THE EULER CHARACTERISTIC OF A NONPOSITIVELY

CURVED, PIECEWISE EUCLIDEAN MANIFOLD

Ruth Charney and Michael Davis

0. Introduction.

A conjecture of H. Hopf states that if M^{2n} is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic, χ(M^{2n}), should satisfy (−1)^{n}χ(M^{2n}) ≥ 0. In this paper, we investigate the conjecture in the context of piecewise Euclidean manifolds having “nonpositive curvature” in the sense of Gromov’s CAT(0) inequality. In this context, the conjecture can be reduced to a local version which predicts the sign of a “local Euler characteristic” at each vertex. In the case of a manifold with cubical cell structure, the local version is a purely combinatorial statement which can be shown to hold under appropriate conditions.

The original conjecture of Hopf, and a similar conjecture for nonnegative curvature (which we shall not be concerned with here), are true in dimensions 2 and 4, by the Chern-Gauss-Bonnet Theorem: in both cases the curvature condition forces the Gauss-Bonnet integrand to have the correct sign. This is immediate in dimension 2. Chern [Ch] gives a proof in dimension 4 and attributes the result to Milnor. A result of [Ge] shows that, in dimensions ≥ 6, the curvature condition does not force the Gauss-Bonnet integrand to have the correct sign; hence, the same argument does not work in higher dimensions. (However, the hypothesis that the curvature operator is negative semidefinite does force the integrand to have the correct sign).

Here we are concerned with the analogous conjecture for piecewise Euclidean manifolds. In order to make sense of this, two points bear further discussion: (1) the meaning of “nonpositive curvature” for piecewise Euclidean spaces, and (2) the combinatorial analogue of the Gauss-Bonnet Theorem for such spaces.

As for the first point, “nonpositive curvature” makes sense for a more general class of metric spaces than Riemannian manifolds, namely, it makes sense for “geodesic spaces” (also called “length spaces”). For geodesic spaces the notion of nonpositive curvature is defined via Gromov’s “CAT(0) inequality” (see [G: p.106]). A piecewise Euclidean cell

Typeset by AMS-TEX

Both authors were partially supported by NSF Grant DMS 9208071 and the Institute for Advanced Study

complex naturally has the structure of a geodesic space (cf. (1.3)) so nonpositive curvature makes sense in this context. Associated to a vertex v of such a cell complex, there is a piecewise spherical complex called the “link of v”. It is homeomorphic to the topological link; its metric is determined by the solid angles at v in the cells which contain v. There is an infinitesimal version of nonpositive curvature, due to Gromov, which is equivalent to the previous version: the link of each vertex must be “large”. (By definition, a piecewise spherical space is large if there is a unique geodesic between any two points of distance less than π).

In dimension 2, in the case of a piecewise Euclidean surface M^{2}, these ideas are quite well-known. For a vertex v of M^{2}, let θ(v) denote the sum, over all 2-cells containing v, of the interior angles at v. The link of v is then a circle of length θ(v); it is large if and only if θ(v) ≥ 2π.

As for the second point, there is a well-known analogue of the Gauss-Bonnet formula for piecewise Euclidean cell complexes. It has the form χ(X) = ∑ κ(v), where the summation is over the vertices of X and where κ(v) depends only on the link of v. (The precise definition of κ(v) is given in (3.4.3)). It is shown, rather convincingly, in [CMS] that κ(v) is the correct analogue of the Gauss-Bonnet integrand in this context.

In contrast with the smooth case, we conjecture that, in all dimensions, the infinitesimal version of nonpositive curvature should force κ(v) to have the correct sign.

For a vertex v of a piecewise Euclidean surface, κ(v) = 1 − (2π)^{−1}θ(v). Hence, in dimension 2, our conjecture is true: θ(v) ≥ 2π implies κ(v) ≤ 0.

In general, it is difficult to check that a piecewise spherical space L is large. However, if L is a simplicial complex and if each edge of L has length π/2, then Gromov has given a simple combinatorial condition which is necessary and sufficient for L to be large: it must be a “flag complex”. (The definition is given in (2.7)). Such L (i.e., simplicial complexes with edge lengths π/2) arise as links in any piecewise Euclidean cell complex whose cells are regular cubes. Furthermore, for such L, the quantity κ(L) is easily seen to be given by the formula:

κ(L) = 1 + ∑(−\frac{1}{2})^{i+1}f_{i},

where f_{i} is the number of i-simplices in L. Hence, the local form of the Hopf Conjecture for piecewise Euclidean cubical manifolds is the following, purely combinatorial conjecture, called Conjecture D in Section 4. Conjecture D. Suppose that L is a simplicial complex homeomorphic to S^{2n−1} and that κ(L) is defined by the formula above. If L is a flag complex, then (−1)^{n}κ(L) ≥ 0.

This conjecture is first interesting in dimension 3. For any triangulation L^{3} of S^{3}, the Lower Bound Theorem of [W], states that f_{1} ≥ 4f_{0} − 10. In this dimension, Conjecture D is equivalent to the statement that if, in addition, L^{3} is a flag complex, then f_{1} ≥ 5f_{0} − 16.

Given an arbitrary simplicial complex L, we construct, in Section 6, a finite, piecewise Euclidean, cubical complex so that the link of each vertex is L. If L is a flag complex, then the cubical complex is nonpositively curved. If L is homeomorphic to a sphere, the cubical complex is a manifold. Hence, Conjecture D is equivalent to the Hopf Conjecture for piecewise Euclidean manifolds cellulated by cubes (cf. Proposition 6.5).

There is a beautiful generalization, due to G. Moussong [M], of Gromov’s combinatorial condition for largeness of piecewise spherical simplicial complexes with edges of length π/2. Moussong gives an analogous condition in the case where the edge lengths are ≥ π/2 which is partly combinatorial and partly metric (see (2.9), (2.10)). This leads to a conjecture for such L, analogous to Conjecture D, called Conjecture C in Section 4.

Section 5 provides some evidence for Conjectures C and D. First, we show that Conjecture D holds for some wide classes of flag complexes which arise as subdivisions. Then we show in Proposition (5.7) that Conjecture D implies Conjecture C (in the case where the underlying simplicial complex is a flag complex). The proof uses a formula of [CMS] for the first derivative of κ(t), where κ(t) = κ(L_{t}) for some 1-parameter family L_{t} of piecewise spherical simplicial complexes.

Finally, in Section 7, we discuss an equivalent form of Conjecture D, of interest to combinatorialists.

