Boundaries of right angled Coxeter groups with manifold nerves · Hanspeter Fischer 5 Theorem 3.1. [21] Let L 0 ←α 1 L 1 ←α 2 L 2 ←···α 3 be an inverse sequence of connected

Boundaries of right angled Coxeter groupswith manifold nerves

Hanspeter Fischer

September 1998

Abstract

All abstract reflection groups act geometrically on non-positively curvedgeodesic spaces. Their natural space at infinity, consisting of (bifurcating)infinite geodesic rays emanating from a fixed base point, is called a boundaryof the group.

We will present a condition on right angled Coxeter groups under whichthey have topologically homogeneous boundaries. The condition is that theyhave a nerve which is a connected closed orientable PL manifold.

In the event that the group is generated by the reflections of one of Davis’exotic open contractible n-manifolds (n ≥ 4), the group will have a boundarywhich is a homogeneous cohomology manifold. This group boundary can thenbe used to equivariantly Z-compactify the Davis manifold.

If the compactified manifold is doubled along the group boundary, one ob-tains a sphere if n ≥ 5. The system of reflections extends naturally to thissphere and can be augmented by a reflection whose fixed point set is the groupboundary. It will be shown that the fixed point set of each extended originalreflection on the thusly formed sphere is a tame codimension-one sphere.

1. Introduction

Unlike boundaries of negatively curved groups, the boundary of a non-positivelycurved group cannot be defined independent of the underlying CAT (0) space on whichit acts geometrically [11]. In this article we shall discuss right angled Coxeter groups.Such a group is non-positively curved since it acts geometrically on its CAT (0) Davis-Vinberg complex. It is the visual boundary of this complex that is often referred toas the boundary of the Coxeter group.

2 Boundaries of right angled Coxeter groups with manifold nerves

An important feature of every Coxeter group is its nerve: the abstract simplicialcomplex with one simplex for every set of generators that spans a finite subgroup.

In Section 3 we will show that the boundary of a right angled Coxeter group ishomogeneous, provided its nerve is a connected closed orientable PL manifold. Ourproof will be based on an inverse limit construction of W. Jakobsche’s, Theorem 3.1[21]. Although closely related, the inverse sequence defining the boundary of ourgroup will not be exactly of the required form. The right balance between adjustingthis deficiency and not altering the limit of the sequence simultaneously will be foundin M. Brown’s result, Theorem 3.3 [9]. Some of the more technical details are exportedinto the appendix, Section 7.

In Section 4 we specialize to right angled reflection groups of open contractibleDavis manifolds of dimensions 5 and higher [12]. For this, the nerve will be assumedto be a PL manifold with the integral homology groups of a sphere.

Davis manifolds, by construction, cannot be the interior of any compact manifoldwith boundary. This peculiarity will be contrasted by Theorem 4.1, which shows thatthe considered manifolds can be equivariantly compactified to objects, which enjoymany of the fundamental properties of manifolds and their boundaries. To provethis, we combine the results of Section 3 with a construction of M. Bestvina’s [6],in order to mount Jakobsche’s homogeneous cohomology manifolds (disguised as thereflection groups’ boundaries) equivariantly onto the Davis manifolds. One propertyof the thusly compactified manifolds is that their doubles along the group boundaryare homeomorphic to actual spheres. This latter fact is a theorem of F. Ancel and C.Guilbault’s [4], which applies to doubles of Z-compactifications of open contractiblemanifolds of dimension at least 5.

In Section 6, we further investigate the action of the reflection group when ex-tended to the sphere. There is a canonical augmentation of the right angled Coxetersystem by a reflection whose fixed point set is the group boundary. We will showthat the fixed point set of each extended original reflection is a tame codimension-onesphere.

The analysis of Section 6 requires a special version of the doubling theorem of[4] in dimension 4, namely Theorem 5.1. We will give a prove of this theorem inall dimensions 4 and higher. It will yield the result that we need in dimension 4,and an alternative proof of the theorem in [4] for the special case when non-positivecurvature is present.

The results in this article mainly represent the authors dissertation [18]. Theauthor wishes to express his sincere gratitude to his supervisor Fredric D. Ancel forhis guidance.

Hanspeter Fischer 3

2. Definitions and notation

Let V be a finite set and m : V × V −→ {∞} ∪ {1, 2, 3, 4, · · ·} a function with theproperty that m(u, v) = 1 if and only if u = v, and m(u, v) = m(v, u) for all u, v ∈ V .Then the group Γ = 〈V | (uv)m(u,v) = 1 for all u, v ∈ V 〉 defined in terms of generatorsand relations is called a Coxeter group. The pair (Γ, V ) is called a Coxeter system. Ifmoreover m(u, v) ∈ {∞, 1, 2} for all u, v ∈ V then (Γ, V ) is called right angled. Theabstract simplicial complex N = N(Γ, V ) = {∅ 6= S ⊆ V | 〈S〉 is finite} is the nerveof (Γ, V ) (where 〈S〉 is the subgroup of Γ generated by S). A geometric realizationof N will be denoted by |N |.

An abstract simplicial complex N is called a flag complex if for every non-emptyset S, ({u, v} ∈ N ∀ u, v ∈ S) =⇒ S ∈ N, so that it is determined by its 1-skeleton.For example, the first barycentric subdivision of any finite simplicial complex is aflag complex. If N is a finite flag complex with vertex set V, it follows that Γ =〈V | v2 = (uv)2 = 1 for all v ∈ V and {u, v} ∈ N〉 is a right angled Coxeter systemwhose nerve is N (Lemma 11.3 of [12]).

In turn, the nerve of every right angled Coxeter system is a flag complex, so thatthe above describes a one-to-one correspondence between finite flag complexes N andright angled Coxeter systems (Γ, V ). For the remainder of this section, let us fix aright angled Coxeter system (Γ, V ).

The Davis-Vinberg complex A = A(Γ, V ) is formed as in [12]: its fundamentalchamber is the cone Q = x0 ∗ |N ′| with panels {Qv = |star(v,N ′)| | v ∈ V } in its base(N ′ denotes the first barycentric subdivision of N). We give Γ the discrete topologyand put A = Γ×Q/ ∼ where (g, x) ∼ (h, y)⇔ x = y and g−1h ∈ 〈v | x ∈ Qv〉. Notethat A is contractible (Corollary 10.3 of [12]).

Alternatively, A(Γ, V ) can be viewed as a geometric realization of a first simplicialsubdivision of the cell-complex Σ(Γ, V ) = {g 〈S〉 | g ∈ Γ, S ⊆ V, 〈S〉 finite}, in whichincidence is by inclusion.

Finally, we cubify the complex A: Let σ ∈ N . We identify |x0σ′| with the cube

[0, 1]σ ⊆ [0, 1]V as follows. The cone point x0 corresponds to 0 and the barycenter ofa face {vi1 , vi2 , · · · , vik} of σ to ei1 + ei2 + · · · + eik , where e1, e2, · · · is the standardbasis for Euclidean space. The cubical complex Q ⊆ [0, 1]V , and subsequently thecomplex A, are then given the induced path length metric.

Since N is a flag complex, this turns A into a CAT (0) geodesic space (cf. [19]pp. 120–122, [13] §3, and [2]). Now, Γ acts geometrically on A, that is, properlydiscontinuously by isometry with compact quotient Q.

The visual boundary of a CAT (0) geodesic space X, denoted bdyX, is defined tobe the inverse limit of concentric metric spheres and geodesic retraction, i.e. the spaceof geodesic rays emanating from a fixed base point with the compact open topology,


which in turn is independent of the choice of base point [15].For x0, x1 ∈ X we denote the unique geodesic segment from x0 to x1 by [x0, x1].

If X denotes the inverse limit of closed concentric metric balls and geodesic retrac-tion, then X embeds in X by means of geodesic segments, and we may identifyX ≡ X ∪ bdyX.

A sequence {Tk | k ≥ 0} of compact subsets of a CAT (0) geodesic space X isexhausting from x0 ∈ X if (i) x0 ∈ intT0 and Tk−1 ⊆ Tk for all k, (ii) ∪k∈IN intTk = X,and (iii) for all k and x ∈ Tk \ int Tk−1, [x, x0] ⊆ Tk and [x, x0] ∩ bdy Tk−1 is a singlepoint (where T−1 = {x0}). Then the geodesic retraction map rk : Tk −→ Tk−1,x 7−→ max ([x0, x] ∩ Tk−1) towards x0 is well-defined and automatically continuous.Also notice that each Tk must be contractible. Since some subsequence of {Tk | k ≥ 0}is cofinal in the collection of metric balls around x0, we get that

X ∪ bdyX ≈ lim←−

(T0

r1←− T1r2←− T2

r3←− · · ·)

and

bdyX ≈ lim←−

(bdy T0

r1|←− bdy T1r2|←− bdy T2

r3|←− · · ·).

For every geodesic ray p0 emanating from x0 there is a unique parallel geodesic rayp1 emanating from x1, namely the limit of the segments [x1, p(t)] as t → ∞. Hence,the action of any subgroup G ≤ Isom(X) extends continuously to X.

In this article we define the boundary of Γ to be bdyA, and denote it by bdyΓ. Notethat this definition is independent of the particular choice of V in the presentation ofΓ, because any two nerves are isomorphic [26].

