Approximating L 2 -signatures by their compact analogues

Transcription

1 Approximating L 2 -signatures by their compact analogues Wolfgang Lück Fachbereich Mathematik Universität Münster Einsteinstr Münster Thomas Schick Fakultät für Mathematik Universität Göttingen Bunsenstrasse Göttingen Germany Last edited Jan 29, 2003 or later last compiled: April, 2003 Abstract Let Γ be a group together with a sequence of normal subgroups Γ Γ Γ 2... of finite index [Γ : Γ k ] such that k Γ k = {}. Let X, Y be a compact 4n-dimensional Poincaré pair and p : X, Y X, Y be a Γ-covering, i.e. normal covering with Γ as deck transformation group. We get associated Γ/Γ k -coverings X k, Y k X, Y. We prove that sign 2 X, Y = lim signx k, Y k, [Γ : Γ k ] where sign or sign 2 is the signature or L 2 -signature, respectively, and the convergence of the right side for any such sequence Γ k k is part of the statement. If Γ is amenable, we prove in a similar way an approximation theorem for sign 2 X, Y in terms of the signatures of a regular exhaustion of X. Our results are extensions of Lück s approximation results for L 2 -Betti numbers [0, Theorem 0.]. Key words: L 2 -signature, signature, covering with residually finite deck transformation group, amenable exhaustion mathematics subject classification: 57P0, 57N65, 58G0 wolfgang.lueck@math.uni-muenster.de www: schick@uni-math.gwdg.de www: Research partially carried out during a stay at Penn State university funded by the DAAD

2 Approximating L 2 -signatures by their compact analogues 2 0 Introduction Throughout most of this paper we will use the following conventions. We fix a group Γ, together with a sequence of normal subgroups Γ Γ Γ 2... of finite index [Γ : Γ k ] such that k Γ k = {}. Provided that Γ is countable, Γ is residually finite if and only if such a sequence Γ k k exists. Moreover, given a Γ-covering p : X X, i.e. a normal covering with Γ as group of deck transformations, we will denote the associated Γ/Γ k -coverings by X k := X/Γ k X and for a subspace Y X let Y X and Y k X k be the obvious pre-images of Y. One of the main results of the paper is 0. Theorem. Let X, Y be a 4n-dimensional Poincaré pair. Then the sequence signx k, Y k /[Γ : Γ k ] k converges and signx k, Y k lim = sign 2 X, Y. [Γ : Γ k ] Some explanations are in order. An l-dimensional Poincaré pair X, Y is a pair of finite CW -complexes X, Y with connected X together with a so called fundamental class [X, Y ] H l X; Q such that for the universal covering, and hence for any Γ-covering p : X X, the Poincaré QΓ-chain map induced by the cap product with a representative of the fundamental class [X, Y ] : C l X, Y C X is a QΓ-chain homotopy equivalence. Because we are working with free finitely generated left QΓ-chain complexes, this is the same as saying that the induced map in homology is an isomorphism. One usually also requires that Y itself is a l-dimensional Poincaré space using the corresponding definition where the second space is empty with [X, Y ] = [Y ], although this is not really necessary for our applications. Here C X is the cellular left QΓ-chain complex and C l X, Y is the dual QΓ-chain complex hom QΓ C l X, Y, QΓ. This is canonically a right QΓ-chain complex. Throughout the paper, we deal with left modules. We turn any right QΓ-module into a left QΓ-module using the involution of QΓ induced by Γ g g. Examples for Poincaré pairs are given by a compact connected topological oriented manifold X with boundary Y or merely by a rational homology manifold. The Poincaré duality chain map of a 4n-dimensional Poincaré pair X, Y induces an isomorphism H p X, Y ; C H 4n p X; C. If we compose the inverse with the map induced in cohomology by the inclusion X X, Y and with the natural isomorphism H p X; C = H p X; C to the dual space H p X; C of H p X; C, we get in the middle dimension 2n a homomorphism A : H 2n X; C H 2n X; C which is selfadjoint. The signature of the oriented pair X, Y is by definition the signature of the in general indefinite form A, i.e. the difference of the

3 Approximating L 2 -signatures by their compact analogues 3 number of positive and negative eigenvalues of the matrix representing A after choosing a basis for H 2n X, C and the dual basis for H 2n X. The L 2 -signature on X, Y is defined similarly, but one has to replace homology by L 2 -homology. L 2 -homology and L 2 -cohomology groups in this paper are always reduced, i.e. we divide by the closure of the image of the differential to remain in the category of Hilbert modules. We get then an operator A : H 2 2n X H2 2n X using the natural isomorphism of a Hilbert space with its dual space. The L 2 -homology is a Hilbert module over the von Neumann algebra N Γ and A is a selfadjoint bounded Γ-equivariant operator. Hence H 2 2n X splits orthogonally into the positive part of A, the negative part of A and the kernel of A. The difference of the N Γ-dimensions of the positive part and the negative part is by definition the L 2 -signature. All this can also be reformulated in terms of cohomology instead of homology, which is convenient e.g. when dealing with de Rham cohomology. An analogue of Theorem 0. for L 2 -Betti numbers has been proved by Lück [0, Theorem 0.]. If X is a smooth closed manifold, Atiyah s L 2 -index theorem [,.] shows that the signature is multiplicative under finite coverings and that sign 2 X = signx k /[Γ : Γ k ] holds for k. There are Poincaré spaces X = X, for which the signature is not multiplicative under coverings by [4, Example 22.28], [24, Corollary 5.4.]. There are also compact smooth manifolds with boundary with the same property, see [3, Proposition 2.2] together with the Atiyah-Patodi-Singer index theorem [2, Theorem 4.4]. This shows that Atiyah s L 2 -signature theorem does not generalize to these situations. Our result says for these cases that the signature is multiplicative at least approximately. For closed topological manifolds, it is known that the signature is multiplicative under finite coverings [8, Theorem 8]. In a companion [2, Theorem 0.2] to this paper, we prove the following theorem, this way apparently filling a gap in the literature: 0.2 Theorem. Let M be a closed topological manifold with normal covering M M. Then sign 2 M = signm. There, we also discuss to what extend Theorem 0.2 can be true for Poincaré duality spaces X = X,. We show [2] that Theorem 0.2 for Poincaré duality spaces X = X; is implied by the L-theory isomorphism conjecture or by a strong form of the Baum-Connes conjecture provided that Γ is torsion-free. Dodziuk-Mathai [9, Theorem 0.] give an analog of Lück s approximation theorem [0, Theorem 0.] for L 2 -Betti numbers to Følner exhaustions of amenable covering spaces. Along the same lines, we compute the L 2 -signature using a Følner exhaustion in Theorem 0.4, proved in Section 2.. The relevant definition is the following: 0.3 Definition. Let X be a connected compact smooth Riemannian manifold possibly with boundary X and X X be a Γ-covering for some amenable

