j. differential geometry 58 (2001) 535-541

RELATIVE HYPERBOLIZATION AND ASPHERICAL BORDISMS: AN ADDENDUM TO “HYPERBOLIZATION OF POLYHEDRA”

MICHAEL W. DAVIS, TADEUSZ JANUSZKIEWICZ & SHMUEL WEINBERGER

Abstract

We give two versions of relative hyperbolization. We use the first version to prove that if (each component of) a closed manifold M is aspherical and if M is a boundary, then it is the boundary of an aspherical manifold.

1. Introduction

In [2, p. 116], Gromov introduced the notion of hyperbolization: It is a procedure for associating to a finite dimensional simplicial complex X a certain nonpositively curved polyhedron H(X). A few pages later [2, pp. 117–118], he discusses the idea of relative hyperbolization: given a subcomplex Y of X, it should produce a new space H(X, Y) which contains Y as a subspace. One of the key properties of such a procedure should be the following:

(∗) If (each component of) Y is aspherical, then so is the relative hyperbolization H(X, Y).

The first author was partially supported by NSF grant DMS 0104026.

The second author was partially supported by KBN grant 5P03A 035 20.

The third author’s work was partially supported by NSF grants DMS 9803633

and DMS 0073812.

Received October 30, 2001.

Gromov points out that it follows from the existence of such a relative hyperbolization procedure that:

• Any (triangulable) closed manifold M is bordant to an aspherical manifold.

• If a closed aspherical manifold M bounds a (triangulable) manifold, then it bounds an aspherical manifold.

The proof of the second claim uses property (∗), but the proof of the first does not. Unfortunately, the details of Gromov’s definition of a relative version of hyperbolization did not quite make sense. In [1, Section 1g], the first two authors described a different version of relative hyperbolization (here denoted by K(X, Y)) and used it to demonstrate Gromov’s first claim, cf. [1, Example 1g.1]. However, they did not know how to prove that their version satisfied property (∗). In fact, it does (as does the simpler version of relative hyperbolization, J(X, Y), defined in Section 2). Our purpose here is to prove that both these relative hyperbolization procedures satisfy (∗) (Theorems 2.5 and 3.2) and to prove Gromov’s second claim, which is stated as the following theorem (and is proved in Section 2).

Theorem 1.1. Suppose that each component of a closed manifold M is aspherical and that M is the boundary of a (triangulable) manifold. Then M bounds an aspherical manifold.

Gromov defined several hyperbolization procedures in [2]. The specific one which we want to relativize is discussed in [1, Section 4c]. It works as follows. Given a finite dimensional simplicial complex X, there is a new polyhedron H(X), called a hyperbolization of X, together with a map c : H(X) → X. Some important properties of the construction are listed below. (Proofs of these properties can be found in [1].)

(1) H(X) is a nonpositively curved cubical cell complex (and hence, is aspherical).

(2) The construction is functorial in the sense that if i : Y → X is a simplicial embedding, then there is an induced isometric embedding H(i) : H(Y) → H(X).

(3) The link of a vertex in H(X) is isomorphic to a subdivision of the link of the corresponding vertex in X.

(4) The map c : H(X) → X induces surjections on integral homology groups and on fundamental groups.

(5) If X is an n-manifold, then so is H(X). If X is a smooth triangulation of a smooth manifold, then H(X) is a smooth manifold. Moreover, c : H(X) → X pulls back the stable tangent bundle of X to that of H(X).

2. Relative hyperbolization

Suppose Y is a subcomplex of X and that {Y_{i}} is the set of path components of Y. Let X ∪ CY denote the simplicial complex formed by attaching to X the cone on each Y_{i}. Let y_{i} denote the cone point corresponding to Y_{i} in the hyperbolization H(X ∪ CY) of X ∪ CY and let L_{i} denote the link of y_{i} in H(X ∪ CY). Then L_{i} is identified with a subdivision of Y_{i}. The relative hyperbolization of X with respect to Y is defined to be the space J(X, Y) formed by removing a small open conical neighborhood of each y_{i} from H(X ∪ CY). Since the boundary of such a neighborhood is L_{i} (= Y_{i}), Y is identified with a subspace of J(X, Y).

Remark 2.1. If X is a manifold with boundary and Y is a union of boundary components, then J(X, Y) is also a manifold with boundary and Y is identified with a union of its boundary components. This gives the proof of Gromov’s first claim: for any closed manifold M, J(M × [0, 1], M × 1) is a bordism between M and H(M).

Let H(X ∪ CY) denote the universal cover of H(X ∪ CY) and let J(X, Y) denote the inverse image of J(X, Y) in H(X ∪ CY).

Lemma 2.2. Let L_{i} be the link of any cone point y_{i} in H(X ∪CY). Then J(X, Y) retracts onto L_{i}. Hence, π_{1}(L_{i}) → π_{1}(J(X, Y)) is an injection.

