On the zero-in-the-spectrum conjecture

ON THE ZERO-IN-THE-SPECTRUM CONJECTURE arXiv:math/9911077v1 [math.DG] 11 Nov 1999 M. FARBER AND S. WEINBERGER Abstract. We prove that the answer to the ”zero-in-the-spectrum” conjecture, in its form, suggested by J. Lott, is negative. Namely, we show that for any n ≥ 6 there exists a closed n-dimensional smooth manifold M n , so that zero does not belong to the spectrum of the Laplace-Beltrami operator acting on the L2 forms of all degrees on the universal covering M˜. 1. The Main result M. Gromov formulated the following conjecture (cf. [G1], p. 120; [G2], p.21, Problem 0.5.F1” , and also [G2], p. 238): Conjecture A. Let M be a closed aspherical manifold; is it true that zero is always in the spectrum of the Laplace-Beltrami operator ∆p , acting on the square integrable p-forms on the universal covering M˜ , for some p? If the Strong Novikov Conjecture holds for the fundamental group π1 (M), then 0 ∈ Spec(∆p ) for some p, cf. [L], p. 371. Hence a conterexample to Conjecture A would be also a conterexample to the Strong Novikov Conjecture. G. Yu obtained in [Yu1], [Yu2] results, confirming Conjecture A under some addi- tional assumptions. In 1991 J. Lott raised a more general ”zero-in-the-spectrum” question: Conjecture B. Is it true, that for any complete Riemannian manifold M zero is always in the spectrum of the Laplace-Beltrami operator ∆p , acting on the square inte- grable p-forms on M, for some p? We refer to the survey articles [L] and [Lu]. J. Lott showed in [L] that Conjecture B is true for manifolds of low dimension and also for some classes of higher dimensional manifolds. In this article we give negative answers to Conjecture B and also to a version of Conjecture A where one removes the assumption of asphericity of M. We prove the following Theorem. Date: October 1, 1999. 1991 Mathematics Subject Classification. Primary 57Q10; Secondary 53C99. Key words and phrases. Zero in the spectrum conjecture, extended L2 cohomology. Partially supported by the US - Israel Binational Science Foundation and by the Minkowski Center for Geometry. 1 2 M. FARBER AND S. WEINBERGER Theorem 1. For any n ≥ 6 there exists a closed n-dimensional smooth manifold M, so that for any p = 0, 1, . . . , n the zero does not belong to the spectrum of the Laplace- Beltrami operator ∆p : Λp (M ˜ ) → Λp (M), ˜ (2) (2) ˜ ) on the universal covering M acting on the space of L2 -forms Λp(2) (M ˜ of M. Our proof of Theorem 1 will be based on the fact that it can be restated in an equivalent form using the notion of extended L2 -homology, introduced in [F1]: Theorem 2. For any n ≥ 6 there exists a closed orientable n-dimensional smooth manifold M, so that extended L2 homology Hp (M; ℓ2(π)) = 0 vanishes for all p. Here π denotes the fundamental group π = π1 (M), and ℓ2(π) denotes the L2 -completion of the group ring C[π]. Equivalence between Theorem 1 and Theorem 2 can be established as follows. Zero ˜ → Λ p (M not in the spectrum of the Laplacian ∆p : Λp(2) (M) ˜ ) for all p is equivalent to (2) vanishing of the extended L2 cohomology H∗ (M; ℓ2(π)), cf. [F1], according to the De Rham Theorem for extended cohomology, cf. section 7 of [F2] and also [S]. Vanishing of the extended L2 -cohomology is equivalent to vanishing of the extended L2 -homology H∗ (M; ℓ2(π)), because of the Poincar´e duality, cf. [F1], Theorem 6.7. The proof of Theorem 2 is based