1. Piecewise Euclidean and piecewise spherical cell complexes.

(1.1) A Euclidean cell is a convex polytope in some Euclidean space. A convex polyhedral cone in ℝ^{n} (with vertex at the origin) is any intersection of a finite number of linear halfspaces which contains no line. The spherical cell associated to a convex polyhedral cone is its intersection with S^{n−1}. A convex polyhedral cone is a simplicial cone if this intersection is a spherical simplex.

(1.1.1) If P is a Euclidean cell and F is a face of P, then the normal cone of F in P is the convex polyhedral cone N(F, P) consisting of all inward pointing normal vectors to F. Its dimension is the codimension of F in P. The associated spherical cell is called the link of F in P and denoted Lk(F, P). Thus, Lk(F, P) is the “solid angle” of P along F. Similarly, if F is a face in a spherical cell P, then Lk(F, P) is defined to be the link of the associated polyhedral cones.

(1.1.2) A piecewise Euclidean cell complex is a space X formed by gluing together Euclidean cells via isometries of their faces together with the decomposition of X into cells. A piecewise spherical cell complex is defined similarly using spherical cells. We assume throughout that all cell complexes are locally finite. We also assume that there is a positive lower bound on the heights of all cells of X. (The height of a cell C is the minimum distance between disjoint faces of C.)

(1.1.3) If F_{1} and F_{2} are faces of a Euclidean (resp. spherical) cell P and F_{1} ⊂ F_{2}, then Lk(F_{1}, F_{2}) is canonically identified with a face of Lk(F_{1}, P). It follows that if F is a cell in a piecewise Euclidean (resp. spherical) complex X, then the set of all Lk(F, P), P a cell of X which contains F, fit together to give a piecewise spherical cell complex

Lk(F, X) = ⋃_{F ⊂P} Lk(F, P)

called the link of F in X.

(1.1.4) An n-cell P is simple if precisely n codimension-one faces of P meet at each vertex. Equivalently, P is simple if Lk(F, P) is a simplex for each proper face F of P. For example, a simplex is simple, so is a cube; an octahedron is not. If each cell of a piecewise Euclidean cell complex X is simple, then Lk(F, X) is a simplicial cell complex for each cell F in X.

(1.1.5) A note on terminology. “Simplicial complex” is not synonymous with “simplicial cell complex”. In a simplicial cell complex the intersection of two simplices is a union of faces, while in a simplicial complex such an intersection is either empty or a single simplex.

(1.2) A complete metric space Y is a geodesic space (or “length space”) if, given points y_{1}, y_{2} in Y, there is a path γ from y_{1} to y_{2} with ℓ(γ) = d(y_{1}, y_{2}). Such a distance minimizing path is called a geodesic.

(1.3) Suppose X is a piecewise Euclidean or piecewise spherical cell complex. Then “arc length” makes sense in each cell; hence, the length ℓ of a path in X is well-defined. Given points x and y in X, define d(x, y) = inf {ℓ(γ)}, where the infimum is over all paths γ from x to y of finite length. (If x and y are in distinct path components, put d(x, y) = ∞). Under the assumptions of (1.1.2), Moussong ([M]) showed that d is a complete, geodesic metric on X. It is locally isometric with the given Euclidean or spherical metrics on the individual cells. A piecewise Euclidean (resp. piecewise spherical) space is the metric space underlying such a piecewise Euclidean (resp. piecewise spherical) cell complex.

2. The infinitesimal version of nonpositive curvature.

(2.1) A geodesic space is nonpositively curved, abbreviated (NP), if, locally, geodesic triangles satisfy the CAT(0)-inequality (see [G, p. 106]). The (NP) condition has some strong consequences. For example, if a space is (NP), then it is locally contractible and its universal covering space is contractible.

(2.1.1) A piecewise spherical space is a large if there is a unique geodesic between any two points of distance < π.

The following result is due to Gromov. For a proof, see [B] or [Br]. (See also Theorem 3.1 and the appendix of [CD1].)

(2.2) Theorem (Gromov). A piecewise Euclidean cell complex is (NP) if and only if the link of each vertex is large.

In some cases, when the cell structure on a piecewise spherical space is simplicial, there are combinatorial conditions which are necessary and sufficient for it to be large. In order to state these conditions, it is first necessary to develop some terminology and elementary facts about spherical simplices.

(2.3) Let σ be a spherical n-simplex in S^{n} with vertex set V, a set of linearly independent unit vectors in ℝ^{n+1}. The associated cosine matrix, c(σ) = (c_{vv′}), v, v′ in V is the symmetric V by V matrix of inner products: c_{vv′} = v · v′. We note that c(σ) is positive definite.

(2.3.1) For v ≠ v′, let ℓ_{vv′} be the length of the edge (v, v′) of σ. Put ℓ_{vv} = 0. Then c_{vv′} = cos(ℓ_{vv′}).

(2.3.2) Conversely, suppose we are given a V by V symmetric matrix (ℓ_{vv′}) satisfying: ℓ_{vv} = 0 and ℓ_{vv′} ∈ (0, π) for v ≠ v′. If the associated cosine matrix c, defined by c_{vv′} = cos(ℓ_{vv′}), is positive definite, then we can find a basis V for ℝ^{n+1}, unique up to isometry, which spans a spherical n-simplex with associated cosine matrix c. Hence, (ℓ_{vv′}) is the matrix of edge lengths of a spherical simplex if and only if the associated cosine matrix is positive definite.

(2.4) A spherical simplex is all right if each of its edges has length π/2; it has size ≥ π/2 if each of its edges has length ≥ π/2. The associated cosine matrix of an all right n-simplex σ is the identity matrix. It follows that σ is isometric to the regular n-simplex in S^{n} spanned by the standard basis of ℝ^{n+1}. The associated cosine matrix of a simplex of size ≥ π/2 is “almost negative” in the sense that its off-diagonal entries are ≤ 0.

The following is a result of [M].

(2.4.1) Lemma. Let τ be a face of a spherical simplex σ.

(i) If σ is all right, then so is Lk(τ, σ).

(ii) If σ has size ≥ π/2, then so has Lk(τ, σ).