However, for our purposes the following description is more practical. Let thelength of an element g ∈ Γ be the minimal number of generators from V needed toexpress g. Then

bdy Γ ≈ lim←−

(M0

r1←M1r2←M2

r3←M3r4← · · ·

), where

Mk = bdy⋃ {gQ | length(g) ≤ k} and rk : Mk −→Mk−1, x 7−→ [x, x0] ∩Mk−1. Note

that Corollary 7.3 (c) below ensures that this is an exhausting sequence.

3. Homogeneity of the boundary

A topological space X is called p-homogeneous if given any two collections{x1, x2, · · · , xp} and {y1, y2, · · · , yp} of p distinct points in X there is a homeomor-phism h : X −→ X such that h(xi) = yi for all i.

We shall show that the boundary of a right angled Coxeter group is p-homogeneousfor every positive integer p, provided its nerve is a connected closed orientable PLmanifold. The proof will be based on the following inverse limit construction:

Hanspeter Fischer 5

Theorem 3.1. [21] Let L0α1← L1

α2← L2α3← · · · be an inverse sequence of connected

closed orientable n-manifolds (n ≥ 2) and Dk finite collections of disjoint collareddisks in Lk such that

(a) each Lk is a connected sum of finitely many copies of L0,

(b) each αk+1 is a homeomorphism over the set Lk \⋃{int D | D ∈ Dk},

(c) each α−1k+1(D) (D ∈ Dk) is homeomorphic to a copy of L0 with the interior of a

collared disk removed,

(d) {αj+1 ◦ αj+2 ◦ · · · ◦ αi(D) | D ∈ Di, i ≥ j} is null and dense in Lj for all j, †

(e) αj+1 ◦ αj+2 ◦ · · · ◦ αi(D) ∩ bdyD′ = ∅ for all D ∈ Di, D′ ∈ Dj, i > j.

Then lim←−

(L0

α1← L1α2← L2

α3← · · ·)

is p-homogeneous for every positive integer p and

depends on L0 only. This space is denoted by X(L0, {L0}). If, moreover, L0 is ahomology sphere, then X(L0, {L0}) is a cohomology n-manifold. ‡

For a discussion of cohomology manifolds see, for example, [7].

Remark 3.2. Observe that if L0 is not simply connected, then X(L0, {L0}) is notlocally simply connected and, hence, not an ANR. On the other hand, if L0 is asphere, then so is X(L0, {L0}).

We will also need

Theorem 3.3. ([9] Theorem 2) For every two finite sequences

X0s1← X1

s2← X2s3← · · ·Xk−1

sk← Xk and X0t1← X1

t2← X2t3← · · ·Xk−1

tk← Xktk+1← Xk+1

of maps between compact metric spaces there is a positive real numbera(s1, s2, · · · , sk, t1, t2, · · · , tk, tk+1) such that whenever two inverse sequences

Y0α1← Y1

α2← Y2α3← · · · and Y0

β1← Y1β2← Y2

β3← · · · have the property that

d(αk, βk) ≤ a(α1, α2, · · · , αk−1, β1, β2, · · · , βk−1, βk)

for all k ≥ 2, then lim←−

(Y0

α1← Y1α2← Y2

α3← · · ·)≈ lim←−

(Y0

β1← Y1β2← Y2

β3← · · ·).

†By an empty composition of maps we mean the identity map.‡Note that we did not state Jakobsche’s results in its full generality.


We now state and prove the main results of this chapter. Verification of sometechnical details will be the contents of Section 7.

Theorem 3.4. Let Γ be a right angled Coxeter group whose nerve N is a connectedclosed orientable PL n-manifold. Then bdy Γ is p-homogeneous for every positiveinteger p (p ≤ 3 if n = 1). Moreover:

(a) If N is not simply connected, then bdy Γ is not locally simply connected.

(b) If N is a homology sphere, then bdy Γ is a cohomology n-manifold and cohomol-ogy n-sphere,

(c) If N is a sphere, then so is bdy Γ.

It follows from elementary considerations that the boundary is a circle if n = 1.

Theorem 3.5. If the nerves of two right angled Coxeter groups are connected closedorientable PL manifolds which are topologically homeomorphic, then the respectivegroup boundaries are homeomorphic.

To prove these assertions we only need to establish

Theorem 3.6. If the nerve N of a right angled Coxeter group Γ is a connected closedorientable PL manifold, then bdy Γ is homeomorphic to Jakobsche’s X(|N |, {|N |})space.

Remark 3.7. The fact that bdy Γ ≈ X(|N |, {|N |}) is a cohomology sphere when Nis a homology sphere, follows from Theorem 4.1(e) and an application of the Mayer–Vietoris sequence.

Proof. (of Theorem 3.6) Assume that N = N(Γ, V ) is a connected closedorientable PL n-manifold. Consider Mk = bdy

⋃ {gQ | length(g) ≤ k} and definerk : Mk −→ Mk−1, x 7−→ [x, x0] ∩Mk−1. Then the sequence {Mk | k ≥ 0} consists ofthe boundaries of an exhausting sequence and

bdy Γ = bdyA(Γ, V ) ≈ lim←−

(M0

r1←M1r2←M2

r3← · · ·).

For g ∈ Γ we put A(g) =⋃{gQv | length(gv) < length(g)}. Then each A(g) is

a PL-disk and Mk+1 = Mk#A(g){g|N | | length(g) = k + 1} (the connected sumMk#A(g1)g1|N |#A(g2)g2|N |# · · ·#A(gs)gs|N | which is independent of the order in whichwe list the elements {g ∈ Γ | length(g) = k + 1} = {g1, g2, · · · , gs} [12].)

Hanspeter Fischer 7

Figure 1: Violation of condition (e)

The inverse sequence M0r1← M1

r2← M2r3← · · · satisfies all but two conditions

of Theorem 3.1: the collections {A(g) | length(g) = k + 1}, being the natural can-didates for the Dk’s, are not disjoint and, worse yet, condition (e) does not holdin the simplest of examples (e.g. if we set V = {v1, v2, v3, v4} and define Γ =〈V | v2

i = 1, (vivj)2 = 1 for all i 6≡ j mod 2〉. Then A is just Euclidean 2-space, tes-

sellated by squares. Obviously, condition (e) is violated significantly. See Figure 1.)

We will solve this problem by adjusting the inverse system just enough to meetall requirements of Theorem 3.1, but not so much as to actually change its limit.

To this end, we metrize |N | as a piecewise spherical all right complex (i.e. eachsimplex is given the angle metric of its corresponding standard simplex: the convexhull of standard basis vectors in Euclidean space). The resulting path length metricon |N | will be denoted by α. In formulas:

α(x, y) = cos−1

∑si=0 xiyi√∑s

i=0 x2i

√∑si=0 y

2i

∈ [0,π

2],

for σ = {u0, u1, · · · , us} ∈ N , x, y ∈ |σ|, x =∑si=0 xiui, y =

∑si=0 yiui, xi, yi ≥ 0, and∑s

i=0 xi =∑si=0 yi = 1. For each g ∈ Γ \ {id} and 0 < η < ε < π we define:


Figure 2: The disks D(g, ε, η)

pg = [gx0, x0] ∩ g|N |◦B(g, ε) = {x ∈ g|N | | α(pg, x) < ε}B(g, ε) = {x ∈ g|N | | α(pg, x) ≤ ε}C(g, ε) = gx0 ∗ {x ∈ g|N | | α(pg, x) = ε}†

D(g, ε, η) = (B(g, ε) ∪ C(g, ε)) \◦B(g, η).

Then each B(g, ε), C(g, ε), and D(g, ε, η) is topologically an n-disk by Proposition 7.7.See Figure 2. Recall that the shadow of the cone point gx0 is described by

{x ∈ g|N | | gx0 ∈ [x, x0]} = g|N |\◦B(g, π) ([14] Lemma 2d.1). (∗)

Note that with ε sufficiently close to π, pg ∈ int A(g) ⊆ A(g) ⊆◦B (g, ε) (cf. Re-

mark 7.6). Here is our strategy: Factor the bonding maps through manifolds of theform

M ′k(εk) = Mk#A(g){B(g, εk) ∪ C(g, εk) | length(g) = k + 1}

with appropriately chosen εk’s:

†We will modify this definition in Remark 7.8 in a way which does not effect what follows.

Hanspeter Fischer 9

M ′k(εk)

↙ ↖Mk ←− Mk+1

All maps in this diagram are geodesic retractions. Then, by Proposition 7.9 below,x 7−→ [x, x0] ∩Mk : M ′

k(εk) −→ Mk will be a near-homeomorphism for each k. Sothat in the inverse sequence

M0 ←−M ′0(ε0)←−M1 ←−M ′

1(ε1)←−M2 ←− · · · ,

whose limit is bdy Γ, we can inductively change every other map to a nearby home-omorphism fk and leave the remaining maps geodesic retractions gk. If this is donecarefully enough, Theorem 3.3 will guarantee that the limit does not change:

M0f0←−M ′

0(ε0)g1←−M1

f1←−M ′1(ε1)

g2←−M2f2←− · · · .

On the other hand, there is enough flexibility in choosing εk and ηk to arrange forcondition (e) of Theorem 3.1 to hold for the sequence

M ′0(ε0)

g1◦f1←− M ′1(ε1)

g2◦f2←− M ′2(ε2)

g3◦f3←− · · ·

and the collection of disks Dk = {D(g, εk, ηk) | length(g) = k + 1}. Note that, becauseof (∗) we can pinpoint the location of these disks. Finally, as all the above sequenceshave homeomorphic limits, the result will follow.

Specifically, define Rk :⋃{gQ | length(g) ≥ k + 1} −→ Mk, x 7−→ [x, x0] ∩Mk.

