In this paper, we investigate a relation between finite graphs, simplicial
flag complexes and right-angled Coxeter groups, and we provide a class of
reconstructible finite graphs. We show that if $\Gamma$ is a finite graph which
is the 1-skeleton of some simplicial flag complex $L$ which is a homology
manifold of dimension $n \ge 1$, then the graph $\Gamma$ is reconstructible.Comment: 7 page

Hosaka, Tetsuya

English

arXiv

Osaka University Knowledge Archive

RECONSTRUCTIBLE GRAPHS, SIMPLICIAL FLAG COMPLEXES OF HOMOLOGY MANIFOLDS AND ASSOCIATED RIGHT-ANGLED COXETER GROUPS

HOSAKA, TETSUYA

Osaka City University Repository

Hosaka, T
Osaka J. Math.
52 (2015), 1173–1180
RECONSTRUCTIBLE GRAPHS,
SIMPLICIAL FLAG COMPLEXES OF
HOMOLOGY MANIFOLDS AND
ASSOCIATED RIGHT-ANGLED COXETER GROUPS
TETSUYA HOSAKA
(Received April 4, 2014, revised October 2, 2014)
Abstract
In this paper, we investigate a relation between finite graphs, simplicial flag com-
plexes and right-angled Coxeter groups, and we provide a class of reconstructible
finite graphs. We show that if 0 is a finite graph which is the 1-skeleton of some
simplicial flag complex L which is a homology manifold of dimension n  1, then
the graph 0 is reconstructible.
1. Introduction
In this paper, we investigate a relation between finite graphs, simplicial flag com-
plexes and right-angled Coxeter groups, and we provide a class of reconstructible finite
graphs. This paper treats only “simplicial” graphs. We show that if 0 is a finite graph
which is the 1-skeleton of some simplicial flag complex L which is a homology mani-
fold of dimension n  1, then the graph 0 is reconstructible.
A graph 0 is said to be reconstructible, if any graph 00 with the following prop-
erty () is isomorphic to 0.
() Let S and S0 be the vertex sets of 0 and 00 respectively. Then there exists a bi-
jection f W S ! S0 such that the subgraphs 0S {s} and 00S0 { f (s)} are isomorphic for any
s 2 S, where 0S {s} and 00S0 { f (s)} are the full subgraphs of 0 and 00 whose vertex sets
are S   {s} and S0   { f (s)} respectively.
The following open problem is well-known as the reconstruction conjecture.
PROBLEM (Reconstruction conjecture). Every finite graph with at least three ver-
tices will be reconstructible?
Some classes of reconstructible graphs are known (cf. [3], [20], [21], [22], [23],
[26]) as follows: Let 0 be a finite graph with at least three vertices.
2010 Mathematics Subject Classification. 57M15, 05C10, 20F55.
Partly supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture,
Sports, Science and Technology, Japan. (No. 25800039).
1174 T. HOSAKA
(i) If 0 is a regular graph, then it is reconstructible.
(ii) If 0 is a tree, then it is reconstructible.
(iii) If 0 is not connected, then it is reconstructible.
(iv) If 0 has at most 11 vertices, then it is reconstructible.
Our motivation to consider graphs of the 1-skeletons of some simplicial flag com-
plexes comes from the following idea on right-angled Coxeter groups and their nerves.
Details of Coxeter groups and Coxeter systems are found in [4], [6] and [19], and
details of flag complexes, nerves, Davis complexes and their boundaries are found in
[8], [9] and [24].
Let 0 be a finite graph and let S be the vertex set of 0. Then the graph 0 uniquely
determines a finite simplicial flag complex L whose 1-skeleton L (1) coincide with 0.
Here a simplicial complex L is a flag complex, if the following condition holds:
() For any vertex set {s0, : : : , sn} of L , if {si , s j } spans 1-simplex in L for any
i, j 2 {0, : : : , n} with i ¤ j then the vertex set {s0, : : : , sn} spans n-simplex in L .
Also every finite simplicial flag complex L uniquely determines a right-angled
Coxeter system (W, S) whose nerve L(W, S) coincide with L (cf. [1], [8], [9], [10],
[12]). Here for any subset T of S, T spans a simplex of L if and only if the para-
bolic subgroup WT generated by T is finite (such a subset T is called a spherical
subset of S).
Moreover it is known that every right-angled Coxeter group W uniquely determines
its right-angled Coxeter system (W, S) up to isomorphisms ([28], [18]).
By this corresponding, we can identify a finite graph 0, a finite simplicial flag
complex L , a right-angled Coxeter system (W, S) and a right-angled Coxeter group W .
Let 0 and 00 be finite graphs, let L and L 0 be the corresponding flag complexes,
let (W, S) and (W 0, S0) be the corresponding right-angled Coxeter systems, and let