Proof. Since H(X ∪ CY) is CAT(0), geodesic contraction provides a deformation retraction of H(X ∪ CY) \ y_{i} onto L_{i}. The restriction of this to J(X, Y) gives the desired retraction. q.e.d.

Corollary 2.3. For each Y_{i}, π_{1}(Y_{i}) → π_{1}(J(X, Y)) is injective.

Remark 2.4. Lemma 2.2 provides a proof of the following theorem of Hausmann [3]. Suppose that a (not necessarily connected) closed manifold M is a boundary. Then M bounds a manifold N such that for each path component M_{i} of M, the homomorphism π_{1}(M_{i}) → π_{1}(N) is injective. Moreover, M_{i} → N is a “pseudo covering projection” in the sense that each M_{i} is a retract of some covering space of N.

Theorem 2.5. J(X, Y) is aspherical if and only if each component of Y is aspherical.

In order to prove this, we need to introduce a space H̃(X ∪ CY), the “universal branched cover of H(X ∪ CY) along the cone points.” Let S denote the union of the set of cone points in H(X ∪ CY). Then H(X ∪ CY) \ S is connected. Let Z be its universal cover. Define H̃(X ∪ CY) to be the metric completion of Z. It is clear that H̃(X ∪ CY) is formed by adjoining to Z a new cone point for each end of Z which corresponds to a copy of the inverse image of a L_{i} in Z. Thus, H̃(X ∪ CY) is homeomorphic to the universal cover of J(X, Y) with each copy of the universal cover of L_{i} coned off. In other words, the universal cover J̃(X, Y) of J(X, Y) can be idenitified with inverse image of J(X, Y) in H̃(X ∪ CY).

Lemma 2.6. H̃(X ∪ CY) is CAT(0).

Proof. Since H(X ∪ CY) is a piecewise Euclidean cubical cell complex, this same type of structure is induced on H̃(X ∪ CY). Moreover, H̃(X ∪CY) is simply connected. So, it suffices to show that H̃(X ∪CY) is locally CAT(0). This is clear except possibly in neighborhoods of the cone points. Here we need to show that the link of each cone point in H̃(X ∪ CY) is CAT(1) (cf. [2, p. 120]). The link of such a cone point is the universal cover of the link of its image in H(X ∪ CY). Since H(X ∪ CY) is CAT(0), the link of each of its cone points is CAT(1). Since any covering space of a CAT(1) piecewise spherical complex is also CAT(1), the cone points in H̃(X ∪ CY) have CAT(1) links. The lemma follows. q.e.d.

Proof of Theorem 2.5. The “only if” part of this theorem follows immediately from Lemma 2.2. So, suppose each Y_{i} is aspherical. The link L̃_{i} of a cone point in H̃(X ∪CY) is the universal cover of Y_{i}; hence, it is contractible. By Lemma 2.6, H̃(X ∪ CY) is contractible. Since H̃(X ∪ CY) is formed from J̃(X, Y) by attaching cones on the L̃_{i}, it follows that J̃(X, Y) is also contractible. Hence, J(X, Y) is aspherical (since J̃(X, Y) is a covering space of it). q.e.d.

We are now in position to prove Theorem 1.1 from the Introduction.

Proof of Theorem 1.1. Suppose M = ∂N. As in Remark 2.1, M is also the boundary of the manifold J(N, M). By Theorem 2.5, J(N, M) is aspherical. q.e.d.

Remark 2.7. Theorem 1.1 is valid for any bordism theory.

3. Another version

When (X, Y) is a manifold with boundary, the construction of the relative hyperbolization J(X, Y) is perfectly adequate. However, in more general situations it has a serious defect: it changes the local topology near Y. A regular neighborhood of Y in J(X, Y) is homeomorhic to Y ×[0, 1]. It would be preferable for this to be homeomorphic to the original regular neighborhood of Y in X. This can be acheived by the procedure of [1]. The details are explained below.

Replace X by its barycentric subdivision. Let R_{i} denote the first derived neighborhood of Y_{i} in X, let R_{i}^{∘} be its relative interior and let ∂R_{i} = R_{i} \ R_{i}^{∘}. Also, let R, R^{∘} and ∂R denote the union of the R_{i}, the R_{i}^{∘} and the ∂R_{i}, respectively. Set X̂ = X \ R^{∘}. Apply the construction of the previous section to the pair (X̂, ∂R) to obtain J(X̂, ∂R). Our second version of relative hyperbolization, is the space K(X, Y) formed by gluing each R_{i} back onto J(X̂, ∂R) along ∂R_{i}. Next, we want to establish that Lemma 2.2 and Theorem 2.5 hold for K(X, Y).

