An Algorithmic Approach to Fundamental Groups
and Covers of Combinatorial Cell Complexes
SARAH REES and LEONARD H. SOICHER
Department of Mathematics, University of Newcastle, Newcastle NE1 7RU, UK,
e-mail: [email protected]
School of Mathematical Sciences, Queen Mary and Westeld College, University of
London, Mile End Road, London E1 4NS, UK, e-mail:[email protected]
Abstract. We rst develop a construction, originally due to Reidemeister, of the
fundamental group and covers of a combinatorial cell complex. Then, we describe
a practical algorithmic approach to the computation of fundamental groups, rst
homology groups, deck groups, and covers of nite simple such complexes. In the
case of clique complexes of nite simple graphs, the algorithms described have been
implemented in GAP, making use of the GRAPE package.
Mathematics Subject Classications (1991): 57M05, 57M10, 57M15, 57M20.
Key words: cell complex, algorithms, homotopy, fundamental group, covers
Acknowledgements
We thank Klaus Lux, Armando Martino and Martin Roller for very
helpful discussions during the preparation of this paper.
1. Introduction
Finite 2-complexes and their covers arise naturally in the studies of
nite geometries, groups and graphs. In particular, we are interested
in problems such as determining the fundamental group of a nite 2complex and classifying the r-fold covers of that complex for a given
r. This has led us to a combinatorial and algorithmic approach to the
study of nite 2-complexes, which is the main subject of this article.
Furthermore, we have been interested in complexes with an associated
group of automorphisms, and the relationship of this group action to
the covers of that complex.
We shall describe a practical algorithmic approach to the computation of fundamental groups, rst homology groups, deck groups, and
covers, in the general context of nite simple 2-dimensional combinatorial cell complexes. For the case of clique complexes of nite simple
graphs, the algorithms described have been implemented in GAP [22],
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SARAH REES and LEONARD H. SOICHER
making use of the GRAPE package [26]. This implementation should
eventually be included in GRAPE.
Our denition of a 2-dimensional combinatorial cell complex follows
Reidemeister's denition of a `Flachenkomplexe' in [19],[23] (the `Surface complex' of the English translation [24]). The denitions and the
basic results of section 3, and the basis of the graph-theoretic imagery
of section 4 are found in chapters 4, 5, and 6 of [19], although our
notation is slightly dierent from Reidemeister's. One of our aims has
been to consolidate and extend Reidemeister's point of view, and we
do this in the later part of section 4. General topological background
is provided by Maunder's book [14].
The remainder of this paper is divided into four parts. Section 2 contains a very brief history of the use of combinatorial topology which
motivated our own study, section 3 contains the basic denitions and
results for the algebraic topology of combinatorial cell complexes necessary for this paper, section 4 describes the view of the fundamental
group and covering spaces of such objects which is used by the algorithms, and section 5 describes the algorithms.
2. Some History
The standard algebraic topological notions of homotopy, coverings, fundamental groups, and homology have been studied in various, essentially equivalent, combinatorial settings.
Reidemeister developed a theory for graphs, and then for `Flachenkomplexe' (and hence simplicial complexes) in [19]. His approach (which
is also described by Seifert and Threlfall in [23], [24]) has descended
into mathematical folklore, and is the basis of our own treatment.
In [28], Tits developed a theory of various types of covers for chamber systems, and in particular locally nite incidence geometries, and
derived a local characterisation of buildings. Further, he proved that
certain local properties of an incidence geometry (that is, the fact that
certain subdiagrams do not occur in its diagram) ensure that its universal (2-) cover is a building, and hence that it can be found as a
quotient of that building. Geometries of this type for various sporadic
and exceptional groups are described in [10],[11],[12],[21]. Tits' theory
of covers for chamber systems was further developed in [20],[27],[17]; in
particular Ronan proved that the systems of covers of a chamber system
correspond to the systems of topological covers of a related simplicial
complex.