on the following Theorem: Theorem 3. There exists a finite 3-dimensional polyhedron Y with fundamental group π1 (Y ) = π = F × F × F , where F denotes a free group with two generators, such that the extended L2 -homology Hp (Y ; ℓ2 (π)) = 0 vanishes for all p = 0, 1, . . . . The strategy of our proof of Theorems 2 and 3 is similar to the method used by M.A. Kervaire [K], who constructed smooth homology spheres with prescribed fundamental groups. Our proof uses L2 -analog of the Hopf exact sequence. The authors are thankful to B. Eckmann and A. Connes for helpful conversations. 2. Proofs of Theorems 2 and 3 A. Let π be a discrete group given by a finite presentation π = |x1 , x2 , . . . , xn : r1 = 1, r2 = 1, . . . , rm = 1| by generators and relations. We will assume that: (a) The extended L2 -homology of π in dimensions 0, 1 and 2 vanishes, i.e. H0 (π; ℓ2(π)) = H1 (π; ℓ2(π)) = H2 (π; ℓ2(π)) = 0. (b) Let X be a finite cell complex with π1 (X) = π having one zero-dimensional cell, n cells of dimension 1 and m cells of dimension two, constructed in the usual way out of the given presentation of π. Then the second homotopy group π2 (X) of X, viewed as a Z[π]-module, is free and finitely generated.  ZERO-IN-THE-SPECTRUM CONJECTURE 3 Our purpose is to show that there exists a 3-dimensional cell complex Y , obtained from X by first taking a bouquet with finitely many copies of S 2 and then adding a finite number of 3-dimensional cells, so that Hi (Y ; ℓ2(π)) = 0 for any i = 0, 1, . . . . (1) B. L2 -Hopf exact sequence. First we will calculate the extended L2 homology of X using the spectral sequence constructed in Theorem 9.7 of [F1]. We will work in the von Neumann category Cπ of Hilbert representations of π, cf. [F2], §2, example 5. We ˜ will denote by E(Cπ ) the corresponding extended abelian category, cf. [F2], §1. Let X be the universal covering of X. We will use the functors ˜ C)) T orpπ (ℓ2(π), Hq (X; with values in the extended abelian category E(Cπ ), which are defined in [F1], page 660 under the assumption that the homology modules Hq (X; ˜ C) of the universal covering admit finite free resolutions. In our case only two of these homology modules can be nonzero (for q = 0 and q = 2), and (since H2 (X;˜ C) = C ⊗ π2 (X)) our assumption (b) ˜ C)) can guarantees this finiteness condition for q = 2. The functor T or0π (ℓ2(π), Hq (X; be denoted by ℓ2(π)⊗ ˜ C). ˜ π Hq (X; (2) It is an analog of the tensor product, cf. [F2], §6. Note that in general it takes values in the extended category E(Cπ ), i.e. it may have a nontrivial torsion part. By Theorem 9.7 of [F1], there exists a spectral sequence in the abelian category E(Cπ ) with the following properties: 2 • the initial term Ep,q ˜ C)). equals T orpπ (ℓ2(π), Hq (X; • The spectral sequence converges to the extended homology Hp+q (X; ℓ2(π)). For q = 0 we have H0 (X; ˜ C) = C, and T or π (ℓ2(π), C) can be also understood as the q extended L2 homology of the Eilenberg - MacLane space K(π, 1). We will use notation T orqπ (ℓ2(π), C) = Hq (π; ℓ2(π)). (3) It is an analog of the group homology. Since X is two-dimensional, the spectral sequence contains only two rows (q = 0 and q = 2) and may have only one nontrivial differential. Hence we obtain the following isomorphisms: H0 (X; ℓ2(π)) ≃ H0 (π; ℓ2(π)) and H1 (X; ℓ2(π)) ≃ H1 (π; ℓ2(π)). (4) These are Hurewicz type isomorphisms. The first nontrivial differential of the E 2 -term 2 is d2 : E3,0 2 → E0,2 2 . Here E3,0 2 = H3 (π; ℓ2(π)) and E0,2 = ℓ2(π)⊗ ˜ C). Using the ˜ π H2 (X; Hurewicz isomorphism H2 (X) ˜ ≃ π2 (X)˜ ≃ π2 (X), we may write 2 ˜ π π2 (X) E0,2 = ℓ2(π)⊗ 4 M. FARBER AND S. WEINBERGER and the above differential is ˜ π π2 (X). d2 : H3 (π; ℓ2(π)) → ℓ2(π)⊗ (5) Note also that this differential must be a monomorphism (viewed as a morphism of the abelian category E(Cπ )), since H3 (X; ℓ2(π)) = 0 (recall that X is two-dimensional). The spectral sequence above yields the following exact sequence d h ˜ π π2 (X) → 0 → H3 (π; ℓ2(π)) →2 ℓ2(π)⊗ H2 (X; ℓ2(π)) → H2 (π, ℓ2(π)) → 0. (6) It is an L2 analog of the Hopf ’s exact sequence. We conclude (using (4) and our assumptions (a)) that H0 (X; ℓ2(π)) = H1 (X; ℓ2(π)) = 0 and H2 (X; ℓ2(π)) can be found from the exact sequence h ˜ π π2 (X) → 0 → H3 (π; ℓ2(π)) → ℓ2(π)⊗ H2 (X; ℓ2(π)) → 0. (7) C. We will now specialize our discussion to the following group π = F × F × F, where F is a free group with two generators. We will denote the free generators of the factor number r (where r = 1, 2, 3) by ar1 , ar2 . We will fix the presentation of π given by 6 generators a11 , a12 , a21 , a22 , a31 , a32 and the following 12 relations (aki , alj ) = 1, for k 6= l, k, l ∈ {1, 2, 3}, i, j ∈ {1, 2}, where (v, w) = vwv −1w −1 denotes the commutator. π satisfies condition (a) above, as follows from the Kunneth theorem for the extended L -cohomology, cf. Appendix, Theorem 6. Here we use that Hj (F ; ℓ2 (F )) is nonzero 2 only for j = 1 and has no torsion; hence the terms containing the periodic product in formula (34), vanish; cf. Proposition 4, statement (b). Let us show that this group π, together with its specified presentation, satisfies condition (b). The two-dimensional complex X constructed out of this presentation will have one zero-dimensional cell e0 , six 1-dimensional cells e1i , e2i , e3i and 12 two- dimensional cells e12 13 23 k ij , eij , eij . Here ei denotes the 1-cell corresponding to the generator aki and ekl k l ij denotes the 2-cell corresponding to the relation (ai , aj ) = 1. Let 0 → C2 → C1 → C0 → 0 denote the chain complex of the universal covering X. ˜ The boundary homomorphism acts as follows ∂eki = (aki − 1)e0 , ∂ekl k l l k ij = (ai − 1)ej − (aj − 1)ei . Using the Hurewicz isomorphisms π2 (X) = π2 (X) ˜ = H2 (X), ˜ we may compute the group π2 (X), viewed as a Z[π]-module, as the kernel of ∂ : C2 → C1 . Let X X X x ∈ C2 , x = λ12 e12 ij ij + λ 13 13 e ij ij + λ23 23 ij eij , λkl ij ∈ Z[π], ij ij ij  ZERO-IN-THE-SPECTRUM CONJECTURE 5 be an element with ∂x = 0. Then the following equations hold P 12 1 P 23 3 i λij (ai − 1) = i λji (ai − 1), λ12 2 λ13 3 P P j ij (aj − 1) + j ij (aj − 1) = 0, λ13 1 λ23 2 P P i ij (ai − 1) + i ij (ai − 1) = 0. Hence we may write λ12 µ12 3 µ12 P ij = k ijk (ak − 1), ijk ∈ Z[π], λ23 µ23 1 µ23 P ij = k ijk (ak − 1), ijk ∈ Z[π], λ13 µ13 2 µ13 P ij = k ijk (ak − 1), ijk ∈ Z[π]. Therefore we obtain µ12 23 13 ijk = µjki = −µikj . (8) Conversely, any system µrsijk ∈ Z[π] satisfying (8) determines a cycle x ∈ C2 , ∂x = 0. This proves that π2 (X) is a free Z[π]-module of rank 8 with the basis xijk = (a1i − 1)e23 2 13 3 12 jk − (aj − 1)eik + (ak − 1)eij , i, j, k ∈ {1, 2}. (9) Note that the Eilenberg-MacLane complex K = K(π, 1) is B ×B ×B, where B is the bouquet of two circles; K is obtained from X by adding 8 three-dimensional cells eijk , where i, j, k ∈ {1, 