Proof. Statement (i) is obvious. To prove (ii) first note that if τ_{1} ⊂ τ_{2} ⊂ σ, the Lk(Lk(τ_{1}, τ_{2}), Lk(τ_{1}, σ)) is isometric to Lk(τ_{2}, σ). Hence, by induction, we can reduce to the case dim τ = 0. So, suppose σ is spanned by {v_{0}, …, v_{n}} and τ is the vertex v_{0}. Let p : ℝ^{n+1} → (ℝv_{0})^{⊥} be orthogonal projection. Then Lk(v_{0}, σ) can be identified with the spherical simplex in (ℝv_{0})^{⊥} spanned by u_{i} = p(v_{i})/|p(v_{0})|, i = 1, …, n. For i ≠ j,

p(v_{i}) · p(v_{j}) = (v_{i} − (v_{i} · v_{0})v_{0}) · (v_{j} − (v_{j} · v_{0})v_{0})
= v_{i} · v_{j} − (v_{i} · v_{0})(v_{j} · v_{0})

which is ≤ 0 since the cosine matrix for σ is almost negative. It follows that u_{i} · u_{j} ≤ 0 so the cosine matrix for Lk(v_{0}, σ) is almost negative.

(2.5) A piecewise spherical simplicial cell complex is all right (resp. has simplices of size ≥ π/2) if the corresponding property holds for each of its simplices.

(2.5.1) If L is all right or if it has simplices of size ≥ π/2, then, by Lemma (2.4.1), the link of any simplex in L has the same property.

(2.6) A piecewise Euclidean cell complex has nonacute, simple cells if each of its cells is simple (cf. (1.1.4)) and the face angles in each cell are all ≥ π/2; or equivalently, if the link of every cell is simplicial with simplicies of size ≥ π/2. It is cubical if each cell is a regular Euclidean cube. In this case, links are all right simplicial cell complexes.

(2.7) Let K be a simplicial complex (no metric structure is assumed). A set V of vertices in K spans a complete graph if any two distinct elements of V span an edge in K. A simplicial cell complex K is a flag complex if it is a (genuine) simplicial complex (cf. 1.1.5) and if any set of vertices which spans a complete graph actually spans a simplex. Thus, a flag complex is a simplicial complex with no “empty simplices”.

(2.7.1) Flag complexes have nice properties. For example,

(a) the link of any simplex in a flag complex is again a flag complex, and

(b) the join of two flag complexes is a flag complex.

The following lemma is proved in [G, p. 122].

(2.8) Gromov’s Lemma. An all right piecewise spherical simplicial cell complex is large if and only if it is a flag complex.

(2.8.1) Hence, a piecewise Euclidean, cubical cell complex is (NP) if and only if the link of each vertex is a flag complex.

(2.9) Suppose L is a piecewise spherical simplicial complex and that V is a set of vertices of L which span a complete graph. As in (2.3), there is associated to V a cosine matrix, (c_{vv′}), where c_{vv′} is the cosine of the length of the edge (v, v′). A cell complex L is a metric flag complex if it is a simplicial complex and if whenever V is a set of vertices such that (i) V spans a complete graph, and (ii) the associated cosine matrix is positive definite, then V spans a simplex of L.

The following difficult generalization of Gromov’s Lemma is proved in [M].

(2.10) Moussong’s Lemma. A piecewise spherical simplicial cell complex with simplices of size ≥ π/2 is large if and only if it is a metric flag complex.

(2.10.1) So, a piecewise Euclidean cell complex with nonacute, simple cells is (NP) if and only if the link of each vertex is a metric flag complex.

3. The Gauss-Bonnet Theorem for piecewise Euclidean spaces (3.1) Let C be an n-dimensional convex polyhedral cone and σ the associated spherical

(n − 1)-cell. The outward pointing normals to the supporting hyperplanes of C generate another convex polyhedral cone C^{∗} called the dual cone. (If u_{1}, …, u_{k} are outward pointing normal vectors to the supporting hyperplanes, then C^{∗} is the set of all nonnegative linear combinations of the u_{i}). The associated spherical cell σ^{∗} = C^{∗} ∩ S^{n−1} is the dual cell to σ.

(3.2) The angle of C at 0, denoted by a(σ), is the (n − 1)-dimensional volume of σ, normalized so that the volume of S^{n−1} is 1, i.e.,

a(σ) = \frac{vol (σ)}{vol (S^{n−1})}.

The exterior angle of C at 0, denoted by a^{∗}(σ) is the angle of C^{∗}, i.e., a^{∗}(σ) = a(σ^{∗}).

(3.3) Let P be a Euclidean n-cell. After choosing an interior point of P as the origin, one can define the “dual cell” P^{∗} whose vertices are the normal vectors to the codimension-one faces of P. For example, the octahedron is dual to the cube. Radial projection of ∂P^{∗} onto S^{n−1} gives a cellulation of S^{n−1}. The (n−1)-cells in this cellulation are associated to the duals of the normal cones at the vertices of P, i.e., each such (n − 1)-cell is of the form Lk(v, P)^{∗} for some vertex v. It follows that ∑ a^{∗}(Lk(v, P)) = 1, where the summation is over all vertices v of P. (If v = P, then Lk(v, P) = ∅, so we adopt the convention a^{∗}(∅) = 1).

(3.4) Now suppose that X is a finite piecewise Euclidean cell complex. Then

χ(X) = ∑_{P}(−1)^{dim P}

= ∑_{P}(−1)^{dim P} ∑_{v} a^{∗}(Lk(v, P))

where P runs over the cells of X and v runs over the vertices of P. Reversing the order of summation gives

(3.4.1)  χ(X) = ∑_{v} ∑_{v∈P}(−1)^{dim P} a^{∗}(Link (v, P))

= ∑_{v}(1 + ∑_{σ}(−1)^{dim σ+1}a^{∗}(σ))

where v runs over the vertices of X and σ over the cells of Lk(v, X). (Note that the 1 in the summation arises from the case P = v). For any finite, piecewise spherical cell complex L, define

(3.4.2)  κ(L) = 1 + ∑_{σ}(−1)^{dim σ+1}a^{∗}(σ)

where σ runs over the cells of L. Then (3.4.1) can be rewritten as

(3.4.3)  χ(X) = ∑_{v} κ(v),

where κ(v) = κ(Lk(v, X)). This is the desired analogue of the Gauss-Bonnet Theorem for piecewise Euclidean spaces.

(3.5) If σ is an all right simplex, then σ^{∗} = σ and a^{∗}(σ) = a(σ) = (\frac{1}{2})^{dim σ+1}. Hence, if L is an all right, piecewise spherical, simplicial cell complex, then

(3.5.1)  κ(L) = 1 + ∑_{σ}(−\frac{1}{2})^{dim σ+1}

(3.5.2) Let K be a simplicial cell complex (without metric structure) and f_{i}(= f_{i}(K)) the number of i-simplices in K. Put

λ(K) = 1 + ∑_{i}(−\frac{1}{2})^{i+1}f_{i}.