Tits developed in [29] a theory of homotopy and covers for partially
ordered sets, and in particular for the system of ags of a nite geome-
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3
try; he extended results of Serre for groups acting on trees, and proved
that a ag-transitive group of automorphisms of a simply connected
nite geometry (that is, one with trivial fundamental group) is a free
product with amalgamation (over ag stabilisers) of vertex stabilisers,
for the vertices in a xed maximal ag. (This result was also proved
independently by Pasini (see [17]) and by Shpectorov.) Subsequently, various sporadic simple groups were described as free amalgams,
through their relationship to simply connected nite geometries, as,
for example in [9],[2]. In [3] a general theory of uniqueness systems is
developed which allows proofs of the uniqueness of many of the sporadic simple groups, through their representation as free amalgams,
associated with a simply connected nite simplicial complex.
The homotopy invariants of the partially ordered sets Sp (G) of psubgroups, and Ap(G) of elementary abelian p-subgroups of a nite
group G were investigated by Quillen in [18]. Quillen's conjecture for
various classes of groups was investigated in [1], [5]; appropriate topological machinery in the context of simplicial complexes is developed
in [4].
We also remark that covers of nite clique complexes arise naturally
in the study of graphs that are locally a given graph and in the study of
distance-regular antipodal covers of distance-regular graphs (see [6],[7]).
An algorithmic approach to combinatorial cell complexes
3. The Algebraic Topology of a Combinatorial Cell Complex
We dene a 2-dimensional combinatorial cell complex ? to be a nonempty undirected graph, possibly with loops or multiple edges, in which
certain circuits, known as the 2-cells, or faces, of ?, are specied. A
vertex is also called a 0-cell, and an edge a 1-cell. The vertex-set of ?
is denoted V (?), and the edge-set E (?). For brevity, we shall often use
the term 2-complex to mean 2-dimensional combinatorial cell complex.
If there are no loops and no multiple edges, then ? is simple. In
sections 4 and 5 we shall assume this to be the case, and also that ? is
connected, which for our purposes means that ? is connected as a graph.
If ? is simple and all 2-cells are triangles, then ? is a simplicial complex, but we shall not require this in general; however, we observe that
at most two barycentric subdivisions will transform any 2-complex ?,
simple or otherwise, into a simplicial complex. The complex ? might be
the 2-skeleton of a higher dimensional complex, but higher dimensional
cells are irrelevant to us, since we are only interested in this paper in
the low dimensional topology of ?.
Throughout this paper we shall study ? through its underlying graph
(or 1-skeleton), and so we shall use the language and notation of graph
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SARAH REES and LEONARD H. SOICHER
theory, alongside more topological language. We shall abuse notation
and use the same label ? for the graph as for the 2-complex.
We dene a path p in ? to be a path in the underlying graph, that is,
a sequence, v0 ; e1 ; v1 ; e2 ; : : : ; en ; vn of vertices vi and edges ei , such that
each ei is incident with the vertices vi?1 and vi ; of course, if ? is simple,
then p is completely determined by the sequence of vertices. We call
v0 the initial vertex and vn the terminal vertex of p. The reverse of p,
p?1 , is dened to be the path vn ; en ; : : : ; e2 ; v1 ; e1 ; v0 , and for any path
q whose initial vertex is the terminal vertex of p, pq is dened to be the
concatenation of p and q (with the terminal vertex of p identied with
the initial vertex of q). If v0 = vn , then p is called a circuit based at v0 .
Let b be a specied vertex of ?, which we shall call the basepoint of
?. Suppose that c and c0 are two circuits based at b. We say that c and
c0 are elementary homotopy equivalent if there are paths p; q; q0 ; r of ?
(not necessarily of non-trivial length) such that c = pqr, c0 = pq0r and
qq0?1 is a 2-cell of ?. More generally, c and c0 are homotopy equivalent
if there is a sequence c0 ; c1 ; : : : ; ck of circuits in ?, each based at b,
such that c0 = c, ck = c0 and for each i = 0; : : : ; k ? 1, ci and ci+1 are
elementary homotopy equivalent.