W and W 0 be the corresponding right-angled Coxeter groups, respectively. Then the
following statements are equivalent:
(1) 0 and 00 are isomorphic as graphs;
(2) L and L 0 are isomorphic as simplicial complexes;
(3) (W, S) and (W 0, S0) are isomorphic as Coxeter systems;
(4) W and W 0 are isomorphic as groups.
Also, for any subset T of the vertex set S of the graph 0, the full subgraph 0T of
0 with vertex set T corresponds the full subcomplex LT of L with vertex set T , the
parabolic Coxeter system (WT , T ) generated by T , and the parabolic subgroup WT of
W generated by T .
Hence we can consider the reconstruction problem as the problem on simplicial
flag complexes and also as the problem on right-angled Coxeter groups.
Moreover, the right-angled Coxeter system (W, S) associated by the graph 0 de-
fines the Davis complex 6 which is a CAT(0) space and we can consider the ideal
boundary 6 of the CAT(0) space 6 (cf. [1], [2], [5], [8], [9], [10], [12], [15], [16],
[24]). Then the topology of the boundary 6 is determined by the graph 0, and the
RECONSTRUCTIBLE GRAPHS, SIMPLICIAL FLAG COMPLEXES 1175
topology of 6 is also a graph invariant.
Based on the observations above, we can obtain the following lemma from results
of F.T. Farrell [13, Theorem 3], M.W. Davis [10, Theorem 5.5] and [17, Corollary 4.2]
(we introduce details of this argument in Section 3).
Lemma 1.1. Let (W, S) be an irreducible Coxeter system where W is infinite and
let L D L(W, S) be the nerve of (W, S). Then the following statements are equivalent:
(1) W is a virtual Poincaré duality group.
(2) L is a generalized homology sphere.
(3) QH i (L S T ) D 0 for any i and any non-empty spherical subset T of S.
Here a generalized homology n-sphere is a polyhedral homology n-manifold with
the same homology as an n-sphere Sn (cf. [10, Section 5], [11], [25, p. 374], [27]).
Also detail of (virtual) Poincaré duality groups is found in [7], [10], [11], [13].
In Lemma 1.1, we particularly note that the statement (3) is a local condition of L
which determines a global structure of L as the statement (2). From this observation,
it seems that the following theorem holds. (However the proof is not so obvious.)
Theorem 1.2. Let 0 be a finite graph with at least 3 vertices and let (W, S)
be the right-angled Coxeter system associated by 0 (i.e. the 1-skeleton of the nerve
L(W, S) of (W, S) is 0). If the Coxeter group W is an irreducible virtual Poincaré
duality group, then the graph 0 is reconstructible. Hence,
(i) if 0 is the 1-skeleton of some simplicial flag complex L which is a generalized
homology sphere, then the graph 0 is reconstructible, and
(ii) in particular, if 0 is the 1-skeleton of some flag triangulation L of some n-sphere
S
n (n  1), then the graph 0 is reconstructible.
Here, based on this motivation, we investigate a finite graph which is the 1-skeleton
of some simplicial flag complex which is a homology manifold as an extension of a
generalized homology sphere, and we prove the following theorem. (Hence as a corol-
lary, we also obtain Theorem 1.2.)
Theorem 1.3. Let 0 be a finite graph with at least 3 vertices.
(i) If 0 is the 1-skeleton of some simplicial flag complex L which is a homology
n-manifold (n  1), then the graph 0 is reconstructible.
(ii) In particular, if 0 is the 1-skeleton of some flag triangulation L of some n-manifold
(n  1), then the graph 0 is reconstructible.
Here detail of homology manifolds is found in [10, Section 5], [11], [25, p. 374], [27].
1176 T. HOSAKA
2. Proof of Theorem 1.3
We prove Theorem 1.3.
Proof of Theorem 1.3. Let 0 be a finite graph with at least 3 vertices which is
the 1-skeleton of some simplicial flag complex L which is a homology manifold of
dimension n  1. Then we show that the graph 0 is reconstructible.
Let 00 be a finite graph and let L 0 be the finite simplicial flag complex associated
by 00. Also let S and S0 be the vertex sets of the graphs 0 and 00 respectively.
Now we suppose that the condition () holds:
() There exists a bijection f W S ! S0 such that the subgraphs 0S {s} and 00S0 { f (s)}
are isomorphic for any s 2 S.
To show that the graph 0 is reconstructible, we prove that the two graphs 0 and
0
0 are isomorphic, i.e., the two simplicial flag complexes L and L 0 associated by 0
and 00 respectively are isomorphic.
Let v0 2 S and let v00 D f (v0). Then the two subgraphs 0S {v0} and 00S0 {v00} are
isomorphic by the assumption (), and the two subcomplexes L S {v0} and L 0S0 {v00} are