If we endow K with the structure of an all right spherical complex by declaring each simplex to be all right, then λ(K) = κ(K).

(3.6) The following facts concerning κ are proved in [CMS, Section 3].

(3.6.1) If the underlying metric spaces of L_{1} and L_{2} are isometric, then κ(L_{1}) = κ(L_{2}). (3.6.2) If L is isometric to the round sphere, then κ(L) = 0. (3.6.3) Suppose σ^{n} ⊂ S^{n} and τ^{k} ⊂ S^{k} are spherical cells; embed S^{n} and S^{k} in orthogonal

linear subspaces of ℝ^{n+k+2}; then the orthogonal join σ ∗ τ is the spherical (n + k + 1)-cell spanned by σ and τ (it is the union of all geodesics in S^{n+k+1} from σ to τ). This extends to a definition of the orthogonal join L_{1} ∗ L_{2} of two piecewise spherical cell complexes L_{1} and L_{2} (see the Appendix of [CD]). By [CMS, p. 442],

κ(L_{1} ∗ L_{2}) = κ(L_{1})κ(L_{2}).

(3.6.4) If L is homeomorphic to an even dimensional sphere, then κ(L) = 0. (Actually for this to be true it is only necessary to assume that χ(L) = 2 and χ(Lk(σ, L)) = 2 for each odd dimensional cell σ).

(3.6.5) Let K be a simplicial cell complex and L_{t} a 1-parameter family of piecewise spherical structures on K. For each edge e in K, let a_{e}(t) be the length of the corresponding edge e_{t} in L_{t}, normalized so that S^{1} has length 1. (Thus, a_{e}(t) = (2π)^{−1}ℓ(e_{t}), where ℓ is the usual edge length). Put κ(t) = κ(L_{t}). The following beautiful formula for the first derivative of κ(t) is given in [CMS, p. 424]

(3.6.6)  κ′(t) = − ∑_{e} a′_{e}(t)κ(Lk(e_{t}, L_{t})).

The proofs of the above facts can be sketched as follows. In [C], Cheeger considers another quantity, let us call it \overline{κ}(L), defined as the difference of the L_{2}-Euler characteristic of the cone of radius 1 on L and \frac{1}{2} the L_{2}-Euler characteristic of L. So \overline{κ}(L) depends only on the metric and not the cell structure. Using heat equation methods he derives a formula for \overline{κ}(L) solely in terms of interior angles and flags of odd dimensional cells (formula 3.35 in [CMS]) and he proves that χ(X) = ∑ \overline{κ}(Lk(v, X)). In [CMS, p. 424] it is proved that \overline{κ}(L) = κ(L). Since \overline{κ}(L) depends only on the metric, (3.6.1) follows. The formulas in (3.6.2), (3.6.4) and (3.6.6) follow easily from the formula for \overline{κ}(L); (3.6.3) is an easy computation.

4. Statements of the conjectures We are now in position to state precisely the various conjectures which were mentioned

in the Introduction. First, there is the analogue of the Hopf Conjecture.

Conjecture A. If M^{2n} is a nonpositively curved, piecewise Euclidean, closed manifold, then (−1)^{n}χ(M^{2n}) ≥ 0.

Next, suppose that L^{2n−1} is a (2n − 1)-dimensional piecewise spherical cell complex which is a generalized homology sphere, in the sense that it has the homology of S^{2n−1} and for each k-cell σ, Lk(σ, L^{2n−1}) has the homology of S^{2n−k−2}. For example, if L^{2n−1} is homeomorphic to S^{2n−1}, then it is a generalized homology sphere. In view of Gromov’s Theorem (2.2), Conjecture A is implied by the following.

Conjecture B. If L^{2n−1} is large (cf. (2.1.1)), then (−1)^{n}κ(L^{2n−1}) ≥ 0, where κ is defined by (3.4.2).

By Moussong’s Lemma (2.10) the following conjecture is a special case of Conjecture B. Conjecture C. Suppose that L^{2n−1} is simplicial and has simplices of size ≥ π/2 (cf.

(2.5)). If L^{2n−1} is a metric flag complex (cf. (2.9)), then (−1)^{n}κ(L^{2n−1}) ≥ 0.

If L is a metric flag complex with simplices of size ≥ π/2, then the same is true for the link of each simplex in L, by Lemma (2.4.1). In the sequel we will want to consider the statement that Conjecture C holds for a particular complex L and for each link in L. It is therefore convenient to state the following equivalent formulation of Conjecture C.

Conjecture C′. If L^{2n−1} is a metric flag complex with simplices of size ≥ π/2, then (−1)^{n}κ(L^{2n−1}) ≥ 0 and for each simplex σ of codimension 2k, (−1)^{k}κ(Lk(σ, L)) ≥ 0.

Finally, we want to consider the special case where L^{2n−1} is all right. In order to deemphasize the metric in this case, we let K^{2n−1} be a simplicial cell complex which is a generalized homology sphere, as above, and λ(K) the quantity defined in (3.5.2).

Conjecture D. If K^{2n−1} is a flag complex, then (−1)^{n}λ(K^{2n−1}) ≥ 0.

We also have an equivalent reformulation.

Conjecture D′. If K^{2n−1} is a flag complex, then (−1)^{n}λ(K^{2n−1}) ≥ 0 and for each simplex σ of codimension 2k, (−1)^{k}λ(Lk(σ, K^{2n−1}) ≥ 0.

5. Evidence

We begin by discussing some partial results for Conjecture D.

(5.1) Let O^{m} be the (m + 1)-fold join of S^{0} with itself. By (2.7.1), O^{m} is a flag complex and by (3.6.3), λ(O^{m}) = 0. It is alternately described as the boundary complex of the (m+1)-dimensional octahedron (or “cross polytope”). It is the simplest flag complex which triangulates the m-sphere. If we endow it with an all right, piecewise spherical structure, then it is isometric to the round sphere. The link of a k-simplex in O^{m} is isomorphic to O^{m−k−1}. It follows that Conjecture D′ holds for O^{2n−1}.

(5.2) Flag complexes arise naturally as derived complexes of posets. (Recall that if A is a poset, then its derived complex A′ is the abstract simplicial complex with simplices the finite chains in A). If A is the poset of cells in a cell complex, then the geometric realization of A′ can be identified with the barycentric subdivision of the cell complex. In particular, if A is the poset of faces of the boundary complex of a 2n-cell, then A′ is a flag complex homeomorphic to S^{2n−1}. Based on work of R. Stanley, E. Babson showed that Conjecture D′ holds for such A′. (For the argument, see (7.3)). Hence, Conjecture D′ holds for the barycentric subdivision of the boundary complex of a Euclidean cell.