The fundamental group of ? based at b, 1 (?; b), is dened to be the
group of all homotopy equivalence classes of circuits based at b, composed by concatenation of equivalence class representatives. Provided
that ? is connected (as a graph), the isomorphism type of the group is
independent of the choice of b. This is our situation, and we shall often
simply denote the fundamental group of the 2-complex ? by G. ? is
said to be simply connected if it is connected as a graph and has trivial
fundamental group.
For our purposes, the rst homology group of ? based at b is dened
to be the abelianized quotient G=[G; G] of the fundamental group G,
and, for p a prime, the rst homology group mod p is dened to be the
quotient of the rst homology group by the subgroup consisting of the
p-th powers of all the elements.
A morphism from a 2-complex ?0 to ? is dened to be a map from ?0
to ? which maps i-cells of ?0 to i-cells of ?, for i = 0; 1; 2, and preserves
incidence between cells. A bijective morphism is called an isomorphism,
and an isomorphism from ? to itself is called an automorphism of ?.
A morphism from a 2-complex ?^ to ? is called a covering map if
maps ?^ onto ? and is locally an isomorphism, equivalently, for any
0,1 or 2-cell of ?, ?1 is a set of cells no two of which are incident
with any common cell 0 of ?^ . The pair (?^ ; ) (or sometimes simply the
2-complex ?^ ) is then called a cover of ?. For any cell of ? the set ?1
of cells of ?^ which map under to is called the bre of ; provided
that ? is connected, the cardinality of the bre of is independent of
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5
the choice of . Where m is an integer, is called an m-fold covering
map and (?^ ; ) an m-fold cover if for each , ?1 has cardinality m.
If (?^ ; ) is a cover of ?, then, where ^b = b, the morphism maps
circuits of ?^ based at ^b to circuits of ? based at b. If c; c0 are distinct
circuits based at ^b, then c and c0 are distinct, and c; c0 are homotopy
equivalent if and only if c and c0 are. Hence clearly the fundamental
group of ?^ is isomorphic to a subgroup of the fundamental group of ?.
More generally, if (?1 ; 1 ) and (?2 ; 2 ) are covers of ?, and b1 ; b2
are vertices in ?1 , ?2 respectively such that b1 1 = b2 2 , then there
is a covering map from ?1 to ?2 with b1 = b2 if and only if the
fundamental group of ?1 relative to b1 is isomorphic to a subgroup of
the fundamental group of ?2 relative to b2 .
A cover (?~ ; ) of ? is said to be a universal cover of ? if ?~ is simply
connected. By the above, ?~ is determined uniquely up to isomorphism,
and covers all other covers of ?. Further, if 1 and 2 are two covering
maps from ?~ to ? then there is an isomorphism of ?~ such that
1 = 2 . In particular this implies that any automorphism of ? lifts
to an automorphism ~ of ?~ with the property that ~ = , since must be a covering map. Note also that if (?^ ; ) is any cover of ?, then
for some , (?~ ; ) is a universal cover of ?^ .
An automorphism of ?^ with the property that = is called a
deck transformation of the cover (?^ ; ) of ?. The deck transformations
of (?^ ; ) form a group. The group of deck transformations of (?~ ; ) is
isomorphic to the fundamental group of ?, provided that ? is connected.
A fundamental region for ? is dened to be a set F of cells of ?~ which
contains a unique element of the bre of , for each cell of ?. The
images of F under the group of deck transformations of ? partition ?~ ;
? can be described as the quotient of ?~ under the action of that group.
An algorithmic approach to combinatorial cell complexes
4. An Edge-Labelled Graph Viewpoint
From now on, we assume that our 2-complex is simple (no loops and
no multiple edges) and connected. The basis of the description of the
fundamental group and covers of ? given in subsections 4.1, 4.2 and
4.3 below is found in [19]; we believe that the further development of
this viewpoint in the remainder of section 4 is new.
