On the simplicial volume and the Euler characteristic of (aspherical) manifolds

A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov's question and collect strategies towards affirmative answers and strategies towards negative answers to this problem. Moreover, we put Gromov's question into context with other open problems in low- and high-dimensional topology. A special emphasis is put on a comparative analysis of the additivity properties of the simplicial volume and the Euler characteristic for manifolds with boundary. We explain that the simplicial volume defines a symmetric monoidal functor (TQFT) on the amenable cobordism category, but not on the whole cobordism category. In addition, using known computations of simplicial volumes, we conclude that the fundamental group of the 4-dimensional amenable cobordism category is not finitely generated. We also consider new variations of Gromov's question. Specifically, we show that counterexamples exist among aspherical spaces that are only homology equivalent to oriented closed connected manifolds.


Introduction
The simplicial volume M is a homotopy invariant of oriented compact manifolds M , defined as the ℓ 1 -semi-norm of the singular R-fundamental class. The simplicial volume is proportional to the Riemannian volume for hyperbolic manifolds and zero in the presence of amenability. By the Gauß-Bonnet theorem and Følner covering towers, a similar behaviour is also exhibited by the Euler characteristic of aspherical manifolds. But understanding the connection between the vanishing behaviour of the simplicial volume and the Euler characteristic of closed aspherical manifolds remains a mystery.
In particular, the following problem by Gromov  The main challenge in answering Question 1.1 is to find a common ground for the various conditions and invariants involved: asphericity, being a closed manifold, the vanishing of the simplicial volume, and the vanishing of the Euler characteristic.
In this article, we explore different approaches to Question 1.1, in search of both positive and negative examples, as well as study its connections with other open problems in low-and high-dimensional topology.
On the one hand, a key direction pursued in this paper is the comparative study of the additivity properties of the simplicial volume and of the Euler characteristic, which naturally leads us to look at the simplicial volume (and stable integral simplicial volume) of compact manifolds with boundary (see also Section 1.1 below).
On the other hand, a different direction considered in this paper is based on the observation that if the answer to Question 1.1 is affirmative, then Property (SVχ) will hold for a general class of oriented closed manifolds which are only homology equivalent to an aspherical one (see Remark 5.1). Based on this, we consider generalized versions of Question 1.1 for aspherical spaces which are only homology equivalent to an oriented closed manifold and for oriented closed manifolds which are only homology equivalent to an aspherical space (see also Section 1.2 below).
A quick summary of various other strategies and examples is provided in Section 1.3.
1.1. Additivity. As the category of closed aspherical manifolds is difficult to handle structurally, we extend the setup to compact aspherical manifolds with π 1 -injective boundary (Section 2.3) and consider an (equivalent) version of Gromov's question in this context.
Question (see Question 2.13). Let (M, ∂M ) be an oriented compact aspherical manifold with non-empty π 1 -injective aspherical boundary. Does the following implication hold? We recall some examples and sufficient conditions for the vanishing of the simplicial volume in Section 3. Moreover, we show an extension of Gromov's vanishing theorem (Theorem 3.5) to the vanishing of the relative simplicial volume M, ∂M for oriented compact connected manifolds (M, ∂M ) which admit open covers with certain amenability properties (Theorem 3.13). Also, in Section 6, we discuss the properties of the relative stable integral simplicial volume and the analogues of the above questions for this invariant.
The extension to manifolds with boundary generally allows us to compare additivity and filling properties of the simplicial volume and of the Euler characteristic more systematically. For example, we show that Question 1.1 is related to Edmonds' problem (Conjecture 3.30) as follows: Proposition (see Proposition 3.31). Suppose that the following hold: (a) Every oriented closed aspherical 3-manifold with amenable fundamental group is the π 1 -injective boundary of an oriented compact aspherical 4-manifold W with W, ∂W = 0. (b) All oriented closed aspherical 4-manifolds satisfy Property (SVχ). Then, there exists an oriented closed aspherical 4-manifold M with χ(M ) = 1.
The notion of a topological quantum field theory (TQFT), as a symmetric monoidal functor on the cobordism category, provides an efficient way of encoding additivity properties. The connections between the simplicial volume and (invertible) TQFTs are discussed in Section 4. Specifically, based on known additivity properties, we explain in Section 4.2 that the simplicial volume defines an invertible TQFT on a suitable amenable cobordism category with values in R. In addition, we show that this functor cannot be extended to a functor on the whole cobordism category (Proposition 4.7). This contrasts the additivity behaviour of the relative Euler characteristic, which is unconditional and thus defines a TQFT on the whole cobordism category (Remark 4.9). We also obtain the following result about the fundamental group of the amenable cobordism category of 4-manifolds and related cobordism categories (see Section 4.2 for the precise definition of Cob G d ): Theorem (see Theorem 4.5). Let G be a class of amenable groups that is closed under isomorphisms and let M be an object of Cob G 4 . Then the group π 1 (B Cob G 4 , [M ]) is not finitely generated. 1.2. Aspherical spaces homology equivalent to closed manifolds. Using the Kan-Thurston theorem, we show in Section 5 that Property (SVχ) fails in general if closed aspherical manifolds are replaced by aspherical spaces homology equivalent to closed manifolds or closed manifolds homology equivalent to an aspherical space (see Section 5.2 for the definition of an acyclic map): Theorem (see Theorem 5.7). Let n ∈ N ≥2 be even.
(1) There exist aspherical spaces X that admit an acyclic map X → M to an oriented closed connected n-manifold M and satisfy X = 0 and χ(X) = 0. In particular, these aspherical spaces do not satisfy Property (SVχ). (2) There exist oriented closed connected n-manifolds M that admit an acyclic map X → M from an aspherical space X and satisfy M = 0 and χ(M ) = 0. In particular, these manifolds do not satisfy Property (SVχ).