isomorphic. Let  be an isomorphism from L S {v0} to L 0S0 {v00}.
If for any a 2 Lk(v0, L)(0), (a) 2 Lk(v00, L 0)(0) then we obtain an isomorphism
N
 W L ! L 0 from NjL S {v0} D  and N(v0) D v00 (since deg v0 D deg v00), hence L and L 0
are isomorphic.
Now we suppose that there exists a0 2 S   {v0} such that a0  Lk(v0, L)(0) and
a00 WD (a0) 2 Lk(v00, L 0)(0).
Here if there does not exist u00 2 S0   St(a00, L 0)(0), then St(a00, L 0)(0) D S0, where
St(a00, L 0) means the closed star of a00 in L 0. Hence [a00,b0] 2 L 0(1) for any b0 2 S0 {a00}.
Since dega0 D dega00 and jSj D jS0j, [a0,b] 2 L (1) for any b 2 S {a0}. This particularly
implies [a0, v0] 2 L (1). This is a contradiction because it means a0 2 Lk(v0, L)(0).
Thus we suppose that there exists u00 2 S0   St(a00, L 0)(0).
Let u0 WD f  1(u00). Then by the assumption (), the two subcomplexes L S {u0} and
L 0S0 {u00} are isomorphic and let  be an isomorphism from L S {u0} to L
0
S0 {u00}
.
Then
Lk(  1(a00), L S {u0})  Lk(a00, L 0S0 {u00})
 Lk(a00, L 0),
since  is an isomorphism and u00  St(a00, L 0). Also we obtain
St(  1(a00), L S {u0})  St(a00, L 0S0 {u00})
 St(a00, L 0).
Then
St(a00, L 0S0 {v00}) ¤ St(a
0
0, L
0)  St(  1(a00), L S {u0}).
RECONSTRUCTIBLE GRAPHS, SIMPLICIAL FLAG COMPLEXES 1177
Here we note that St(  1(a00), L S {u0}) is either
(a) the closed star St(  1(a00), L) of the vertex   1(a00) in the homology n-manifold
L , or
(b) St(  1(a00), L)   u0 where u0 2 Lk(  1(a00), L),
and also note that St(a00, L 0S0 {v00}) D St(a
0
0, L
0)   v00. Hence we obtain that
(I) St(a00, L 0S0 {v00}) is isomorphic to some closed star deleted one or two vertices from
its link in the homology n-manifold L .
On the other hand,
St(a00, L 0S0 {v00})  St(a0, L S {v0})  St(a0, L),
since  is an isomorphism and a0  St(v0, L). Here we note that St(a0, L) is the closed
star in the homology n-manifold L . Hence we obtain that
(II) St(a00, L 0S0 {v00}) is isomorphic to some closed star in the homology n-manifold L .
Then (I) and (II) imply the contradiction. Indeed the following claim holds.
Claim. Let A D St(a) be a closed star of a vertex a in a homology n-manifold
and let B D St(b)  {c1, c2} be a closed star of a vertex b deleted one or two vertices
{c1, c2}  Lk(b) in a homology n-manifold. Then the simplicial complexes A and B
are not isomorphic.
We first note that every triangulated homology n-manifold is a union of n-simplexes
([25, Corollary 63.3 (a)]). Hence A D St(a) and St(b) are unions of n-simplexes contain-
ing a and b respectively. Then there exists an n-simplex 0 such that c1 2 0  St(b).
Here if c1 ¤ c2 then we can take 0 as c2  0. Indeed if c1 ¤ c2 and c2 2 0
then [c1, c2]  0 and we can consider (n   1)-simplex  as  (0) D  (0)0   {c2}. Then
by [25, Corollary 63.3 (b)], there exist precisely two n-simplexes containing  as a
face. Hence we can take an n-simplex  00 containing  as a face and  00 ¤ 0. Then
c1 2 
0
0  St(b) and c2   00. Hence in this case we retake 0 as  00.
Now 0 is an n-simplex such that c1 2 0  St(b) and if c1 ¤ c2 then c2  0.
Let 0 be the (n   1)-simplex as  (0)0 D  (0)0   {c1}. Then we note that 0  St(b)  
{c1, c2} D B.
Now we suppose that A and B are isomorphic and there exists an isomorphism
g W B ! A. Then g(0) is an (n   1)-simplex in A. By [25, Corollary 63.3 (b)],
there exist precisely two n-simplexes N1 and N2 containing g(0) as a face in A. Then
g 1( N1) and g 1( N2) are n-simplexes containing 0 as a face in B, since g W B ! A is
an isomorphism. Here g 1( N1), g 1( N2) and 0 are distinct n-simplexes containing 0
as a face in St(b). This contradicts to [25, Corollary 63.3 (b)].
Thus the simplicial complexes A and B are not isomorphic.
Hence, there does not exist a0 2 S   {v0} such that a0  Lk(v0, L)(0) and (a0) 2
Lk(v00, L 0)(0), that is, for a 2 S {v0}, a 2 Lk(v0, L)(0) if and only if (a) 2 Lk(v00, L 0)(0),
1178 T. HOSAKA
since degv0 D degv00. Hence the map NW S ! S0 defined by NjS {v0} D  and N(v0)D v00
induces an isomorphism of the two graphs 0 and 00.
Therefore the graph 0 is reconstructible.
3. Virtual Poincaré duality Coxeter groups and reconstructible graphs
We introduce a relation of virtual Poincaré duality Coxeter groups and reconstruct-
ible graphs, which is our motivation of this paper.
DEFINITION 3.1 (cf. [7], [10], [11], [13]). A torsion-free group G is called an
n-dimensional Poincaré duality group, if G is of type FP and if
H i (GI ZG) 