4 Approximating L 2 -signatures by their compact analogues 4 group Γ. We lift the metric on X to X and use this metric to measure the volume of submanifolds open as well as of codimension of X. Let X X 2... X with k N X k = X be an exhaustion of X, X by smooth submanifolds with boundary where we don t make any assumptions about the intersection of X k and X. Set Y k := X k X k X i.e. X k = Y k X k X. The exhaustion is called regular if it has the following properties: areay k / volx k 0; 2 The second fundamental forms of X k in X and each of their covariant derivatives are uniformly bounded independent of k; 3 The boundaries X k are uniformly collared and the injectivity radius of X k is uniformly bounded from below always uniformly in k. Regular exhaustions were introduced in [8, p. 52]. The existence of such an exhaustion is equivalent to amenability of Γ provided that the total space X is connected. 0.4 Theorem. In the situation of Definition 0.3 in particular we require that X is connected we get signx k, X k lim volx k = sign2 X, Y, volx where the convergence of the left hand side is part of the assertion. The assumption that the base space X is connected is necessary. For smooth manifolds with boundary, the L 2 -signature of course is defined in terms of the intersection pairing on L 2 -homology. On the other hand, there is the L 2 -index of the signature operator with Atiyah-Patodi-Singer boundary conditions. The latter is computed in [3, Theorem.] in terms of the L 2 -η-invariant and a local integral. It is a nontrivial assertion that the L 2 -index of the signature operator really gives the cohomologically defined L 2 -signature. This is proved in [2, Theorem 3.2], using [23]. In Theorem 2.48 we give a combinatorial version of Theorem 0.4. For this we need the following definition: 0.5 Definition. For a simplicial complex Y let Y be the total number of simplices of Y. Similarly, for any subset W of a simplicial complex which is a union of open simplices, W is the number of open simplices in W. A sequence X X 2... X of finite subcomplexes of a simplicial complex X is called an amenable exhaustion if k N X k = X and if for each R > 0 U R X k X k.

5 Approximating L 2 -signatures by their compact analogues 5 It is called a balanced exhaustion, if for each orbit Γσ of simplices in X X k Γσ X k X. 0.6 Theorem. Assume that X is a compact simplicial complex triangulating a rational homology manifold with boundary the subcomplex X. Assume X is a normal covering of X with amenable covering group Γ. Let X X 2... be subcomplexes forming a balanced amenable exhaustion of X by rational homology manifolds with boundaries Y k. If X is a homology manifold, such an exhaustion does always exist. Then signx k, Y k lim X = sign 2 X, X. X k Acknowledgements: We thank the referee for valuable comments concerning the exposition of the paper. We thank Steve Ferry who explained to us how one can obtain homology submanifolds of codimension zero of a homology manifold by thickening subcomplexes. Organization of the paper: We will prove convergence of the signature for coverings in Section, and in Section 2 the statement about amenable exhaustions. Residual convergence of signatures This section is devoted to the proof of Theorem Abstract QΓ-chain complexes Let C be a finitely generated based free 4n-dimensional left QΓ-chain complex.. Definition. C being finitely generated based free means that each chain module C p is of the shape QΓ r = r i=qγ for some integer r 0. We define its dual QΓ-chain complex C 4n as follows. It has by definition as p-th chain module C 4n p and its p-th differential c 4n p : C 4n p C 4n p is given by the adjoint. c 2d p : C 2d p C 2d p The adjoint f : QΓ s QΓ r of a QΓ-map f : QΓ r QΓ s is given by right multiplication with the matrix A Ms, r, QΓ if f is given by right multiplication with the matrix A ij Mr, s, QΓ and A i,j = A j,i for w Γ λ w w := w Γ λ w w. Note that by this definition f is a left QΓ-module map. Identify hom QΓ QΓ r, QΓ homomorphisms which commute with the left QΓ-module structure with QΓ r using the canonical basis on QΓ r by sending φ to the vector φe i i=,...,r. This is an isomorphism of left QΓ-modules using our convention for the left QΓ-module structure on hom QΓ. Then f corresponds to hom QΓ f, id QΓ. Given a QΓ-chain map f : C 4n C, define its adjoint QΓ-chain map f 4n : C 4n C as given by left multiplication with the adjoint of the matrix representing f.

6 Approximating L 2 -signatures by their compact analogues 6.2 Definition. Define the finitely generated 4n-dimensional Hilbert N Γ-chain complex C 2 by l 2 Γ QΓ C and the finitely generated based free 4n-dimensional Q[Γ/Γ k ]-chain complex C [k] by Q[Γ/Γ k ] QΓ C. Notice that C 2 4n is the same as C 4n 2 and will be denoted by C 4n 2 and similarly for C [k]. If f : C 4n C is a QΓ-chain map, define f 2 : C 4n 2 C 2 as given by left multiplication with the matrix representing f, i.e. f 2 = id l2 Γ f. In a similar way we define f [k]: C 4n [k] C [k]. Let f : C 4n C be a QΓ-chain map such that f and its dual f 4n are QΓ-chain homotopic. Then both H 2 2 2n f and H 2n f [k] are selfadjoint. We want to define the signature of such a chain complex..3 Definition. Given a selfadjoint map g : V V of Hilbert N Γ-modules and an interval I R, let χ I g be the map obtained from g by functional calculus for the characteristic function χ I : R R of I. Define b 2 + g := tr N Γ χ 0, g; b 2 g := tr N Γ χ,0 g; b 2 g := dim N Γ kerg = tr N Γ χ {0} g; sign 2 g := b 2 + g b 2 g. If h : W W is a selfadjoint endomorphism of a finite-dimensional Hermitian complex vector space, define analogously b + h := tr C χ 0, h; bh := dim C kerh = signh := b + h b h. b h := tr C χ,0 h; tr C χ {0} h; Of course, signh is the difference of the number of positive and of negative eigenvalues of h counted with multiplicity..4 Definition. Let f : C 4n C be a QΓ-chain map such that f and its dual f 4n are QΓ-chain homotopic. Then both H 2 2 2n f and H 2n f [k] of Definition.2 are selfadjoint. Using Definition.3 define b 2 2 2n± f := b 2 ± H 2 2n f ; b 2n± f [k] := b 2 ± H 2n f [k]; b 2 2 2n f := b 2 H 2 2n f ; b 2n f [k] := bh 2n f [k]; sign 2 f 2 := sign 2 H 2 2n f ; signf [k] := signh 2n f [k]. The QΓ-chain complex of a Poincaré pair A classical result proved e.g. in [2], or with much more information in [5, 6] says that, given a 4n-dimensional Poincaré pair X, Y with Γ-covering X X, the composition of the Poincaré QΓ-chain map [X, Y ] : C 4n X, Y ; Q

