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Topology of Compact Solvmanifolds

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Topology of Compact Solvmanifolds Topology of compact solvmanifolds Sergio Console Topology of compact solvmanifolds Aims Nilpotent and solvable Nilmanifolds joint work with A. Fino, M. Macrì, G. Ovando, M. Subils Solvmanifolds de Rham Cohomology Nilmanifolds Rosario (Argentina) – July 2012 Solmanifolds Main Theorem Proof of the Main Theorem Applications Nakamura manifold Almost abelian 6 dim almost abelian Kähler and symplectic structures on solvmanifolds Hyperelliptic surface The oscillator group Sergio Console 1 Dipartimento di Matematica Università di Torino Topology of compact solvmanifolds 1 Aims Sergio Console 2 Nilpotent and solvable Lie groups Aims Nilmanifolds Nilpotent and solvable Solvmanifolds Nilmanifolds Solvmanifolds de Rham Cohomology 3 de Rham Cohomology Nilmanifolds de Rham cohomology of nilmanifolds Solmanifolds Main Theorem de Rham cohomology of solvmanifolds Proof of the Main Theorem Applications 4 Main Theorem Nakamura manifold Almost abelian Proof of the Main Theorem 6 dim almost abelian Kähler and symplectic structures on solvmanifolds Hyperelliptic surface 5 Applications The oscillator group Nakamura manifold Almost abelian 6-dimensional almost abelian Kähler and symplectic structures on solvmanifolds Hyperelliptic surface The oscillator group 2 Topology of compact M = G=Γ: solvmanifold solvmanifolds Sergio Console G: real simply connected solvable Lie group Γ: lattice (discrete cocompact subgroup) Aims Nilpotent and solvable Nilmanifolds Solvmanifolds Aims: find lattices, compute de Rham cohomology, de Rham Cohomology study existence of symplectic structures, Nilmanifolds Solmanifolds hard Lefschetz property, formality Main Theorem Proof of the Main Theorem We will explain: Applications Nakamura manifold Almost abelian • the difference with nilmanifolds 6 dim almost abelian Kähler and symplectic structures on solvmanifolds • known results on the computation of Rham cohomology in Hyperelliptic surface special cases (completely solvable, Mostow condition) The oscillator group • a method to compute the de Rham cohomology in general (following results by Guan and Witte) • applications: Nakamura manifold, almost abelian solvable Lie groups, hyperelliptic surface, three families of lattices on the 3 oscillator group Topology of compact Nilpotent and solvable Lie groups solvmanifolds Sergio Console G is k-step nilpotent () Aims the descending chain of normal subgroups Nilpotent and solvable Nilmanifolds G = G ⊃ G = [G; G] ⊃ · · · ⊃ G = [G ; G] ⊃ :::: Solvmanifolds 0 1 i+1 i de Rham Cohomology Nilmanifolds Solmanifolds degenerates, i.e. Gi = feg 8i ≥ k,(e is the identity element). Main Theorem Proof of the Main Theorem G is k-step solvable () Applications Nakamura manifold the derived series of normal subgroups Almost abelian 6 dim almost abelian Kähler and symplectic G(0) = G ⊃ G(1) = [G; G] ⊃ · · · ⊃ G(i+1) = [G(i); G(i)] ⊃ :::: structures on solvmanifolds Hyperelliptic surface The oscillator group degenerates. In particular a solvable Lie group is completely solvable if every eigenvalue λ of every operator Ad g, g 2 G, is real. Note that a nilpotent Lie group is completely solvable. 