Strategies and known examples.
As the following examples show, the hypotheses in Question 1.1 cannot be reasonably weakened or modified in any straightforward way: • In general, non-aspherical oriented closed connected manifolds do not satisfy Property (SVχ): For example, S 2 = 0, but χ(S 2 ) = 2. • The converse implication of Property (SVχ) does not hold in general for aspherical manifolds: For example, oriented closed connected hyperbolic 3-manifolds have vanishing Euler characteristic, but their simplicial volume is non-zero. • In general, Property (SVχ) does not hold for oriented compact connected manifolds with non-empty boundary without imposing additional conditions on the inclusion of the boundary (Remark 2.16). • In general, Property (SVχ) does not hold for aspherical spaces that are only homology equivalent to oriented closed connected manifolds (Theorem 5.7). We also refer to Section 3 for a suvey of examples of oriented closed manifolds with vanishing or non-vanishing simplicial volume.
Various strategies have been developed to handle Question 1.1. In particular, this also led to a wide range of positive examples: Direct computations of both sides. One of Gromov's original motivations to formulate Question 1.1 was the observation that the simplicial volume and the Euler characteristic share some common vanishing properties. Examples of this phenomenon include manifolds that admit non-trivial self-coverings, closed aspherical manifolds with amenable fundamental group, and closed aspherical manifolds that admit small amenable open covers (Section 3).
Boundedness properties of the Euler class. The simplicial volume of an oriented closed connected n-manifold M is closely related to the comparison map from bounded cohomology to singular cohomology in degree n (see Section 2.4 and Proposition 2.18). On the other hand, the Euler characteristic is related to the Euler class by duality. As we explain in Proposition 2.22, an immediate consequence is that Property (SVχ) can be reformulated in terms of the boundedness of the Euler class. The problem of the boundedness of the Euler class is well studied and understood in several cases [84,103,63,17].
L 2 -Betti numbers. Gromov suggested to use the fact that the Euler characteristic can be computed as the alternating sum of the L 2 -Betti numbers and asked the following version of Question 1.1: 53, p. 232]). Let M be an oriented closed aspherical manifold. Does the following implication hold?
Assuming the Singer conjecture on the vanishing of the L 2 -Betti numbers of closed aspherical manifolds outside the middle dimension [35], Question 1.1 and Question 1.2 are equivalent. More concretely, Gromov [54, p. 306] proposed a definition of integral foliated simplicial volume, involving dynamical systems, and then • to establish an upper bound of the L 2 -Betti numbers in terms of integral foliated simplicial volume (via Poincaré duality), and • to investigate whether the vanishing of the simplicial volume of closed aspherical manifolds implies the vanishing of integral foliated simplicial volume.
The first step has been carried out by Schmidt [98]. The second step is an open problem, which is known to have a positive answer in many cases, e.g., for oriented closed aspherical manifolds • that have amenable fundamental group [46], • that carry a non-trivial smooth S 1 -action [39], • that are generalised graph manifolds [40], • that are smooth and have trivial minimal volume [11, (proof of) Corollary 5.4]. In particular, Question 1.1 with the simplicial volume replaced by the minimal volume has a positive answer [97].
Moreover, the integral foliated simplicial volume is related to the cost of the fundamental group [73] and to the stable integral simplicial volume [78,46] (Section 6). In turn, the stable integral simplicial volume gives upper bounds for homology growth, torsion homology growth [46], and the rank gradient [72].
Functorial semi-norms. If the integral foliated simplicial volume is a functorial semi-norm on aspherical closed manifolds, then Question Geometric positivity results. Conversely, it is known that many examples of closed aspherical manifolds with potentially non-zero Euler characteristic have positive simplicial volume. Examples include oriented closed connected hyperbolic manifolds [101,51], closed manifolds with negative sectional curvature [59], closed irreducible locally symmetric spaces of higher rank [69], closed manifolds with non-positive sectional curvature and sufficiently negative intermediate Ricci curvature [25], and closed manifolds with non-positive sectional curvature and strong enough conditions at a single point [26]. We refer to Section 3.1 for further examples of manifolds with (non-)vanishing simplicial volume.
Outlook. Supported by the wealth of positive examples, and in view of the existence of "exotic" aspherical manifolds, it seems plausible that Question 1.1 has a positive answer in the following special case: Let M be an oriented closed aspherical n-manifold whose universal covering is homeomorphic to R n . Does M satisfy Property (SVχ)?
Organisation of the article. In Section 2, we collect the definitions of simplicial volume (Section 2.1) and bounded cohomology (Section 2.4) as well as the duality principle which connects these (Section 2.4). Moreover, we discuss the behaviour of the simplicial volume with respect to glueings (Section 2.2) and introduce a relative version of Question Known results and open problems about the behaviour of the simplicial volume with respect to products are recalled in Section 3.4 and these are then discussed in connection with Question 1.1 (Proposition 3.24). Finally, Section 3.5 explains a connection between Question 1.1 and a conjecture of Edmonds in four-dimensional topology (Conjecture 3.30) via "fillings" of closed manifolds.
In Section 4, we define the amenable cobordism category and explain how to interpret the simplicial volume as an invertible TQFT on this cobordism category. Using the simplicial volume, we prove that the fundamental group of the 4-dimensional amenable cobordism category is not finitely generated (Theorem 4.5). Also, using known results about cobordism categories, we show that the simplicial volume does not extend to the whole cobordism category (Proposition 4.7).
Section 5 is concerned with the study of Question 1.1 using known constructions that produce aspherical spaces. More precisely, in Sections 5.1-5.3, we recall the Kan-Thurston theorem and explain how to use this to prove the result stated in Section 1.2 (see Theorem 5.7). Then, in Section 5.4, we briefly review known constructions of closed aspherical manifolds, Davis' reflection group trick and Gromov's hyperbolization, in the context of Question 1.1.
Finally, Section 6 surveys the approach to Question 1.1 via the stable integral simplicial volume.

Simplicial volume
We recall the definition of the simplicial volume, basic glueing properties, and the role of the comparison map for bounded cohomology. Moreover, we consider and study a relative version of Question 1.1, and discuss the connection of Question 1.1 with (the boundedness of) the Euler class.
2.1. Simplicial volume. The simplicial volume originally appeared in Gromov's proof of Mostow rigidity as a homotopy invariant replacement of the hyperbolic volume [90,51].  In the oriented, compact, non-connected case, we define the (relative) simplicial volume as the sum of the (relative) simplicial volumes of the components. In particular, ∅ = 0. In particular, M, ∂M = 0 implies ∂M = 0. Note that in the case of compact 3-manifolds better estimates are available [15].

