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L2-Betti Numbers L2-Betti numbers Wolfgang Lück Bonn Germany email [email protected] http://131.220.77.52/lueck/ Bonn, 24. & 26. April 2018 Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 1 / 66 Outline We introduce L2-Betti numbers. We present their basic properties and tools for their computation. We compute the L2-Betti numbers of all 3-manifolds. We discuss the Atiyah Conjecture and the Singer Conjecture. Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 2 / 66 Basic motivation Given an invariant for finite CW -complexes, one can get much more sophisticated versions by passing to the universal covering and defining an analogue taking the action of the fundamental group into account. Examples: Classical notion generalized version Homology with coeffi- Homology with coefficients in cients in Z representations Euler characteristic Z Walls finiteness obstruction in 2 K0(Z⇡) Lefschetz numbers Z Generalized Lefschetz invari- 2 ants in Z⇡φ Signature Z Surgery invariants in L (ZG) 2 ⇤ — torsion invariants Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 3 / 66 We want to apply this principle to (classical) Betti numbers bn(X) := dimC(Hn(X; C)). Here are two naive attempts which fail: dimC(Hn(X; C)) dimC⇡(Hn(X; C)), e where dimC⇡(M) for a C[⇡]-module could be chosen for instance as dim (C eG M). C ⌦C The problem is that C⇡ is in general not Noetherian and dimC⇡(M) is in general not additive under exact sequences. We will use the following successful approach which is essentially due to Atiyah [1]. Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 4 / 66 Group von Neumann algebras Throughout these lectures let G be a discrete group. Given a ring R and a group G, denote by RG or R[G] the group ring. Elements are formal sums r g, where r R and only g G g · g 2 finitely many of the coefficients2r are non-zero. P g Addition is given by adding the coefficients. Multiplication is given by the expression g h := g h for g, h G · · 2 (with two different meanings of ). · In general RG is a very complicated ring. Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 5 / 66 2 Denote by L (G) the Hilbert space of (formal) sums g G λg g 2 2 · such that λg C and g G λg < . 2 2 | | 1 P P Definition Define the group von Neumann algebra weak (G):= (L2(G), L2(G))G = CG N B to be the algebra of bounded G-equivariant operators L2(G) L2(G). ! The von Neumann trace is defined by tr (G) : (G) C, f f (e), e L2(G). N N ! 7! h i Example (Finite G) 2 If G is finite, then CG = L (G)= (G). The trace tr (G) assigns to N N g G λg g the coefficient λe. 2 · P Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 6 / 66 Example (G = Zn) Let G be Zn. Let L2(T n) be the Hilbert space of L2-integrable functions T n C. Fourier transform yields an isometric Zn-equivariant ! isomorphism = L2(Zn) ⇠ L2(T n). ! n Let L1(T ) be the Banach space of essentially bounded measurable functions f : T n C. We obtain an isomorphism ! n ⇠= n L1(T ) (Z ), f Mf ! N 7! 2 n 2 n n where Mf : L (T ) L (T ) is the bounded Z -operator g g f . ! 7! · Under this identification the trace becomes n tr (Zn) : L1(T ) C, f fdµ. N ! 7! n ZT Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 7 / 66 von Neumann dimension Definition (Finitely generated Hilbert module) A finitely generated Hilbert (G)-module V is a Hilbert space V N together with a linear isometric G-action such that there exists an isometric linear G-embedding of V into L2(G)n for some n 0. ≥ A map of finitely generated Hilbert (G)-modules f : V W is a N ! bounded G-equivariant operator. Definition (von Neumann dimension) Let V be a finitely generated Hilbert (G)-module. Choose a 2 n N 2 n G-equivariant projection p : L (G) L (G) with im(p) = (G) V . ! ⇠N Define the von Neumann dimension of V by n 0 dim (G)(V ) := tr (G)(p):= tr (G)(pi,i ) R≥ . N N N 2 Xi=1 Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 8 / 66 Example (Finite G) For finite G a finitely generated Hilbert (G)-module V is the same as N a unitary finite dimensional G-representation and 1 dim (G)(V )= dimC(V ). N G · | | Example (G = Zn) Let G be Zn. Let X T n be any measurable set with characteristic ⇢n 2 n 2 n function χX L1(T ). Let MχX : L (T ) L (T ) be the n 2 ! Z -equivariant unitary projection given by multiplication with χX . Its image V is a Hilbert (Zn)-module with N dim (Zn)(V )=vol(X). N 0 In particular each r R≥ occurs as r = dim (Zn)(V ). 