7 Approximating L 2 -signatures by their compact analogues 7 C X; Q with the QΓ-chain map induced by the inclusion yields a QΓ-chain map f : C 4n X, Y ; Q C X, Y ; Q of finitely generated based free 4n-dimensional QΓ-chain complexes such that f is QΓ-chain homotopic to f 4n. The normal subgroups Γ k Γ correspond to Γ/Γ k -coverings X k, Y k of X, Y as explained in the introduction. Use Definition.2 and Definition.4 to define b 2 2 2n± X, Y := b2 2n± f ; b 2n± X k, Y k := b 2n± f [k]; b 2 2n 2 X, Y := b2 2n f ; b 2n X k, Y k := b 2n f [k]; sign 2 X, Y := sign 2 f 2 ; signx k, Y k := signf [k]. Note that C 4n [k] and C [k] are the cellular Q[Γ/Γ k ]-cochain and chain complexes of X k, Y k, and f[k] its Poincaré duality map. Therefore the definitions above coincide with the usual definitions of Betti numbers and signature for the compact Poincaré duality pairs X k, Y k. Theorem 0. is an immediate consequence of.5 Theorem. Let f : C 4n X, Y ; Q C X, Y ; Q be the QΓ-chain map introduced above. Then b 2 2n± 2 f = lim b 2n± f [k]. [Γ : Γ k ] The proof of Theorem.5 is split into a sequence of lemmas..6 Lemma. Let A : l 2 Γ n l 2 Γ n be a selfadjoint Hilbert N Γ-module morphism. Let q j : R R be a sequence of measurable functions converging pointwise to the function q such that q j x C on the spectrum of A, where C does not depend on j. Then tr N Γ q j A j tr N Γ qa. Proof. By the spectral theorem, q j A converges strongly to qa. Moreover, q j A C for j Z. By [7, p. 34] q j A converges ultra-strongly and therefore ultra-weakly to qa. Since l 2 Γ n is a finite Hilbert-N Γ-module : l 2 Γ n l 2 Γ n is of Γ-trace class. Normality of the Γ-trace implies the conclusion compare [7, Proposition 2 on p. 82] or [20, Theorem 2.34]. Let p := c p+ c p+ + c p c p : C p C p be the combinatorial Laplacian on X, where we abbreviate C p := C p X, A; Q. Using a cellular basis of C p coming from C p X, Y ; Z this is given by a matrix over ZΓ. Then 2 p = c 2 p+ p+ c2 + c 2 2 p c p : C p 2 C p 2 is the Laplacian of C 2 and p [k] = c p+ [k]c p+ [k] + c p [k] c p [k] is the Laplacian on C p [k], i.e. the cellular Laplacian of X k. Let f : C 4n C be homotopic to its adjoint as introduced in the beginning of this section. The next lemma follows from [0, Lemma 2.5].

8 Approximating L 2 -signatures by their compact analogues 8.7 Lemma. There is K such that for all k.8 Definition. In the sequel we write tr k := 2 2n, 2n[k], f 2 2n, f 2n[k] K. tr Q [Γ : Γ k ] ; dim k := dim Q [Γ : Γ k ] ; sign k := sign Q [Γ : Γ k ], and denote by pr 2 2n : C2 2n C2 2n and pr 2n[k] : C 2n [k] C 2n [k] the orthogonal projection onto the kernel of 2 2n and 2n[k], respectively..9 Definition. For each ɛ > 0 fix a polynomial p ɛ x R[x] with real coefficients satisfying p ɛ 0 =, 0 p ɛ x + ɛ for x ɛ and 0 px ɛ for ɛ x K where K is the constant of Lemma.7. Such polynomials exist by the Weierstrass approximation theorem [7, Theorem 7.26]..0 Lemma. For each p and k we have and hence in particular dim k C p [k] = dim N Γ C p 2 X, lim dim k C p [k] = dim N Γ C p 2 X. Proof. For every k, dim k C p X k is equal to the number of p-cells in X, and the same is true for dim N Γ C p 2 X.. Lemma. For QΓ-linear maps h,..., h d : QΓ r QΓ r and a polynomial px,..., x d in non-commuting variables x,..., x d we have tr N Γ ph 2,..., h2 d = lim tr k ph [k],..., h d [k]. Proof. By linearity it suffices to prove this for monomials p = x i... x id, and since the h j are not assumed to be different, without loss of generality we can assume p = x... x d. The proof of [0, Lemma 2.6] applies and shows that there is L > 0 such that tr N Γ h 2 h 2 d = tr kh [k] h d [k] for k L. The lemma is formulated in a way that it can be applied if the assignment h h[k] is not a homomorphism. This is unnecessary here, but will be needed in Section 2..2 Lemma. There is a constant C > 0 independent of k such that for 0 < ɛ < and k tr k χ0,ɛ] 2n [k] C lnɛ..3 Proof. This is part of the conclusion of[0, Lemma 2.8].

9 Approximating L 2 -signatures by their compact analogues 9.4 Lemma. There is a constant C > 0 independent of k such that for all k and 0 < ɛ < 0 tr k p ɛ 2n [k] pr 2n [k] C ɛ + C lnɛ. Recall that p ɛ was fixed in Definition.9, and pr[k] is defined in Definition.8. Moreover we have p lim tr N Γ ɛ 2 2n ɛ 0 pr2 2n = 0. Proof. First observe that by our construction p ɛ 2n [k] pr 2n [k] is non-negative since 0 p ɛ χ {0} on the spectrum of 2n [k]. We also have p ɛ χ {0} ɛ+χ 0,ɛ] on the spectrum of the operators. Since the trace is positive, we get 0 tr k p ɛ 2n [k] pr 2n [k] ɛ tr k id C2n[k] + tr k χ 0,ɛ] 2n [k]. Now the first inequality follows from Lemma.0 and Lemma.2. The second one follows from tr N Γ p ɛ 2 2n pr2 2n ɛ tr N Γ id 2 C + tr N Γ χ 0,ɛ] 2 2n 2n and the fact that because of Lemma.6 lim ɛ 0 tr N Γ χ 0,ɛ] 2 2n = 0. We also cite the following result [0, Theorem 2.3]:.5 Theorem. The normalized sequence of Betti numbers converges, i.e. for each p lim dim kker p [k] = dim N Γ ker 2 p. For the proof of Theorem.5 eventually we want to approximate χ a,b by polynomials. Next we check that for a fixed polynomial we can replace pr 2n [k] in the argument by p ɛ 2n [k]..6 Lemma. Fix a polynomial q R[x]. Then we find a constant D > 0 independent of k such that for all k and 0 < ɛ < tr k q p ɛ 2n [k] f 2n [k] p ɛ 2n [k] Moreover, we have q lim ɛ 0 tr N Γ tr k q pr 2n [k] f 2n [k] pr 2n [k] D ɛ + p ɛ 2 2n f 2 2n pɛ 2 2n = tr N Γ q pr 2 2n D lnɛ. 2 f 2n pr2 2n. Proof. By linearity it suffices to prove the statement for all monomials qx = x n. Obviously it suffices to consider n. In the sequel we abbreviate x = p ɛ 2n [k], f = f 2n [k] and y = pr 2n [k]. Notice that x + ɛ, f K and y holds for the constant K appearing in Lemma.7. We estimate