2.2.
Simplicial volume and glueings of manifolds. In general, the simplicial volume is not additive with respect to the glueing of manifolds along submanifolds. However, (sub)additivity does hold in the case of amenable glueings: Theorem 2.5 (Simplicial volume and glueings [51,13][45, Theorem 7.6]). Let I be a finite set and let (M i , ∂M i ) i∈I be a family of oriented compact connected manifolds of the same dimension. Assume that all the boundary components have amenable fundamental group. Moreover, let (M, ∂M ) be obtained from (M i , ∂M i ) i∈I by a pairwise glueing (along orientation reversing homeomorphisms) of a set of boundary components. Then, we have If all glued boundary components are π 1 -injective in their original manifold, then Remark 2.6. Theorem 2.5 allows that boundary components of the given manifolds (M i , ∂M i ) are glued to boundary components of the same manifold (M i , ∂M i ); i.e., self-glueings are included. Furthermore, not all boundary components need to be glued (so that some of the components remain boundary components of M ).
Remark 2.7. It is well-known that the Euler characteristic is always additive with respect to glueings: given oriented compact connected n-manifolds (M, ∂M ) and (N, ∂N ) with homeomorphic (or just homotopy equivalent) boundary components M 1 ⊆ ∂M and N 1 ⊆ ∂N , we set Z = M ∪ M 1 ∼ =N1 N . Then we have: and similarly: Here χ(W, ∂W ) := χ(W ) − χ(∂W ) denotes the relative Euler characteristic of the compact manifold (W, ∂W ).
Assuming that M 1 is aspherical and has amenable fundamental group, then both the simplicial volume and the Euler characteristic of M 1 vanish (Example 3.2(1) and Theorem 3.6). In particular, the last formula simplifies in this case to a formula analogous to the one in Theorem 2.5:  where −M denotes a copy of M with the opposite orientation. It is easily seen that we always have subadditivity of the simplicial volume in this case: Indeed, given a relative fundamental cycle c of M , we can set c to be the relative fundamental cycle of −M corresponding to −c. Then, c ′ = c + c is in fact a fundamental cycle of D(M ) with norm Then the subadditivity of the simplicial volume follows from taking the infimum over all such c. The same computation also works for integral coefficients.
Remark 2.9 (Doubles and asphericity). In general, the double of an oriented compact aspherical manifold with boundary is not necessarily aspherical; prototypical examples of this kind are the 3-ball or S 1 × D 2 .
Let (M, ∂M ) be a compact n-manifold, where M is aspherical and ∂M is connected, and let F denote the homotopy fibre of the inclusion ∂M ⊂ M . For simplicity, we write H := π 1 (∂M, x) and G := π 1 (M, x) and denote by ι : H → G the induced homomorphism. Moreover, let G ′ := im(ι) be the image of ι, and consider the corresponding diagram induced by ι and the inclusion maps. Suppose that the (homotopy) pushout is aspherical. Then the induced map between the homotopy pushouts is a homotopy equivalence by the Seifert-van Kampen theorem. Passing to the homotopy fibres of the diagram above, regarded as a diagram over K(G, 1), we obtain the following diagram (up to canonical homotopy where D is discrete with cardinality equal to the index [G : G ′ ]. We recall that the homotopy fibre of a map from a homotopy pushout is canonically identified up to homotopy equivalence with the homotopy pushout of the respective homotopy fibres. Thus, since g is a homotopy equivalence, the induced map g ′ : ΣF → ΣD, between the homotopy fibres of the source and target of g as spaces over K(G, 1), where Σ denotes here the (unreduced) suspension, is again a homotopy equivalence. Therefore, π 0 (F ) ∼ = [G : G ′ ] and each path-component of F must have trivial integral homology. In addition, the homotopy fibre of q : is identified with a path-component of F , so the map q is acyclic (see Section 5.2). In particular, q is an integral homology equivalence and arises as the plus construction associated to the kernel of ι. As a consequence, if ι is injective, then q : ∂M → K(G ′ , 1) is a homotopy equivalence, so ∂M is again aspherical. Conversely, it is well known that the double is aspherical if M and ∂M are aspherical and ι is injective.
On the other hand, we do not know if the injectivity of ι is necessary for the asphericity of the double.
An interesting example of amenable glueings is given by connected sums: Remark 2.11. Note that the connected sum of aspherical manifolds in dimension ≥ 3 is never aspherical [81,Lemma 3.2]. Thus, Proposition 2.10 cannot be used to produce counterexamples to Question 1.1.

A relative version of Gromov's question.
We consider the following version of Question 1.1 for manifolds with boundary and show that it is a consequence of Property (SVχ) (Proposition 2.15). In order to disprove Property (SVχ) it suffices to find an example that does not satisfy Property (SVχ, ∂): Proposition 2.15. Let n ≥ 1. If all oriented closed aspherical manifolds of dimension n or n − 1 satisfy Property (SVχ), then all oriented compact aspherical n-manifolds with non-empty π 1 -injective aspherical boundary satisfy Property (SVχ, ∂).
Proof. Let (M, ∂M ) be an oriented compact aspherical n-manifold with nonempty π 1 -injective aspherical boundary. Remark 2.17. Note that by allowing the case of empty boundary, Proposition 2.15 can be formulated as an equivalence between Question 2.13 and Question 1.1. The extension of Gromov's question to manifolds with boundary allows us in particular to explore Question 1.1 by studying the properties of (vanishing of) the simplicial volume and the Euler characteristic along glueings of manifolds and compare their respective additivity properties. This viewpoint will also be explored in Section 4.
2.4. The comparison map. Dually, simplicial volume can be expressed in terms of bounded cohomology. Bounded cohomology ) is the cohomology of the topological dual C * b ( · ; R) of the singular chain complex, where the dual is taken with respect to the ℓ 1 -norm. Bounded cohomology is then endowed with the ℓ ∞ -seminorm, denoted by · ∞ .
, which is called the comparison map. A straightforward application of the Hahn-Banach Theorem shows: Proposition 2.18 (Duality principle [51]). Let (X, A) be a pair of spaces, let k ∈ N, and let α ∈ H k (X, A; R). Then The usual explicit formula for the cup-product on singular cohomology shows that the cup-product lifts to a cup-product on bounded cohomology. Then The vanishing of the simplicial volume thus implies that not too many classes can be bounded; dually, the vanishing of the simplicial volume causes that there are many other classes with vanishing ℓ 1 -semi-norm.
Proof. On the one hand, by the duality principle (Proposition 2.18), we have On the other hand, Poincaré duality and Proposition 2.19 imply that Combining both estimates gives the claim. The norm of the Euler class has been studied extensively in the literature, especially, in connection with the existence of flat structures (a detailed account of results in this direction is given in Frigerio's book [45]). The boundedness of the Euler class is also closely related to Question 1.1.  Example 2.25. Assuming that Question 1.1 has an affirmative answer, then Proposition 2.22 has interesting implications for the existence of tangential maps between smooth manifolds. We recall that a map f : M → N between closed smooth manifolds is called tangential if the vector bundles T M and f * T N are isomorphic. As a consequence, a tangential map f : M → N between oriented closed connected smooth manifolds preserves the Euler class up to sign. Assuming Question 1.1, it follows that there cannot exist tangential maps f : M → N if χ(M ) = 0 and N is aspherical with zero simplicial volume. Indeed, assuming that N satisfies Property (SVχ), it follows that e(N ) = 0 (since χ(N ) = 0). Then, given a tangential map f : M → N , the classes f * (e(N )) and e(M ) agree up to sign, so e(M ) = 0 (and therefore also χ(M ) = 0).