Choose ε0 ∈ (0, π) and a non-degenerate closed interval I00 ⊆ (0, ε0) such that

R0(D(g, ε0, η)) ⊆ int A(g) for all g ∈ Γ with length(g) = 1 and all η ∈ I00 (this is

where we use (∗) ). Put M ′0 = M ′

0(ε0) and define g1 : M1 −→M ′0, x 7−→ [x, x0] ∩M ′

0.By Proposition 7.9 we may choose a homeomorphism f0 : M ′

0∼−→ M0 such that

f0(D(g, ε0, η)) ⊆ int A(g) for all g ∈ Γ with length(g) = 1 and all η ∈ I00 .

Fix m ≥ 0 and assume inductively that we have already chosen positive realnumbers ε0, ε1, · · · , εm, non-degenerate closed intervals {Ik2k1 | 0 ≤ k1 ≤ k2 ≤ m} and

homeomorphisms f0, f1, · · · , fm such that (i) εk ∈ (0, π) and I ik ⊆ Ijk ⊆ (0, εk) forall indices 0 ≤ k ≤ j ≤ i ≤ m, (ii) fk : M ′

k∼−→ Mk has the property that

d(fk ◦ gk+1, rk+1) ≤ a(f0 ◦ g1, f1 ◦ g2, · · · , fk−1 ◦ gk, r1, r2, · · · , rk+1) (if k ≥ 1) andfk(D(g, εk, η)) ⊆ int A(g) for all g ∈ Γ with length(g) = k + 1 ≤ m + 1 and allη ∈ Ikk , where M ′

k = M ′k(εk) and gk+1 : Mk+1 −→ M ′

k, x 7−→ [x, x0] ∩M ′k, and (iii)

if 1 ≤ length(h) = k1 + 1 < k2 + 1 = length(g) ≤ m + 1, η ∈ Ik2k1 , ξ ∈ Ik2k2 , thenbdyD(h, εk1 , η) ∩ gk1+1 ◦ fk1+1 ◦ · · · ◦ gk2 ◦ fk2(D(g, εk2 , ξ)) = ∅.


Then choose (i) for each 0 ≤ k ≤ m a non-degenerate closed interval Im+1k ⊆ Imk

such that gk+1 ◦ fk+1 ◦ · · · ◦ fm ◦ gm+1 ◦ Rm+1(gx0) 6∈ bdy D(h, εk, η) wheneverlength(h) = k + 1, length(g) = m + 2, and η ∈ Im+1

k , (ii) εm+1 ∈ (0, π) and a non-degenerate closed interval Im+1

m+1 ⊆ (0, εm+1) such that Rm+1(D(g, εm+1, η)) ⊆ int A(g)for all g ∈ Γ with length(g) = m+ 2 and all η ∈ Im+1

m+1 , and such that

bdyD(h, εk, η) ∩ gk+1 ◦ fk+1 ◦ · · · ◦ fm ◦ gm+1 ◦Rm+1(D(g, εm+1, ξ)) = ∅

whenever 1 ≤ length(h) = k + 1 < m + 2 = length(g), η ∈ Im+1k , ξ ∈ Im+1

m+1 , and(iii) a homeomorphism fm+1 : M ′

m+1∼−→Mm+1 such that

d(fm+1 ◦ gm+2, rm+2) ≤ a(f0 ◦ g1, f1 ◦ g2, · · · , fm ◦ gm+1, r1, r2, · · · , rm+2)

and (ii) holds with Rm+1 replaced by fm+1, where M ′m+1 = M ′

m+1(εm+1) andgm+2 : Mm+2 −→M ′

m+1, x 7−→ [x, x0] ∩M ′m+1.

This completes the induction. Finally, for each m ≥ 0, we pick an ηm ∈⋂k≥m

Ikm.

The goal is accomplished and the proof complete.//

4. Equivariant Z-compactifications of Davis manifolds

by homogeneous group boundaries

Some Coxeter groups are generated by affine orthogonal reflections in the sidesof a compact convex polyhedron P in some Euclidean space which produce an exacttessellation under the group action. The list of these crystallographic Coxeter groupsis well-known. The occurring orders m(u, v) must necessarily be in {∞, 1, 2, 3, 4, 6}.(See [20] for sufficient conditions.) Observe that, in this case, |N(Γ, V )| = ∂P is asphere.

In [12], this situation is generalized to the case when N(Γ, V ) is a generalizedhomology sphere. Since we will only be concerned with right angled Coxeter sys-tems whose nerves are PL manifolds, we review the construction of Davis’ reflectionmanifolds from [12] for this case:

Let N be a flag complex which is a PL n-manifold with the integral homology ofan n-sphere. We then call N a homology sphere. For n 6= 2, N is the boundary of aunique compact contractible (n + 1)–manifold P (see [23] for n ≥ 4, [17] for n = 3,and [3] for an alternative proof if n ≥ 5). Let (Γ, V ) be the right angled Coxetersystem with nerve N . Divide ∂P = |N | into panels {Pv = |star(v,N ′)| | v ∈ V }. Asin the definition of the Davis-Vinberg complex A(Γ, V ), we call P a chamber, anddefine M(Γ, V ) analogously (with Q replaced by P ).

Hanspeter Fischer 11

Observe that Γ acts on M(Γ) = M(Γ, V ) by left multiplication on the firstcoordinate. It is shown in [12], that M(Γ) is an open contractible manifold onwhich Γ acts properly discontinuously. In fact, if n ≥ 4, then M(Γ) is the uniqueopen contractible manifold on which Γ acts properly discontinuously with compactquotient (combining Proposition 4.3 and [4], Theorem 18).

The fixed-point set of every reflection v ∈ V is a codimension-one flat submanifoldwhich separatesM =M(Γ) into two components that are interchanged by v [12]. Infact, since we restrict ourselves to right angled Coxeter systems with manifold nerves,these fixed point sets are homeomorphic to Euclidean space (cf. Lemma 6.2).

By Selberg’s Lemma, Γ has a torsion-free (normal) subgroup Γ′ of finite index.Since Γ acts properly discontinuously on M, Γ′ will act fixed-point free. Therefore,M covers the closed manifold M/Γ′. But, as was shown in [12], if N(Γ, V ) is notsimply connected (and there are such examples in all dimensions 3 and higher), thenM is not homeomorphic to Euclidean space – disproving a conjecture of Johnson’s[22]. In fact, M will not be the interior of any compact manifold with boundary,because its π1-system is not stable at infinity [28]. (On the other hand, if N(Γ, V ) isa sphere (and P a ball if n = 2), then M is homeomorphic to Euclidean space.)

However, there are compactifications of Davis’ reflection manifolds which enjoymost of the properties of manifolds and their boundaries. In order to describe suchcompactifications we review the following terminology (discussed, for example, in [32]and [16]). A closed subset Z of a compact (separable metric) ANR X is a Z-set inX (and X a Z-compactification of X = X \ Z) if any one of the following equivalentconditions holds:

(1) There is a homotopy H : X×[0, 1] −→ X such that H0 = id and instantaneouslypushes X off Z: Ht(X) ∩ Z = ∅ for all t > 0.

(2) For all ε > 0 there is a map f : X −→ X with d(f, id) < ε and f(X) ∩ Z = ∅.

(3) For all open subsets U of X, U \ Z ↪→ U is a homotopy equivalence.

(4) Z is k-LCC in X for all k ≥ 0. (Recall, that a subset A of an ANR Y islocally k-coconnected in Y , denoted k-LCC, if for every y ∈ cl A and everyneighborhood U of y in Y there is a neighborhood V of y in Y such that everymap Sk −→ V \ A extends to Bk+1 −→ U \ A.)

The standard examples of Z-sets are boundaries of compact manifolds.We now propose a compactification of M(Γ), which enjoys many of the funda-

mental properties of manifolds and their boundaries.

Theorem 4.1. Let Γ be a right angled Coxeter group whose nerve N is a connectedclosed orientable PL manifold of dimension at least 4 and with the integral homology


groups of a sphere. Then there is a compactification M =M∪ Z of M =M(Γ) byZ = bdy Γ such that

(a) M is an AR and Z is a Z-set in M,

(b) Z is a cohomology n-manifold and cohomology n-sphere.

(c) Z is p-homogeneous for every positive integer p,

(d) If N is not simply connected, then Z is not locally simply connected,

(e) M∪ZM≈ Sn+1,

(f) the action of Γ on M extends continuously to M∪ZM.

Remark 4.2. Observe that Z will be an ANR if and only if N is a sphere, in whichcase M is homeomorphic to Euclidean space. This is the only item in this list ofproperties which clearly distinguishes Z from a manifold boundary.

Theorem 4.1 improves an unpublished 1984 result of F. Ancel and L. Sieben-mann’s, in which Jakobsche’s homogeneous cohomology manifolds are used directlyto construct compactifications satisfying properties (a) through (e). Their methodsare based on a slick rearrangement of the chambers. Although this approach ap-plies to all of Davis’ reflection manifolds (with manifold chambers), the constructionprohibits the action of Γ from extending to the compactification.

The proof of Theorem 4.1 is based on Theorem 3.6. This result opens up anew, indirect way of equivariantly compactifying reflection manifolds by homogeneouscohomology manifolds using the notion of Z-structures: A Z-compactification X =X ∪ Z of X is a Z-structure [6] on a group G if

(1) G acts properly discontinuously, fixed-point free and cocompactly on the finitedimensional AR X, and

(2) for all compact subsets K of X and all open covers U of X there is a finitesubset F of G, such that for all g ∈ G \ F there is a U ∈ U with g(K) ⊆ U .