10 Approximating L 2 -signatures by their compact analogues 0 using the trace property trab = trba and the trace estimate trab A tr B which also holds for the normalized traces tr k and for tr N Γ by [7, p. 06] since all the traces we are considering are normal, tr k p ɛ 2n [k] f 2n [k] p ɛ 2n [k] n tr k pr 2n [k] f 2n [k] pr 2n [k] n = tr k xfxxfx... xfx yfyyfy... yfy = tr k x yfxxfx... xfx + yfx yxfx... xfx = + yfyx yfxxfx... fx yfyyfy... yfx y 2n + ɛ 2n K n tr x y = 2n + ɛ 2n K n tr k p ɛ 2n [k] pr 2n [k]. Exactly the same reasoning applies if 2n [k] and pr 2n [k] is replaced by 2 2n and pr 2 2n, respectively, to give the corresponding estimate in this case. The assertion of the lemma now follows from Lemma.4..7 Lemma. Fix a polynomial qx R[x]. Then lim tr k q pr 2n [k] f 2n [k] pr 2n [k] = tr N Γ q pr 2 2n 2 f 2n pr2 2n. Proof. Fix δ > 0. By Lemma.6 we find ɛ > 0 such that for all k trk q p ɛ 2n [k] f 2n [k] p ɛ 2n [k] trn Γ q p ɛ 2 2n f 2 2n pɛ 2 2n Hence it suffices to show for each fixed ɛ lim tr k q p ɛ c p+ [k]c p+ [k] + tr k q pr 2n [k] f 2n [k] pr 2n [k] δ/3; tr N Γ q pr 2 2n f 2 2n pr2 2n δ/3. c p [k] c p [k] f 2n [k] p ɛ c p+ [k]c p+ [k] + c p [k] c p [k] = tr N Γ q p ɛ c 2 p+ c2 p+ + c 2 p c 2 p f 2 2n pɛ c 2 p+ c2 p+ + c 2 p c 2. p Since q and p ɛ are fixed, we deal with a fixed polynomial expression in c p, c p, c p+, c p+, and f 2n. Therefore the last claim follows from Lemma.. This finishes the proof of Lemma.7..8 Lemma. We have for a, b R with a < b tr N Γ χ a,b H p 2 f 2 lim inf tr k χa,b H p f [k].

11 Approximating L 2 -signatures by their compact analogues Proof. We approximate χ a,b by polynomials. Namely, for 0 < ɛ < b a/2 and K as above let q ɛ R[x] be a polynomial with q ɛ x χ a,b x for x K; q ɛ x χ a,b x ɛ for x [ K, a] [a + ɛ, b ɛ] [b, K]. Under the identification of impr 2n [k] and H p C [k] coming from the combinatorial Hodge decomposition the operator pr 2n [k] f 2n [k] pr 2n [k] restricted to impr 2n [k] becomes H p f [k] which is selfadjoint because of f f 4n. Hence pr 2n [k] f 2n [k] pr 2n [k] and also the operator q ɛ pr 2n [k] f 2n [k] pr 2n [k] are selfadjoint. Exactly the same is true on the L 2 -level and we conclude tr k χa,b pr 2n [k] f 2n [k] pr 2n [k] = tr k χa,b H p f [k] ;.9 tr N Γ χ a,b pr 2 2 f 2n pr2 2n = tr N Γ χ a,b H p 2 f n Positivity of the trace and q ɛ x χ a,b x for all x in the spectrum of pr 2n [k] f 2n [k] pr 2n [k] implies tr k q ɛ pr 2n [k] f 2n [k] pr 2n [k] tr k χa,b pr 2n [k] f 2n [k] pr 2n [k]. Note that for fixed q ɛ the left hand side converges for k by Lemma.7. For the right hand side this is not clear, but in any case we get tr N Γ q ɛ pr 2 2n 2 f 2n pr2 2n lim inf n tr k χa,b pr 2n [k] f 2n [k] pr 2n [k]..2 On the spectrum of the operator in question, the functions q ɛ are uniformly bounded and converge pointwise to χ a,b if ɛ 0. By Lemma.6 lim tr N Γ ɛ 0 q ɛ pr 2 2n 2 f 2n pr2 2n = tr N Γ χ a,b pr 2 2n Since inequality.2 holds for arbitrary ɛ > 0, we conclude tr N Γ χ a,b pr 2 2n 2 f 2n pr2 2n lim inf tr k Now the claim follows from.9 and f 2n pr2 2n. χa,b pr 2n [k] f 2n [k] pr 2n [k]..22 Lemma. Let f : C D be a QΓ-chain map of finitely generated based free QΓ-chain complexes. Then we get for all p lim dim k ker H p f [k] = dim N Γ ker H p 2 f 2.

12 Approximating L 2 -signatures by their compact analogues 2 Proof. We can assume without loss of generality that C and D are p + - dimensional. Consider the long exact sequence of left QΓ-chain complexes 0 D conef ΣC 0, where conef is the mapping cone of f and ΣC the suspension of C. It is a split exact sequence in each dimension and thus remains exact after applying l 2 Γ QΓ. The weakly exact long homology sequence yields a weakly exact sequence of Hilbert N Γ-modules 0 H 2 p+2 conef 2 H 2 p+ C2 H2 2 p+ f H 2 H 2 p+ conef 2 kerh p f 2 0. This implies dim N Γ kerh p f 2 = dim N Γ H 2 p+ conef 2 Analogously we get + dim N Γ H 2 p+ C2 dim k kerh p f [k] dim N Γ dim N Γ p+ D2 H 2 p+ D2 H 2 p+2 conef 2 = dim k H p+ conef [k] dim k H p+ D [k] dim k H p+ C [k] dim k H p+2 conef [k]..24 We conclude from Theorem.5 dim N Γ H 2 p+ conef 2 dim N Γ H 2 p+ D2 dim N Γ H 2 p+ C2 dim N Γ H 2 p+2 conef 2 = lim dim k H p+ conef [k] ;.25 = lim dim k H p+ D [k] ;.26 = lim dim k H p+ C [k] ;.27 = lim dim k H p+2 conef [k]..28 Now the claim follows from equations Now we are ready to prove Theorem.5. Proof of Theorem 0.. We get from Lemma.8 and Lemma.22 b 2 2n+ g2 b 2 p g 2 lim inf = lim b 2n+ g [k] ; b 2 [Γ : Γ k ] b p g [k] [Γ : Γ k ]. 2n g2 lim inf b 2n g [k] ; [Γ : Γ k ]