2}, which correspond to different triple products of 1-dimensional cells of B. It is easy to see that the boundary of eijk is given by ∂eijk = xijk ∈ π2 (X). The chain complex of the universal covering K ˜ is 0 → C3 → C2 → C1 → C0 → 0, where C3 is the free Z[π]-module generated by the cells eijk and the rest is the same as the chain complex of X. ˜ ∗ For a discrete group π we will denote by CR (π) ⊂ Cr∗ (π) the real part of the reduced C ∗ -algebra, i.e. the norm closure of the real group ring R[π] ⊂ C[π]. D. Proposition. Let F be the free group with generators a1 , a2 . Then there exist ∗ u 1 , u 2 ∈ CR (F ) ⊂ Cr∗ (F ) such that (i) u1 (a1 − 1) + u2 (a2 − 1) = 0, (ii) for any pair v1 , v2 ∈ ℓ2 (π) with v1 (a1 − 1) + v2 (a2 − 1) = 0 (10) there exists a unique w ∈ ℓ2 (π) such that v1 = wu1, v2 = wu2 . 6 M. FARBER AND S. WEINBERGER Here we consider F as a subgroup of π = F × F × F identifying it with one of the factors.The reduced C ∗ -algebra Cr∗ (F ) ⊂ Cr∗ (π) acts in the usual way on ℓ2 (π). Proof. For convenience, we will assume in the proof that F is the third factor in π. Consider the standard complex d ℓ2 (F ) ⊕ ℓ2 (F ) → ℓ2 (F ), (v1 , v2 ) 7→ v1 (a1 − 1) + v2 (a2 − 1). (11) calculating the extended L2 homology of the bouquet S 1 ∨ S 1 of two circles with coefficients in ℓ2 (F ). Since F is not amenable, we know from Brooks’ theorem that d is an epimorphism, i.e. H0 (S 1 ∨ S 1 ; ℓ2 (F )) = 0. The Euler characteristic calculation shows that ker d = H1 (S 1 ∨ S 1 ; ℓ2 (F )) is one dimensional, i.e. it is isomorphic to ℓ2 (F ). Here we use the fact that the von Neumann algebra N (F ) is a factor. Let P : ℓ2 (F ) ⊕ ℓ2 (F ) → ℓ2 (F ) ⊕ ℓ2 (F ) be the orthogonal projection onto ker d. We claim that the element P (1, 0) belongs to ∗ CR ∗ (F ) ⊕ CR (F ) ⊂ Cr∗ (F ) ⊕ Cr∗ (F ) ⊂ ℓ2 (F ) ⊕ ℓ2 (F ). (12) Let d∗ be the adjoint of d. Then ker d = ker(d∗ d). Moreover, the image of d∗ d is closed and thus zero is an isolated point in the spectrum of d∗ d. Hence we may use the holomorphic functional calculus (Cauchy’s formula) in order to express the projector P as 1 Z P = (z − d∗ d)−1 dz, 2πi Γ where Γ is a small circle around the origin. This explains that P (v1 , v2 ) belongs to Cr∗ (F ) ⊕ Cr∗ (F ) (cf. (12)), assuming that v1 , v2 lie in the reduced C ∗ -algebra Cr∗ (F ). Moreover, since the operator d∗ d is real, we obtain that P (v1 , v2 ) ∈ CR ∗ ∗ (F ) ⊕ CR (F ), ∗ for v1 , v2 ∈ CR (F ). We will set now (u1 , u2) = P (1, 0). Then (i) is clearly satisfied. We want to show that the restriction of P on the first summand ℓ2 (F ) in (11) gives an isomorphism P : ℓ2 (F ) → ker d. Since both ker d and ℓ2 (F ) have von Neumann dimension one, and the spectrum of P contains only 0 and 1, we conclude that it is enough to show that P (v, 0) = 0 for v ∈ ℓ2 (F ) implies v = 0. If P (v, 0) = 0 i.e. (v, 0) ∈ (ker d)⊥ then hv, ker di = 0, i.e. v is orthogonal to the projection of ker d on the first summand ℓ2 (F ). From this we will obtain that necessarily v = 0 if we will show that the projection of ker d on the first summand is dense. Let fi : ℓ2 (F ) → ℓ2 (F ) be operator x 7→ x(ai − 1), where i = 1, 2. It is clear that f1 and f2 are injective and hence their images are dense. We claim that f1−1 (im f2 ) is dense. If not, let H denote the orthogonal complement to the closure of f1−1 (im f2 ). Then we may apply Proposition 2.4 from [F2]; it implies that H must intersect im f2 , which is impossible. Hence it follows that the projection of ker d on the first summand ℓ2 (F ) (which coincides with f1−1 (im f2 )) is dense.  