Vanishing of the simplicial volume
In this section, we collect some known results on the simplicial volume. We will be mainly interested in describing sufficient conditions for the vanishing of the simplicial volume. We also compare those situations with the respective behaviour of the Euler characteristic.
3.1. Computations of the simplicial volume. In general, computing exact values of the simplicial volume is difficult. For example, the problem of determining whether a given (triangulated) manifold has vanishing simplicial volume or not is undecidable [102, Chapter 2.6]. The two major sources for (non-)vanishing results are amenability (which leads to vanishing) and negative curvature (which leads to non-vanishing). (1) Oriented closed connected hyperbolic manifolds [101,51]; (2) The compactification of oriented connected complete finite-volume hyperbolic manifolds [51,49]; (3) Oriented closed connected manifolds with negative sectional curvature [59]; (4) Oriented closed connected locally symmetric spaces of non-compact type [18,69]; In this paper, we focus our attention on vanishing results for the simplicial volume. The following example contains some known vanishing results: Example 3.2 (Vanishing). The following manifolds have zero simplicial volume: (1) Oriented closed connected n-manifolds with amenable fundamental group and n > 0 [51] or, more generally, with n-boundedly acyclic fundamental group [89,51]; finitely presented non-amenable boundedly acyclic groups have been recently constructed [42,87]; (2) Oriented compact connected n-manifolds M with non-empty boundary such that both the fundamental groups of M and of ∂M are amenable [51]; (3) More generally, oriented compact connected n-manifolds M with non-empty boundary such that ∂M = 0 and the connecting ho- Manifolds satisfying the latter condition can be constructed by taking manifolds whose boundary inclusion is π 1 -surjective and such that their fundamental group lies in Lex [10]. Recall that Lex groups are those groups Γ such that every epimorphism Λ ։ Γ induces an injective map in bounded cohomology in every degree. Examples of Lex groups contain free groups, amenable groups [10], boundedly acyclic groups and certain extensions of these [42, Remark 3.8].
(4) Oriented compact connected manifolds with (possibly empty) boundary that admit a self-map f of degree deg(f ) ∈ {0, 1, −1} [51]; (5) Oriented closed connected manifolds that are the boundary of an oriented compact connected manifolds with zero simplicial volume (Remark 2.3); (6) Oriented closed connected n-manifolds that admit a smooth nontrivial S 1 -action [104]. More generally, manifolds admitting an Fstructure also have zero simplicial volume [24,91]; (7) Oriented closed aspherical manifolds supporting an affine structure whose holonomy map is injective and contains a pure translation [14]; (8) Oriented closed connected smooth manifolds with zero minimal volume [51,8] or zero minimal volume entropy [51, p. 37][4]; (9) Oriented closed connected graph 3-manifolds [99,51]; (10) All mapping tori of oriented closed connected 3-manifolds [16]; however, the general behaviour of simplicial volume of general mapping tori is very diverse [65]. (1) Let X be a topological space and let U be a subset of X. We say is amenable. We say that an open cover of X is amenable, if it consists of amenable sets. (2) The amenable category of X, denoted by cat Am (X), is the minimal integer n such that X admits an open amenable cover with cardinality n. If no such integer exists, we simply set cat Am (X) = +∞.
The vanishing results for open amenable covers are usually stated in terms of assumptions on the multiplicity of the cover instead of the cardinality. These assumptions essentially are the same when working with paracompact Hausdorff spaces [21,Remark 3.13].
The importance of amenable covers in our setting is demonstrated by the following two results: A similar result for the Euler characteristic has been proved by Sauer [97]: Theorem 3.6 (Euler characteristic and amenable covers). Let M be an oriented closed aspherical n-manifold. Then, Remark 3.7. In Theorem 3.6, the asphericity assumption is crucial: every even-dimensional sphere provides a counterexample in the non-aspherical setting.
In particular, Theorems 3.5 and 3.6 show that all oriented closed aspherical manifolds M with cat Am (M ) ≤ dim(M ) satisfy Property (SVχ). Recently, Gromov's vanishing theorem was extended to weaker situations, such as weakly boundedly acyclic open covers and other more general homotopy colimit decompositions [62,93]. This new context suggests in particular the following question: Question 3.9. Let M be an oriented closed aspherical n-manifold. Assuming there exists a weakly boundedly acyclic open cover of M in the sense of Ivanov [62] with cardinality at most n, does it follow that χ(M ) = 0 ?
A negative answer to Question 3.9 would also produce closed aspherical examples that do not satisfy Property (SVχ). In fact, no example of an oriented closed aspherical manifold M with vanishing simplicial volume and cat Am (M ) = dim(M ) + 1 seems to be known.