0 (i ¤ n),
Z (i D n).
Also a group G is called a virtual Poincaré duality group, if G contains a torsion-free
subgroup of finite-index which is a Poincaré duality group.
On Coxeter groups and (virtual) Poincaré duality groups, the following results
are known.
Theorem 3.2 (Farrell [13, Theorem 3]). Suppose that G is a finitely presented
group of type FP, and let n be the smallest integer such that H n(GI ZG) ¤ 0. If
H n(GIZG) is a finitely generated abelian group, then G is an n-dimensional Poincaré
duality group.
REMARK. It is known that every infinite Coxeter group W contains some torsion-
free subgroup G of finite-index in W which is a finitely presented group of type FP
and H(GIZG) is isomorphic to H(W IZW ). Hence if n is the smallest integer such
that H n(W I ZW ) ¤ 0 and if H n(W I ZW ) is finitely generated (as an abelian group),
then W is a virtual Poincaré duality group of dimension n.
Theorem 3.3 (Davis [10, Theorem 5.5]). Let (W, S) be a Coxeter system. Then
the following statements are equivalent:
(1) W is a virtual Poincaré duality group of dimension n.
(2) W decomposes as a direct product W D WT0  WT1 such that T1 is a spherical
subset of S and the simplicial complex LT0 D L(WT0 , T0) associated by (WT0 , T0) is a
generalized homology (n   1)-sphere.
Theorem 3.4 ([17, Corollary 4.2]). Let (W, S) be an infinite irreducible Coxeter
system, let L D L(W, S) and let 0  i 2 Z. Then the following statements are equivalent:
(1) H i (W I ZW ) is finitely generated.
RECONSTRUCTIBLE GRAPHS, SIMPLICIAL FLAG COMPLEXES 1179
(2) H i (W I ZW ) is isomorphic to QH i 1(L).
(3) QH i 1(L S T ) D 0 for any non-empty spherical subset T of S.
Here L S T D L(WS T , S   T ).
We obtain the following lemma from results above.
Lemma 3.5. Let (W, S) be an irreducible Coxeter system where W is infinite and
let L D L(W, S). Then the following statements are equivalent:
(1) W is a virtual Poincaré duality group.
(2) L is a generalized homology sphere.
(3) QH i (L S T ) D 0 for any i and any non-empty spherical subset T of S.
Proof. (1) , (2): We obtain the equivalence of (1) and (2) from Theorem 3.3,
since (W, S) is irreducible.
(1) ) (3): We obtain this implication from Theorem 3.4, because if W is a virtual
Poincaré duality group then H i (W I ZW ) is finitely generated for any i .
(3) ) (1): Suppose that QH i (L S T ) D 0 for any i and any non-empty spherical
subset T of S. Then by Theorem 3.4, H iC1(W I ZW ) is finitely generated for any i .
Since W is infinite, H i0 (W I ZW ) is non-trivial for some i0 (cf. [7], [14]). Hence by
Theorem 3.2, W is a virtual Poincaré duality group.
We obtain Theorem 1.2 from Theorem 1.3. In particular, we obtain the following.
Theorem 3.6. Let 0 be a finite graph with at least 3 vertices and let (W, S)
be the right-angled Coxeter system associated by 0. If the Coxeter group W is an
irreducible virtual Poincaré duality group, then the graph 0 is reconstructible.
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Department of Mathematics
Shizuoka University
Suruga-ku, Shizuoka 422-8529
Japan
e-mail: hosaka.tetsuya@shizuoka.ac.jp


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Reconstructible graphs, simplicial flag complexes of homology manifolds and associated right-angled Coxeter groups

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