Mike Davis. July 4, 2006

Transcription

1 July 4, 2006

2 1 Properties 2 3 The Euler Characteristic Conjecture Singer Conjecture

3 Properties Last time L 2 C i (X ) := {ϕ : {i-cells} R ϕ(e) 2 < } L 2 H i (X ) := Z i (X )/B i (X ) L 2 H i (X ) := Z i (X )/B i (X ).

4 Properties A harmonic 1-cycle 1/4 1/4 1/4 1/4 1/2 1 1/2 1/4 1/2 1/2 1/4 1/4 1/ ( ) 1 2 ( ) = 1 + ( 1 2 n = 1 + ( 1 2 ) 2n ) n 1 <

5 Properties (X, Y ) a pair of CW-complexes. Γ acts properly and cellularly on X. Y is a Γ-stable subcx. Reduced L 2 -(co)homology groups L 2 H i (X, Y ) are defined in the usual manner by completing of C i (X, Y ). Versions of most of the Eilenberg-Steenrod Axioms hold for L 2 H (X, Y ). Some standard properties. Functorality f : (X 1, Y 1 ) (X 2, Y 2 ) a Γ-map. There is an induced map f : L 2 H i (X 1, Y 1 ) L 2 H i (X 2, Y 2 ) giving a functor from pairs of Γ-complexes and Γ-homotopy classes of maps to Hilbert Γ-modules.

6 Properties Properties Exact sequence of a pair The sequence, L 2 H i (Y ) L 2 H i (X ) L 2 H i (X, Y ) is weakly exact. Excision U is a Γ-stable subset of Y s.t. Y U is a subcx. Then (X U, Y U) (X, Y ) induces an iso: L 2 H i (X U, Y U) = L 2 H i (X, Y ).

7 Properties Mayer-Vietoris sequence X = X 1 X 2, with X 1, X 2 Γ-stable subcxes. The M-V sequence, L 2 H i (X 1 X 2 ) L 2 H i (X 1 ) L 2 H i (X 2 ) L 2 H i (X ) is weakly exact.

8 Twisted products Properties H a subgp of Γ and Y is a space with H-action. The twisted product: Γ H Y := (Γ Y )/H where the H-action is defined by h (g, y) = (gh 1, hy). It is a left Γ-space and a Γ-bundle over Γ/H. Since Γ/H is discrete, Γ H Y is a disjoint union of copies of Y, one for each element of Γ/H. If Y is an H-CW-complex, then Γ H Y is a Γ-CW-complex.

9 More properties Properties Twisted products and the induced representation L 2 H i (Γ H Y ) = Ind Γ H (L2 H i (Y )). Künneth Formula Γ = Γ 1 Γ 2 and X j is a Γ j -CW-cx, j = 1, 2. Then X 1 X 2 is a Γ-CW-cx and L 2 H k (X 1 X 2 ) = L 2 H i (X 1 ) L 2 H j (X 2 ), i+j=k where denotes the completed tensor product.

10 Properties Reduced homology of Euclidean space Example We know for X = E 1 (= R) with standard action of Γ = Z that L 2 H k (E 1 ) = 0 for k = 0, 1. By the Künneth Formula, L 2 H k (E n ) = 0, k.

11 Review of dim Γ ( ) The von Neumann dimension of V (or its Γ-dimension) is defined by dim Γ (V ) := tr Γ (p V ). Properties dim Γ (V ) [0, ) and dim Γ (V ) = 0 iff V = 0. Γ = {1} = dim Γ (V ) = dim R (V ). dim Γ (L 2 (Γ)) = 1. dim Γ (V W ) = dim Γ (V ) + dim Γ (W ).

12 More properties of dim Γ ( ) f : V W a map of Hilbert Γ-modules, then dim Γ (V ) = dim Γ (Ker f ) + dim Γ (Im f ) = dim Γ (Ker f ) + dim Γ (Im f ). H Γ index m = dim H (V ) = m dim Γ (V ). Γ finite = dim Γ (V ) = 1 Γ dim(v ). H Γ, W then dim Γ (Ind Γ H (W )) = dim H(W ). dim Γ1 Γ 2 (V 1 V 2 ) = dim Γ1 (V 1 ) dim Γ2 (V 2 ).