Amenable covers:
The case with boundary. The vanishing theorem can also be extended to the relative setting via Gromov's vanishingfiniteness theorem [51,47,61]. We quickly recall the formulation of the vanishing-finiteness theorem and mention two convenient special cases for compact manifolds with boundary: the relative vanishing theorem and the case of locally co-amenable subcomplexes.
For an oriented connected (possibly non-compact) manifold M without boundary, the locally finite simplicial volume is defined by denotes the (singular) locally finite chain complex of M . Because locally finite chains are not necessarily ℓ 1 , the locally finite simplicial volume is not always finite. (i) For each i ∈ N, the subset U i ⊂ M is amenable and relatively compact. (ii) The sequence (U i ) i∈N is amenable at infinity, i.e., there exists an increasing sequence (K i ) i∈N of compact subsets of M such that: The multiplicity of (U i ) i∈N is at most n. Then M lf = 0.
Remark 3.11. Properties (a) and (b) in (ii) show that every sequence (U i ) i∈N which is amenable at infinity is necessarily locally finite.
Vanishing of the locally finite simplicial volume leads to vanishing results for the relative simplicial volume:  (i) The multiplicity of (U i ) i∈I is at most n.
(ii) The multiplicity of (U i ∩ ∂M ) i∈I is at most n − 1.
(iii) For each i ∈ I, the set U i ∩ ∂M is amenable in ∂M . Then M, ∂M = 0.
Proof. In view of Remark 3.12, it suffices to show that the vanishing-finiteness theorem (Theorem 3.10) can be applied to int(M ). To this end, we modify the given open cover (U i ) i∈I as follows. Up to homeomorphism, we can write We extend (U i ) i∈I to the right-hand side by replacing the sets V of (U i ) i∈I intersecting ∂M with V ∪ ((V ∩ ∂M ) × [0, ∞)). Let (V i ) i∈I be the resulting open cover of int(M ). We now upgrade this cover (V i ) i∈I to a locally finite cover made of relatively compact sets without increasing the multiplicity; because the intersection of (U i ) i∈I with ∂M has multiplicity at most n − 1, this is indeed possible by a standard procedure [ Amenable covers as in the vanishing-finiteness theorem (Theorem 3.10) appear naturally in the presence of locally co-amenable subcomplexes [51,47]: Assume that there exists a simplicial complex K ⊂ P such that |K| ⊂ int(M ) and M is homeomorphic to a closed regular neighbourhood of K inside P [47, Definition 11.1.4]. Suppose also that K has codimension at least 2 in M . Then, K is called locally co-amenable in P (or in M ) if for each vertex v ∈ (K ′′ ) 0 of the second barycentric subdivision K ′′ of K we have that π 1 (|S v \ (S v ∩ K)|) is amenable. Here, S v denotes the simplicial sphere in P ′′ centered at v. Remark 3.16. If K is locally co-amenable in M , then M is homotopy equivalent to K.
However, as M is homotopy equivalent to K (Remark 3.16), also K is aspherical and thus which is a contradiction. 3.4. Products of manifolds. While the Euler characteristic is multiplicative with respect to products, the product behaviour for the simplicial volume is more delicate. If one of the factors is closed, the vanishing behaviour of simplicial volume is controlled by the factors: The exact values in general are unknown; the only known non-zero computation is the product of two closed surfaces [19].
On the other hand, the product of at least three compact manifolds with non-empty boundary always has vanishing simplicial volume: ) is an isomorphism in every degree [51,89]. The long exact sequence of the pair then implies that H n A nice application of the previous result is the following: Example 3.21. Let (T 2 ) × denote a 2-dimensional torus with an open disk removed. Then the simplicial volume of the product of (at least) three copies of (T 2 ) × vanishes. On the other hand, χ((T 2 ) × × (T 2 ) × × (T 2 ) × ) = −1.
While triple products have zero relative simplicial volume, the situation remains undecided in the following case of products of two compact manifolds [47, Question 8]: The following proposition shows an interesting connection between Property (SV×) and Question 1.1. 3.5. Small aspherical fillings. We now come to a higher order version of vanishing, which asks for "small" aspherical fillings of the given manifold with vanishing Euler characteristic/simplicial volume. We will be mainly interested in filling aspherical 3-manifolds with amenable fundamental group.  In the same spirit, we could ask the corresponding question for the simplicial volume:  Question 3.28 can be interpreted as a manifold variant of the uniform boundary condition (UBC) [82]. Recall that a space X satisfies UBC in dimension n if there exists a constant K > 0 such that every boundary c ∈ im ∂ n+1 ⊂ C n (X; R) can be filled K-efficiently, i.e., there exists a chain b ∈ C n+1 (X; R) such that ∂ n+1 b = c and |b| 1 ≤ K · |c| 1 . Spaces with amenable fundamental groups satisfy UBC in all dimensions [82]. Therefore, in a similar way, Question 3.28 asks whether the small fundamental cycles of oriented closed connected 3-manifolds M with amenable fundamental group can be filled efficiently using relative fundamental cycles of 4-manifolds with M as π 1 -injective boundary.
Similar quantified bordism problems have been successfully studied in more geometric contexts [22]. Remark 3.29. Property (Fill) does not hold in dimension 1. Indeed, the only surfaces that have the circle as π 1 -injective boundary are hyperbolic surfaces with totally geodesic boundary. All these have uniformly positive simplicial volume (Example 3.1(2)).

Simplicial volume and cobordism categories
In this section, we will introduce the amenable cobordism category and explain how the simplicial volume extends to a symmetric monoidal functor on this cobordism category. In other words, the simplicial volume defines an invertible TQFT in this restricted sense. Interestingly, it will be shown that the simplicial volume does not extend to a functor on the whole cobordism category of smooth oriented manifolds. This fact reflects the (non-)additivity properties of the simplicial volume.
Viewing the simplicial volume as an invertible TQFT will allow us to obtain some interesting information about the fundamental group of the amenable cobordism category and its variations. Specifically, we will show that this fundamental group is not finitely generated (Theorem 4.5). This result is based on the following computations of simplicial volume in dimension 4: If n ≥ 4, then SV(n) has no gap at zero [58] and thus is non-discrete. Moreover, SV(3) contains the set of all volumes of oriented closed connected hyperbolic 3-manifolds (scaled by 1/v 3 ) and thus is non-discrete [101]. Therefore, if n ≥ 3, then the additive monoid SV(n) is not finitely generated.
Moreover, SV(4) contains an infinite family of values that are linearly independent over Q [57].
We first consider the simpler case of the connected sum monoid and prove that it is not finitely generated (Section 4.1). In Section 4.2, we explain how to view the simplicial volume as a symmetric monoidal functor on the amenable cobordism category and use this description to deduce the nonfinite generation of the fundamental group of the 4-dimensional amenable cobordism category (Theorem 4.5). Finally, we prove that the functor of simplicial volume cannot be extended to an invertible TQFT on the whole cobordism category (Proposition 4.7).

4.1.
The connected sum monoid. For n ∈ N, let Mfd # n denote the monoid, whose elements are diffeomorphism classes of oriented closed connected smooth n-manifolds and whose operation is given by connected sum. By the classification of surfaces, the monoid Mfd # 2 is generated by the 2torus. This finite generation fails in higher dimensions: Proposition 4.2. Let n ∈ N ≥3 . Then the monoid Mfd # n is not finitely generated.
Proof. As simplicial volume is additive in dimension ≥ 3 with respect to connected sums (Proposition 2.10), we can view the simplicial volume as a monoid homomorphism S : Mfd # n → R ≥0 from Mfd # n to the additive monoid (R ≥0 , +). The submonoid S(Mfd # n ) is not finitely generated (Remark 4.1). Because finite generation is preserved by monoid homomorphisms, we conclude that Mfd # n is not finitely generated.