ZERO-IN-THE-SPECTRUM CONJECTURE 7 As a result we obtain from the above arguments that for any pair (v1 , v2 ) ∈ ker d (i.e. which is a solution of (10)) there exists w ∈ ℓ2 (F ), so that P (w, 0) = (v1 , v2 ), i.e. v1 = wu1 and v2 = wu2. This is in fact a part of our statement (ii). In order to prove (ii) in full generality, observe that ℓ2 (π) = ℓ2 (F )⊗ℓ ˆ 2 (F )⊗ℓ ˆ 2 (F ), (13) (cf. appendix) and thus (using the Kunneth theorem for extended L2 homology, cf. Theorem 5) we find that the kernel of the operator d : ℓ2 (π) ⊕ ℓ2 (π) → ℓ2 (π), (v1 , v2 ) 7→ v1 (a1 − 1) + v2 (a2 − 1), ˆ 2 (F ) ⊗ equals ℓ2 (F )⊗ℓ ˆ H1 (S 1 ∨ S 1 ; ℓ2 (F )). (ii) now follows. E. Now we describe the kernel of the Hurewicz homomorphism ˜ π π2 (X) → H2 (X; ℓ2(π)). h : ℓ2(π)⊗ Let usi ∈ Cr∗ (π), where s = 1, 2, 3 and i = 1, 2, denote the element given by Proposition D applied to the factor F ⊂ π number s = 1, 2, 3. Here we consider Cr∗ (F ) as being canonically embedded into the von Neumann algebra N (π). We claim that the kernel of the Hurewicz homomorphism h is generated by the ele- ment X y= u1i u2j u3k xijk ∈ CR ∗ ˜ π π2 (X). (π)⊗ (14) ijk More precisely, our statement is that any element x ∈ ℓ2(π)⊗ ˜ π π2 (X) with h(x) = 0 2 has the form x = µy for some µ ∈ ℓ (π). Note that the product µy has sense because the coefficients of y in the basis xijk ∗ belong to CR (π) ⊂ Cr∗ (π). First, it is easy to check (using (9)) that h(y) = 0. Let X x= ˜ π π2 (X), h(x) = 0, µijk xijk ∈ ℓ2(π)⊗ ijk be an arbitrary element of ker h, where µijk ∈ ℓ2 (π). Using (9), we obtain (equating to zero the coefficients of the cells e23 jk ) that for any pair of indices j, k holds 2 X µijk (a1i − 1) = 0. i=1 Hence, applying Proposition D, we conclude that there exist µjk ∈ ℓ2 (π) such that µijk = µjk u1i . (15) 8 M. FARBER AND S. WEINBERGER We write again h(x) = 0, equating to zero the coefficients of the cells e13 ik and using (15). We obtain that for any pair of indices i, k holds " # X X 1 2 µjk ui (aj − 1) = µjk (aj − 1) u1i = 0. 2 (16) j j Note that wus1 = 0 for w ∈ ℓ2 (π) implies wus2 = 0 (using (10)) and from the uniqueness statement in Proposition D, (ii), we obtain that w = 0. Therefore (16) implies X µjk (a2j − 1) = 0 j and hence using Proposition D, µjk = µk u2j , where µk ∈ ℓ2 (π). Substitute again µijk = µk u1i u2j into h(x) = 0 and equating to zero the coefficients of the cells e13 ik we obtain " # X X µk (ak − 1) u1i u2j = 0, and hence 3 µk (a3k − 1) = 0. (17) k k Using Proposition D as above we finally obtain µk = µu3k , where µ ∈ ℓ2 (π). Therefore, we find that µijk = µu1i u2j u3k and x = µy. F. Our goal is to show that one may add 8 cells of dimension 3 to the bouquet X ∨ S 2 such that the obtained 3-dimensional complex Y will have all trivial extended L2 homology Hj (Y ; ℓ2(π)) = 0, j = 0, 1, . . . . For the proof, let’s examine again the exact sequence (7): φ h ˜ π π2 (X) → H2 (X; ℓ2(π)) → 0. 