13 Approximating L 2 -signatures by their compact analogues 3 Since b 2 2n+ g2 + b 2 2n g2 + b 2 2n g2 = dim N Γ b 2n+ g [k] + b 2n g [k] Γ/Γ k Γ/Γ k dim N Γ C 2 = lim 2n + b 2ng [k] Γ/Γ k C 2 2n = dim k C 2n [k]; ; dim kc 2n [k] = dim QC 2n [k], [Γ : Γ k ] Theorem.5 and thus Theorem 0. follow from Lemma Remark. Theorem 0. can be applied to a 4n-dimensional Riemannian manifold X with boundary Y. In this case, the Atiyah-Patodi-Singer theorem [2, Theorem 4.4] and [5, 0.9] and the L 2 -signature theorem of [2] imply signx k, X k volx k sign 2 X, X volx = volx k = volx X X k LX k + η X k volx k + volx k LX + η2 X volx + volx X X k Π L X k, Π L X. Here LX k and LX denote the Hirzebruch L-polynomial, and Π L X k and Π L X are a local correction terms which arises because the metric is not a product near the boundary. Being local expressions, the first and the third summand does not depend on k. It follows that the sequence of η-invariants converges. In fact, even without the assumption that Y 4n is a boundary of a suitable manifold X, in [23, Theorem 3.2] it is proved lim ηy k [Γ : Γ k ] = η2 Y. Key ingredients are on the one hand the analysis of Cheeger-Gromov in [5, Section 7] of the formulas 2.24 and 2.25 which holds for operators different from the signature operator. We present similar considerations in Section 2.. The second key ingredient is Lück s approximation result for L 2 -Betti numbers [0, Theorem 0.] which is special to the Laplacian, the square of the signature operator..30 Remark. The normalized signatures signx k,y k Γ/Γ k are the L 2 -signatures sign 2 X k, Y k of the Γ/Γ k -coverings X k, Y k X, Y. With this reformulation, one may ask whether Theorem 0. holds if Γ/Γ k is not necessarily finite. This is indeed the case if the groups Γ/Γ k belong to a large class of groups G defined in [9, Definition.]. The corresponding question for L 2 -Betti numbers is answered affirmatively in [9, Theorem 6.9] whenever Γ/Γ k G. As just mentioned, Theorem 0. extends to this situation as well, and the proof we have given is formally unchanged, using the generalization of Lemma.7 and Lemma. given in [9, Lemma 5.5 and 5.6]. It only remains to establish Lemma.2, which is not done in [9]. We do this in the following Lemma.3, which applies because of [9, 6.9] and because of Lemma.7.

14 Approximating L 2 -signatures by their compact analogues 4.3 Lemma. If [X k ] K and then ln det 2 [X k ] := 0 + lnλ df [Xk ]λ 0.32 tr k χ 0,ɛ] [X k ] d lnk lnɛ..33 Here F [Xk ]λ := tr k χ [0,λ] [k] is the spectral density function of the operator [k] computed using tr k instead of dim C, and d = F [k] K is the number of rows and columns of the matrix. Proof. We argue as follows with F := F [k] : ɛ [k] lnλ df λ = lnλ df λ + lnλ df λ ɛ lnɛ F ɛ F 0 + ln [k] F [k]. }{{} =tr k χ 0,ɛ] [k] For 0 < ɛ <, using the bound [k] K of the generalization of Lemma.7, Inequality.32 immediately gives Amenable convergence of signatures 2. Analytic version In this subsection we want to prove Theorem 0.4. We will use the following notion of manifold with bounded geometry compare e.g. [, Definition 2.24]. 2. Definition. A Riemannian manifold M, g the boundary may or may not be empty is called a manifold of bounded geometry if bounded geometry constants C q for q N and R I, R C > 0 exist, so that the following holds: The geodesic flow of the unit inward normal field induces a diffeomorphism of [0, 2R C M onto its image, the geodesic collar; 2 For x M with dx, M > R C /2 the exponential map T x M M is a diffeomorphism on B RI 0; 3 The injectivity radius of M is bigger than R I ; 4 For every q N we have i R Ck and i l Cl for 0 i q, where R is the curvature tensor of M, l the second fundamental form tensor of M, and i and i are the covariant derivatives of M and M. By [2, Theorem 2.4] this is equivalent to [, Definition 2.24]. Every compact manifold, or more generally every covering of a compact manifold, is a manifold with bounded geometry. We now repeat a few well known facts about manifolds of bounded geometry.

15 Approximating L 2 -signatures by their compact analogues Proposition. Let M be a compact smooth Riemannian manifold. There is a constant A > 0, depending only on the bounded geometry constants and the dimension of M, such that exp t p Mx, x A for t, x M; b p M A volm; b p M, M A volm, where the Laplacian can be taken with either relative or absolute boundary conditions. Proof. The first inequality is proved in [, Theorem 2.35]. The claim for the Betti numbers is a consequence of the fact that the Betti number b p M or b p M, M can be written as lim t M tr x exp t p Mx, x dx for the Laplacian with absolute or relative boundary conditions, respectively. 2.3 Theorem. Let M, N be Riemannian manifolds without boundary which are of bounded geometry and with a fixed set of bounded geometry constants. Let U be an open subset of M which is isometric to a subset of N which we identify with U. For R > 0 set U R := {x U dx, M U R and dx, N U R}. Let D[M] and D[N] be the tangential signature operators on M and N, respectively; and similarly [M] and [N] the Laplacian on differential forms. Let e t x, y and De td2 x, y be the integral kernels which are smooth of the operators e t and De td2. Then there are constants C, C 2 > 0 which depend only on the dimension and the given bounded geometry constants such that for t > 0, x U R e t [M] x, x e t [N] x, x C e R2 C 2/t ; 2.4 D[M]e td[m]2 x, x D[N]e td[n]2 x, x C e R2 C 2/t. 2.5 Proof. This follows by a standard argument of Cheeger-Gromov-Taylor [6] from unit propagation speed and local elliptic estimates here the bounded geometry constants come in. A detailed account is given in the proof of [, Theorem 2.26] which yields immediately 2.4. Replacing by D which is possible since we are looking for manifolds without boundary, so that we do not have to worry about the non-locality of boundary conditions and therefore have unit propagation speed for D, too, the proof also applies to the tangential signature operator to give Proposition. Let M m be a manifold of bounded geometry with fixed bounded geometry constants and with M =. Let D be the tangential signature operator on M. Then there is a function A : [0, 0, which depends only on the bounded geometry constants and the dimension m, such that for T 0 tr x De td2 x, x AT t /2 for 0 t T, x M.

16 Approximating L 2 -signatures by their compact analogues 6 Proof. One can use the proof of [3, Lemma 3.. on p. 324] where a slightly different statement is proved. The proposition is also implicit in [5, Proof of Theorem 0. on p. 40]. The proof uses the cancellation of the coefficients of negative powers of t in the local asymptotic expansion due to Bismut and Freed [4, Theorem 2.4] and a localization argument based on elliptic estimates here the local geometry comes in, together with the finite propagation speed method of Cheeger-Gromov-Taylor [6]. We fix the following notation. 2.7 Notation. In the situation of Definition 0.3 put for r 0 U r Y k := {x X; dx, Y k r}, where two points y, z X have distance dy, z = d if there is a geodesic of length d in X joining y and z and d = if there is no such geodesic. In particular dy, z < implies that y and z lie in the same path component of X. Let F be a compact connected simplicial fundamental domain for X in X such that F X is a fundamental domain for X. We can construct F as a union of lifts of the top-dimensional simplices in a smooth triangulation of X and achieve F to be connected, since X is connected by assumption. For r 0 let N k r be the number of translates of F contained in X k U r Y k and n k r the number of translates of F which have a non-trivial intersection with U r Y k. Set N k := N k 0; n k := n k 0. The next lemma shows that our Definition 0.3 of a regular exhaustion coincides with the one given by Dodziuk and Mathai [8], with one exception: we require a lower bound on the injectivity radius of the boundaries X k and control of the covariant derivatives of the second fundamental form, what they seem to have forgotten but also use. 2.8 Lemma. If X k k is a regular exhaustion of X as in Definition 0.3, then for each r 0 volu r Y k lim = 0. volx k Proof. To obtain this we discretize: Choose ɛ > 0 such that 4ɛ is smaller than the injectivity radius, and choose sets of points P k Y k such that the balls of radius ɛ around x P k are mutually disjoint, but the balls of radius 4ɛ are a covering of Y k. Because of bounded geometry compare the proof of [22, Lemma.2 in Appendix ], we find c, c 2 > 0 independent of k such that c P k areay k c 2 P k. The triangle inequality implies U r Y k k B r+4ɛx k. Therefore volu r Y k C r+4ɛ P k C r+4ɛ c areay k, where C r+4ɛ is a uniform upper bound for the volume of balls of radius r + 4ɛ in X which exists because of bounded geometry. Since we have by assumption areay lim k volx k = 0, Lemma 2.8 follows.