If a group G acts geometrically and fixed point free on a CAT (0) space X, thenX = X ∪ bdyX is a Z-structure on G. This enables us to use

Proposition 4.3. ([6] Lemma 1.4) Let G be a group acting properly discontinuously,cocompactly and fixed-point free on two finite dimensional AR’s X1 and X2. Assumethat X2 = X2 ∪ Z is a Z-structure on G, X1 = X1 ∪ {∞} the one-point compacti-fication of X1, f : X1 −→ X2 a G-equivariant homotopy equivalence. Define an


embedding e : X1 −→ X1 ×X2, x 7−→ (x, f(x)). Then X1 ∪ Z ≡ e(X1) ∪ Z ′ is a Z-compactification of X1 and a Z-structure on G, where Z ′ = cle(X1)\e(X1) = {∞}×Z.

Proof. (of Theorem 4.1) LetA be the Davis-Vinberg complexA(Γ). First recall thatΓ has a torsion free finite index subgroup Γ′. Then Γ′ acts properly discontinuouslycocompactly and fixed point free on bothM andA. Choose a Γ-equivariant homotopyequivalence f : M −→ A. (Notice that their chambers are homotopy equivalentrelative to their identical boundaries.)

Consider the embedding e : M −→ (M∪ {∞}) × (A ∪ bdy A), x 7−→ (x, f(x))and define Z = cl e(M) \ e(M) = {∞} × bdy A as in Proposition 4.3 (where weuse Γ′ for G). The canonical action of Γ on e(M), which mimics that on M (i.e.g(x, f(x)) = (gx, gf(x)) = (gx, f(gx)) for x ∈M and g ∈ Γ), is well-defined, becausef is Γ-equivariant and not only Γ′-equivariant. Also, Γ acts on Z, by g(∞, z) =(∞, gz) for g ∈ Γ and z ∈ bdyA.

Part (f) is now automatic. Moreover, (a) follows from Proposition 4.3, (b), (c),and (d) follow from Theorem 3.4, and (e) follows from Theorem 4.4 below.//

Theorem 4.4. [4] If M = M ∪ Z and N = N ∪ Z are two Z-compactifications ofopen manifolds M and N of dimension at least 5 by the same Z-set, then M ∪Z N isa manifold. If, moreover, M and N are contractible, then M ∪Z N is a sphere.

5. Doubling CAT(0) manifolds along their visual boundaries

The following theorem is a special case of Theorem 4.4 in dimensions n ≥ 5. Sincewe will need this result in the next section in dimension n = 4, we devote this sectionto proving it. The proof of Theorem 5.1 given here, also provides an alternative tothe proof of Theorem 4.4 in [4] for this particular case.

Theorem 5.1. Let M be an open topological manifold of dimension n ≥ 4, equippedwith a CAT (0) metric and an exhausting sequence of compact n-manifolds. Then thedouble of M = M ∪ bdyM along its visual boundary bdyM is a sphere.

Remark 5.2. It was shown in [2], that given any compact contractible (n + 1)–manifold P with PL boundary ∂P (necessarily a homology sphere) and n ≥ 4,there is always a flag triangulation N of ∂P such that M(Γ) supports a CAT (0)metric with respect to which Γ (the right angled Coxeter group with nerve N)acts by isometry. Notice that for these CAT (0) reflection manifolds the subspaces


Tk =⋃{gP | length(g) ≤ k} define an exhausting sequence of compact manifolds, so

that Theorem 5.1 applies.

Our strategy for proving Theorem 5.1 will be based on the cell-like approximationtheorem. A compact metric space is cell-like if it contracts within any ANR that itis embedded in. A subset A of an n-manifold M is cellular if for every neighborhoodV of A in M there is a subset B of M such that A ⊆ int B ⊆ B ⊆ V and B ishomeomorphic to [0, 1]n. Cellular subsets of manifolds are cell-like, but the converseis false.

A map f : X −→ Y between topological spaces is proper if f−1(K) is compactfor all compact subsets K ⊆ Y . A proper surjection f : X −→ Y is cell-like iff−1(y) = f−1({y}) is cell-like for all y ∈ Y . A proper surjection f : M −→ Y froma manifold onto a topological space is cellular if f−1(y) is a cellular subset of M forall y ∈ Y . A map f : X −→ Y between topological spaces is a near-homeomorphismif for every open cover U of Y there is a homeomorphism h : X −→ Y such that forevery x ∈ X there is a U ∈ U with {f(x), h(x)} ⊆ U . Here is our main tool:

Theorem 5.3. A cell-like map f : Mn −→ Nn (read cellular if n = 3) betweenn-manifolds without boundary is a near-homeomorphism.

This theorem is due to [24], [27] and [33] in dimension 2, [5] in dimension 3, [25]in dimension 4, and [29] in dimension 5 and above.

Proof. (of Theorem 5.1) Choose an exhausting sequence {Tk | k ∈ IN} of compactn-manifolds for M from x0 ∈ int T1. We may assume that Tk−1 ⊆ int Tk for all k.Let rk : Tk −→ Tk−1, x 7−→ max ([x0, x] ∩ Tk−1) be the retraction map. For eachk ∈ IN let T ′k be a homeomorphic copy of Tk and r′k be defined accordingly. LetDk = Tk ∪∂k T ′k be the double of Tk along ∂k = ∂Tk ≈ ∂T ′k. Let dk : Dk −→ Dk−1

denote the map with dk|Tk = rk and dk|T ′k

= r′k.

Since each Tk is a compact contractible manifold, each ∂Tk is a homology sphereby Lefschetz duality. Then each Dk is a homology sphere by Mayer–Vietoris. Itis a homotopy sphere by Van Kampen’s Theorem and the Hurewicz IsomorphismTheorem, and is consequently homeomorphic to a sphere by the Poincare Conjecture

in dimension 4 and higher. Observe that the limit of D1d2←− D2

d3←− D3d4←− · · · is

homeomorphic to the double of M along bdy M . By Theorem 5.3 it will suffice toshow that dk : Dk −→ Dk−1 are cell-like for all k ≥ 2. Because then, by Theorem 3.3,

the limit of D1d2←− D2

d3←− D3d4←− · · · is homeomorphic to the limit of an inverse

sequence in which each dk is replaced by a homeomorphism, and the latter limit ishomeomorphic to D1 ≈ Sn. To this end we fix a k ≥ 2 and x ∈ Dk−1. In showingthat d−1

k (x) is cell-like we may assume that x ∈ ∂Tk−1. We break the argument into


six lemmas, from which the theorem will follow.//

Lemma 5.4. The reduced integral Cech cohomology ˇH∗(d−1k (x)) ≡ 0.

Proof. Similar to [30], we get:

0 = Hi(Tk \ r−1k (x)) (since Tk \ r−1

k (x) is contractible)

' Hi(int Tk \ r−1k (x))

= Hi(Dk \ (r−1k (x) ∪ T ′k))

' ˇHn−i−1(r−1k (x) ∪ T ′k) (by Alexander Duality)

' Hn−i−1(r−1k (x) ∪ T ′k, T ′k) (since T ′k is contractible)

' Hn−i−1(r−1k (x), r−1

k (x) ∩ ∂Tk) (by excision)

' ˇHn−i−2(r−1k (x) ∩ ∂Tk) (since r−1

k (x) is contractible)

for all i. So, by Mayer–Vietoris, ˇH∗(d−1k (x)) ≡ 0.//

Lemma 5.5. Let C and D be two chain complexes of free modules of finite rank over

a PID R, f# : C −→ D a chain map with f ∗ : Hk(D)0−→ Hk(C) for some k. Then

(a) f∗(Hk(C)) ⊆ Hk(D)t (the torsion module of Hk(D) ) and

(b) f∗(Hk−1(C)t) = 0.

(All groups are understood to have R coefficients.)

Proof. All homomorphisms are R-module homomorphisms. Consider the followingshort exact rows and induced homomorphisms:

0 −→ Ext(Hk−1(D), R) −→ Hk(D) −→ Hom(Hk(D), R) −→ 0↓ ↓ 0 ↓

0 −→ Ext(Hk−1(C), R) −→ Hk(C) −→ Hom(Hk(C), R) −→ 0.

Then, in fact, all vertical homomorphisms are trivial. Hence, (i) the composition

Hk(C)f∗−→ Hk(D)

φ−→ R is trivial for all homomorphisms φ : Hk(D) −→ R and

(ii) the composition Bk−1(C) f#−→ Bk−1(D)φ−→ R extends over Zk−1(C) for all ho-

momorphisms φ : Bk−1(D) −→ R (where B denotes the boundaries and Z the cy-cles). In order to prove (a), express Hk(D) = Fk(D)⊕Hk(D)t with a free R-moduleFk(D). Let [z] ∈ Hk(C). Choose a basis {b1, b2, · · · , bs} for Fk(D) and define for eachj ∈ {1, 2, · · · , s} a homomorphism φj : Hk(D) −→ R such that φj(bj) = 1, φj(bi) = 0


for all i 6= j and φj(Hk(D)t) = {0}. Then, by (i), φj ◦ f∗([z]) = 0 for all j. Hence,f∗([z]) ∈ Hk(D)t. For (b), define

Wk−1(D) = {d ∈ Dk−1 | ∃ r ∈ R \ {0} such that rd ∈ Bk−1(D)} ⊆ Zk−1(D).