4 Topology of compact Nilmanifolds solvmanifolds Sergio Console G simply connected nilpotent Lie group Aims Nilpotent and solvable Recall: exp : g ! G is a diffeomorphism Nilmanifolds Solvmanifolds de Rham Cohomology For nilpotent Lie groups there is a simple criterion for the Nilmanifolds existence of lattices: Solmanifolds Main Theorem Proof of the Main Theorem Theorem (Malcev)ˇ Applications Nakamura manifold G simply connected nilpotent Lie group Almost abelian 6 dim almost abelian 9 Γ lattice on G () Kähler and symplectic structures on solvmanifolds the Lie algebra g of G has a basis such that the structure Hyperelliptic surface constants in this basis are rational () The oscillator group 9 gQ such that g = gQ ⊗ R If g = gQ ⊗ R, one also says that g has a rational structure A nilmanifold is the a quotient M = G=Γ, where G is a real 5 simply connected nilpotent Lie group and Γ is a lattice. Topology of compact Solvmanifolds solvmanifolds There is no simple criterion for the existence of a lattice in a Sergio Console connected and simply-connected solvable Lie group G. Aims Here are some necessary criteria. Nilpotent and solvable Nilmanifolds Solvmanifolds Proposition (Milnor) de Rham Cohomology Nilmanifolds If G admits a lattice then it is unimodular. [tr adX = 0; 8X 2 g] Solmanifolds Main Theorem Proof of the Main Theorem Applications The Mostow bundle Nakamura manifold Almost abelian Let G=Γ be a solvmanifold that is not a nilmanifold. 6 dim almost abelian Kähler and symplectic N = nilradical of G =largest connected nilpotent normal subgroup of G. structures on solvmanifolds Hyperelliptic surface Then ΓN := Γ \ N is a lattice in N, ΓN = NΓ is closed in G and The oscillator group G=(NΓ) =: Tk is a torus. =) we have the fibration: k N=ΓN = (NΓ)=Γ ,! G=Γ −! G=(NΓ) = T 6 Much of the rich structure of solvmanifolds is encoded in this bundle. The nilradical has an important rôle in the study of solvmanifolds. Topology of compact Solvmanifolds solvmanifolds Sergio Console G connected and simply connected solvable Lie group Aims diffeo n =) G ' R Nilpotent and solvable Nilmanifolds (BUT exp : g ! G is not necessarily injective or surjective. ) Solvmanifolds ∼ de Rham Cohomology =) solvmanifolds G=Γ are aspherical and π1(G=Γ) = Γ. Nilmanifolds Solmanifolds The fundamental group plays an important rôle: Main Theorem Proof of the Main Theorem Diffeomorphism Theorem Applications Nakamura manifold G =Γ and G =Γ solvmanifolds and Almost abelian 1 1 2 2 6 dim almost abelian ' :Γ ! Γ isomorphism. Kähler and symplectic 1 2 structures on solvmanifolds =) 9 diffeomorphism Φ G1 ! G2 such that Hyperelliptic surface The oscillator group (i) ΦjΓ1 = '; (ii) 8γ2Γ1 8p2G1 Φ(pγ) = Φ(p)'(γ): Corollary 7 Two solvmanifolds with isomorphic π1 are diffeomorphic. Topology of compact de Rham Cohomology solvmanifolds Sergio Console Aims Nilpotent and solvable G=Γ solvmanifold (nilmanifold) Nilmanifolds Solvmanifolds de Rham Cohomology General question: can one compute Dolbeault cohomology of Nilmanifolds M by invariant forms, Solmanifolds i.e., using the Chevalley-Eilenberg complex: Main Theorem Proof of the Main Theorem Applications k−1 ∗ d k ∗ d k+1 ∗ Nakamura manifold ···! Λ g ! Λ g ! Λ g ! ::: Almost abelian 6 dim almost abelian Kähler and symplectic X i+j ^ ^ structures on solvmanifolds dα(x1;:::; xk+1) = (−1) α([xi ; xj ]; x1;::: xi ;:::; xj ;:::; xk+1) Hyperelliptic surface i<j The oscillator group ∗ ∼ ∗ So, when is HdR(G=Γ) = H (g)? 