17 Approximating L 2 -signatures by their compact analogues Lemma. If X k k is a regular exhaustion of X as in Definition 0.3, then area X k X lim volx k = area X volx. Proof. Obviously volf = volx and areaf X = area X. Recall that F is connected. Suppose that F X k and F X k U r Y k. Then, for each r, F must intersect U r Y k because otherwise we can find a path in F connecting a point in X k to a point in X X k and this path must meet Y k. Hence we get for r 0, using Notation 2.7 N k r volx volx k N k r + n k r volx; 2.0 N k r area X area X k X N k r + n k r area X. 2. If follows that N k area X N k + n k volx area X k X volx k N k + n k area X. 2.2 N k volx Since F U r Y k implies F U r+diamf Y k, we have n k r volx volu r+diamf Y k. Therefore 2.0 implies n k r n k r + N k r = n k r volx n k r + N k r volx From Lemma 2.8 we conclude Now Lemma 2.9 follows from 2.2 and 2.3. volu r+diamfy k. volx k n k r lim = N k r 2.4 Theorem. If X k k is a regular exhaustion of X as in Definition 0.3, then, using Notation 2.7, b p X k b p X k volx lim = lim N k volx k = b 2 p X. Proof. Let V k X k X be the union of translates gf X for g Γ such that gf X k Y k. The number of these translates gf X is just N k. The number Ṅm,δ of boundary pieces appearing in [9] is bounded by C δ n k for a constant C δ which does not depend on k. Because of Inequality 2.3, V k k is a regular exhaustion of X in the sense of [9] by 2.3. We conclude from [9, Theorem 0.] b p V k lim = b 2 p X. 2.5 N k We can thicken V k inside of X to a regular neighborhood V k. From Proposition 2.2 we obtain a constant A independent of k such that b p X k intv k, V k A vol X k intv k A voly k + n k vol X F. 2.6

18 Approximating L 2 -signatures by their compact analogues 8 We have by excision b p X k, V k = b p X k intv k, V k and by homotopy invariance b p V k = b p V k. From 2.6 and the long exact homology sequence of the pair X k, V k we conclude b p X k b p V k 2A voly k + n k vol X F. 2.7 We get from 2.0 and 2.3 since voly k / volx k 0 by assumption that 2A voly k + n k vol X F lim = N k We conclude from 2.5 and 2.7 and 2.8 that b p X k lim = b 2 p X. 2.9 N k Now Theorem 2.4 follows from 2.0, 2.3 and 2.9. Remember that the Atiyah-Patodi-Singer index theorem [2, Theorem 4.4] and [5, 0.9] and its L 2 -version compare e.g. [2] imply for manifolds as in Definition 0.3 signx k, X k volx k sign 2 X, X volx = volx k = volx X X k LX k + η X k volx k + volx k LX + η2 X volx + volx X X k Π L X k, Π L X. Here LX k and LX denote the Hirzebruch L-polynomial, and Π L X k and Π L X are local correction terms which arises because the metric is not a product near the boundary. We want to show that each of the individual summands converges for k to the corresponding term for X. The L-polynomial is given in terms of the curvature, Π L in terms of the second fundamental form, therefore both are uniformly bounded independent of k by some constant C. Moreover, because these are local expressions, the integral over each translate of the connected fundamental domain F which is contained in X k Y k coincides with the corresponding integral on X or X. Then by splitting the domain of integration appropriately as done in the proofs above, and using Notation 2.7, LX k N k LX n k volx C; 2.20 X k X Π L X k N k Π L X areay k C X k X +n k area X C. 2.2

19 Approximating L 2 -signatures by their compact analogues 9 We conclude from 2.0 and 2.20 volx k LX k X k volx LX X volx k LX k N k volx X k + N k volx LX k N k LX X k X 2.20 volx k N k volx volx k C + n k C 2.0 N k 2C nk N k We conclude from 2.0, 2., 2.2 and Lemma 2.9 volx k Π L X k X k volx Π L X X volx k Π L X k N k volx X k + N k volx Π L X k N k Π L X X k X 2.2 volx k N k volx area X k C + area Y k N k volx C + n k C area X N k volx 2.0 C n k + N k volx N K volx areay k + area X k X + areay k volx k C N k + n k + n k C area X N k N k volx 2. n k C N k n k + N k volx areay k + n k + N k area X areay k volx k C N k + n k + n k C area X N k N k volx n k areay k N k volx k volx + n k area X N k volx + areay k volx k C N k + n k + n k C area X. N k N k volx 2.23 areay Since lim k volx k = 0 by assumption, from 2.3, 2.22 and 2.23 follows lim volx k LX k X k volx LX = 0; X lim volx k Π L X k Π L X X k volx = 0. X

20 Approximating L 2 -signatures by their compact analogues 20 It remains to consider the eta-invariants. Because of their non-local nature this is the most difficult task. The strategy of the proof of the next proposition is similar to the proof of Remark.29. We first recall a few facts about the η-invariant. Let D be the tangential signature operator of a 4n -dimensional Riemannian manifold M, and M a Γ-covering with lifted signature operator D. Then and ηm = η 2 M := Γ/2 Γ/2 0 0 t /2 trde td2 dt t /2 tr N Γ De td2 dt, 2.25 where with a fundamental domain F of the covering M M tr N Γ De td2 = tr x De td2 x, x dx; 2.26 F trde td2 = tr x De td2 x, x dx M Following Cheeger and Gromov [5, Section 7] we give an a priori estimate for the large time part of the integrand defining the η-invariant. First observe that for x 0 we have the following inequality of functions: xt /2 e tx2 dt = e T x2 x t /2 e x2 t T dt T Making the substitution t = u x 2 + T we obtain xt /2 e tx2 dt = e T x2 x e u u x 2 + T /2 x 2 du T 0 T = e T x2 e u u + T x 2 /2 du 0 e T x2 χ {0} π, where we used u /2 e u du = π and for the last inequality that T x For x = 0 obviously T xt /2 e tx2 dt = 0. Hence we get for all x R xt /2 e tx2 dt e T x2 χ {0} x π, T where χ {0} is the characteristic function of the set {0}. Applying the functional calculus with x = D we get t /2 tr N Γ De td2 dt π tr N Γ e T pr ker, 2.28 T