Remark 4.3.
As suggested by the referee, the previous result also admits a more geometric proof: In every dimension n ≥ 3 there exist infinitely many hyperbolic n-manifolds (and none of them is a non-trivial connected sum).

4.2.
Simplicial volume as a TQFT. The simplicial volume can be viewed as an (invertible) TQFT defined on an appropriate cobordism category of oriented smooth manifolds. This is essentially a basic consequence of known additivity properties of the simplicial volume. For background material about cobordism categories and TQFTs, we refer the interested reader to the work of Abrams [1] and the book by Kock [68], both of which focus especially on the 2-dimensional case, and to the lecture notes of Debray, Galatius, and Palmer [34], which contain an excellent exposition of the classification of invertible TQFTs following major recent developments in the field. → ∂ in W (incoming boundary) and N ∼ = → ∂ out W (outgoing boundary). The equivalence relation is given by orientation-preserving diffeomorphisms that preserve the boundary pointwise. Composition of morphisms in Cob d is given by glueing of cobordisms, using the given identifications of the boundary components. The category Cob d is a symmetric monoidal category under the operation of disjoint union.

4.2.2.
Amenability conditions. Let G be a class of groups that is closed under isomorphisms. We consider the subcategory Cob G d ⊂ Cob d defined as follows: The objects are those manifolds with fundamental group in G (for each component). The morphisms are the cobordisms (W ; M, N ) such that M ֒→ W and N ֒→ W are π 1 -injective (for all components). It should be noted that Cob G d is indeed a subcategory of Cob d , i.e., that Cob G d is closed under composition. To see this, we only need to check that the π 1injectivity of the boundary components is preserved under composition of cobordisms. This can be shown inductively by glueing one pair of components at a time and applying the Seifert-van Kampen theorem as well as the normal form theorems for amalgamated free products and HNN extensions [96,Chapter 11]. These guarantee at each stage that the remaining boundary components are π 1 -injective in the resulting manifold.
The symmetric monoidal pairing of Cob d clearly restricts to a symmetric monoidal pairing on Cob G d . When G = Am is the class of all amenable groups, we will refer to Cob Am d as the amenable cobordism category.

Simplicial volume as a TQFT on the amenable cobordism category.
Let R = (R, +) denote the additive (abelian) group of real numbers, regarded as a symmetric monoidal groupoid with one object. Moreover, let G be a class of amenable groups that is closed under isomorphisms. The additivity of the simplicial volume with respect to amenable glueings (Theorem 2.5) and disjoint union shows that we obtain a symmetric monoidal functor with values in the abelian group R (regarded as a symmetric monoidal category): is an infinite loop space, therefore, all of its path components have the same homotopy type. After passing to the classifying spaces, the functor − induces a group homomorphism: Here, we have used the homotopy equivalence ΩBΓ ≃ Γ for groups Γ. The group homomorphisms ϕ M (for all basepoints M ) uniquely determine the functor − ; similar facts hold more generally for functors whose target is a groupoid (see, for example, [34]).  Proof. The relative simplicial volume of 4-manifolds induces a group homomorphism S : The image of this group homomorphism contains the subset SV(4) (Remark 4.4), which contains an infinite family of elements that are linearly independent over Q (Remark 4.1). Therefore, the abelian group im S is not finitely generated and so π 1 (B Cob G 4 , [∅]) is not finitely generated. As explained above, π 1 (B Cob Remark 4.6. We also expect corresponding results in higher dimensions. However, currently, not enough is known about the structure of SV(d) for d ≥ 5.

4.2.5.
Non-extendability to Cob d . Since the simplicial volume does not satisfy additivity in general [45,Remark 7.9], it does not define a functor on Cob d . However, it is still interesting to ask whether there might be a different extension of the simplicial volume to general oriented compact manifolds with boundary which is always additive. This question is closely related to the problem of extending the functor − : Cob G d → R to the whole cobordism category Cob d . Based on the classification of functors with values in a groupoid (see, e.g., [34]), this problem is essentially equivalent to the question of extending the homomorphism ϕ ∅ to π 1 (B Cob d , ∅).
In contrast to π 1 (B Cob G 4 , [M ]) (Theorem 4.5), the fundamental group of the d-dimensional cobordism category π 1 (B Cob d , ∅) is well known and simpler to describe. We first note that it agrees with the fundamental group of the standard topologised cobordism category C d [34,Section 2.4]. This is again independent of the choice of basepoint and can be identified with the Reinhart bordism group R d [95][36, Appendix A]. We recall that R d can be described as the group of equivalence classes of oriented closed d-manifolds where the equivalence relation is defined by cobordisms whose tangent bundle is equipped with a nowhere-vanishing vector field that extends the normal fields on the boundary components. This refined bordism group is known to be a split extension of the usual oriented bordism group Ω SO d by a cyclic group whose generator is represented by the d-sphere. More precisely, there is a split exact sequence: where Eul n = {0} if n is odd, Eul n = 2Z if n ≡ 2 (mod 4), and Eul n = Z if n is a multiple of 4. We refer to the literature [95][36, Appendix A] [9] for the properties of the bordism group R d and the description of the homotopy groups of BC d in terms of bordism classes. Using this description of R d ∼ = π 1 (B Cob d , [∅]), we conclude below that the simplicial volume of oriented closed d-manifolds cannot be extended to a functor on the cobordism category Cob d , i.e., there is no additive extension of the simplicial volume − (analogous to Theorem 2.5) to all oriented compact d-manifolds.
Let M d denote the monoid of endomorphisms of ∅ in Cob d , that is, the monoid of diffeomorphism classes of oriented closed d-manifolds under the operation of disjoint union. (1) There is no functor Cob d → R that extends the restriction of the simplicial volume − |M d : M d → R to oriented closed d-manifolds.
(2) Let G ⊂ Am be a class of groups that is closed under isomorphisms.
The functor − : Cob G d → R does not admit an extension to a functor on Cob d .
Proof. For (1), note that such an extension of Cob d → R would imply a factorization of − |M d : Remark 4.4). In particular, this would imply that − is invariant under the Reinhart bordism relation. Moreover, since S d = 0, it would further follow from the exact sequence ( * ) that − is invariant under oriented bordism. This is obviously false in general, e.g., note that M ⊔(−M ) is null-bordant as oriented closed d-manifold, but its simplicial volume is non-trivial in general. Claim (2) follows directly from (1).
Remark 4.8. The fact that − is not invariant under oriented bordism can also be shown as follows. Note that for d ∈ {2, 3}, this fails because − is non-trivial but Ω SO d ∼ = 0. Then the result follows in all dimensions by taking suitable products. We note also that for d = 4, this property can be shown to fail also because the oriented bordism group Ω SO 4 ∼ = Z is finitely generated, whereas SV(4) is not finitely generated by Remark 4.1.