4.1. The Fundamental Group
Let T be a spanning tree of the connected graph ?. To each edge
e = fv; wg of ? not in T , associate a pair of directed edges (v; w) and
(w; v), and a circuit of ? based at b, formed by joining b by a simple
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SARAH REES and LEONARD H. SOICHER
path in T to v and then w and then back to b by a simple path in T .
It is well-known (see for example [19]) that the homotopy equivalence
classes of these circuits form a generating set for the fundamental group
G. Let gv;w be the generator of G corresponding to the circuit through
(v; w). Now where each (directed) edge (v; w) is assigned the label gv;w
?1 ), and each edge in T is labelled with the
(and its reverse the label gv;w
trivial word, any circuit in ? is naturally labelled by a word in the gv;w 's
and their inverses which is formed by composing the labels on the edges
of the directed circuit; this gives an expression for the corresponding
element of the fundamental group in terms of the generators gv;w . The
set of all labels of circuits of the form pcp?1 , for which p is any path
with initial vertex b, and c a 2-cell, forms a full set of relators for the
group.
Where b0 is a vertex distinct from b, and q is a path in T from b
to b0 , there is a natural correspondence between circuits based at b
and circuits based at b0 , which matches a circuit c based at b with the
circuit q?1 cq based at b0 . Hence it is clear not only that the fundamental
groups based at b and b0 are isomorphic but that the set of generators
of the form ge described above is natural for both groups, in the sense
that it depends only on T and not on b or b0 .
4.2. The Covers of ?
Suppose that (?^ ; ) is a cover of ?. Let b be the basepoint of ?, and
label the elements of the bres of b in ?^ by pairs (b; i) for i in some
indexing set I . Let T be a spanning tree of ?. Then the vertices of ?^ can
be partitioned by trees Ti such that bi 2 Ti , Ti = T , and restricted
to Ti is an isomorphism. For each vertex v of ?, let (v; i) label the single
vertex in Ti \ v?1.
Now if fv; wg is an edge of ?, let v;w be the function from I to
I dened by iv;w = j if f(v; i); (w; j )g is an edge of ?^ . It follows
from the fact that is a covering map that v;w is a permutation
of I . If v0 ; v1 ; : : : ; vk , where vk = v0 , is a circuit c in a 2-cell, then
c?1 is a disjoint union of circuits in ?^ , and so v0 ;v1 v1 ;v2 : : : vk?1 ;vk
is the identity permutation. From this it is clear that the map : G !
= v;w extends to a homomorphism from G
Sym(I ) dened by gv;w
to Sym(I ). The cover ?^ is completely dened by the homomorphism
; every homomorphism from G to Sym(I ) denes a cover. Connected
covers correspond to transitive permutation representations of G, and
for intransitive permutation representations, the number of orbits is
equal to the number of connected components of the corresponding
cover. From now on, given a specied homomorphism : G ! Sym(I )
we shall label the associated cover of ? by ? .
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Where H is the fundamental group 1 (? ; (b; i)) of ? based at the
vertex (b; i), H is isomorphic to the preimage in G of the stabiliser in
Sym(I ) of i. Hence ? can be dened in terms of the right cosets of H
in G. The vertices of ? correspond to pairs (v; Hx), for v a vertex of ?
and x 2 G. Whenever v and w are joined in ?, (v; Hx) and (w; Hxgv;w )
are joined in ? .
An algorithmic approach to combinatorial cell complexes
4.3. The Universal Cover
The right regular representation of G denes the universal cover of ?,
that is, ?~ is isomorphic to the cover ?reg whose vertices correspond to
pairs (v; x), where v is a vertex of ? and x 2 G. Edges of ?reg join pairs
(v; x) and (w; xgv;w ) where v and w are joined in ? and gv;w is the label
on that edge of ?. For any g 2 G, the product of the labels of a path
from (v; x) to (v0 ; xg) must be a word equal to g in G.