21 Approximating L 2 -signatures by their compact analogues 2 and analogously with x = D t /2 trde td2 dt π tre T pr ker T 2.30 Proposition. If X k k is a regular exhaustion of X as in Definition 0.3 then η X k lim volx k = η2 X volx. Proof. In the sequel D[k] or D is the tangential signature operator and [k] or is the differential form Laplacian on X k or X, respectively. Fix ɛ > 0. Choose T such that tr N Γ e T pr ker Γ/2 volx ɛ π Put X R k := g G s.t. U R gf X k gf X. By Theorem 2.3 for the given T > 0 and ɛ > 0 we find R > 0 independent of k such that for 0 t T and x X R k. tr x D[k]e td[k]2 x, x tr x De td2 x, x tr x e t [k] x, x tr x e t x, x Γ/2 volx ɛ 4 ; 2.32 T vol X Γ/2 volx ɛ π vol X Notice that U R gf X k F X k U R Y k. Hence Xk R consists of N k R translates of F X. This implies vol Xk R = N kr vol X. From Proposition 2.2 and 2.33 we get for a constant A independent of k using the fact that and its kernel are Γ-equivariant tre T [k] tr N Γ e T N k R = N k R tr x e T [k] x, x dx tr x e T x, x dx X k F X N k R tr x e T [k] x, x tr x e T x, x dx X k R + N k R tr x e T [k] x, x dx X k X R k = Γ/2 volx ɛ vol XR k N k R 8 π vol X Γ/2 volx ɛ 8 π + A vol X k X R k N k R + A vol X k Xk R. N k R 2.34

22 Approximating L 2 -signatures by their compact analogues 22 We conclude from 2.28, 2.29, 2.3 and 2.34 using D 2 = and D[k] 2 = [k] N k R Γ/2 t /2 trd[k]e td[k]2 dt T Γ/2 t /2 tr N Γ De td2 N k R T π Γ/2 tre T [k] pr ker [k] π + Γ/2 tr N Γe T pr ker 2 π Γ/2 tr N Γe T pr ker π + Γ/2 N k R tre T [k] tr N Γ e T π + Γ/2 tr N Γpr ker N k R trpr ker [k] 2 volx ɛ + volx ɛ π A vol X k Xk R Γ/2 N k R π + Γ/2 tr N Γpr ker N k R trpr ker [k] From 2.26, 2.27, 2.32, and Proposition 2.6 we obtain a constant A 2 independent of k such that the following holds: N k R T Γ/2 t /2 trd[k]e td[k]2 dt 0 T t /2 tr N Γ De td2 Γ/2 = N k R 0 T Γ/2 t /2 tr x D[k]e td[k] 2 x, x dx dt 0 X k T Γ/2 t /2 0 F X tr x De td2 x, x dx dt N k R Γ/ T t /2 tr x D[k]e td[k]2 x, x tr x De td2 x, x dx dt 0 X k R T + t /2 tr x D[k]e td[k]2 x, x dx dt 0 X k X R k

23 Approximating L 2 -signatures by their compact analogues 23 T t /2 volx ɛ N k R 0 4 T vol X vol XR k A 2 Γ/2 t/2 vol X k Xk R dt. volx ɛ 8 + A 2 T Γ/2 vol X k Xk R N k R We conclude from 2.24, 2.25, 2.35 and 2.38 η2 X N k R η X k volx ɛ 8 + A 2 T Γ/2 vol X k Xk R + 3 volx ɛ + N k R 8 π A vol X k Xk R Γ/2 N k π + Γ/2 tr N Γpr ker N k R trpr ker [k].2.39 We get from 2.0 η 2 X η X k volx volx k N kr volx k η2 X N k R η X k + Nk R volx k η 2 X volx volx η2 X N k R η X k + η2 X nkr volx N k R We conclude from 2.39 and 2.40 η 2 X η X k volx volx k ɛ 8 + A 2 T volx Γ/2 vol X k Xk R + 3ɛ N k R 8 π A + volx Γ/2 vol X k Xk R N k π + volx Γ/2 tr N Γpr ker N k R trpr ker [k] + η2 X volx nkr N k R. 2.4 Recall that Xk R consists of N kr translates of F X. The same arguments as above using 2.3 imply vol X k Xk R lim = N k

24 Approximating L 2 -signatures by their compact analogues 24 We get from Theorem 2.4 trpr ker [k] b X k lim = lim N k R N k R = b2 X = tr N Γ pr ker 2.43 with the convention b := p 0 b p. From 2.3, 2.4, 2.42 and 2.43 we get the existence of a constant Kɛ such that for all k Kɛ η 2 X volx η X k volx k ɛ Now Proposition 2.30 follows. This finishes the proof of Theorem Remark. In the proof of Proposition 2.30, we were using general properties of the traces and of generalized Dirac operators, which apply not only to the tangential signature operator together with one additional ingredient, the convergence result for the kernel of the tangential signature operator of Theorem 2.4. Using the symmetry of the tangential signature operator one can restrict to 2n -forms on the boundary, as explained in [3, Proposition 4.20]. In particular, 2.43 has to hold only for b 2n, and we still could prove Proposition 2.30 compare also [5, 23]. 2.2 Combinatorial version In this subsection, we prove a combinatorial version of Theorem 0.4. It uses the more algebraic techniques employed in Section rather than the heat kernel analysis of Subsection 2.. This way, the result applies to triangulated rational homology manifolds with boundary. Throughout this subsection, let X be a compact triangulated rational homology manifold with boundary L, and of dimension 4n. Let X be a regular covering of X with finitely generated amenable covering group Γ. We start by describing the type of exhaustion we are going to use Definition. Let F be a fundamental domain for the covering X X, i.e. F contains exactly one lift of each top-dimensional simplex of X. For each simplex σ in X choose a lift σ in F. Let S be a finite system of generators of Γ. It gives rise to a left invariant word metric on Γ. For a subcomplex Z X and R > 0, define U R Z := σsimplex of X {γσ γ Γ and γ Γwith dγ, γ < R, γ σ Z }. This depends on the choice of S as well as the lifts σ. For each g Γ, U R gz = gu R Z.