For the analog of Lemma 2.2 we need to define a covering space K(X, Y) of K(X, Y) which retracts onto each R_{i}. If ∂R_{i} is connected, then K(X, Y) is defined to be H(X̂ ∪ C(∂R)) with a neighborhood of each cone point removed and replaced by a copy of the appropriate R_{i}. If the ∂R_{i} are not connected, then the definition of H(X̂ ∪C(∂R)) needs to be modified. For each path component Y_{i}, define a graph Ω_{i}: it is the suspension of π_{0}(∂R_{i}). Denote the suspension points by v_{i} and x_{i}. Let Ω be the wedge of the Ω_{i} (i.e., identify the x_{i} to a common point x). There is a continuous map K(X, Y) → Ω which collapses J(X̂, ∂R) to x, collapses Y_{i} to v_{i} and which takes each component of ∂R_{i} to the midpoint of the corresponding edge of Ω_{i}. A map H(X̂ ∪ C(∂R)) → Ω is defined in a similar fashion. Define a graph of groups on Ω by putting the group π_{1}(H(X̂ ∪ C(∂R)) on the vertex x, the trivial group on each of the other vertices and the trivial group on each edge. Let T be the universal cover of this graph of groups. (T is a tree.) The space H(X̂ ∪C(∂R)) is defined by gluing together copies of the universal cover of H(X̂ ∪ C(∂R)) in a pattern given by T. There is one such copy for each vertex lying above x. Two copies are glued together at a common cone point whenever the corresponding vertices of T are each connected by an edge to a vertex lying over some v_{i}. So, the link of a cone point in H(X̂∪C(∂R)) is isomorphic to some ∂R_{i} (which need not be connected). This version of H(X̂ ∪ C(∂R)) is clearly simply connected and CAT(0). Using the tree T, a covering space K(X, Y) of K(X, Y) is defined in a similar fashion. Alternatively, K(X, Y) is formed from H(X̂ ∪ C(∂R)) by removing a neighborhood of each cone point and replacing it with a copy of the appropriate R_{i}.

Lemma 3.1. K(X, Y) retracts onto R_{i}.

Proof. Fix a cone point y_{i} in H(X̂ ∪ C(∂R)) and identify ∂R_{i} with the link of y_{i}. Let J(X̂, ∂R) denote the inverse image of J(X̂, ∂R) in H(X̂ ∪ C(∂R)). As in the proof of Lemma 2.2, geodesic contraction from H(X̂ ∪C(∂R)) onto y_{i}, induces a retraction of J(X̂, ∂R) onto ∂R_{i}. Under this retraction each of the other boundary components is taken to ∂R_{i} by a map which is null-homotopic. Hence, we can extend it to a retraction K(X, Y) → R_{i} by mapping the copy of R_{i} corresponding to y_{i} via the identity map and all other R_{j} inessentially. q.e.d.

For the analog of Theorem 2.5, we want to relate the universal covering space K̃(X, Y) of K(X, Y) to a branched covering space H̃(X̂ ∪ C(∂R)) of H(X̂ ∪ C(∂R)). To this end, we define a new graph of group structure on Ω. The vertex group corresponding to x is π_{1}(J(X̂, ∂R)), the vertex group corresponding to v_{i} is π_{1}(R_{i}) and the edge group corresponding to an edge e of Ω_{i} is the image of π_{1}(∂R_{i,e}) in π_{1}(R_{i}), where ∂R_{i,e} denotes the component of ∂R_{i} corresponding to e. The inclusions of edge groups in vertex groups are the obvious ones. (By the previous lemma, the map from an edge group to the vertex group for x is an inclusion.) Let T̃ be the tree corresponding to this graph of groups. Let H̃(X̂ ∪ C(∂R)) be the branched covering space of H(X̂ ∪ C(∂R)) corresponding to T̃ and let K̃(X, Y) be the covering space K(X, Y) corresponding to T̃. Then H̃(X ∪ CY) and K̃(X, Y) are simply connected. Moreover, K̃(X, Y) can be constructed from H̃(X̂ ∪ C(∂R)) by removing a neighborhood of each cone point and replacing it with a copy of the universal cover R̃_{i} of the appropriate R_{i}.

Theorem 3.2. K(X, Y) is aspherical if and only if each component of Y is aspherical.

Proof. As before, the “only if” part follows from Lemma 3.1. As in the proof of Theorem 2.5, H̃(X̂ ∪ C(∂R)) is simply connected and locally CAT(0). Hence, it is contractible. Supposing each Y_{i} to be aspherical, we have that each R̃_{i} is contractible. Since K̃(X, Y) is formed from H̃(X̂ ∪ C(∂R)) by replacing (contractible) neighborhoods of cone points by (contractible) copies of R̃_{i}, K̃(X, Y) and H̃(X̂ ∪ C(∂R)) are homotopy equivalent. So, K̃(X, Y) is contractible and hence, K(X, Y) is aspherical. q.e.d.

References

[1] M.W. Davis & T. Januszkiewicz, Hyperbolization of polyhedra J. Differential Geometry 34 (1991) 347–388.

[2] M. Gromov, Hyperbolic groups, Essays in Group Theory (ed. S.M. Gersten) M.S.R.I. Publ., 8, Springer–Verlag, Berlin, Heidelberg and New York (1987) 75–319.

[3] J-C. Hausmann, On the homotopy of non-nilpotent spaces, Math. Z. 178 (1981) 115–123.

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