Choose d1, d2, · · · dm ∈ Dk−1 and r1, r2, · · · rm ∈ R such that {d1, d2, · · · , dm} is a basisfor Wk−1(D) and {r1d1, r2d2, · · · , rmdm} is a basis for Bk−1(D). This is possible sincewe have finitely generated modules over a PID. Let [z] ∈ Hk−1(C)t. Say, rz ∈ Bk−1(C)with r ∈ R \ {0}. Then rf#(z) ∈ Bk−1(D) so that f#(z) ∈ Wk−1(D). Express

f#(z) = α1d1 + α2d2 + · · ·αmdm with αi ∈ R

andrf#(z) = β1r1d1 + β2r2d2 + · · ·+ βmrmdm with βi ∈ R.

Define homomorphisms

φj : Bk−1(D) −→ Rrjdj 7−→ 1ridi 7−→ 0 for i 6= j.

Then the composition Bk−1(C) f#−→ Bk−1(D)φj−→ R extends over Zk−1(C) to φj,

using (ii). Set sj = φj(z). Then rsj = φj(rz) = φj(f#(rz)) = βj for all j =1, 2, · · · ,m. Hence, rf#(z) = rs1r1d1+rs2r2d2+· · ·+rsmrmdm. Consequently, we havef#(z) = s1r1d1 + s2r2d2 + · · · + smrmdm ∈ Bk−1(D). This implies that f∗([z]) = 0 inHk−1(D).//

Lemma 5.6. For all neighborhoods U of d−1k (x) in Dk there is a neighborhood V of

d−1k (x) in Dk such that d−1

k (x) ⊆ V ⊆ U and H∗(V )incl∗−→ H∗(U) is trivial.

Proof. Choose a sequence {Pm | m ∈ IN} of nested compact polyhedra in Dk such

that d−1k (x) =

⋂m∈IN

Pm and d−1k (x) ⊆ int Pm ∀m ∈ IN. Then,

lim−→

(H i(P1) −→ H i(P2) −→ H i(P3) −→ · · ·

)' ˇH i(d−1

k (x)) = 0

by continuity and Lemma 5.4. Hence, there is a subsequence {P ′m | m ∈ IN} of

{Pm | m ∈ IN} such that H i(P ′m)incl∗−→ H i(P ′m+1) is trivial for all i and m. Then,

by Lemma 5.5, there is a subsequence {P ′′m | m ∈ IN} of {P ′m | m ∈ IN} such that

Hi(P′′m+1)

incl∗−→ Hi(P′m) is trivial for all i and m.//


Lemma 5.7. For all neighborhoods U of d−1k (x) in Dk there is a neighborhood V of

d−1k (x) in Dk such that d−1

k (x) ⊆ V ⊆ U and π1(V ) = 1.

Proof. Choose an open (n − 1)-disk B around x in ∂Tk−1 and collarsc : ∂Tk−1 × [0, ε) −→ Tk−1 and c′ : ∂T ′k−1 × [0, ε) −→ T ′k−1, small enough so thatV = c(B × [0, ε)) ∪ d−1

k (B) ∪ c′(B′ × [0, ε)) ⊆ U. It will suffice to show thatπ1(d−1

k (B)) = 1.By the proof of Lemma 5.4, rk|∂Tk is monotone. Also, rk|∂Tk is a closed map, so

that (rk|∂Tk)−1 (B) is an open connected subset of ∂Tk and hence path connected.

Since r−1k (B) is clearly contractible (first to B and then to {x}), d−1

k (B) is simplyconnected by Van Kampen’s Theorem.//

Finally, we use the following well-known neighborhood version of the HurewiczTheorem.

Lemma 5.8. Let X be a closed subset of an ANR Y . Suppose that for all neighbor-hoods U of X in Y , there is a neighborhood V of X in Y such that π1(V ) −→ π1(U)and Hi(V ) −→ Hi(U) are trivial for 0 ≤ i ≤ n. Then for all neighborhoods U of Xin Y , there is a neighborhood V of X in Y such that πi(V ) −→ πi(U) are trivial for0 ≤ i ≤ n.

Since this lemma is not found in standard text books, we include a proof.

Proof. We proceed by induction on n. The case n = 1 is obvious. Assume thetheorem holds with n replaced by n− 1. Let U be a neighborhood of X in Y . Thenthere are neighborhoods U = Un ⊇ Un−1 ⊇ · · · ⊇ U1 ⊇ U0 of X in Y such thatπi(Ui) −→ πi(Ui+1) is trivial for 0 ≤ i ≤ n − 1. There is also a neighborhood V ofX in Y such that Hn(V ) −→ Hn(U0) is trivial. We will prove that πn(V ) −→ πn(U)is trivial. Let f : Sn −→ V be any map. Since U0 is an ANR, there are maps

U0φ−→ K

ψ−→ U0, where K is a CW-complex and ψ ◦ φ ' idU0 in U0. Observe thatψ : K −→ U0 extends to a map ψ : K ∪ C(Kn−1) −→ U because πi(Ui) −→ πi(Ui+1)is trivial for 0 ≤ i ≤ n−1. Here Kn−1 denotes the (n−1)–skeleton of K and C(Kn−1)the cone on Kn−1. Consider

πn(Sn)(φ◦f)#−→ πn(K ∪ C(Kn−1))

↓ ↓Hn(Sn)

(φ◦f)∗−→ Hn(K ∪ C(Kn−1)),

where the vertical arrows are Hurewicz homomorphisms.Since πi(K ∪ C(Kn−1)) = 0 for 0 ≤ i ≤ n − 1, the Hurewicz homomorphism

πn(K ∪C(Kn−1)) −→ Hn(K ∪C(Kn−1)) is an isomorphism by the classical Hurewicz


Isomorphism Theorem.Since Hn(V ) −→ Hn(U0) is trivial, so is (φ ◦ f)∗ : Hn(Sn) −→ Hn(K ∪C(Kn−1)).

It follows that (φ ◦ f)# : πn(Sn) −→ πn(K ∪ C(Kn−1)) is trivial, so thatφ ◦ f : Sn −→ K ∪ C(Kn−1) extends to a map F : Bn+1 −→ K ∪ C(Kn−1). So,ψ ◦ F : Bn+1 −→ U extends ψ ◦ φ ◦ f : Sn −→ U0 ⊆ U . Since ψ ◦ φ ' idU0 in U0,f : Sn −→ V is homotopically trivial in U . QED.//

Lemma 5.9. d−1k (x) is cell-like.

Proof. Since d−1k (x) has arbitrarily small polyhedral neighborhoods, the above

lemmas show that for all neighborhoods U of d−1k (x), there is a neighborhood V of

d−1k (x) such that V contracts within U .//

6. Extension to tame reflection systems on spheres

In this last section we will explore the group action on the (n + 1)-sphere ofTheorem 4.1 in more detail. Here is what we shall prove:

Theorem 6.1. Let (Γ, V ) be a right angled Coxeter system whose nerve is a non-simply connected homology sphere of dimension n ≥ 4. Then there exists an involutiong : Sn+1 −→ Sn+1 such that

(a) The right angled Coxeter group Γ × {g, 1} acts effectively on Sn+1 by homeo-morphism.

(b) The fixed point set Fix(v) = {x ∈ Sn+1 | v(x) = x} of any v ∈ V is a tamecodimension-one sphere.

(c) The fixed point set Fix(g) = {x ∈ Sn+1 | g(x) = x} of g is homeomorphic tobdyΓ, a cohomology n-manifold and cohomology n-sphere that is p-homogeneousfor every positive integer p but not locally simply connected.

(d) Sn+1 \ Fix(g) has two components, both of which are copies of the reflectionmanifold M(Γ) and are interchanged by g.

(e) Sn+1/g is a Z-compactification of M(Γ) and a Z-structure on Γ.

We first analyze the fixed point sets.


Lemma 6.2. Let M(Γ, V ) be a right angled reflection manifold, v ∈ V ,(Γ, V ) the right angled Coxeter system with nerve N(Γ, V ) = lk(v,N(Γ, V )),Fix(v|M) = {x ∈ M(Γ) | v(x) = x}. Then Fix(v|M) is canonically isomorphicto A(Γ, V ) and hence homeomorphic to a codimension-one Euclidean space.

Proof. Let P be the chamber of M. Then |N(Γ, V )| = ∂P is by definitiona PL manifold. We have Fix(v|M) = {gPv | g ∈ Γ}, because (vg, x) ∼ (g, x) inΓ × P ⇐⇒ g−1vg ∈ 〈u | x ∈ Pu〉 ⇐⇒ x ∈ Pv and g ∈ Γ. Now, N ′(Γ, V ) iscanonically isomorphic to lk(v,N ′(Γ, V )) (a barycenter of a simplex σ in N(Γ, V )corresponds to the barycenter of vσ in N(Γ, V )). Under this isomorphism, the panelPv = |star(v,N ′(Γ, V ))| of P in M(Γ) corresponds to v ∗ |N ′(Γ, V )|, the chamber ofA(Γ, V ). The panels (Pv)u = |star(u,N ′(Γ, V ))| of Pv in A(Γ, V ) in turn correspondto Pu ∩ Pv = |star(u′, lk(v,N ′(Γ, V )))| for all u ∈ V = V ∩ lk(v,N(Γ, V )) (where u′

is the barycenter of {u, v} in N(Γ, V )). We have the correct identifications, since(i) (g, x) ∼ (h, x) ⇐⇒ g−1h ∈ 〈u | x ∈ Pu〉 = 〈u | x ∈ Pu ∩ Pv〉 ∀ x ∈ Pv, g, h ∈ Γ,(ii) (g, x) ∼ (h, x) ⇐⇒ g−1h ∈ 〈u | x ∈ (Pv)u〉 ∀ x ∈ v ∗ |N ′(Γ, V )|, g, h ∈ Γ. Finally,A(Γ, V ) is homeomorphic to Euclidean space as its chamber is a ball.//

Remark 6.3. Notice that by construction, A(Γ, V ) is a convex subset of A(Γ, V ).Also, by choosing the vertex x0 = v as the base point in A(Γ, V ) ⊆ A(Γ, V ), we mayassume that bdyA(Γ, V ) ⊆ bdyA(Γ, V ).