To verify that this construction really does yield the universal cover,
we need only to see that ?reg is simply connected. Let be the covering
map from ?reg to ?, dened by (v; x) = v. Now suppose that c is a
circuit in ?reg based at (b; 1). Then, as a path from (b; 1) to (b; 1), c
can only be labelled by a word equal to the identity element. The same
is true of c. Since G is the fundamental group of ? relative to b, this
implies that c is homotopically equivalent to the trivial loop, Since
homotopies of ? lift to homotopies of ?reg , the same is true of c.
4.4. Lifting an Automorphism to the Universal Cover
Suppose that is an automorphism of ?. Then for each element of G
a lift of to Aut(?~ ) is dened. More precisely, the following is true.
PROPOSITION 4.4.1. Each automorphism of ? lifts to an automorphism of ?~ . For each automorphism 2 Aut(?) and for each element
g 2 G, there is a unique automorphism ~ of ?~ such that
(b; 1)~ = (b; g)
More precisely, ~ is dened by the rule
(v; x)~ = (v; gx tv; )
(1)
for each (v; x) 2 ?~ , where x is the label on the image under of any
circuit based at b with label x, and tv; is the label on the image under
of the path in T from b to v.
Proof: The fact that lifts to an automorphism of ?~ is simply a
consequence of the universality of ?~ .
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SARAH REES and LEONARD H. SOICHER
Uniqueness is determined once the rule (1) has been veried.
First we observe that induces an automorphism of the fundamental
group G, viewed as an abstract group with the sets of generators and
relators described, rather than as the fundamental group based at the
specic basepoint b. For induces an action on the set of circuits of
?, and preserves homotopy between circuits. Circuits based at b are
mapped under to circuits based at b. If x is the label of a circuit
based at b (reading from b to b), and x0 the label of the image of that
circuit (reading from b to b), then the map induced by which takes
x to x0 is certainly a homomorphism from the free group on the set of
all generators ge to itself, since the concatenation of two circuits is
clearly mapped to the concatenation of their images. Since preserves
homotopy, it xes the set of relators described above, and so induces
a homorphism from G to itself. Finally, since is invertible, it must
act as an automorphism of G. x is then the image of x under this
automorphism.
Now suppose that p and p0 are two paths from b to some vertex v,
both labelled by the same element x of G. Then pp0?1 is a circuit based
at b and labelled with the identity element 1 of G, and hence is mapped
under to a circuit based at b also labelled by the identity element,
since 1 = 1. So p and p0 carry the same label, which we shall call
xv .
Let q be the path in T from b to v. Then tv; is dened to be the
label of q. (Note that tv; = 1v .) So the circuit p(q)?1 has label
xv t?v;1 . However p(q)?1 is the image of the circuit pq?1 under ,
and so must have label x . Hence
xv = x tv; :
Let ~ be a lift of dened by (b; 1)~ = (b; g).
Now any edge or path in ?~ has the same label as its projection in
?. Hence if v and w are adjacent vertices of ?, then (v; x)~ is joined to
(w; xgv;w )~ by an edge with the label gv;w . So
(v; x)~ = (v; y) implies that (w; xgv;w )~ = (w; ygv;w ): (2)
Any path c from (b; 1) to (v; x) has label x, and xv is the label of its
image under ~ . Hence it follows from (2) that
(v; x)~ = (v; gxv ) = (v; gx tv; ):
(3)
2
As an immediate corollary of Proposition 4.4.1 we have the following:-
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An algorithmic approach to combinatorial cell complexes
COROLLARY 4.4.2. For any g 2 G, the rule
9
(v; x)g = (v; gx);
for all v; x, denes an automorphism g of ?~ with the property that
g = . Any automorphism of ?~ with the property that = must
be of this form for some g 2 G.
Proof: Any automorphism of ?~ with the property that = is a
lift of the trivial automorphism of ?. By Proposition 4.4.1, for any
g 2 G, there is a unique lift g of with the property that
(b; 1)g = (b; g);
and by (3), for all v; x,
(v; x)g = (v; gx 1) = (v; gx):
2
This veries that the fundamental group G of ? is isomorphic to the
full group of deck transformations of the universal cover.
