphism Γ → Γ may not extend to a Lie group homomorphism G → G. • In general, the de Rham cohomology of a solvmanifold G/Γ for a discrete pre- sentation (G, Γ) is not isomorphic to the Lie algebra cohomology of the Lie algebra g of G. Assuming some conditions, we can get a ”linearization formula” (see for examples [12] and [10]). However on general solvmanifolds, it seems to be difficult to get a ”linearization formula”. In this paper we consider the more general setting. An infra-solvmanifold is a man- ifold of the form G/∆, where G is a simply connected solvable Lie group, and ∆ is a torsion-free subgroup of Aut(G) ⋉ G such that the closure of h(∆) in Aut(G) is com- pact where h : Aut(G) ⋉ G → Aut(G) is the projection. An infra-solvmanifold is a generalization of a solvmanifold. See [2] and [4] for properties of infra-solvmanifolds. The purpose of this paper is to give a ”linearization formula” for any diffeomorphism of any infra-solvmanifold by using the twisted Lefschetz number and algebraic groups. Let M be an infra-solvmanifold with the fundamental group Γ and ϕ : M → M a diffeomorphism. Denote by f : Γ → Γ the homomorphism induced by ϕ. Then, by methods of algebraic groups given in [16, 2], we can say that f : Γ → Γ induces an explicit isomorphism on a certain Lie algebra uΓ. For a matrix presentation A of such isomorphism, we will prove that by taking some E and Ξ, the formula L(ϕ, E, Ξ) = det(I − A) holds. 2. Algebraic groups and representations An algebraic group G is a Zariski-closed subgroup of GLn(C). We denote by U(G) the unipotent radical of G. If U(G) is trivial, an algebraic group G is called reductive. Denote by Rep(G) the category of finite-dimensional rational representations Vρ of G. Vρ ∈ Rep(G) says Vρ is a C-vector space with a rational representation ρ : G → GL(Vρ). For Vρ ∈ Rep(G), we denote by Vρ the C-vector space Vρ with the trivial representation. 3. Representations of reductive algebraic groups Let G be a reductive algebraic group and Vρ ∈ Rep(G). We assume that we have an algebraic group isomorphism F : G → G and a linear isomorphism Φ : Vρ → Vρ satisfying ρ(g) ◦ Φ=Φ ◦ ρ(F (g)) for every g ∈ G. TWISTED LEFSCHETZ NUMBERS OF INFRA-SOLVMANIFOLDS 3 We take a finite set Λρ = {Vα} of irreducible representations of G such that we have an irreducible decomposition Vρ = Λρ Eα and Eα is a non-trivial irreducible component of Vρ corresponding to an irreducibleL Vα ∈ Rep(G). Denote Eα◦F = Φ(Eα). Then we have the irreducible decomposition Vρ = Λρ Eα◦F . Consider Vα◦F ∈ Rep(G) as the vector space Vα with the representation αL◦ F . Then, Eα◦F is a non-trivial irreducible component of Vρ corresponding to an irreducible Vα◦F ∈ Rep(G). Comparing two irreducible decomposition Vρ = Λρ Eα = Λρ Eα◦F , for each α ∈ Λρ, we have a ∼ unique β ∈ Λρ such that we haveL an isomorphismL ψα : Vα◦F = Vβ. ∗ ∗ Let us consider the linear map Φα : Vρ ⊗ Vα ⊗ Vα → Vρ ⊗ Vα◦F ⊗ Vα◦F defined by ∗ Φα(fα ⊗ vα)=Φ ◦ fα ⊗ vα for fα ∈ Vρ ⊗ Vα , vα ∈ Vα (we should notice Vα = Vα◦F ). ∗ G Then this linear map is G-equivariant and hence we can restrict Φα : (Vρ ⊗Vα ) ⊗Vα → ∗ G −1 ∗ ∗ ∗ (Vρ ⊗ Vα◦F ) ⊗ Vα◦F . Consider the isomorphism (ψα ) ⊗ ψα : Vα◦F ⊗ Vα◦F → Vβ ⊗ Vβ. −1 ∗ ∗ G ∗ G Then we have the linear map ((ψα ) ⊗ ψα) ◦ Φα : (Vρ ⊗ Vα ) ⊗ Vα → (Vρ ⊗ Vβ ) ⊗ Vβ. ∗ G We consider Vφ ∈ Rep(G) as Vφ = Λρ (Vρ ⊗ Vα ) ⊗ Vα. Then we have the linear −1 ∗ LG G map Λρ ((ψα ) ⊗ ψα) ◦ Φα : (Vρ ⊗ Vφ) → (Vρ ⊗ Vφ) . L G ∼ Lemma 3.1. There exists an linear isomorphism (Vρ ⊗ Vφ) = Vρ which identifies the linear map −1 ∗ G G with . Λρ ((ψα ) ⊗ ψα) ◦ Φα : (Vρ ⊗ Vφ) → (Vρ ⊗ Vφ) Φ : Vρ → Vρ L Proof. By [17, Theorem 27.3.6], we have the isomorphism ∗ G (Vρ ⊗ Vα ) ⊗ Vα ∋ fα ⊗ vα 7→ f(vα) ∈ Eα. G ∼ Thus we have the isomorphism (Vρ ⊗ Vφ) = Λρ Eα = Vρ. Since we have −1 ∗ L −1 ((ψα ) ⊗ ψα) ◦ Φα(fα ⊗ vα)=Φ ◦ fα ◦ ψα ⊗ ψα(vα), the lemma follows from this isomorphism. 4. Rational cohomology of algebraic groups Let G be an algebraic group. In this section, we denote U = U(G) Let F : G → G be an isomorphism of algebraic groups. Consider the extension (1) 1 / U / G / G/U / 1 . It is known that this extension splits ([13]). The restriction FU : U →U is also an isomorphism and F induces an isomorphism F¯ : G/U → G/U. ∗ ∗ C For Vφ ∈ Rep(G), we define the rational cohomology H (G,Vφ) = ExtG( ,Vφ) as ∗ ∗ ∗ in [7] and [9]. Consider the induced map H (F ) : H (G,Vφ) → H (G,Vφ◦F ). Let ∗ Ψ : Vφ◦F → Vφ be an isomorphism in Rep(G). We consider the linear map Ψ ◦ H (F ) : ∗ ∗ H (G,Vφ) → H (G,Vφ). It is known that we have a natural isomorphism ∗ ∼ ∗ G/U ∼ ∗ G/U H (G,Vφ) = H (U,Vφ) = H (u,Vφ) induced by the extension (1) and the exponential map u → U (see [7],[5]) where u ∗ is the Lie algebra of U and H (u,Vφ) is the Lie algebra cohomology and we regard ∗ ∗ H (U,Vφ),H (u,Vφ) ∈ Rep(G/U) via the adjoint action. This isomorphism identifies ∗ ∗ ∗ ∗ the linear map Ψ ◦ H (F ) : H (G,Vφ) → H (G,Vφ) with the linear map Ψ ◦ H (FU∗) : ∗ G/U ∗ G/U H (u,Vφ) → H (u,Vφ) where FU∗ : u → u is the derivation of FU : U→U. Denote by H∗(u) the Lie algebra cohomology with values in the 1-dimensional trivial representation. We apply the arguments in the last section to the objects: • The reductive algebraic group G/U. 4 HISASHI KASUYA • The representation H∗(u) ∈ Rep(G/U). • The isomorphism F¯ : G/U → G/U. ∗ ∗ ∗ • The linear isomorphism H (FU∗) : H (u) → H (u).
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