0 → H3 (π; ℓ2(π)) → ℓ2(π)⊗ (18) As we know, φ maps the generator y of H3 (π; ℓ2(π)) according to formula (14), i.e. φ ∗ is given by a matrix with entries in CR (π) ⊂ Cr∗ (π). Let ˜ π π2 (X) → ℓ2(π)⊗ Q : ℓ2(π)⊗ ˜ π π2 (X) denote the orthogonal projection onto (im φ)⊥ , the orthogonal complement of the image of φ. Since X is two-dimensional, H2 (X; ℓ2(π)) in has no torsion and therefore im φ is closed. Note that (im φ)⊥ coincides with ker(φφ∗ ). Since the image of φφ∗ is closed we conclude that zero is an isolated point in the spectrum of φφ∗ and hence we may write 1 Z Q= (z − φφ∗ )−1 dz, 2πi Γ  ZERO-IN-THE-SPECTRUM CONJECTURE 9 where Γ is a small circle round zero. Therefore, in the basis xijk the projector Q is ∗ given a (8 × 8)-matrix with entries in CR (π). ∗ The projective CR (π)-module determined by Q is stably free; we know that adding a free one-dimensional module (generated by y) makes it free. Therefore we may consider the bouquet X1 = X ∨ S 2 so that H2 (X1 ; ℓ2(π)) = H2 (X; ℓ2(π)) ⊕ ℓ2(π) and π2 (X1 ) = π2 (X) ⊕ Z[π]. Thus, the exact sequence (18) for X1 ψ h ˜ π π2 (X1 ) → 0 → H3 (π; ℓ2(π)) → ℓ2(π)⊗ H2 (X1 ; ℓ2(π)) → 0. (19) will have the following property: the orthogonal projection ˜ π π2 (X1 ) → ℓ2(π)⊗ Q1 : ℓ2(π)⊗ ˜ π π2 (X1 ) onto (im ψ)⊥ is given by a (9 × 9)-matrix with entries in CR ∗ (π) which determines a ∗ free CR (π)-module of rank 8. We may reformulate the last statement as follows: there exists a Z[π]-homomorphism γ : (Z[π])8 → CR ∗ ˜ π π2 (X1 ) (π)⊗ (20) such that the following composite ˜ π (Z[π])8 1⊗γ ℓ2(π)⊗ → ℓ2(π)⊗ h ˜ π π2 (X1 ) → H2 (X1 ; ℓ2(π)) (21) is an isomorphism. Now we will use the fact that the rational group ring Q[π] is dense ∗ in CR (π) with respect to the operator norm topology. Hence we may approximate γ by a Z[π]-homomorphism γ1 : (Z[π])8 → Q[π] ⊗π π2 (X1 ) so that the similar composition (21) is an isomorphism. Finally, we may multiply γ1 by a large integer N to obtain a Z[π]-homomorphism γ2 : (Z[π])8 → Z[π] ⊗π π2 (X1 ) = π2 (X1 ) such that the composition ˜ π (Z[π])8 1⊗γ (ℓ2(π))8 = ℓ2(π)⊗ 2 → ℓ2(π)⊗ h ˜ π π2 (X1 ) → H2 (X1 ; ℓ2(π)) (22) is an isomorphism. Let z1 , . . . , z8 ∈ π2 (X1 ) be images of a free basis of (Z[π])8 under γ2 . Realize each zj by a continuous map fj : S 2 → X1 , where j = 1, . . . , 8, and let Y = X1 ∪ e31 ∪ · · · ∪ e38 be obtained from X1 by glueing 8 three-dimensional cells to X1 along f1 , . . . , f8 . We claim that Hj (Y ; ℓ2(π)) = 0 for all j = 0, 1, . . . . (23) In order to show this, we note that Hj (Y, X; ℓ2(π)) vanishes for all j 6= 3 and the 3-dimensional extended L2 homology H3 (Y, X; ℓ2(π)) equals (ℓ2 (π))8 . The boundary homomorphism ∂ : H3 (Y, X; ℓ2(π)) → H2 (X; ℓ2(π)) is an isomorphism since it coincides 10 M. FARBER AND S. WEINBERGER with (22). Hence (23) follows from the homological exact sequence of the pair (Y, X). This completes the proof of Theorem 3. G. Now we may complete the proof of Theorem 2. We have constructed above a finite 3-dimensional polyhedron Y . For any n ≥ 6 we may embed Y into Rn+1 as a subpolyhedron. Let N ⊂ Rn+1 be the regular neighborhood of Y ⊂ Rn+1. We will define M as the boundary of N, i.e. M = ∂N. First note that the inclusion M → N induces an isomorphism of the fundamental groups and thus π1 (M) = π = F × F × F, where F is a free group in two generators. We want to show that Hj (M; ℓ2(π)) = 0, for all j = 0, 1, . . . (24) In the exact homological sequence · · · → Hj (M; ℓ2(π)) → Hj (N; ℓ2(π)) → Hj (N, M; ℓ2(π)) → . . . we have Hj (N; ℓ2(π)) = 0. Also, Hj (N, M; ℓ2(π)) ≃ Hn+1−j (N; ℓ2(π)) by the Poincar´e duality (cf. [F1]) and Hn+1−j (N; ℓ2(π)) = 0 because of (23) using the Universal Co- efficients Theorem (cf. [F1]). Hence, (24) follows. Appendix: Kunneth theorem for extended L2 cohomology 1. A Hilbert category