Remark 6.4. Recall that Coxeter groups have a very simple solution to the wordproblem [31]. A word (finite sequence of generators) is reduced if and only if it cannotbe shortened by a combination of the following operations:

(i) The obvious cancelation of a subword of the form vv,

(ii) Replacement of a subword of the form uvuvuv · · · (of length m) by vuvuvu · · ·(of length m), where m is the order of the element uv in the group.

This becomes especially easy to check in a right angled Coxeter group!

For the fixed point sets in the boundary we have

Lemma 6.5. Let M(Γ, V ) be a right angled reflection manifold, v ∈ V , (Γ, V )the right angled Coxeter system with nerve N(Γ, V ) = lk(v,N(Γ, V )). Consider theaction of Γ on bdy A(Γ, V ) and let Fix(v|bdy ) = {p ∈ bdy A(Γ, V ) | vp = p}. Then

Fix(v|bdy ) = bdyA(Γ, V ), which is homeomorphic to a codimension-two sphere.

Proof. Clearly bdy A(Γ, V ) ⊆ Fix(v|bdy ), since this reflection fixes A(Γ, V )


pointwise (following the proof of Lemma 6.2). To obtain the reverse inclusion, letp ∈ bdy A(Γ, V ) with vp = p (i.e. v ◦ p and p are parallel rays). Choose a mini-mal gallery Q, v1Q, v1v2Q, v1v2v3Q, · · · in A(Γ, V ) covering p [8]. Then the sequence(length(vk · · · v2v1vv1v2 · · · vk))k∈IN is bounded. But all words (v1v2 · · · vk) are re-duced. So, by Remark 6.4, there are u1, u2 · · · , uk0 ∈ V and a sequence (wk)k∈INin V ∩ lk(v,N(Γ, V )) such that v1v2 · · · vk = w1w2 · · ·wk−k0u1u2 · · ·uk0 for all k ≥ k0.Therefore, the limit of the segments [x0, w1w2 · · ·wkx0] is a ray in A(Γ, V ) parallel top. This places p in bdyA(Γ, V ).

Since A(Γ, V ) is a PL manifold (all links are PL homeomorphic to iterated sus-pensions of links in N(Γ, V )), bdy A(Γ, V ) is a codimension-two sphere ([14] Theo-rem 3b.2).//

We will make use of the 1-LCC taming theorem:

Theorem 6.6. [10] Let Sn ⊆ Sn+1 be the standard embedding and n ≥ 4.If e : Sn −→ Sn+1 is any embedding of Sn onto a 1-LCC subset of Sn+1, then thereis a homeomorphism h : Sn+1 −→ Sn+1 which extends e.

The following two lemmas are joint work with F. Ancel.

Lemma 6.7. Let Z be a cohomology n-manifold. Assume, further, that Z is a finitedimensional locally connected separable metric space. Then any Sn−1 ⊆ Z locallyseparates Z into two components: for every x ∈ Sn−1 and every neighborhood U ofx in Z there is a neighborhood V of x in Z such that V ⊆ U, V is connected, andV \ Sn−1 has precisely two components.

We adapt the proof from [1], Theorem VI.3:

Proof. For this proof, Hc∗ will denote Borel-Moore homology and H∗c Cech-

cohomology with compact supports over the constant sheaf ZZ2 as discussed, forexample, in [7]. We will denote singular homology with ZZ2 coefficients by H∗.

Choose open disks D0, D1 ⊆ Sn−1 and open connected subsets U0 and U1 of Z suchthat (i) x ∈ D0 ⊆ D1 ⊆ Sn−1, (ii) x ∈ U0 ⊆ U1 ⊆ V , (iii) Di = Ui ∩ Sn−1 (i = 0, 1),

and (iv) Hc1(U0)

0−→ Hc1(U1). (Notice that Z is homologically locally connected over

ZZ2.) Put A = D1 \D0. Then A is closed in D1, D1 is closed in U1, and D0 is closedin U0. We claim that Hc

1(U0, U0 \ D0)∼−→ Hc

1(U1, U1 \ D1) is an isomorphism. Itwill suffice to show that the map α in the following commutative triangle, in which εdenotes the excision of (U1 \ A) \ U0 , is an isomorphism:


α -

HHHHY 6

Hc1(U1 \ A,U1 \D1) Hc

1(U1, U1 \D1)

Hc1(U0, U0 \D0)

ε

To this end, consider the commutative diagram

- -β

? ? ?∆ ∆ ∆

Hc1(D1, D1 \ A) Hc

0(D1 \ A) Hc0(D1) Hc

0(D1, D1 \ A)

-

? ? ?∆ ∆ ∆

Hn−2c (A) Hn−1

c (D1, A) Hn−1c (A)

- -α -Hc2(U1, U1 \ A) Hc

1(U1 \ A,U1 \D1) Hc1(U1, U1 \D1) Hc

1(U1, U1 \ A)

Here, ∆ denotes Alexander duality isomorphisms. Since D1 and D0 = D1 \ A arepath connected, β is an isomorphism and Hc

0(D1, D1 \A) = 0. It follows that α is anisomorphism.

Consequently, in the following commutative diagram, we must have γ = 0, so that∂ is an isomorphism:

0 o

-γ -∂ -Hc1(U0) Hc

1(U0, U0 \D0) Hc0(U0 \D0) Hc

0(U0) = 0

-? ?

Hc1(U1) Hc

1(U1, U1 \D1)

Since, Hc1(U0, U0 \ D0)

∆−→ Hn−1c (D0)

∆−→ Hc0(D0)

∼−→ H0(D0) ' ZZ2, this impliesthat Hc

0(U0 \D0) ' ZZ2, so that U0 \Sn−1 = U0 \D0 has precisely two components.//

Lemma 6.8. Let Σ ⊆ Sn+1 be homeomorphic to Sn, and let Z be a closed connectedand locally connected subset of Sn+1, separating Sn+1 into the two components M1

and M2. Assume, further, that (i) Mi ∪ Z is a Z-compactification of Mi (i = 1, 2),(ii) Σ ∩Mi is locally flat in Mi (i = 1, 2), (iii) Σ ∩ Z locally separates Z into twocomponents, one on each side of Σ in Sn+1, and (iv) Σ ∩ Z is 1-LCC in Σ. Then Σis 1-LCC in Sn+1.


Proof. Let C1 and C2 be the two closed complementary domains of Σ in Sn+1.We need to show that Σ is 1-LCC in Sn+1 at every point of Σ ∩ Z. To this end, letx ∈ Σ ∩ Z and U a neighborhood of x in Sn+1. Choose neighborhoods V1, V2 andV of x in Sn+1 and a neighborhood V3 of x in Z such that (i) V ⊆ V2 ⊆ V1 ⊆ Uand V ∩ Z ⊆ V3 ⊆ V2 ∩ Z, (ii) every map S1 → (V1 ∩ Σ) \ Z extends to a mapB2 → (U ∩Σ) \Z, (iii) every map S1 → V2 \Z extends to a map B2 → V1 \Z (noticethat Z is in particular 1-LCC in each Mi ∪ Z), and (iv) V3 is connected and V3 \ Σhas precisely two components (one in each Ci).

Let α : S1 → V \ Σ be any map. If α(S1) ∩ Z = ∅, we can extend α to a mapα : B2 → V1 \ Z. Say, α(B2) ⊆ M1. Make α transverse to Σ ∩M1. Then α−1(Σ) isa finite union of simple closed curves in B2. Since for every outer most componentC of α−1(Σ), we can extend α|C : C → (V1 ∩ Σ) \ Z to a map α : D → (U ∩ Σ) \ Z(where D is the disk in B2 with ∂D = C), we may assume that α : B2 → (U ∩Ci)\Zfor some i ∈ {1, 2}. By pushing it off Σ slightly (which we can do because Σ ∩Mi isbicollared in Mi), we may, in fact, assume that α : B2 → U \ Σ and we are done.

Otherwise, there is a sequence (ak, bk) of pairwise disjoint open subintervals of S1

with⋃k(ak, bk) = α−1(V \ Σ). Let Jk denote the straight line segment in B2 joining

ak to bk. Then [ak, bk]∪Jk bounds a disk Ek ⊆ B2. Put J = (S1 \ ⋃k(ak, bk))∪⋃k Jk.Since V3 \ Σ has one component in each Ci, and Z is locally connected, we candefine β : J → V3 \ Σ such that β(x) = α(x) for all x ∈ α−1(Z) and such thatβ|Jk is a sequence of maps whose image diameters converge to zero. (Use a sequenceof coverings by open connected sets whose meshes converge to zero.) Then β iscontinuous.

