The spectrum of simplicial volume of non-compact manifolds

We show that, in dimension at least $4$, the set of locally finite simplicial volumes of oriented connected open manifolds is $[0, \infty]$. Moreover, we consider the case of tame open manifolds and some low-dimensional examples.


Introduction
Simplicial volumes are invariants of manifolds defined in terms of the ℓ 1 -seminorm on singular homology [Gro82]. where C lf * denotes the locally finite singular chain complex. If M is compact, then we also write M := M lf . Using relative fundamental cycles, the notion of simplicial volume can be extended to oriented manifolds with boundary.
Simplicial volumes are related to negative curvature, volume estimates, and amenability [Gro82]. In the present article, we focus on simplicial volumes of non-compact manifolds. Only few concrete results are known in this context: There are computations for certain locally symmetric spaces [LS09a,LS09b,BKK14,KK15] as well as the general volume estimates [Gro82], vanishing results [Gro82,FM18], and finiteness results [Gro82,Löh08]. It is known that SV(d) is countable and that this set has no gap at 0 if d ≥ 4: Theorem 1.2 ([HL20, Theorem A]). Let d ∈ N ≥4 . Then SV(d) is dense in R ≥0 and 0 ∈ SV(d).
In contrast, if we allow non-compact manifolds, we can realise all nonnegative real numbers: The proof uses the no-gap theorem Theorem 1.2 and a suitable connected sum construction.
If we restrict to tame manifolds, then we are in a similar situation as in the closed case: As an explicit example, we compute SV lf (2) and SV lf tame (2) (Proposition 4.2) as well as SV lf tame (3) (Proposition 4.3). The case of non-tame 3-manifolds seems to be fairly tricky.
From a geometric point of view, the so-called Lipschitz simplicial volume is more suitable for Riemannian non-compact manifolds than the locally finite simplicial volume. It is therefore natural to ask the following:

Organisation of this article
Section 2 contains the proof of Theorem A. The proof of Theorem B is given in Section 3. The low-dimensional case is treated in Section 4.

Construction
We first describe the construction of a corresponding oriented connected open manifold M : For each n ∈ N, we choose an oriented closed connected dmanifold M n with M n = α n . Moreover, for n > 0, we set where B n,− = i n,− (D d ) and B n,+ = i n,+ (D d ) are two disjointly embedded closed d-balls in M n . Similarly, we set W 0 := M 0 \ B • 0,+ . Furthermore, we choose an orientation-reversing homeomorphism f n : S d−1 → S d−1 . We then consider the infinite "linear" connected sum manifold ( Figure 1) where ∼ is the equivalence relation generated by for all n ∈ N and all x ∈ S d−1 ⊂ D d ; we denote the induced inclusion W n → M by i n . By construction, M is connected and inherits an orientation from the M n .

Computation of the simplicial volume
We will now verify that M lf = α: Proof. The proof is a straightforward adaption of the chain-level proof of subadditivity of simplicial volume with respect to amenable glueings. In particular, we will use the uniform boundary condition [MM85] and the equivalence theorem [Gro82, BBF + 14]: UBC The chain complex C * (S d−1 ; R) satisfies (d − 1)-UBC, i.e., there is a constant K such that: EQT Let N be an oriented closed connected d-manifold, let B 1 , . . . , B k be disjointly embedded d-balls in N , and let W := N \(B • 1 ∪. . . , ∪B • 1 ). Moreover, let ǫ ∈ R >0 . Then where Z(W ; R) ⊂ C d (W ; R) denotes the set of all relative fundamental cycles of W .
We now use UBC to construct a locally finite fundamental cycle of M out of these relative cycles: For n ∈ N, the boundary parts C d−1 (i n ; R)(∂ d z n | Bn,+ )

Computation of the simplicial volume
A straightforward computation shows that is a locally finite d-cycle on M . Moreover, the local contribution on W 0 shows that c is a locally finite fundamental cycle of M . By construction, Thus, taking ǫ → 0, we obtain M lf ≤ α.
Proof. Without loss of generality we may assume that M lf is finite. Let c ∈ C lf d (M ; R) be a locally finite fundamental cycle of M with |c| 1 < ∞. For n ∈ N, we consider the subchain c n := c| W (n) of c, consisting of all simplices whose images touch W (n) := n k=0 i k (W k ) ⊂ M . Because c is locally finite, each c n is a finite singular chain and (|c n | 1 ) n∈N is a monotonically increasing sequence with limit |c| 1 .
Let ǫ ∈ R >0 . Then there is an n ∈ N >0 that satisfies |c − c n | 1 ≤ ǫ and α − n k=0 α k ≤ ǫ. Let be the map that collapses everything beyond stage n + 1 to a single point x. Then is a cycle on W ; because z and z have the same local contribution on W 0 , the cycle z is a fundamental cycle of the manifold As d > 2, the construction of our chains and additivity of simplicial volume under connected sums [Gro82, BBF + 14] show that Thus, taking ǫ → 0, we obtain |c| 1 ≥ α; hence, M lf ≥ α.
This completes the proof of Theorem A.
Remark 2.3 (adding geometric structures). In fact, this argument can also be performed smoothly: The constructions leading to Theorem 1.2 can be carried out in the smooth setting. Therefore, we can choose the (M n ) n∈N to be smooth and equip M with a corresponding smooth structure. Moreover, we can endow these smooth pieces with Riemannian metrics. Scaling these Riemannian metrics appropriately shows that we can turn M into a Riemannian manifold of finite volume.