C is defined as an additive subcategory of the category of Hilbert spaces and bounded linear maps, such for any morphism f : H → H ′ of C the inclusion ker(f ) ⊂ H belongs to C and also the adjoint map f ∗ : H ′ → H belongs to C, cf. [F1]. It is shown in [F1] that any Hilbert category can be canonical embedding into an abelian category E(C), called the extended abelian category. Let C, C ′ and C ′′ be three Hilbert categories and let ˆ : C × C ′ → C ′′ ⊗ (25) be a covariant functor of two variables (the ”tensor product”) such that (a) for H ∈ Ob(C) and H ′ ∈ Ob(C ′ ) the image H ⊗ ˆ H ′ has as the underlying Hilbert space the tensor product of Hilbert spaces H and H ′ ; (b) if f : H → H1 is a morphism of C and f ′ : H ′ → H1′ is a morphism of C ′ then f⊗ ˆ f′ : H ⊗ˆ H ′ → H1 ⊗ˆ H1′ is the tensor product of bounded linear maps f and ′ f. Recall that the tensor product if Hilbert spaces H ⊗ ˆ H ′ is defined as the Hilbert space completion of the algebraic tensor product H ⊗ H ′ with respect to the following scalar product hv ⊗ w, v ′ ⊗ w ′ i = hv, v ′i · hw, w ′i. Suppose that (C, d) and (C ′ , d) are chain complexes in C and C ′ correspondingly. We assume that all chain complexes are graded by non-negative integers and have a finite length. Their tensor product (C, d) ⊗(C ˆ ′ , d) (defined in the usual way) is a chain ′′ complex in C . (C, d) ⊗(C ˆ ′ , d) is a projective chain complex in the abelian category ′′ E(C ) and its extended homology H∗ (C ⊗ ˆ C ′ ) is an object of the extended category  ZERO-IN-THE-SPECTRUM CONJECTURE 11 ˆ ′ , d) in terms of E(C ′′ ). Our purpose is to express the extended homology of (C, d) ⊗(C the extended homology H∗ (C) of (C, d) and H∗ (C ′ ) of (C ′ , d). 2. Example. Suppose that G and H are discrete groups. Let CG denote the category of Hilbert representations of G. Recall, that an object of C is a Hilbert space with a unitary G-action which can be continuously and G-equivariantly imbedded into a finite direct sum ℓ2 (G) ⊕ · · · ⊕ ℓ2 (G); morphisms of C are bounded linear maps commuting with the G-action. Then we have the tensor product functor ˆ : CG × CH → CG×H ⊗ (26) which is of a primary interest for us. 3. Tensor and periodic products. Given a tensor product (25), it defines two bifunctors E(C) × E(C) → E(C), which we now describe. Let X = (α : A′ → A) ∈ Ob(E(C)) and Y = (β : B ′ → B) ∈ Ob(E(C ′ )) be two objects with α and β injective. Consider the following chain complex in C ′′ ˆβ   −1 ⊗ ˆ   α⊗1 ˆ 1, 1 ⊗ (α ⊗ ˆ β) ˆ B′ 0 → A′ ⊗ −→ ˆ B) ⊕ (A ⊗ (A′ ⊗ ˆ B′) −→ A ⊗ B → 0. (27) In other words, we view the objects X and Y as chain complexes of length 1 and then (27) is the tensor product of these chain complexes. The extended homology of (27) in dimension 0 will be called the tensor product of X and Y: ˆ Y = ((α ⊗ X⊗ ˆ 1, 1 ⊗ ˆ β) : (A′ ⊗ ˆ B) ⊕ (A ⊗ ˆ B′) → A ⊗ ˆ B). (28) The extended homology of (27) in dimension 1 will be called the periodic product of X and Y: ˆβ    −1 ⊗ ′ ˆ ′ X ∗Y = ˆ1 : A ⊗B → Z , (29) α⊗ where ˆβ    −1 ⊗ ′ ˆ B) ⊕ (A ⊗ ˆ B ) → A⊗ ′ ˆ B) . Z = ker ˆ : (A ⊗ (30) α⊗1 ˆ Y and X ∗ Y are covariant functors of two variables. It is easy to see that X ⊗ Proposition 4. Let ⊗ ˆ : C × C ′ → C ′′ be a tensor product functor (25). Let X ∈ Ob(E(C)) and Y ∈ Ob(E(C ′ )). Then (a) X ⊗ˆ Y is projective if both X and Y are projective; (b) X ∗ Y = 0 if X or Y is projective; (c) X ⊗ˆ Y is torsion if X or Y is torsion; (d) If