8 Topology of compact de Rham Cohomology of nilmanifolds solvmanifolds Sergio Console Aims Nilpotent and solvable G=Γ nilmanifold Nilmanifolds Solvmanifolds de Rham Cohomology Theorem (Nomizu) Nilmanifolds Solmanifolds H∗ (G=Γ) =∼ H∗(g). Main Theorem dR Proof of the Main Theorem Applications Nakamura manifold V∗ Almost abelian g is a minimal model of G=Γ (in the sense of Sullivan). 6 dim almost abelian Kähler and symplectic structures on solvmanifolds ∗ V g is formal () G is abelian and G=Γ is a torus. Hyperelliptic surface The oscillator group If a nilmanifold is Kählerian, then it is a torus. [Benson & Gordon, Hasegawa] 9 Topology of compact Idea of the proof of Nomizu’s Theorem solvmanifolds Sergio Console Aims Suppose G is k-step nilpotent. g` Lie algebra of G`. Nilpotent and solvable Nilmanifolds Let H := simply-connected Lie subgroup of G with Lie algebra Solvmanifolds ∼ n de Rham Cohomology h = gk−1 =) H central and H = R . Nilmanifolds Solmanifolds We have the fibration: Main Theorem Proof of the Main Theorem π n = H=H \ Γ ,! M = G=Γ ! M = G=HΓ Applications T Nakamura manifold Almost abelian p;q 6 dim almost abelian E∗ := Leray-Serre spectral sequence associated with the Kähler and symplectic structures on solvmanifolds fibration: Hyperelliptic surface The oscillator group q p;q p q n ∼ p ^ n E2 = HdR(M; HdR(T )) = HdR(M) ⊗ R ; p;q p+q E1 ) HdR (M) : 10 Topology of compact solvmanifolds Sergio Console Main idea: construct a second spectral sequence Aims ~ p;q Nilpotent and solvable E∗ = Leray-Serre spectral sequence for the complex of V∗ ∗ Nilmanifolds G-invariant forms g Solvmanifolds de Rham Cohomology V∗ ∗ V∗ ~ p;q p;q g subcomplex of M =) E∗ ⊆ E∗ & Nilmanifolds Solmanifolds Main Theorem p;q ^q ~ p n Proof of the Main Theorem E2 = H (g=h) ⊗ R ; Applications Nakamura manifold E~ p;q ) Hp+q(g) : Almost abelian 1 6 dim almost abelian Kähler and symplectic structures on solvmanifolds induction on dim p ∼ p Hyperelliptic surface dim M < dim M, =) HdR(M) = H (g=h) for any p. The oscillator group ~ ~ =) E2 = E2 & E1 = E1. ` ∼ ` i.e., HdR(M) = H (g) for any `. 11 Topology of compact de Rham Cohomology of solvmanifolds solvmanifolds Sergio Console Aims Nilpotent and solvable G=Γ solvmanifold Nilmanifolds Solvmanifolds adX ∼ Ad G(G) = fe j X 2 gg solvable and Aut (G) = Aut (g). de Rham Cohomology Nilmanifolds Solmanifolds A(Ad G(G)) and A(Ad G(Γ)): real algebraic closures of Ad G(G) Main Theorem and Ad G(Γ) (respectively) Proof of the Main Theorem Applications Nakamura manifold Theorem (Borel density theorem) Almost abelian 6 dim almost abelian Kähler and symplectic Let Γ be a lattice of a simply connected solvable Lie group G, structures on solvmanifolds =) 9 a maximal compact torus ⊂ A(Ad (G)), such that Hyperelliptic surface Tcpt G The oscillator group A(Ad G(G)) = Tcpt A(Ad G(Γ)): 12 Topology of compact When is the de Rham comology of a solvmanifold given by solvmanifolds the Chevalley-Eilenberg complex? Sergio Console There are 2 important cases: Aims Nilpotent and solvable Nilmanifolds 1 Hattori: If G is completely solvable, i.e., Solvmanifolds if the linear map adX : g ! g has only real eigenvalues de Rham Cohomology Nilmanifolds 2 Mostow condition: If A(Ad G(G)) = A(Ad G(Γ)) Solmanifolds Main Theorem Proof of the Main Theorem Remarks Applications Nakamura manifold Almost abelian • (1) is a particular case of (2). 6 dim almost abelian Kähler and symplectic Indeed if all eigenvalues of Ad are real, then A(Ad G(G)) has no non structures on solvmanifolds Borel density Hyperelliptic surface trivial conn.
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