We now extend each loop α|[ak,bk]∪β|Jk over Ek in two steps. First we extend overan annular neighborhood of the boundary of Ek: using the fact that Z is a Z-set wepush the loop off Z into either (V2 ∩M1) \ Z or (V2 ∩M2) \ Z, whichever intersectsα|[ak,bk]. We then extend over the remaining subdisk of Ek to a map Ek → V1 \Z. Aswe did for α above, we may assume that these maps extend to Ek → U \ Σ. Again,we may arrange that the image diameters of these extensions converge to zero (usingappropriately chosen, fine coverings). Because we can do the same for β itself, thereis, in fact, a map B2 → U \ Σ extending α.//

Proof. (of Theorem 6.1) We only have to prove (b). Let v ∈ V and (Γ, V ) bethe right angled Coxeter system with nerve N(Γ, V ) = lk(v,N(Γ, V )). Then byLemma 6.2, we may identify Fix(v|M) = {x ∈ M | v(x) = x} with A(Γ, V ) ≈ IRn.We can choose f in the proof of Theorem 4.1 such that it restricts to the identity on|N(Γ, V )|. Then the double S of clA(Γ, V ) along Z ∩ (clA(Γ, V )) within M∪ZMmay be identified with the double of A(Γ, V ) ∪ bdy A(Γ, V ) along its visual sphere.By Lemma 6.2 and Lemma 6.5, S = Fix(v), and by Theorem 5.1 (which holds one


dimension lower than Theorem 4.4), S is an n-sphere. Tameness of this sphere inSn+1 follows from Lemma 6.7, Lemma 6.8, and Theorem 6.6. (Recall that Fix(v|M)is locally flat in M [12].)//

7. Appendix

The goal of this section is to prove Proposition 7.9. We assume the conditions andnotation of the proof of Theorem 3.6. In order to simplify notation we will write |x0σ|for the cube |x0σ

′| = [0, 1]σ. For v ∈ σ, let πv : |x0σ| −→ |x0v| denote orthogonalprojection. A directed maximal non-degenerate straight line segment I ⊆ |x0σ| iscalled a segment. We say that σ carries I if I 6⊆ |x0τ | for all proper faces τ of σ.Let I0 and I1 denote initial and terminal point of I respectively. Put ∆(I, v) =πv(I1)− πv(I0). Also, for τ ⊆ σ and x ∈ |x0σ| \ {x0} we define

β(x, τ) =∑v∈τ

πv(x)∑w∈τ πw(x)

v ∈ |τ | (in barycentric coordinates).

Let ασ denote the path length metric on |lk(σ,N)| (when viewed as an all rightpiecewise spherical complex). Then α∅ agrees with α defined earlier. Note that Nand all its links are flag complexes. Observe that the fundamental chamber Q is aconvex subset of A, because the panels Qv cover |N ′| and are subsets of the fixedpoint sets of the reflections v ∈ Γ. (Here we use the fact that a CAT (0) space has nonon-trivial bi-gons.)

We begin by making two simple geometric observations, Lemma 7.1 and 7.2.These lemmas describe the type of branching that geodesics in A are subject to.

Lemma 7.1. Let I and J be segments carried by σ and τ respectively such thatI1 = J0. Then I ∪ J is geodesic in Q (and hence in A) if and only if

(a) τ 6⊆ σ and σ 6⊆ τ ,

(b) ∆(I, v) < 0 for all v ∈ σ \ τ and ∆(J, v) > 0 for all v ∈ τ \ σ,

(c)∆(I, v)

length I=

∆(J, v)

length Jfor all v ∈ σ∩τ (i.e. I ∪ J is geodesic in |x0σ|∪|x0τ |),

(d) ασ∩τ (β(I0, σ \ τ), β(J1, τ \ σ)) ≥ π (i.e. I ∪ J is geodesic in Q also).


And, in the same spirit:

Lemma 7.2. Let I and J be segments carried by σ and τ respectively, η ∈ N ,g =

∏v∈η v ∈ Γ with I1 = gJ0 (so that η ⊆ σ ∩ τ). Then I ∪ gJ is geodesic in A if

and only if

(a) Neither σ is a proper face of τ nor τ a proper face of σ,

(b) ∆(I, v) < 0 for all v ∈ σ \ τ and ∆(J, v) > 0 for all v ∈ τ \ σ,

(c)∆(I, v)

length I= − ∆(J, v)

length J∀ v ∈ η and

∆(I, v)

length I=

∆(J, v)

length J∀ v ∈ (σ ∩ τ) \ η,

(d) ασ∩τ (β(I0, σ \ τ), β(J1, τ \ σ)) ≥ π provided τ 6= σ.

The next corollary relates the global direction of a geodesic segment, which iscontained in some cubical part of some chamber, to a local sense of direction. As onewould expected, the segment is increasing with respect to those coordinates of thecube, which shorten the length of the word labeling the chamber.

Corollary 7.3. Let I be a segment carried by σ and g ∈ Γ such that gI ⊆ [gI0, x0].Then for all v ∈ σ:

(a) ∆(I, v) > 0 if and only if length(gv) < length(g),

(b) ∆(I, v) < 0 if and only if length(gv) > length(g),

(c) ∆(I, v) 6= 0.

Proof. We may assume that g 6= id. Every geodesic in A is covered by a minimalgallery [8]. Choose η1, η2, · · · , ηk ∈ N, integers 1 ≤ n0 < n1 < · · · < nk = m andsegments P (1), P (2), · · · , P (m) = I. such that if we put

gi =

∏w∈η1

w

∏w∈η2

w

· · · ∏w∈ηi

w

∈ Γ (reduced!)

then

(i) g = gk,

(ii) P (j) ∩ P (j+1) = P(j)0 = P

(j+1)1 for all 1 ≤ j ≤ m− 1,


(iii) P (1)∪P (2)∪· · ·∪P (n0)∪k⋃i=1

gi(P (ni−1+1) ∪ · · · ∪ P (ni)

)= [x0, gI0].

(If one of the words representing a gi were not reduced, then the deletion condition [8](p.37) would imply that [gI0, x0] crosses the fixed point set of some reflection huh−1

(u ∈ V, h ∈ Γ), a so-called wall, twice ([8] Theorem III 4B(d)). This cannot happenin a CAT (0) space.)

First assume that ∆(I, v) > 0.Since ∆(P (m), v) = ∆(I, v) ≥ 0, v ∈ carrier(P (m))∩carrier(P (m−1)). So, by Lemma 7.1,

∆(P (m), v)

length P (m)=

∆(P (m−1), v)

length P (m−1).

Thus, ∆(P (m−1), v) > 0. Repeat this argument and conclude that ∆(P (nk−1+1), v) > 0.Then again, v ∈ carrier(P (nk−1+1)) ∩ carrier(P (nk−1)), because ∆(P (nk−1+1), v) ≥ 0.Also, ηk ⊆ carrier(P (nk−1+1)) ∩ carrier(P (nk−1)) so that (vw)2 = id for all w ∈ ηk. Weconsider two cases:

(1) v ∈ ηk. Then clearly length(gv) < length(g).

(2) v 6∈ ηk. Then, by Lemma 7.2,

∆(P (nk−1+1), v)

length P (nk−1+1)=

∆(P (nk−1), v)

length P (nk−1).

Hence, ∆(P (nk−1), v) > 0.

So, if we had length(gv) > length(g), we would always be in Case 2 all the way downto P (1) (Note that length(giv) < length(gi) if and only if length(gv) < length(g) aslong as v ∈ carrier(P (ni−1+1)) ∩ carrier(P (ni−1)), because this implies that (vw)2 = id

for all w ∈ ηi). Therefore, ∆(P (1), v) > 0. But P(1)1 = x0, so that ∆(P (1), v) < 0.

Contradiction.Now assume that ∆(I, v) = 0. Then, by the first part, ∆(P (j), v) = 0 for all

1 ≤ j ≤ m (use Lemma 7.2(c) for either Case 1 or Case 2). But this is impossible!Finally, assume that ∆(I, v) < 0. Extend P (1), P (2), · · · , P (m) to an infinite se-

quence (P (j))∞j=1 so as to form a ray⋃∞j=1 P

(j). Define the ni, ηi, and gi accordingly.Traveling along this ray traverses I in opposite direction and we can use the argu-ment of part one, where ∆(I, v) > 0: First we claim that v ∈ ηi0 for some i0 > k(i0 chosen minimal). Indeed, otherwise, by Lemma 7.1 and Lemma 7.2,

∆(P (j), v)

length P (j)=

∆(P (j+1), v)

length P (j+1)for all j > m.


But∑∞j=m+1 length P (j) =∞, so that

∑∞j=m+1 ∆(P (j), v) diverges. This is absurd.

Also, (vw)2 = id for all w ∈ ⋃i0j=k+1 ηj (as we saw in part one). If now length(gv) <length(g), then

gi0 =

∏w∈η1

w

· · · ∏w∈ηk

w

∏w∈ηk+1

w

· · · ∏w∈ηi0

w

is not reduced.

Contradiction.//

The direction principle of the previous corollary is “locally greedy”:

Corollary 7.4. Let I be a segment carried by σ and g ∈ Γ such that gI ⊆ [gI0, x0].Then for all v ∈ lk(σ,N): length(gv) > length(g).