13 Definition The i th L 2 -Betti number of X is: L 2 b i (X ; Γ) := dim Γ L 2 H i (X ). If X is contractible (and the Γ-action is proper and cocompact), then L 2 b i (X ; Γ) is an invariant of Γ. Denote it L 2 b i (Γ) and call it the L 2 -Betti number of Γ.

14 Properties of L 2 -Betti numbers L 2 b i (X ; Γ) = 0 = L 2 H i (X ) = 0. H Γ index m = L 2 b i (X ; H) = m(l 2 b i (X ; Γ)). Künneth Formula: L 2 b k (X 1 X 2 ; Γ 1 Γ 2 ) = i+j=k L 2 b i (X 1 ; Γ 1 )L 2 b j (X 2 ; Γ 2 ) Suppose Γ 1, Γ 2 both infinite. Then { L 2 L 2 b i (Γ 1 ) + L 2 b i (Γ 2 ), if i > 1, b i (Γ 1 Γ 2 ) = L 2 b 1 (Γ 1 ) + L 2 b 1 (Γ 2 ) 1 if i = 1 (Mayer-Vietoris sequence).

15 Orbihedral Euler characteristic χ orb (X /Γ) := orbits of cells ( 1) dim c Γ c Q, where Γ c is the order of the stabilizer of the cell c. If Γacts freely, then χ orb (X /Γ) is the ordinary Euler characteristic χ(x /Γ). If H Γ is index m, then χ orb (X /H) = mχ orb (X /Γ). χ orb (X 1 /Γ 1 X 2 /Γ 2 ) = χ orb (X 1 /Γ 1 )χ orb (X 2 /Γ 2 )

16 The L 2 -Euler characteristic L 2 χ(x ; Γ) := ( 1) i L 2 b i (X ; Γ). i=0 Theorem (Atiyah) χ orb (X /Γ) = L 2 χ(x ; Γ). Lemma C a chain complex of Hilbert Γ-modules. H i (C ) = reduced homology. Then ( 1) i dim Γ C i = ( 1) i dim Γ H i (C ). i i

17 Proof of Lemma Proof. Put Z i := Ker(C i C i 1 ), B i := Im(C i+1 C i ) and c i := dim Γ (C i ), h i := dim Γ (H i (C )) z i := dim Γ (Z i ), b i := dim Γ (B i ). Weak short exact sequences: 0 Z i C i B i B i Z i H i 0. So, c i = z i + b i 1 and z i = h i + b i.

18 ( 1) i c i = ( 1) i (z i + b i 1 ) = ( 1) i (h i + b i + b i 1 ) = ( 1) i h i. Proof of. c i := dim Γ (C i (X )) = = 1 Γ c. orbits of i-cells dim Γ (L 2 (Γ/Γ c )) So, ( 1) i c i = χ orb (X /Γ) and Lemma = Formula.

19 Free groups Example Y = a figure 8. T its universal cover (a regular 4-valent tree). F 2 = free group of rank 2. L 2 b 0 (T ; F 2 ) = 0 (because F 2 is infinite). So, L 2 b 1 (T ; F 2 ) = L 2 χ(t ; F 2 ) = χ(y ) = 1. t s -1 1 s t -1

20 Surface groups Example Y = closed surface of genus g (> 0), X its univ cover, Γ = π 1 (Y ). Showed previously L 2 b 0 = 0 = L 2 b 2. So, L 2 b 1 (X ; Γ) = L 2 χ(x : Γ) = χ(y ) = 2g 2 Notation BΓ := K(Γ, 1) and EΓ := its univ cover.

21 2-dimensional groups Example Suppose BΓ is a finite 2-dim cx (e.g., Γ is a small cancellation gp). g = #{1-cells} = #{generators} r = #{2-cells} = #{relations} χ(γ) = 1 g + r and L 2 χ(γ) = L 2 b 2 (Γ) L 2 b 1 (Γ). So, r g = χ(γ) > 0 = L 2 b 2 (Γ) > 0 r < g 1 = χ(γ) < 0 = L 2 b 1 (Γ) > 0.

