L 2 -cohomology of hyperplane complements

Transcription

1 (work with Tadeusz Januskiewicz and Ian Leary) Oxford, Ohio March 17, 2007

2 1 Introduction 2 The regular representation L 2 -(co)homology Idea of the proof 3 Open covers Proof of the Main Theorem

3 Statement of Main Theorem A a collection of affine hyperplanes in C n Definition rk(a), the rank, of A is the maximum codimension of a nonempty intersection of hyperplanes in A. Usually we denote rk(a) by l. A is essential if rk(a) = n. Σ(A) := H H A M(A) := C n Σ(A)

4 Statement of Main Theorem Fact H (C n, Σ) vanishes except in dimension l (= rk(a)). In fact, Σ S l 1. The number α(a) α(a) := dim H l (C n, Σ) := b l (C n, Σ) = the number of spheres in the wedge

5 Statement of Main Theorem Example Suppose A R is an essential hyperplane arrangement in R n. It divides R n into convex regions. dim H n (R n, Σ(A R )) is the number of bounded components of R n Σ(A R ). If A is complexification of A R, then (C n, Σ(A)) (R n, Σ(A R )). So α(a) is the number of bounded components of R n A R.

6 Statement of Main Theorem Main Theorem Suppose A is an arrangement of a finite number of affine hyperplanes in C n with rk(a) = l. Then the of M(A) (the complement of the hyperplanes) are all 0, except in dimension l, where β l (M(A)) = α(a). Here β i ( ) denotes the i th L 2 -Betti number (to be defined later).

7 Statement of Main Theorem Theorem Suppose L is a generic flat complex line bundle over M(A). Then H (M(A); L) vanishes except in dimension l and dim C H l (M(A); L) = α(a). Basic idea If L is a flat line bundle over S 1 giving a nonconstant local coefficient system, then H (S 1 ; L) = 0 for = 0, 1. Similarly, the basic idea for the Main Theorem is that the of S 1 vanish.

8 Introduction The regular representation L 2 -(co)homology Idea of the proof The regular representation π is a countable discrete gp. L 2 π := {f : π C f (x) 2 < }, where the sum is over all x π. L 2 π is a Hilbert space with Hermitian inner product: f f := f (x)f (x). x π There are unitary π-actions on L 2 π by left or right translation.

9 The regular representation L 2 -(co)homology Idea of the proof von Neumann dimension Using the von Neumann algebra N π is of π-equivariant bounded linear operators on L 2 π, it is possible to attach a dimension to any π-stable closed subspace of L 2 π. dim π V is a nonnegative real number. It is = 0 iff V = 0. Also, dim π L 2 π = 1.

10 The regular representation L 2 -(co)homology Idea of the proof (Co)homology with local coefficients X a CW complex X its universal cover C i ( X) the cellular i-chains on X π = π 1 (X). Suppose M is a π-module. C i (X; M) := C i ( X) π M C i (X; M) := Hom π (C i ( X), M) are the (co)chains with local coefficients in M, H (X; M) and H (X; M) are the corresponding (co)homology groups.

11 The regular representation L 2 -(co)homology Idea of the proof L 2 -(co)homology To fix ideas, let s stick to cohomology. At first approximation L 2 -cohomology means local coefficients in L 2 π, i.e., H (X; L 2 π). C (X; L 2 π) is a Hilbert space but H (X; L 2 π) need not be. Ker δ is a closed subspace but Im δ need not be. Define H (X; L 2 π) := Ker δ/im δ. H (X; L 2 π) is a closed, π-stable subspace of C (X; L 2 π). (It is = Ker δ (Im δ).)

12 The regular representation L 2 -(co)homology Idea of the proof If X is a finite complex, then C i (X; L 2 π) is a direct sum of finitely many copies of L 2 π (one for each i-cell of X). So the closed, π-stable subspace H i (X; L 2 π) has a well-defined von Neumann dimension called the i th L 2 -Betti number β i (X) := dim π H i (X; L 2 π). If X is a finite complex then C (X; L 2 π) can be identified with the square summable cochains on X (denoted by L 2 C ( X)). The corresponding (reduced) cohomology groups are denoted L 2 H ( X).

13 The regular representation L 2 -(co)homology Idea of the proof Lemma The of S 1 vanish. Corollary All of S 1 B vanish. Proof. Künneth Formula.

14 The regular representation L 2 -(co)homology Idea of the proof Rough idea of proof of theorems Suppose U = {U i } is a cover of X by connected open subsets and V is a subcover s.t. V = {U i U π 1 (U i ) 1}. σ N(U), π 1 (U σ ) π 1 (X) (= π) is injective. (Here N(U) denotes the nerve of U and U σ = U i1 U ik, where σ = {i 1,... i k }.) σ N(U) N(V), U σ is contractible. σ N(V), U σ = S 1 (something).

15 The regular representation L 2 -(co)homology Idea of the proof There is a Mayer-Vietoris spectral sequence converging to H (X; L 2 π) with E 2 -term E p,q 2 = H p (N(U); H q (U σ ; L 2 (π 1 (U σ )), where the coefficient system is the functor σ H q (U σ ; L 2 (π 1 (U σ )). Hypotheses = E p,q 2 is concentrated on the bottom row q = 0 and E p,0 2 = H p (N(U), N(V)) L 2 π. (We are ignoring terms with vanishing.) So, β p (X) = b p (N(U), N(V)).

16 Open covers Proof of the Main Theorem More on hyperplane arrangements A is a hyperplane arrangement in C n and Σ is the union of hyperplanes. A subspace of A is a nonempty intersection of hyperplanes in A. L(A) is the poset of subspaces of A. A is central if the intersection of all hyperplanes in A is nonempty. Given G L(A), we have a central subarrangement A G := {H A G H}.

17 Open covers Proof of the Main Theorem Definition An open convex subset U C n is small (wrt A) if {G L(A) G U } has a minimum element U min. Given H A, H U H U min. Given a small U, put Û := U Σ. So, Û = M(A Umin ), the complement of a central arrangement.

18 Open covers Proof of the Main Theorem Choose a cover U of C n by small open sets. For each σ N(U), A σ is the corresponding central arrangement. V := {U U A σ is not trivial}. V is an open cover of a nbhd of Σ homotopy equiv to Σ. Since each element of U is convex, H (N(U), N(V)) = H (C n, Σ).

19 Open covers Proof of the Main Theorem Similarly, we have open covers Û := {Û} U U and V := {Û} U V. (Û is an open cover of M(A).) Key point N( V) = N(V) and N(Û) = N(U). Lemma Suppose A is a nonempty central arrangement. Then M(A) = S 1 (something).

20 Open covers Proof of the Main Theorem Main Theorem The of M(A) are all 0, except in dimension l, where β l (M(A)) = α(a). Proof. Û is open cover of M(A). σ N(Û) N( V), U σ. σ N( V), U σ = S 1 (something). Use spectral sequence and fact that H (N(Û), N( V)) = H (C n, Σ) to complete the proof.