Proof. Without loss of generality g 6= id. Assume to the contrary that thereis a v0 ∈ lk(σ,N) with length(gv0) < length(g). Choose η1, η2, · · · , ηk ∈ N , integers1 ≤ n0 < n1 < · · · < nk = m, segments P (1), P (2), · · · , P (m) = I, and g1, g2, · · · , gk in Γas in the proof of Corollary 7.3. Let i0 = max{i ∈ {1, · · · , k} | v0 ∈ ηi}. We claim thatv0 ∈ lk(carrier(P (j)), N) for all ni0−1 < j ≤ m. In particular, v0 6∈ carrier(P (ni0−1+1)).This will be a contradiction because v0 ∈ ηi0 .

We induct on j. The case j = m follows by assumption. Assuming the claimfor j + 1 we prove it for j. Put ξ = carrier(P (j+1)) and τ = carrier(P (j)). Theinduction hypothesis is that v0 ∈ lk(ξ,N). We may assume that ξ 6= τ . Let u ∈ τ \ ξ.Then ∆(P (j), u) > 0 by Lemma 7.1 or Lemma 7.2. Choose i1 ∈ {i0, · · · , k} such thatni1−1 < j ≤ ni1 . Then length(gi1u) < length(gi1) by Corollary 7.3. Hence, v0u = uv0

(since gi1 =(∏

v∈η1 v)· · ·

(∏v∈ηi1

v)

is reduced and length(gv0) < length(g)). Also,

v0 6∈ τ (otherwise v0 ∈ τ \ ξ and v0 ∈ lk(ξ,N) so that

αξ∩τ (β(P(j+1)0 , ξ \ τ), β(P

(j)1 , τ \ ξ))

≤ αξ∩τ (β(P(j+1)0 , ξ \ τ), v0)︸︷︷︸

=π2

(since v0ξ ∈ N)

+ αξ∩τ (v0, β(P(j)1 , τ \ ξ))︸︷︷︸

<π2

(since πv0(P(j)1 ) > 0)

< π.

This would contradict Lemma 7.1 or Lemma 7.2.) Therefore, v0 ∈ lk(τ,N) andthe claim is proved.//

Note that if g ∈ Γ \ {id} and σ = {v ∈ V | length(gv) < length(g)}, then σ ∈ N([12], Lemma 7.12). Corollary 7.4 identifies the possible singularities of the geodesicflow through the chamber gQ towards the base point x0. Branching occurs in subsets


of the form g (x0 ∗ bdy |Ni(σ,N)|), where we define Ni(σ,N) = {η ∈ N | η ⊆ ξ ∈N and dim(ξ ∩ σ) ≥ i} (−1 ≤ i ≤ dim σ). That these subsets are disks, is the whatwe prove next.

Lemma 7.5. Let L be an abstract simplicial flag complex which is a closed PL n-manifold. Then Ni(σ, L) is a PL-n-ball for all 0 ≤ i ≤ dim σ. In particular, thesimplicial neighborhood N0(σ, L) is a regular neighborhood of σ in L.

Proof. We induct on dim σ. If dim σ = 0, then N0(σ, L) = star(σ, L) is an n-ball.Assume dim σ ≥ 1 and fix 0 ≤ i ≤ dim σ. Without loss of generality we may assumethat i < dim σ, because Ndim σ(σ, L) = σlk(σ, L) is also an n-ball. Say σ = vγ. Weclaim that

(i) Ni(σ, L) = vNi−1(γ, lk(v, L)) ∪Ni(γ, L),

(ii) vNi−1(γ, lk(v, L)) ∩Ni(γ, L) = vNi(γ, lk(v, L)),

(iii) lk(v,Ni(γ, L)) = lk(v, L) ∩Ni(γ, L) = Ni(γ, lk(v, L)).

(i) “⊆” Let η ∈ Ni(σ, L). Then η ⊆ ξ ∈ L and dim(ξ ∩ σ) ≥ i. And there-fore dim(ξ ∩ γ) ≥ i − 1. Case 1: dim(ξ ∩ γ) ≥ i. Then η ∈ Ni(γ, L). Case 2:Assume dim(ξ ∩ γ) = i − 1. Then v ∈ ξ, say ξ = vτ with τ ∈ lk(v, L) anddim(τ ∩ γ) ≥ i − 1. If v 6∈ η, then η ⊆ τ so that η ∈ Ni−1(γ, lk(v, L)). If onthe other hand v ∈ η, say η = vη′, then η′ ⊆ τ and η′ ∈ Ni(γ, lk(v, L)). Con-sequently, η ∈ vNi−1(γ, lk(v, L)). “⊇” Notice that we have Ni(γ, L) ⊆ Ni(σ, L)and vNi−1(γ, lk(v, L)) ⊆ Ni(σ, L). (ii) “⊆” Let η ∈ vNi−1(γ, lk(v, L)) ∩ Ni(γ, L).Case 1: η = v. Done. Case 2: η ∈ Ni−1(γ, lk(v, L)) ∩ Ni(γ, L). Thenη ⊆ ξ ∈ lk(v, L) and dim(ξ ∩ γ) ≥ i − 1. Also η ⊆ τ ∈ L and dim(τ ∩ γ) ≥ i.If dim(ξ ∩ γ) ≥ i, then η ∈ Ni(γ, lk(v, L)). If dim(ξ ∩ γ) = i − 1, pick aw ∈ (τ ∩ γ) \ ξ. Then η ⊆ w(ξ ∩ γ)η ∈ lk(v, L) (since L is a flag complex!), anddim(w(ξ ∩ γ)η ∩ γ) ≥ i so that η ∈ Ni(γ, lk(v, L)). Case 3: η = vη′ withη′ ∈ Ni−1(γ, lk(v, L)). Then η′ ∈ Ni(γ, L). By Case 2, η′ ∈ Ni(γ, lk(v, L)).Hence, η ∈ vNi(γ, lk(v, L)). “⊇” Clearly Ni(γ, lk(v, L)) ⊆ Ni−1(γ, lk(v, L)) andvNi(γ, lk(v, L)) ⊆ Ni(γ, L). (iii) “⊆” Let η ∈ lk(v, L) ∩ Ni(γ, L). Then η ⊆ ξ ∈ Land dim(ξ ∩ γ) ≥ i. Since η, γ ∈ lk(v, L) and L is a flag complex, we haveη ⊆ η ∪ (ξ ∩ γ) ∈ lk(v, L). Also, dim((η ∪ (ξ ∩ γ)) ∩ γ) ≥ dim(ξ ∩ γ) ≥ i. Hence,η ∈ Ni(γ, lk(v, L)). “⊇” This inclusion is immediate. Hence the claim.

Now we argue as follows. Ni(γ, L) and Ni(γ, lk(v, L)) are n and (n− 1)-ballsrespectively by induction hypothesis (Note that lk(v, L) is a flag complex whichis a sphere). Also, if i ≥ 1 then Ni−1(γ, lk(v, L)) is an (n− 1)-ball. Moreover,N−1(γ, lk(v, L)) = lk(v, L). Consequently, Ni(γ, L)∪ star(v, L) shells to Ni(γ, L) and


Ni(γ, L) ∪ star(v, L) shells to Ni(σ, L) if i ≥ 1 (or else is equal to it). Thus, Ni(σ, L)is homeomorphic to the n-ball Ni(γ, L).//

Remark 7.6. By Corollaries 7.3 and 7.4, pg ∈ int g|σ| ⊆ int A(g).

Proposition 7.7. ([14] Lemma 3b.1) Suppose L is a piecewise spherical flag complex(metrized by d) which is a PL-n-manifold. Then for any x ∈ |L| and ε ∈ (0, π),{y ∈ |L| | d(x, y) ≤ ε} is topologically a closed n-disk. Hence, {y ∈ |L| | d(x, y) < π}is topologically an open n-disk.

Remark 7.8. We now give the “correct” definition of C(g, ε) from Section 3;a definition of this cone which insures that the retraction maps are well-defined (i.e.every geodesic ray emanating from x0 will intersect C(g, ε) in at most one point). Weobtain such a cone by lowering the open annulus

◦B(g, π) \B(g, ε)

towards gx0 along the geodesic retraction lines. This works because there are nobifurcations in this region, as we saw above.

Proposition 7.9. The geodesic retractions x 7−→ [x, x0] ∩Mk : M ′k(εk) −→Mk are

near-homeomorphisms.

Proof. Fix k ≥ 0 and list {g ∈ Γ | length(g) = k + 1} = {g1, g2, · · · , gs}. For1 ≤ i ≤ s put σi = {v ∈ V | length(giv) < length(gi)} ∈ N and define spheres Eji

i =bdy gi|x0Nji(σi, N)| with faces F ji

i = A(gi) ∩ gi (Nji(σi, N)) for all 0 ≤ ji ≤ dim σi.By Corollary 7.3 and Corollary 7.4, the retraction M ′

k(εk) −→ Mk factors throughmanifolds of the form Mk#F

jii{Eji

i | i = 1, 2, · · · , s} in which the indices j1, j2, · · · , jscan be increased independently.

By Lemmas 7.1, 7.2, and Proposition 7.7 the point pre-images of each factor mapare homeomorphic to complements of open disks in links of N . Since these linksare spheres of various dimensions, the point pre-images are cellular subsets of thesespheres. Therefore each map is cellular and can be approximated by homeomorphismsusing Theorem 5.3.//


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Department of Mathematics

Brigham Young University

Provo, Utah 84602

U.S.A.

E-Mail: [email protected]

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Boundaries of right angled Coxeter groups with manifold nerves · Hanspeter Fischer 5 Theorem 3.1. [21] Let L 0 ←α 1 L 1 ←α 2 L 2 ←···α 3 be an inverse sequence of connected