25 Approximating L 2 -signatures by their compact analogues Definition. For a simplicial complex Y let Y be the total number of simplices of Y. Similarly, for any subset W of a simplicial complex which is a union of open simplices, W is the number of open simplices in W. A sequence X X 2... X of finite subcomplexes is called an amenable exhaustion if k N X k = X and if for each R > 0 U R X k X k. It is called a balanced exhaustion, if for each orbit Γσ of simplices in X X k Γσ X k X. Denote tr k := tr C X k X ; dim k := dim C X k X Theorem. Assume that X is a compact simplicial complex triangulating a rational homology manifold with boundary the subcomplex X. Assume X is a normal covering of X with amenable covering group Γ. Let X X 2... be subcomplexes forming a balanced amenable exhaustion of X by rational homology manifolds with boundaries Y k. If X is a homology manifold, such an exhaustion does always exist. Then signx k, Y k lim X = sign 2 X, X. X k Before proving this, we investigate the relation between the Poincaré duality maps of one homology manifold being a codimension-zero subcomplex of another homology manifold. As an illustration we consider the following diagram. Let U M be codimension zero submanifold with boundary U of a compact manifold M. For the moment assume M is empty. H p M H p M, M U [M] = H p U, U [U, U] H n p M = H n p M H n p M, M U H n p U = H n p U, U. In this diagram, the maps without labels are induced by inclusions and the isomorphisms are given by excision. The diagram commutes because the fundamental class of M is mapped to the fundamental class of U, U under the composition of the maps in the lowest row for p = 0. A corresponding result holds if M itself has a boundary.

26 Approximating L 2 -signatures by their compact analogues 26 Because we have to apply this in the L 2 -setting, we give a chain-level description of this diagram. For this, let X, L be an n-dimensional pair of simplicial complexes triangulating an oriented rational homology n-manifold X with boundary L. Let X, L be the lifted triangulation of a normal covering of X L is the inverse image of L in X with covering group Γ. Without loss of generality we assume X and X are connected we can deal with one component of X at a time, and then the L 2 -signature is unchanged if we consider only one component of X. We first want to describe the simplicial L 2 -chain- and cochain complexes of X. Set π := π X. We have by definition C 2 X, L = hom ZπC X, L, l 2 Γ, C 2 X = hom ZπC X, l 2 Γ, and C 2 X = l 2 Γ Zπ C X Here X is the induced triangulation of the universal covering of X, L is the inverse image of L in X, and we always use the simplicial cochain complexes Convention. There are canonical identifications of the simplicial L 2 -chain and L 2 -cochain complexes C p 2 X, C p 2 X with the spaces of L2 -summable functions on the set of p-dimensional simplices of X, and of C p 2 X, L and C p 2 X, L with L2 -summable functions on the set of p-dimensional simplices of X which do not belong to L, respectively. These identifications are used in the sequel. We write L 2 -functions on the set of simplices of X as formal sums λ σ σ. Then the identification of cochains with L 2 -functions is the anti-linear isomorphism given by a σ X, a σ σ, where σ is an arbitrary lift of σ to the universal covering. Note that there is a given projection p: X X since X is a connected normal covering of X. The identification of chains with L 2 -functions on the set of simplices is given by g Γ gg λ σ g Γ λ ggσ, where σ is a simplex in X or, for C X, L of X \ L and σ = p σ. 2.5 Remark. Note that this way, in particular we identify the L 2 -chain- and cochain groups with each other via an anti-linear isomorphism. However, this is nothing but the usual isomorphism of a Hilbert space with its dual. Note that this is not an isomorphism of chain complexes. Under the identifications, the chain- and cochain maps induced from the inclusion of X in X, L become the obvious inclusion and orthogonal projection, respectively Lemma. Under the identification of Convention 2.50 of the chain- and cochain complexes with spaces of L 2 -functions on the sets of simplices of X, cap-product with the fundamental class defined on the cochain level using the Alexander-Whitney diagonal map gives a map g: C 2 X, L C2 X

27 Approximating L 2 -signatures by their compact analogues 27 which sends an L 2 -function a on the set of p-simplices in X \ L to f n p σ a, b p σ l 2 X. σ n-simplex of X Here f q, b p are the front- and back-faces of the corresponding dimension, as usual in the Alexander-Whitney diagonal approximation. To be able to define this, we choose also a Γ-invariant local ordering of the vertices of X, e.g. by lifting such a local ordering from X. Proof. Using the notation introduced above, C X, C can be identified with C Zπ C X. For each simplex σ of X choose a lift σ in X. Then the fundamental class of X can be written as σ X n σ, where X p denotes the collection of p-simplices in X. The Alexander-Whitney cap-product of a hom Zπ C p X, L, l 2 Γ with the fundamental class is then given by σ X n ab p σ f n p σ Here : l 2 Γ l 2 Γ is the anti-linear isomorphism induced from g g and from complex conjugation of the coefficients. Now observe that the function a = σ X a σσ is mapped to the cochain α: σ g Γ a g p σg, and a g p σ = g p σ, a l 2. By 2.53, capping this cochain α with the fundamental class gives the chain g pb p σ, a g f n p σ. σ X n g Γ Under our identification, this chain becomes the function σ X n a, b p σ f n p σ, where we use the fact that the family g p σ for g Γ and σ X n is exactly the family of all n-simplices of X, and the fact that the front- and back-face maps commute with the action of π and Γ Lemma. Composing the cap-product with the fundamental class with the map induced from the inclusion X X, L we get a map g X : C2 X, L C2 X, L

28 Approximating L 2 -signatures by their compact analogues 28 which under the identifications of C2 X, U as well as C2 X, U with the space of L 2 -functions on simplices in X \ L maps such a function a on p- simplices to f n p σδ L f n p σ a, b p σ l2 X. σ n-simplex of X Here δ L σ is if σ is not contained in L, and is 0 if σ L. Proof. This is an immediate consequence of the first lemma Definition. Assume now that U is a compact subcomplex of X which has itself a subcomplex V not necessarily contained in L such that U, V triangulates an oriented homology n-manifold with boundary i.e. U is a codimension 0 submanifold with boundary. The above identifications and formulas apply to the chain- and cochain complexes of U and V with complex coefficients. Moreover, observe that these identifications give canonical embeddings PU of C U, V, C in C2 X, L and of C U, V ; C in C 2 X, L. The corresponding orthogonal projection is the adjoint P U Proposition. In the situation of Definition 2.55, with all the identifications described, g 2 U = P U g 2 P X U. In other words, the Poincaré duality operator on U is obtained from the one on X by compression onto the chain and cochain complex of U, considered as subcomplex of the ones of X. Proof. This is implied by the formula of Lemma We only have to make the simple but key observation that a top-dimensional simplex of X which is not contained in U has no face contained in U \ V since in the star of an interior point, any two top-dimensional simplices can be joined by a sequence of top-dimensional simplices having pairwise a common face of codimension. Therefore the star in X of an interior point of U can not be bigger than the star in U. Now we are ready to prove Theorem We start with some auxiliary results we will use. As before, let F be a fundamental domain for the covering X X. Remember that for each simplex σ in X we have chosen a lift σ in F Definition. We write l 2 X for the space of L 2 -functions on the set of simplices of X with point measure, each simplex being one element. This way, we get an identification C 2 X = l2 X = σ X l 2 Γ σ. Assume that we have an exhaustion X X 2 X as in Definition These define subspaces l 2 X k l 2 X. Let Pk σ be the orthogonal projection P σ k : l 2 Γ σ l 2 Γ σ l 2 X k = l 2 X k Γσ. 2.58