Clearly we have shown that the lifts of are all composites with
deck transformations of a lift ~ 1 which maps (b; 1) to (b; 1), and hence
satises
(v; x)~1 = (v; x tv; ):
We shall call this lift the principal lift of to ?~ .
The principal lifts of automorphisms of ? do not form a subgroup of
Aut(?~ ). For, if a lift ~ of and its inverse ~ ?1 are dened by the rules
(v; x)~ = (v; g1 x tv; ); (w; y)~?1 = (w?1 ; g2 y?1 tw;?1 )
then g1 and g2 are related by the equation
g1 g2 tb?1 ; = 1:
(4)
For any deck transformation g of ?~ , the conjugate of g by any lift ~
of is an automorphism of ?~ which induces the identity automorphism
on ?, and so is itself a deck transformation. So ~ is an element of
the subgroup of automorphisms of ?~ which normalise the subgroup
TG = fg : g 2 Gg of Aut(?~ ).
In fact, where ~ and its inverse are dened as above, using (4), we
see that
~ ?1 g ~ = g1 g g1?1
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SARAH REES and LEONARD H. SOICHER
and so in particular, when ~ is the principal lift of ,
~ ?1 g ~ = g ;
that is, the action by conjugation of the principal lift of on G as a
group of deck transformations of ?~ is identical to the action of itself
on G as the fundamental group of ? based at b. The action of any other
lift of is conjugate to the action of the principal lift.
4.5. Mapping Down an Automorphism of the Universal
Cover.
The preceding analysis gives a precise formula for the lift to ?~ of any
automorphism of ?, and hence gives a precise condition for an automorphism of ?~ to induce an automorphism of ?.
An automorphism of ?~ induces an automorphism of ? if and only
if it permutes the orbits of the deck group TG of ?~ ; in that case is a
lift of the automorphism dened by the rule
v = (v; x)
and the action of is completely dened once one of its images is
dened, using the above analysis.
We have already shown that such an automorphism must normalise the subgroup TG of deck transformations. Conversely, if is
an automorphism of ?~ which normalises TG , then must permute the
orbits of TG . Thus the automorphisms of ?~ which induce automorphisms of ? are precisely those which normalise the subgroup TG of
deck transformations.
Suppose that H is a subgroup of G, and that H is the image of H
under the action of described above. Then induces an isomorphism
0 between the two covers ?1 = ?~ =H and ?2 = ?~ =H of ?, mapping,
for all v; x, (v; Hx) to (v; H x ). Where 1 is the covering map from
?1 to ? dened by (v; Hx)1 = v and 2 the covering map from ?2 to ?
dened by (v; H x )2 = v, then 2 = 0 1 . If H = H , then induces
an automorphism of ?~ =H .
4.6. Lifting an Automorphism to an Intermediate Cover
Let ?^ = ? be an intermediate cover of ?, with fundamental group H ;
then the vertices of ?^ can be described as pairs (v; Hx) for v a vertex
of ?, and x 2 G, as described above. Thus the vertices of ?^ are seen
naturally as the orbits of H as a group of deck transformations on the
vertices of ?~
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PROPOSITION 4.6.1. Suppose that is an automorphism of ?, and
that b is a vertex of ? and g an element of G. For any x 2 G, dene
x to be the label of the image under of any loop based at b with label
x. Let ?^ be a cover of ? with fundamental group H G.
Then lifts to an automorphism ^ of ?^ which maps (b; H ) to
(b; Hg) if and only if H H g .
An algorithmic approach to combinatorial cell complexes
Proof: The automorphism lifts to an automorphism of ?^ precisely
when one of its lifts ~ in Aut(?~ ) maps down to an automorphism of ?^ .
Clearly ~ as above maps down to an automorphism of ?^ provided
it permutes the orbits of H acting as a group of deck transformations.