Proof of Theorem B
In this section, we prove Theorem B, i.e., that the set of simplicial volumes of tame manifolds is countable.
Definition 3.1. A manifold M without boundary is tame if there exists a compact connected manifold W with boundary such that M is homeormorphic to W • := W \ ∂W .
As in the closed case, our proof is based on a counting argument: Proposition 3.2. There are only countably many proper homotopy types of tame manifolds.
As we could not find a proof of this statement in the literature, we will give a complete proof in Section 3.1 below. Theorem B is a direct consequence of Proposition 3.2: Proof of Theorem B. The simplicial volume · lf is invariant under proper homotopy equivalence (this can be shown as in the compact case). Therefore, the countability of SV lf (d) follows from the countability of the set of proper homotopy types of tame d-manifolds (Proposition 3.2).

Counting tame manifolds
It remains to prove Proposition 3.2. We use the following observations: Definition 3.4 (models of tame manifolds).
• A model of a tame manifold M is a finite CW-pair (X, A) (i.e., a finite CW-complex X with a finite subcomplex A) that is homotopy equivalent (as pairs of spaces) to (W, ∂W ), where W is a compact connected manifold with boundary whose interior is homeomorphic to M .
• Two models of tame manifolds are equivalent if they are homotopy equivalent as pairs of spaces. Proof. It should be noted that we work with topological manifolds; hence, we cannot argue directly via triangulations. Of course, the main ingredient is the fact that every compact manifold is homotopy equivalent to a finite complex [Sie68,KS69]. Hence, there exist finite CW-complexes A and Y with homotopy equivalences f : A → ∂W and g : Y → W . Let j := g • i • f , where i : ∂W ֒→ W is the inclusion and g is a homotopy inverse of g. By construction, the upper square in the diagram in Figure 2 is homotopy commutative.
As next step, we replace j : A → Y by a homotopic map j c : A → Y that is cellular (second square in Figure 2).
The mapping cylinder Z of j c has a finite CW-structure (as j c is cellular) and the canonical map p : Z → Y allows to factor j c into an inclusion J of a subcomplex and the homotopy equivalence p (third square in Figure 2).
We thus obtain a homotopy commutative square where the vertical arrows are homotopy equivalences, the upper horizontal arrow is the inclusion, and the lower horizontal arrow is the inclusion of a subcomplex. Using a homotopy between i • f and F • J and adding another cylinder to Z, we can replace Z by a finite CW-complex X (that still contains A as subcomplex) to obtain a strictly commutative diagram whose vertical arrows are homotopy equivalences and whose horizontal arrows are inclusions.  By the topological collar theorem [Bro62,Con71], we have homeomorphisms Moreover, the homotopy of pairs between (f • g, f ∂ • g ∂ ) and (id V , id ∂V ) glues into a proper homotopy between F • G and id M . In the same way, there is a proper homotopy between G • F and id N . Hence, the spaces M and N are properly homotopy equivalent.
Lemma 3.7 (countability of models). There exist only countably many equivalence classes of models.
Proof. There are only countably many homotopy types of finite CW-complexes (because every finite CW-complex is homotopy equivalent to a finite simplicial complex). Moreover, every finite CW-complex has only finitely many subcomplexes. Therefore, there are only countably many homotopy types (of pairs of spaces) of finite CW-pairs. Proof of Proposition 3.2. We only need to combine Lemma 3.5, Lemma 3.6, and Lemma 3.7.