(5.3) Suppose K is a simplicial complex (no metric assumed). Recall that for any simplex σ of K, Lk(σ, K) can be identified with the subcomplex of K consisting of all closed simplices τ, disjoint from σ, such that the join of σ and τ is a simplex of K containing σ.

(5.3.1) Let e be an edge of K. We shall now define a subdivision of K, called the edge subdivision and denoted by Sub_{e}(K). Introduce a new vertex v_{e} as the midpoint of e, subdividing e into two new edges, say e_{+} and e_{−}. Let σ be a k-simplex containing e. Then σ = e ∗ τ for some simplex τ in Lk(e, K). Introduce a new (k − 1)-simplex σ_{0} = v_{e} ∗ τ, subdividing σ into two new k-simplices σ_{+} = e_{+} ∗ τ and σ_{−} = e_{−} ∗ τ. The simplices which do not contain e are not changed.

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τ eσ  τ v_{e} σ^{+} ∪ σ^{−}^{•}

(5.3.2) If K is a flag complex, then so is Sub_{e}(K). Of course, we could have subdivided K by introducing a barycenter of some simplex of dimension > 1 and then coning off; however, this does not preserve the property of being a flag complex.

(5.3.3) How are the links in Sub_{e}(K) related to links in K? Let σ be a simplex in Sub_{e}(K). There are four cases to consider:

(i) σ corresponds to a simplex of K which is not in the closed star of e, (ii) σ corresponds to a simplex in the link of e, (iii) there is a simplex σ̂ of K which contains e and σ = σ̂_{0} (or σ = v_{e}), (iv) there is a simplex σ̂ of K which contains e and σ = σ̂_{+} or σ = σ̂_{−}. Then

Lk(σ, Sub_{e}(K)) = \{ \begin{matrix}Lk(σ, K) & ; in case (i) \\ Sub_{e}(Lk(σ, K)) & ; in case (ii) \\ S^{0} ∗ Lk(σ̂, K) & ; in case (iii) \\ Lk(σ̂, K) & ; in case (iv) \end{matrix}

(5.3.4) For any set of simplices E, put

λ̃(E) = ∑_{σ∈E}(−\frac{1}{2} )^{dim σ+1}.

If σ is an open simplex and M a simplicial complex, then let E(σ, M) be the set of simplices in σ ∗ M which do not lie in M. Clearly,

λ̃(E(σ, M)) = (−\frac{1}{2} )^{dim σ+1}λ(M).

(5.3.5) Lemma: λ(Sub_{e}(K)) = λ(K) − \frac{1}{4}λ(Lk(e, K)).

Proof. To construct Sub_{e}(K) from K one removes the open star of e and replaces it by the open star of v_{e} in v_{e} ∗ S^{0} ∗ Lk(e, K). The contribution of the simplices in the open star of e to λ(K) is λ̃(E(e, LK(e, K)) = \frac{1}{4}λ(Lk(e, K)). The contribution of those in the open star of v_{e} is λ̃(E(v_{e}, S^{0} ∗ Lk(e, K)) = −\frac{1}{2}λ(S^{0} ∗ Lk(e, K)) = −\frac{1}{2}λ(S^{0})λ(Lk(e, K)) = 0. The formula follows.

(5.4) Now suppose K^{2n−1} is a flag complex and a generalized homology sphere. Let e be an edge of K.

(5.4.1) Proposition. If Conjecture D holds for K and for Lk(e, K), then it also holds for Sub_{e}(K).

Proof. If (−1)^{n}λ(K) ≥ 0 and (−1)^{n−1}λ(Lk(e, K)) ≥ 0, then (−1)^{n}λ(Sub_{e}(K)) = (−1)^{n}λ(K) + \frac{1}{4}(−1)^{n−1}λ(Lk(e, K)) ≥ 0.

Using this, (5.3.2) and (5.3.3), we immediately deduce the following.

(5.4.2) Proposition. If Conjecture D′ holds for K, then it also holds for any complex obtained from K by repeated edge subdivisions.

(5.4.3) Corollary. Conjecture D′ holds for any complex obtained from O^{2n−1} by repeated edge subdivisions.

(5.4.4) Proposition (5.4.2) also follows from formula (3.6.6). Let L_{0} be K with an all right piecewise spherical structure. Let L_{t} be the 1-parameter family where the edge corresponding to e has length (1+t)π/2 and all other edges have length π/2. By induction on dimension, (−1)^{n−1}κ(Lk(e_{t}, L_{t})) ≥ 0. Hence, by (3.6.6), (−1)^{n}κ(L_{t}) is a non-decreasing function of t. But L_{1} is just the all right structure on Sub_{e}(K). Proposition (5.4.2) follows.

(5.5) A k-circuit in K is a circuit of k edges. A 3-circuit is empty if it is not the boundary of a 2-simplex. Similarly a 4-circuit is empty if it is not the boundary of the union of two adjacent 2-simplices.

(5.5.1) The inverse operation of edge subdivision is “edge collapse.” Suppose e is an edge of K which is not part of an empty 3-circuit. (This is automatic if K is a flag complex). To collapse e, one first removes the open star of e. The resulting boundary is isomorphic to S^{0} ∗ Lk(e, K). One then collapses this to Lk(e, K). The result is a simplicial complex \overline{K}_{e} which is a quotient space of K. If K is PL-homeomorphic to S^{2n−1}, then so is \overline{K}_{e}. If K is a flag complex, then \overline{K}_{e} is also a flag complex if and only if e is not part of an empty 4-circuit. Hence, one can try to simplify K by performing edges collapses on edges which are not part of empty 4-circuits.

(5.5.2) At one point we thought it might be possible, given a flag complex K^{m}, PLhomeomorphic to S^{m}, to perform edge collapses to reduce it to O^{m}. This is true for m = 1, 2, however, there is a counterexample for m = 3. (Conjecture D holds for this counterexample).

We turn now to Conjecture C′.

(5.6) Suppose L_{0} and L_{1} are two piecewise spherical structures on a simplicial cell complex K. For a simplex σ in K, let σ_{i}, i = 0, 1, be the corresponding spherical simplex in L_{i}. Let c(σ_{i}), i = 0, 1, be the associated cosine matrix (cf. (2.3)).