2 N Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 9 / 66 Definition (Weakly exact) i p A sequence of Hilbert (G)-modules U V W is weakly exact at N ! ! V if the kernel ker(p) of p and the closure im(i) of the image im(i) of i agree. A map of Hilbert (G)-modules f : V W is a weak isomorphism if it N ! is injective and has dense image. Example The morphism of (Z)-Hilbert modules N 2 2 1 2 2 1 Mz 1 : L (Z)=L (S ) L (Z)=L (S ), u(z) (z 1) u(z) − ! 7! − · is a weak isomorphism, but not an isomorphism. Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 10 / 66 Theorem (Main properties of the von Neumann dimension) 1 Faithfulness We have for a finitely generated Hilbert (G)-module V N V = 0 dim (G)(V )=0; () N 2 Additivity If 0 U V W 0 is a weakly exact sequence of finitely ! ! ! ! generated Hilbert (G)-modules, then N dim (G)(U)+dim (G)(W )=dim (G)(V ); N N N 3 Cofinality Let V i I be a directed system of Hilbert (G)- submodules { i | 2 } N of V , directed by inclusion. Then dim (G) Vi = sup dim (G)(Vi ) i I . N { N | 2 } i I ! 2 Wolfgang Lück (MI, Bonn) [ L2-Betti numbers Bonn, 24. & 26. April 2018 11 / 66 L2-homology and L2-Betti numbers Definition (L2-homology and L2-Betti numbers) Let X be a connected CW -complex of finite type. Let X be its universal covering and ⇡ = ⇡1(M). Denote by C (X) its cellular Z⇡-chain ⇤ complex. e Define its cellular L2-chain complex to bee the Hilbert (⇡)-chain N complex (2) 2 C (X) := L (⇡) Z⇡ C (X)=C (X). ⇤ ⌦ ⇤ ⇤ Define its n-th L2-homology to be the finitely generated Hilbert e e e (G)-module N (2) (2) (2) Hn (X) := ker(cn )/im(cn+1). 2 Define its n-th L -Betti numbere (2) (2) 0 bn (X):=dim (⇡) Hn (X) R≥ . N 2 Wolfgang Lück (MI, Bonn) e L2-Betti numbers e Bonn, 24. & 26. April 2018 12 / 66 Theorem (Main properties of L2-Betti numbers) Let X and Y be connected CW -complexes of finite type. Homotopy invariance If X and Y are homotopy equivalent, then (2) (2) bn (X)=bn (Y ); Euler-Poincaré formula e e We have χ(X)= ( 1)n b(2)(X); − · n n 0 X≥ Poincaré duality e Let M be a closed manifold of dimension d. Then (2) (2) bn (M)=bd n(M); − e e Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 13 / 66 Theorem (Continued) Künneth formula b(2)(X^Y )= b(2)(X) b(2)(Y ); n ⇥ p · q p+q=n X e e Zero-th L2-Betti number We have 1 b(2)(X)= ; 0 ⇡ | | Finite coverings e If X Y is a finite covering with d sheets, then ! b(2)(X)=d b(2)(Y ). n · n e e Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 14 / 66 Example (Finite ⇡) If ⇡ is finite then b (X) b(2)(X)= n . n ⇡ | e| e Example (S1) (2) Consider the Z-CW -complex S1. We get for C (S1) ⇤ Mz 1 ... 0 L2(Zf) − L2(Z) 0 f ... ! ! −− ! ! ! and hence H(2)(S1)=0 and b(2)(S1)=0 for all 0. n n ≥ f f Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 15 / 66 Example (⇡ = Zd ) Let X be a connected CW -complex of finite type with fundamental group Zd . Let C[Zd ](0) be the quotient field of the commutative integral domain C[Zd ]. Then (2) d (0) bn (X)=dim d (0) C[Z ] d Hn(X) C[Z ] ⌦Z[Z ] ⇣ ⌘ Obviously this impliese e (2) bn (X) Z. 2 e Wolfgang Lück (MI, Bonn) L2-Betti numbers Bonn, 24. & 26. April 2018 16 / 66 For a discrete group G we can consider more generally any free finite G-CW -complex X which is the same as a G-covering X X over a finite CW -complex X. (Actually proper finite ! G-CW -complex suffices.) The universal covering p : X X over a connected finite ! CW -complex is a special case for G = ⇡1(X).
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