22 Deficiency of a finitely presented group Definition The deficiency of a presentation of Γ is g r = #{generators} #{relations}. The deficiency of a gp Γ, denoted def(γ), is the maximum of g r over all presentations of Γ. Let Y be presentation cx with χ(y ) minimum. Since Y can be completed to BΓ by attaching cells of dim 3, b 1 (Y ) = b 1 (Γ) and b 2 (Y ) b 2 (Γ). So, def(γ) = 1 χ(y ) = b 1 (Y )) b 2 (Y ) b 1 (Γ) b 2 (Γ). Similarly, def(γ) L 2 b 1 (Γ) L 2 b 2 (Γ) + 1. So, for example, L 2 b 1 (Γ) = 0 = def(γ) 1.

23 Theorem X n an n-mfld, then L 2 b i (X n ; Γ) = L 2 b n i (X n ; Γ). There is a nonsingular pairing: L 2 H i (X ) L 2 H n i (X ) R, defined by α β α β, [X ]. Point is the cup product of 2 L 2 -classes is L 1, [X ] is a bounded class and you can evaluate an L 1 -cohomology class on a bounded homology class.

24 Remark Suppose X is the univ cover of a cx. Same argument shows L 2 H i (X ) = L 2 H n i (X ). Example Suppose Γ is a PD 2 -gp (i.e., the fund gp of a 2-dim PD cx whose univ cover X is contractible). This implies Γ is infinite. So, L 2 b 0 = 0. By L 2 b 2 = 0. So, χ(γ) = χ(x /Γ) = L 2 b 1 (X ; Γ) 0. So, b 1 (Γ) 2 = b 0 (Γ) + b 1 (Γ) b 2 (Γ) 0. So, b 1 (Γ) = rk(γ ab ) 2. (This fact was important in proof that PD 2 -gps are surface gps.)

25 The Euler Characteristic Conjecture The Euler Characteristic Conjecture Singer Conjecture A space Y is aspherical if its univ cover is contractible. Example A complete Riemannian mfld M of nonpositive sectional curvature is aspherical. (Pf: exp : T x M M is a diffeomorphism.) Conjecture If M 2k is a closed aspherical mfld, then ( 1) k χ(m 2k ) 0. In nonpositively curved context this is called the Chern Hopf Conj or Hopf Conj. Conj doesn t follow from the Gauss Bonnet Theorem.

26 Euler Char Conj The Euler Characteristic Conjecture Singer Conjecture For odd-dimensional mflds, χ = 0. (Pf: ). Conj true for surfaces: M 2 is aspherical iff χ(m 2 ) 0. (Pf: χ = 0 univ cover = E 2. χ < 0 univ cover = H 2.) Conj true for product of surfaces: if M 2k is product of k surfaces of nonpositive Euler char, then ( 1) k M 2k 0 (because χ is multiplicative for products). True for closed hyperbolic mflds and other locally symmetric mflds.

27 Other versions The Euler Characteristic Conjecture Singer Conjecture Conjecture Suppose Γ acts properly and cocompactly on contractible (i.e., M 2k /Γ is an aspherical orbifold). Then M 2k Conjecture ( 1) k χ orb ( M 2k /Γ) 0. Suppose Γ is a PD 2k -gp. Then ( 1) k χ(γ) 0.

28 The Dodziuk Singer Conjecture The Euler Characteristic Conjecture Singer Conjecture Conjecture M n a contractible mfld with cocompact proper Γ-action. Then L 2 b i ( M n ; Γ) = 0, i n 2. If n is odd, this means all L 2 -Betti numbers are 0. Theorem Singer Conj. = Euler Char. Conj.

29 The Euler Characteristic Conjecture Singer Conjecture Proof. Suppose n = 2k, Γ = π 1 (M n ). Singer Conj = only L 2 b k 0. gives: ( 1) k L 2 b k ( M 2k ; Γ) = χ orb ( M 2k /Γ). So, ( 1) k χ orb ( M 2k /Γ) 0.