(5.6.1) We shall now define a 1-parameter family L_{t} of piecewise spherical simplicial cell complexes L_{t} which interpolates between L_{0} and L_{1}. Put c_{t}(σ) = tc(σ_{1}) + (1 − t)c(σ_{0}). Since a convex combination of positive definite matrices is positive definite, c_{t}(σ) > 0. It follows that, for each t ∈ [0, 1], there is a corresponding spherical simplex σ_{t}. These fit together to give L_{t}. The family L_{t} is called the canonical deformation from L_{0} to L_{1}. If e is an edge of K, then let ℓ_{e}(t)(= ℓ(e_{t})) be the length of the corresponding edge in L_{t}. Then cos(ℓ_{e}(t)) = t cos(ℓ_{1}) + (1 − t) cos(ℓ_{0}), where ℓ_{i} = ℓ_{e}(i), i = 0, 1.

(5.6.2) In particular, suppose L_{0} is all right and L_{1} has simplices of size ≥ π/2. Then for each edge e, ℓ_{e}(0) = π/2 and ℓ_{e}(t) ∈ [π/2, π). So, ℓ_{e}(t) = cos^{−1}(ct), where c = cos(ℓ_{1}). Hence, ℓ′_{e}(t) = −c(1 − c^{2}t^{2})^{− \frac{1}{2}} , which is ≥ 0 (since c ≤ 0). Similarly, if a_{e}(t) = (2π)^{−1}ℓ_{e}(t) is the normalized length, then a′_{e}(t) ≥ 0.

(5.7) Proposition. Let L^{2n−1} be a piecewise spherical simplicial complex with simplices of size ≥ π/2 which is a generalized homology sphere, and let K^{2n−1} be the underlying simplicial complex. Suppose K^{2n−1} is a flag complex. If Conjecture D′ holds for K^{2n−1}, then Conjecture C′ holds for L^{2n−1}.

Proof. Let L_{0} be the all right structure on K, let L_{1} = L, and let L_{t} be the canonical deformation. We assume that Conjecture D′ holds for K, i.e., that Conjecture C′ holds for L_{0}. We may also assume, by induction on n, that Conjecture C holds for the link of each odd dimensional simplex in L_{t}. In particular, for each edge e_{t}, (−1)^{n−1}κ(Lk(e_{t}, L_{t})) ≥ 0. By (3.6.6),

(−1)^{n}κ′(t) = (−1)^{n−1} ∑ a′_{e}(t)κ(Lk(e_{t}, L_{t}))

which is ≥ 0 (using 5.6.2). Hence, (−1)^{n}κ(t) is nondecreasing. Thus, (−1)^{n}κ(0) ≥ 0 implies (−1)^{n}κ(1) ≥ 0.

(5.8) Remark. In a forthcoming paper [CD2]. we find further evidence for Conjecture B. Suppose that X^{2n} is a convex polytope of finite volume in hyperbolic 2n-space. Its “completed polar dual” L^{2n−1} is large. Moreover, (−1)^{n}κ(L^{2n−1}) is twice the volume of X^{2n} (suitably normalized). Hence, Conjecture B holds in this case.

6. Examples from Coxeter groups In this section we review the work of G. Moussong [M] (also, see [D1], [G, §4.6]). In

particular, we show how to associate to each Coxeter system (W, S) a piecewise spherical, simplicial complex Nerve (W, S) and a contractible, nonpositively curved, piecewise Euclidean cell complex Ỹ with W -action so that the link of each vertex of Ỹ is Nerve (W, S). Moreover, any finite, all right, flag complex can be realized as Nerve (W, S) for some (W, S).

(6.1) Suppose S is a finite set and m = (m_{ss′}) is a symmetric S by S matrix, with entries positive integers or ∞, satisfying m_{ss} = 1 and m_{ss′} ≥ 2, for s ≠ s′. This gives a presentation for a group W = ⟨S|(ss′)^{m_{ss′}} = 1⟩. The pair (W, S) is a Coxeter system and W is a Coxeter group. The rank of (W, S) is the cardinality of S. The group W is right-angled if m_{ss′} = 2 or ∞ for all s ≠ s′. Associated to m, there is an S by S cosine matrix c = (c_{ss′}) defined by c_{ss′} = − cos(π/m_{ss′}). (Here cos(π/∞) = cos(0) = 1).

(6.2) The group W is finite if and only if the cosine matrix is positive definite (cf. [Bo]). Suppose this is the case. Then W can be represented as a group generated by orthogonal linear reflections on ℝ^{n}, where n is the rank of (W, S). We recall how this works. Since c is positive definite, one can find a basis (u_{s})_{s∈S} for ℝ^{n} such that u_{s} · u_{s′} = c_{ss′}. Let r_{s} be reflection across the linear hyperplane H_{s} normal to u_{s}. Then W is identified with the group ⟨r_{s}⟩ generated by the r_{s} and this group is finite. The fundamental cone C(= C(W, S)) is the simplicial cone defined by u_{s} · x ≤ 0, s ∈ S. (So the H_{s} are the supporting hyperplanes of C). Then C is a fundamental domain for W on ℝ^{n}. Its intersection with S^{n−1} is the fundamental simplex σ(= σ(W, S)). As in (3.1), define C^{∗}(= C^{∗}(W, S)), the fundamental dual cone, and its intersection with S^{n−1}, σ^{∗}(= σ^{∗}(W, S)), the fundamental dual simplex. Thus, σ^{∗} is the spherical simplex spanned by the u_{s} and its associated cosine matrix (cf. 2.3) is c. Let σ_{s} = σ ∩ H_{s}. Note that the dihedral angle of σ along σ_{s} ∩ σ_{s′} is π/m_{ss′} and that the length of the edge in σ^{∗} from u_{s} to u_{s′} is π(1 − (m_{ss′})^{−1}).

(6.2.1) The W -translates of σ give a triangulation of S^{n−1}. The underlying simplicial complex is called the Coxeter complex of W.

(6.2.2) Choose a function ϵ : S → (0, ∞). Let x be the interior point of C defined by x · u_{s} = −ϵ(s), i.e., the Euclidean distance from x to H_{s} is ϵ(s). The convex hull of the W -orbit of x is denoted by P (= P (W, S)) and called a Coxeter cell. One checks easily that

(i) the vertices of P are the W -translates of x,

(ii) P is a simple n-cell and the normal cone at any vertex is C^{∗}(W, S), and (iii) ∂P is the dual of the Coxeter complex. The vertices wx and w′x are connected by

an edge if and only if w′ = ws, for some s in S, and the Euclidean length of such an edge is 2ϵ(s).