Asphericalisations
The construction of aspherical closed manifolds with vanishing simplicial volume is a key problem for Question 1.1. There are several known constructions of aspherical closed manifolds from non-aspherical or non-closed manifolds. Important examples of such constructions are Davis' reflection group trick [27,28,30] and Gromov's hyperbolization [52,32,33]. The general difficulty with using these constructions to obtain (counter)examples to Question 1.1 has to do with the difficulty of computing the simplicial volume of the resulting aspherical closed manifolds.
In this section, we consider extensions of the class of aspherical closed manifolds and look for interesting (counter)examples in these contexts. In particular, we will prove that the class of aspherical spaces that are homology equivalent to closed manifolds, as well as the class of closed manifolds that are homology equivalent to an aspherical space, do not satisfy Property (SVχ) in general (Theorem 5.7). We introduce the simplicial volume of such spaces in Section 5.1. The proof of Theorem 5.7 will be given in Section 5.3; the proof is based on the Kan-Thurston theorem [64], which we recall in Section 5.2. Finally, we end with some brief comments on known constructions of aspherical closed manifolds and their possible connections with Question 1.1 (Section 5.4). Besides their independent interest, we hope that the results of this section, especially, combined with the aforementioned constructions of aspherical closed manifolds, might provide useful tools for promoting non-aspherical or non-closed examples to closed aspherical (counter)examples to Question 1.1.

Simplicial volume of spaces homology equivalent to manifolds.
Our goal in this section is to extend the definition of simplicial volume to spaces that are only homology equivalent to an oriented closed manifold and discuss the main properties of this invariant. This is motivated by the following basic observation: Remark 5.1. Suppose that M is an oriented closed connected manifold with M = 0. Let f : M → N be a homology equivalence to an oriented closed connected manifold N ; in particular, this map has degree ±1 and so N = 0 (this conclusion holds more generally if the degree of f is non-zero). Moreover, because f is a homology equivalence, it follows that χ(M ) = χ(N ). In this sense, Property (SVχ) is inherited under homology equivalences between oriented closed connected manifolds.
Thus, in connection with Question 1.1, it would be interesting to understand the class of manifolds which are homology equivalent to an oriented closed aspherical manifold with vanishing simplicial volume.
As in the manifold case, we clearly have |deg h| · Y ≤ X and it follows that the simplicial volume of X is homotopy invariant. Moreover, if X admits a self-map h : X → X with |deg h| ≥ 2, then X = 0. Furthermore, if h : X → Y is a (finite) covering map, then |deg h| · Y = X , as can be seen from the same argument as in the manifold case.
This extension of the simplicial volume to a homotopy invariant of spaces that are only homology equivalent to an oriented closed manifold should not be confused with the fact that the simplicial volume is not invariant under homology equivalences:

Acyclic maps and plus contructions.
We review briefly the definition and basic properties of acyclic maps and refer to the literature [56,92] for more details. A map f : X → Y is acyclic if the induced homomorphism is an isomorphism for every local coefficient system A of abelian groups on Y ; in particular, f induces isomorphisms on singular homology and cohomology with both integral and real coefficients. Equivalently, a map f : X → Y is acyclic if its homotopy fibers have trivial integral homology. Every acyclic map f : X → Y between path-connected based spaces arises up to weak homotopy equivalence as the plus construction ι P : X → X + P with respect to a normal perfect subgroup P π 1 (X). In this case, we have π 1 (X + P ) ∼ = π 1 (X)/P. Theorem 5.6 (Kan-Thurston [64,5,83]). For every path-connected based topological space X, there is a group G X together with an acyclic (based) map f X : K(G X , 1) → X. Moreover, G X and f X can be chosen to be natural in X.
Proof. The original functorial construction of (G X , f X : K(G X , 1) → X) is due to Kan and Thurston [64]. Alternative constructions and refinements were obtained by Baumslag, Dyer, Heller [5], and Maunder [83]. (These constructions are also shown to preserve properties of homotopy finiteness, but they satisfy weaker functoriality properties in general.) 5.3. Using the Kan-Thurston theorem. The Kan-Thurston theorem (Theorem 5.6) has the following consequence in connection with Question 1.1.
(1) There exist aspherical CW-complexes X that admit an acyclic map X → M to an oriented closed connected n-manifold M , and satisfy both X = 0 and χ(X) = 0. In particular, these aspherical spaces do not satisfy Property (SVχ). (2) There exist oriented closed connected n-manifolds M that admit an acyclic map X → M from an aspherical CW-complex X, and satisfy both M = 0 and χ(M ) = 0. In particular, these manifolds do not satisfy Property (SVχ).
Proof. Let M be an oriented closed connected n-manifold that has a (based) self-map h : M → M with |deg h| ≥ 2 and satisfies χ(M ) = 0. For example, as n is even, we may choose M = S n . Ad 1. By Theorem 5.6 (and the functoriality of the construction), there exists an aspherical CW-complex X with an acyclic map f : X → M and a map H : X → X that makes the following square commutative: It follows that |deg H| = |deg h| ≥ 2. Thus, X = 0 (Remark 5.4). Moreover, as f is a homology equivalence, it follows that χ(X) = χ(M ) = 0.
Ad 2. It is sufficient to apply Theorem 5.6 to M .
We give some further context on possible improvements of Theorem 5.7: Remark 5.8 (Poincaré duality). Hausmann [55] proved that the group G X in the Kan-Thurston theorem (Theorem 5.6) can be chosen to be a duality group when X is homotopy finite. (A related interesting refinement of the Kan-Thurston theorem has also been obtained more recently by R. Kim [66].) We do not know whether we can obtain homotopy finite examples in Theorem 5.7 and whether Hausmann's construction can also be made sufficiently functorial for the purpose of the proof above. Thus, it remains open whether Theorem 5.7 can be strengthened to produce examples where the fundamental group is a duality group. We note that it remains an open problem whether every finitely presented Poincaré duality group is the fundamental group of a closed aspherical manifold [31]. In this connection, we also recall the following question [64, Question (ii) p. 254]: Is every oriented closed connected n-manifold, n ≥ 4, homology equivalent to an oriented closed aspherical manifold? If this question has an affirmative answer, then there exist aspherical homology spheres in all high dimensions. This would contradict a version of the Hopf conjecture, which claims that the Euler characteristic of every oriented closed aspherical 2k-manifold is either zero or its sign is (−1) k [29]. 5.4. Further comments. Davis' reflection group trick [27,28,30] takes an oriented compact aspherical n-manifold (W, ∂W ) and constructs an oriented closed aspherical n-manifold M by reflecting W along so-called pieces of the boundary of W . The construction also yields a retraction r : M → W i.e., r • i = id W , where i : W → M is the inclusion. In particular, π 1 (r) and H * (r) are epimorphisms; H * b (r) and H * (r) are monomorphisms. Starting from"exotic" W , this method can be used to construct "exotic" closed aspherical manifolds. Note that we may choose W to be an oriented compact manifold that is homotopy equivalent to X as in Theorem 5.7(1), assuming that the space X can also be chosen to be homotopy finite [5].
There exists an explicit formula for computing the Euler characteristic of the manifold M in terms of the Euler characteristic of the input manifold W and the combinatorics of the pieces of ∂W [28, p. 218]. However, in the case of the simplicial volume, the situation is more delicate. The Davis reflection group trick can be viewed as a refined version of doubling manifolds with boundary, where the refinement is given by the combinatorics of the pieces of ∂W . In order to generalise Example 2.8 to this setting, it would be desirable to find input manifolds W with W, ∂W = 0 and where additionally the simplicial volume of W can be realised by small relative fundamental cycles, whose behaviour on ∂W is adapted to the combinatorics on the pieces of ∂W . Particularly interesting input candidates would be the examples from Remark 2.16 or Proposition 3.20.
On the other hand, Gromov's hyperbolization [52,32,33] is a construction that takes an oriented closed connected triangulated manifold N and produces an oriented closed aspherical manifold h(N ) together with a degree 1 map In particular, In addition, h(N ) is a smooth manifold if N is smooth, and h preserves the stable tangent bundle, i.e., the vector bundles T (h(N )) and c * T N are stably isomorphic. This implies that hyperbolization preserves the characteristic classes and numbers of closed smooth manifolds. Also, h(N ) and N are (oriented) cobordant. Since the mod 2 Euler characteristic of N is determined by the bordism class of N , it is natural to consider the hyperbolization in connection with the following weak version of Question 1.1: Question 5.9. Let M be an oriented closed aspherical manifold. Does the following implication hold?
Assuming that M is smooth, the property "χ(M ) is even" is equivalent to the vanishing of the top Stiefel-Whitney class of M .
It would be interesting to find N as above with the property h(N ) = 0. We note here that the simplicial volume is always positive in the case of strict hyperbolization (in the sense of Charney and Davis [23]). A relative version of this construction, which might still be relevant in connection with Question 2.13, has also been studied by Belegradek [6].