Low dimensions 4.1 Dimension 2
We now compute the set of simplicial volumes of surfaces. We first consider the tame case: Example 4.1 (tame surfaces). Let W be an oriented compact connected surface with g ∈ N handles and b ∈ N boundary components. Then the proportionality principle for simplicial volume of hyperbolic manifolds [Gro82, p. 11] (a thorough exposition is given, for instance, by Fujiwara and Manning [FM11, Appendix A]) gives Proposition 4.2. We have SV lf (2) = 2 · N ∪ {∞} and SV lf tame (2) = 2 · N. Proof. We first prove 2 · N ⊂ SV lf tame (2) ⊂ SV lf (2) and ∞ ∈ SV lf (2), i.e., that all the given values may be realised: 1. There exists an N ∈ N such that for all n ∈ N ≥N the inclusion M n ֒→ M n+1 is a homotopy equivalence.
2. For each N ∈ N there exists an n ∈ N ≥N such that the inclusion M n ֒→ M n+1 is not a homotopy equivalence.
In the first case, the classification of compact surfaces with boundary shows that M is tame. Hence M lf ∈ 2 · N (Example 4.1).
In the second case, the manifold M is not tame (which can, e.g., be derived from the classification of compact surfaces with boundary). We show that M lf = ∞. To this end. we distinguish two cases: a. The sequence (h(M n )) n∈N is unbounded, where h( · ) denotes the number of handles of the surface.
b. The sequence (h(M n )) n∈N is bounded.
In the unbounded case, a collapsing argument (similar to the argument for T 2 # T 2 # . . . and Claim 2.2) shows that M lf = ∞.
We claim that also in the bounded case we have M lf = ∞: Shifting the sequence in such a way that all handles are collected in M 0 , we may assume without loss of generality that the sequence (h(M n )) n∈N is constant. Thus, for each n ∈ N, the surface M n+1 is obtained from M n by adding a finite disjoint union of disks and of spheres with finitely many (at least two) disks removed; we can reorganise this sequence in such a way that no disks are added. Hence, we may assume that M n is a retract of M n+1 for each n ∈ N. Furthermore, because we are in case 2, the classification of compact surfaces shows (with the help of Example 4.1) that lim n→∞ M n = ∞.
Let c ∈ C lf 2 (M ; R) be a locally finite fundamental cycle of M and let n ∈ N. Because c is locally finite, there is a k ∈ N such that c| Mn is supported on M n+k ; the restriction c| Mn consists of all summands of c whose supports intersect with M n . Because M n is a retract of M n+k , we obtain from c| Mn a relative fundamental cycle c n of M n by pushing the chain c| Mn to M n via a retraction M n+k → M n . Therefore, Taking n → ∞ shows that |c| 1 = ∞. Taking the infimum over all locally finite fundamental cycles c of M proves that M lf = ∞.

Dimension 3
The general case of non-compact 3-manifolds seems to be rather involved (as the structure of non-compact 3-manifolds can get fairly complicated). We can at least deal with the tame case: Proof. Clearly, SV(3) ⊂ SV lf tame (3) and ∞ ∈ SV lf tame (3) (Remark 3.3). Conversely, let W be an oriented compact connected 3-manifold and let M := W • . We distinguish the following cases: • If at least one of the boundary components of W has genus at least 2, then the finiteness criterion [Gro82, p. 17][Löh08, Theorem 6.4] shows that M lf = ∞.
• If the boundary of W consists only of spheres and tori, then we proceed as follows: In a first step, we fill in all spherical boundary components of W by 3-balls and thus obtain an oriented compact connected 3-manifold V all of whose boundary components are tori. In view of considerations on tame manifolds with amenable boundary [KK15] and glueing results for bounded cohomology [Gro82][BBF + 14], we obtain that By Kneser's prime decomposition theorem [AFW15, Theorem 1.2.1] and the additivity of (relative) simplicial volume with respect to connected sums [Gro82][BBF + 14] in dimension 3, we may assume that V is prime (i.e., admits no non-trivial decomposition as a connected sum). Moreover, because S 1 ×S 2 = 0, we may even assume that V is irreducible [AFW15, p. 3].
By geometrisation [AFW15, Theorem 1.7.6], then V admits a decomposition along finitely many incompressible tori into Seifert fibred manifolds (which have trivial simplicial volume [Thu97, Corollary 6.5.3]) and hyperbolic pieces V 1 , . . . , V k . As the tori are incompressible, we can now again apply additivity [Gro82][BBF + 14] to conclude that Let j ∈ {1, . . . , k}. Then the boundary components of V j are π 1 -injective tori (as the interior of V j admits a complete hyperbolic metric of finite volume) [BP92, Proposition D.3.18]. Let S be a Seifert 3-manifold whose boundary is a π 1 -injective torus (e.g., the knot complement of a nontrivial torus knot [Mos71, Theorem 2][Lüc02, Lemma 4.4]). Filling each boundary component of V j with a copy of S results in an oriented closed connected 3-manifold N j , which satisfies (again, by additivity) Therefore, the oriented closed connected 3-manifold N := N 1 # · · · # N k satisfies In particular, M lf = V = N ∈ SV(3).