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 P  x  C^{•}^{•}  ^{•}  ^{•}  ^{•}^{•}^{•}^{•}^{•}

(6.2.3) Examples

(i) Suppose S = {s, s′}, m = m_{ss′}, and W is the dihedral group of order 2m. Then P is a 2m-gon and it is regular if and only if ϵ(s) = ϵ(s′).

(ii) Suppose W = (ℤ/2)^{n}. Then P is a product of intervals ∏[−ϵ(s), ϵ(s)], s ∈ S; P is a regular n-cube if and only if ϵ is constant.

(iii) If W is the symmetric group on n + 1 letters, then the n-cell P is sometimes called

a “permutohedron.” (6.2.4) A Coxeter block B(= B(W, S)) is the convex n-cell defined by, B = P ∩ C. It is

combinatorially equivalent to an n-cube. Call the origin the inside vertex of B; its link is σ. Call x the outside vertex of B; its link is σ^{∗}. Note that W B = P.

(6.3) We return to the general situation where W may be infinite. For any subset T of S, let W_{T} denote the subgroup generated by T. (In fact, (W_{T} , T) is a Coxeter system). Associate to (W, S) an abstract simplicial complex Nerve (W, S) as follows: the vertex set is S; a proper subset T of S is a simplex if and only if W_{T} is finite.

(6.3.1) There is a natural piecewise spherical structure on Nerve (W, S) defined as follows. The simplex T is given the structure of the spherical simplex associated to the cosine matrix of (W_{T} , T). Thus, the simplices of Nerve (W, S) are isometric to the fundamental dual simplices σ_{T} = σ^{∗}(W_{T} , T) corresponding to the finite Coxeter subgroups of W. It follows that Nerve (W, S) has simplices of size ≥ π/2 and, by (6.2), that it is a metric flag complex.

(6.3.2) If (W, S) is right-angled, then Nerve (W, S) is an all right flag complex. Conversely, given any finite, all right flag complex K, there is a right-angled Coxeter system with Nerve (W, S) = K: simply, let S be the vertex set of K and define m_{ss′}, for s ≠ s′, by

m_{ss′} = \{ \begin{matrix}2; & if (s, s′) is an edge \\ ∞; & if (s, s′) is not an edge. \end{matrix}

(6.4) We defined a fundamental chamber Y (= Y (W, S)). Let N = Nerve(W, S), N′ the barycentric subdivision of N, and Y the cone on N′ with cone point x. For each simplex T in N, let v_{T} be its barycenter. Each point y in N′ lies in the open star of a (unique) v_{T}. Put W_{y} = W_{T}. For y in Y − N′, let W_{y} be the trivial subgroup.

(6.4.1) Define an equivalence relation ∼ on W × Y by: (w_{1}, y_{1}) ∼ (w_{2}, y_{2}) if and only if y_{1} = y_{2} and w_{1}W_{y_{1}} = w_{2}W_{y_{2}}. Let Ỹ (= Ỹ (W, S)) be the quotient space (W × Y)/ ∼. Then Ỹ is naturally a simplicial complex, and W acts properly on Ỹ with orbit space Y.

(6.4.2) Put piecewise Euclidean structures on Y and Ỹ as follows. Choose a function ϵ : S → (0, ∞). For each simplex T in N, let B_{T} be the Coxeter block defined as in (6.2.4) by using ϵ|_{T}. (So the length of the edge from the outside vertex of B_{T} to the hyperplane corresponding to t is ϵ(t).) We can identify the subcomplex Cone (σ′_{T}) of Y with B_{T} in such a fashion that v_{T} corresponds to the inside vertex of B_{T} and x corresponds to the outside vertex. This defines a piecewise Euclidean structure on Y such that Lk(x, Y) is Nerve (W, S) with its natural piecewise spherical structure. The space Ỹ inherits a piecewise Euclidean structure from Y.

(6.4.3) The map y → (1, y) from Y to W × Y induces an identification of Y with a subcomplex of Ỹ. Then W_{T} B_{T} is a subcomplex of Ỹ isometric to the Coxeter cell P (W_{T} , T). It follows that Ỹ is cellulated by the Coxeter cells wP (W_{T} , T), where w ∈ W and T is a simplex in N. The vertices of these cells are the W -translates of the cone point x and the link of each vertex is isometric to Nerve (W, S). By Moussong’s Lemma (2.10) such links are large. This gives the following result of [M].

(6.4.4) Theorem (Moussong). For any Coxeter system (W, S), the piecewise Euclidean cell complex Ỹ (W, S) is (NP). Moreover, the link of each vertex is isometric to Nerve(W, S).

(6.4.5) We can find finite, piecewise Euclidean cell complexes with the same property by taking the quotient of Ỹ by a torsion-free subgroup Γ of finite index in W. Thus, X = Ỹ /Γ also satisfies the conclusions of Theorem (6.4.4).

(6.4.6) Since (σ_{T})^{∗} is a fundamental simplex for W_{T} , a^{∗}(σ_{T}) = |W_{T} |^{−1}. Hence,

(6.4.7)  κ(Nerve(W, S)) = ∑(−1)^{Card(T)}|W_{T} |^{−1}

where the summation is over all subsets T of S such that W_{T} is finite. We note that the right hand side of (6.4.7) is the usual expression for the Euler characteristic of W (that is, the “orbihedral Euler characteristic” of Y) and that χ(X) = [Γ : W] ∑(−1)^{Card(T)}|W_{T} |^{−1}. (Compare [D2].)

(6.4.8) Let L be any all right flag complex and let (W, S) be the right-angled Coxeter system corresponding to L (cf. (6.3.2)). Let Ỹ and X be as above. The conclusion is that there is a finite, piecewise Euclidean, cubical cell complex X such that the link of each vertex is isometric to L. This gives the following.

(6.5) Proposition. Conjecture A holds for all piecewise Euclidean manifolds cellulated by regular cubes if and only if Conjecture D holds.

7. The h-polynomial (7.1) Combinatorialists, interested in the poset of faces associated to a convex cell, have

made the following definitions. Suppose K is a finite simplicial complex of dimension m−1, that f_{i} is the number of i-simplices in K, and that f_{−1} = 1. The f-vector of K is the m-tuple (f_{−1}, f_{0}, …, f_{m−1}) and the h-vector (h_{0}, …, h_{m}) is defined by the equation

(7.1.1)  ∑_{i=0}^{m} f_{i−1}(t − 1)^{m−i} = ∑_{i=0}^{m} h_{i}t^{m−i}

Polynomials f(t)(= f_{K}(t)) and h(t)(= h_{K}(t)) are defined by

(7.1.2)  f(t) = ∑_{i=0}^{m} f_{i−1}t^{i}

(7.1.3)  h(t) = ∑_{i=0}^{m} h_{i}t^{i}

Formula (7.1.1) can then be rewritten as

(7.1.4)  h(t) = (1 − t)^{m}f(\frac{t}{1 − t})

(7.2) Let λ(K) be the quantity defined in (3.5.2). Clearly, λ(K) = f_{K}(−\frac{1}{2}). By substituting t = −1 into (7.1.4) we get

(7.2.1)  h(−1) = 2^{m}f(− \frac{1}{2}) = 2^{m}λ(K).