Stable integral simplicial volume
Stable integral simplicial volume and integral foliated simplicial volume are versions of the simplicial volume that admit Poincaré duality estimates for Betti numbers and for the Euler characteristic.
In this section, we recall definitions, basic properties, and known examples of stable integral simplicial volume, with a focus on the relative case and the connection with Property (SVχ, ∂). Moreover, we quickly outline the relation with the integral foliated simplicial volume. The key observation is that Poincaré duality leads to Betti number estimates for simplicial volumes with respect to sufficiently integral coefficient rings:  Since the Euler characteristic is multiplicative with respect to finite coverings, this estimate also implies corresponding estimates for the stable integral simplicial volume [43, Proposition 6.1]: 6.2. Integral approximation problems. This estimate for the Euler characteristic (Corollary 6.3) suggests the following question: Question 6.4. Let M be an oriented closed aspherical manifold with residually finite fundamental group. Does the following implication hold?
The corresponding approximation question for non-zero values in general has a negative answer. For example, oriented closed connected hyperbolic manifolds M of dimension at least 4 satisfy M < M ∞ Z [43]. Moreover, Proposition 2.10 and Corollary 6.3 show that Property (SV ∞ Z ) does not hold for the connected sum of oriented closed aspherical manifolds with zero simplicial volume and zero Euler characteristic (of even dimension at least 4). However, these manifolds are never aspherical (Remark 2.11).
In Question 6.4, one usually adds the hypothesis of residual finiteness to ensure the existence of "enough" finite coverings. However, there are also no known examples of oriented closed aspherical manifolds with nonresidually-finite fundamental group such that the vanishing behaviour of the ordinary simplicial volume is different from that of the stable integral simplicial volume. One possible strategy to produce such examples is to use Davis' reflection group trick (Section 5.4) to construct oriented closed aspherical manifolds whose fundamental group is not residually finite. However, as explained in Section 5.4, it seems to be difficult to gain enough control on the (stable integral) simplicial volume when performing this construction.
What about Question 6.4 for manifolds with boundary? Similarly to Property (SVχ, ∂), also in the case of the stable integral simplicial volume, we need to impose additional boundary conditions: Hence, Corollary 6.3 implies that M, ∂M ∞ Z = 0, even though π 1 (M ) is residually finite.
Remark 6.10. The arguments discussed in this section can be extended to principal ideal domains with norm bounded from below by 1. Interesting examples of such rings include, for example, the finite fields with the trivial norm [75]. In this setting, the proof of Proposition 6.7 also applies verbatim.
6.3. The dynamical version. The Poincaré duality arguments also apply to the dynamical version of the (integral) simplicial volume, i.e., to the integral foliated simplicial volume: Definition 6.11 (Integral foliated simplicial volume [54,98,40]). Let M be an oriented compact connected n-manifold with (possibly empty) boundary ∂M .
• If α = π 1 (M ) (X, µ) is a probability measure-preserving action on a standard Borel probability space, then we set Proof. This is a relative version of the L 2 -Betti number estimate which was shown in the closed case by Schmidt [98] based on ideas of Gromov [54, p. 306]. When phrasing the proof in terms of L 2 -Betti numbers of standard equivalence relations, literally the same proof as in the closed case [74,Theorem 6.4.5] can be applied to the twisted Poincaré-Lefschetz duality isomorphism.
Many positive examples for Question 1.1 have been established using the integral foliated simplicial volume (Section 1.3) and most of the known computations of the stable integral simplicial volume are based on ergodic theoretic methods and the fact that holds [41, Proposition 2.12] [78,46], where π 1 (M ) denotes the dynamical system given by the profinite completion of π 1 (M ).
A summary of computations of integral foliated simplicial volume and of these ergodic theoretic methods can be found in the literature [74,Chapter 6].