C ′′ is a finite von Neumann category then X ∗ Y is torsion for any X and Y. 12 M. FARBER AND S. WEINBERGER Proof. Statements (a) and (b) follow directly from the definitions. Let’s prove (c) assuming that X = (α : A′ → A) is torsion, i.e. im α ⊂ A is dense. From the definition of the tensor product ⊗ ˆ it follows that then the image of ˆ ′ ˆ ˆ α ⊗ 1 : A ⊗ B → A ⊗ B is dense and hence from (28) we see that X ⊗ ˆ Y is torsion. It is enough to prove (d), assuming that both X and Y are torsion. Let X = (α : A → A) and Y = (β : B ′ → B) with α and β injective and having dense images. ′ Then A′ is isomorphic to A and B ′ is isomorphic to B (cf. [F2], §2). Therefore (d) will follow if we can show that Z (given by (30)) is isomorphic to A ⊗ˆ B. The projection of ′ ˆ Z on the first coordinate gives a morphism Z → A ⊗ B which is injective (obviously) and has a dense image (this follows from Proposition in §2 of [F2]). Hence we obtain (using Lemma in §2 of [F2]) that Z is isomorphic to A′ ⊗ ˆ B ≃ A⊗ ˆ B. Theorem 5 (Kunneth formula). Extended homology H∗ (C ⊗ ˆ C ′ ) of a tensor product, where (C, d) is a chain complex in C and (C ′ , d) is a chain complex in C ′ , equals M M ˆ C ′) = Hn (C ⊗ ˆ Hj (C ′ ) ⊕ Hi (C) ⊗ Hi (C) ∗ Hj (C ′ ). (31) i+j=n i+j=n−1 Proof. Let Zi ⊂ Ci and Zi′ ⊂ Ci′ denote the subspaces of cycles. We have the decomposition Ci = Zi ⊕ Zi⊥ ; the boundary homomorphism vanishes on Zi and maps Zi⊥ into Zi−1 . Let’s denote by Di the short chain complex Di = ⊥ ⊥ i+1 → Zi ), where Zi stands in degree i and Zi+1 stands in degree i + 1. Then (d : ZL ∞ C ≃ i=0 Di , i.e. C is isomorphic to the direct sum of the chain complexes Di . ⊥ L∞ Similarly, we define chain complexes Dj′ = (d : Zj+1′ → Zj′ ) and C ′ ≃ ′ j=0 Dj . Hence we obtain M M C⊗ˆ C′ ≃ ˆ Dj′ ), Hn (C ⊗ (Di ⊗ ˆ C ′) = Hn (Di ⊗ˆ Dj′ ). (32) i,j i,j Now we observe that Di has nontrivial homology only in dimension i and Hi (Di ) = Hi (C); similarly, Dj′ has nontrivial homology only in dimension j and Hj (Dj′ ) = ˆ Dj′ has nontrivial homology only in dimensions i + j and Hj (C ′ ). Therefore Di ⊗ i + j + 1, and ˆ Dj′ ) = Hi (C) ⊗ Hi+j (Di ⊗ ˆ Hj (C ′ ), ˆ Dj′ ) = Hi (C) ∗ Hj (C ′ ) Hi+j+1 (Di ⊗ (33) according to our definition of the tensor and periodic products. Formula (31) now follows by combining (32) and (33).  ZERO-IN-THE-SPECTRUM CONJECTURE 13 Theorem 6 (Kunneth formula for extended L2 homology). Let X, X ′ be finite cell complexes with π = π1 (X), π ′ = π1 (X ′ ). Then Hn (X × X ′ ; ℓ2 (π × π ′ )) ≃ M Hi (X; ℓ2 (π)) ⊗ ˆ Hj (X ′ ; ℓ2 (π ′ )) ⊕ (34) i+j=n M Hi (X; ℓ2 (π)) ∗ Hj (X ′ ; ℓ2 (π ′ )), i+j=n−1 where the tensor and periodic products are understood with respect to functor (26). Proof. Let C∗ (X) ˜ and C∗ (X ˜ ′ ) be the cell chain complexes of the universal coverings X ˜ and X ˜ ′ . We apply the previous Theorem to chain complexes C = ℓ2 (π)⊗ ˜ and ˜ π C∗ (X) ′ 2 ′ ˜ ′ C = ℓ (π )⊗π C∗ (X ˜ ). Note that C is a chain complex in category Cπ (cf. Example ′ above) and Hn (C) = Hn (X; ℓ2 (π)). Similarly C ′ is a chain complex in Cπ′ and Hn (C ′ ) = Hn (X ′ ; ℓ2 (π ′ )). Formula (34) follows from (31) using the isomorphism ℓ2 (π) ⊗ ˆ ℓ2 (π ′ ) = ℓ2 (π × π ′ ) and the fact that the chain complex C∗ (X) ˜ ⊗Z C∗ (X ˜ ′ ) over Z[π × π ′ ] is isomorphic to C∗ (X^ × X ′ ), where we consider the obvious product cell structure on ′ X ×X . References [G1] M. 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