Therefore, λ(K) and h(−1) have the same sign. So Conjecture D is equivalent to the following.

Equivalent Form of Conjecture D. Suppose K^{2n−1} is a generalized homology sphere (as in Section 4). If K is a flag complex, then (−1)^{n}h_{K}(−1) ≥ 0.

(7.3) In this paragraph, we discuss an observation of E. Babson (as communicated to us by L. Billera) which proves (5.2). For any “Eulerian poset” A, combinatorialists have defined the “cd-index” Φ_{A}(c, d), a certain polynomial in two (noncommuting) variables (e.g. see [S2]). It is a refinement of the h-vector of the derived complex A′. In particular, it follows directly from its definition that

(7.3.1)  Φ_{A}(0, −2) = h_{A′}(−1).

Stanley [S2] has proved (in slightly greater generality) that if A is the boundary complex of a convex m-cell, then the coefficients of Φ_{A} are all nonnegative. In particular, for m = 2n, Φ_{A}(0, −2) is (−2)^{n} times the coefficient of d^{n}. It follows that Conjecture D holds for the barycentric subdivision of the boundary complex of a 2n-cell.

(7.4) The following well-known formulas hold for any K^{m−1} which is a generalized homology (m − 1)-sphere.

(7.4.1)  h(t) = t^{m}h(t^{−1})
(7.4.2)  h_{i} ≥ 0, for 0 ≤ i ≤ n.

Formula (7.4.1) (which means that h_{i} = h_{m−i}) is equivalent to the Dehn-Sommerville Relations. These relations are a consequence of the facts that χ(K) = χ(S^{m−1}) and that for any i-simplex σ, χ(Lk(σ, K)) = χ(S^{m−i−2}). Thus, (7.4.1) holds for any K which is an “Euler sphere” in the above sense. The inequalities in (7.4.2) hold whenever K is a “Cohen-Macaulay complex” in the sense of [S1]. In particular, both formulas hold for generalized homology spheres.

Conjecture E. If K^{m−1} is a generalized homology sphere and a flag complex, then h(t) has no roots of modulus 1, except possibly −1.

(7.5) Lemma. Conjecture E implies Conjecture D.

Proof. Suppose that m = 2n. Factor h as a product of monic polynomials, h = h_{1}h_{2}, where the roots of h_{1} are real and those of h_{2} are not. We list the non-real roots of h as: β_{1}, … β_{k}, \overline{β}_{1}, …, \overline{β}_{k}. Since h_{2}(t) = ∏(t − β_{i})(t − \overline{β}_{i}), h_{2}(−1) = ∏(−1 − β_{i})(−1 − \overline{β}_{i}) = ∏ | − 1 − β_{i}|^{2} which is ≥ 0. Hence, we need only show h_{1}(−1) has the correct sign. By (7.4.1), if γ is a root then so is γ^{−1}. If |β| ≠ 1, then β^{−1} ≠ \overline{β}. Supposing that Conjecture E holds, we see that k is even. By (7.4.2) the real roots of h are negative. If h(−1) = 0, then Conjecture D holds. So, suppose h(−1) ≠ 0. List the real roots of h as: α_{1}, …, α_{n−k}, (α_{1})^{−1}, … (α_{n−k})^{−1}, where −1 < α_{i} < 0. Since

h_{1}(−1) = ∏(−1 − α_{i}) ∏(−1 − (α_{i})^{−1}),

the sign of h_{1}(−1) is (−1)^{n−k} which is equal to (−1)^{n} (since k is even).

References

[B] W. Ballman, Singular spaces of non-positive curvature, Chapitre 10 in “Sur les Groupes Hyperboliques d’apres Mikhael Gromov,” edited by E. Ghys and P. de la Harpe, Progress in Math. 81, Birkhäuser, Boston, Basel, Berlin, 1990.

[Bo] N. Bourbaki, “Groupes et Algebrès de Lie,” Chapitres IV-VI, Hermann, Paris, 1968.

[Br] M.R. Bridson, Geodesics and curvature in metric simplicial complexes, in “Group Theory from a Geometrical Viewpoint,” edited by E. Ghys, A. Haefliger, and A. Verjovsky, World Scientific, Singapore, 1991, 373-463.

[CD1] R. Charney and M.W. Davis, Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds, Amer.J. Math. 115 (1993), 929-1009.

[CD2] , The polar dual of a convex polyhedral set in hyperbolic space, to appear in Michigan Math. J.

[C] J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geometry 18 (1983), 575-657.

[CMS] J. Cheeger, W. Müller, and R. Schrader, On the curvature of piecewise flat spaces, Commun. Math. Phys. 92 (1984), 405-454.

[Ch] S.S. Chern, On curvature and characteristic classes of a Riemannian manifold, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 117-126.

[D1] M.W. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math., 117 (1983), 293-324.

[D2] , Coxeter groups and aspherical manifolds, in “Algebraic Topology Aarhus 1982,” edited by I. Madsen and B. Oliver, Springer Lec. Notes in Math., 1051, Springer-Verlag, New York and Berlin, 1984, 197-221.

[Ge] R. Geroch, Positive sectional curvature does not imply positive Gauss-Bonnet integrand, Proc. Am. Math. Soc., 54 (1976), 267-270.

[G] M. Gromov, Hyperbolic groups, in “Essays in Group Theory,” edited by S.M. Gersten, M.S.R.I. Publ. 8, Springer-Verlag, New York and Berlin, 1987, 75-264.

[M] G. Moussong, Hyperbolic Coxeter groups, Ph.D. thesis, The Ohio State University, 1988.

[S1] R. Stanley, “Combinatorics and Commutative Algebra,” Progress in Math. 41, Birkhäuser, Boston, 1983.

[S2] , Flag f-vectors and the cd-index, Math. Zeitschrift, 216 (1994), 483-499.

[W] D.W. Walkup, The lower bound conjecture for 3- and 4-manifolds, Acta Math., 125 (1970), 75-107.