More precisely, ~ maps down if and only if, given (v; x) 2 ?~ , for each
h 2 H , there is h0 2 H such that
(v; hx)~ = (v; x)~h0
By Proposition 4.4.1 above, we see that, where (b; 1)~ = (b; g), this
condition translates as
(v; g(hx) tv; ) = (v; h0 gx tv; );
or equivalently
gh = h0 g;
Hence ~ maps down to ?^ if and only if H H g .
Thus has a lift which is an automorphism mapping (b; H ) to
(b; Hg) for each g 2 G such that H H g . When ^ is such a lift,
then
(v; Hx)^ = (v; Hgx tv; )
2
4.7. Equivalence of Covers
Suppose that (?1 ; 1 ) and (?2 ; 2 ) are two covers of ?, associated with
subgroups H1 and H2 of the fundamental group G. We need to decide
under what circumstances we should consider (?1 ; 1 ) and (?2 ; 2 ) to
be equivalent. Let 1 be the covering map from ?~ to ?1 and let 2 be
the covering map from ?~ to ?2 .
First observe that if ?1 and ?2 are isomorphic as 2-complexes, related by an isomorphism : ?1 ! ?2 , then since 1 and 2 are then
both covering maps from ?~ to ?2 , and ?~ is a universal cover for ?2 ,
lifts to an automorphism ~ of ?~ .
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SARAH REES and LEONARD H. SOICHER
It is standard to consider (?1 ; 1 ) and (?2 ; 2 ) to be equivalent as
covers when ?1 and ?2 are related by an isomorphism with the property that 1 = 2 . From a computational point of view it is relevant
also to consider other, broader, notions of equivalence.
We nd isomorphisms with 1 = 2 using deck transformations. For suppose that H1 and H2 are subgroups of G such that
H2 = gH1 g?1 . Let (?1 ; 1 ) and (?2; 2 ) be the covers of ? associated
with H1 and H2 . Then the deck transformation g induces an isomorphism between ?1 and ?2 such that (v; H1 x) maps to (v; H2 gx). Clearly
1 = g 2 . We shall say that ?1 and ?2 are equivalent under the action
of the group of deck transformations.
In fact, any : ?1 ! ?2 with 1 = 2 , must arise in this way. For
such a induces the trivial automorphism on ?. Hence the lift ~ of
to ?~ acts as a deck transformation by Corollary 4.4.2.
We nd a more general notion of equivalence, by allowing to induce
a (possibly non-trivial) automorphism on ?. In that case 1 = 2 .
Then is induced by the action of a lift = ~ of to ?~ .
Suppose that is the (principal) lift of dened by the rule
(v; x) = (v; x tv; )
(Any other lift is a composite of this with a deck transformation.)
Now let H1 be any subgroup of G and H2 its image under the action
of .
Then the set of vertices of ?~ of the form (v; hx) for h 2 H1 maps
under to the set of vertices of the form (v; h x tv; ) So induces an
isomorphism between ?1 and ?2 with the property that 1 = 2 .
Conversely, suppose that is an isomorphism between covers ?1
and ?2 with the property that 1 = 2 for some automorphism of
?. Suppose that
(b; H1 ) = (b; H2 )
(This can be achieved if necessary by composing with the isomorphism induced by a deck transformation, and replacing H2 by a conjugate.) Then arises exactly as described above.
We see therefore that we can classify covers of ? up to the standard
notion of equivalence (1 = 2 ) by enumerating the conjugacy classes of subgroups of G. If the r-fold covers of ? are required, for r at
most some xed positive integer m, then we may apply the low-index
subgroup algorithm with input m and a nite presentation for G, to
determine up to conjugacy the subgroups of G of index of at most m.
However, this can be an extremely time-consuming process. The lowindex subgroup algorithm is described in [25, section 5.6] and in [16].
A method for the case when G is polycyclic is described in [13].
homotopy.tex; 15/05/1998; 17:39; no v.; p.12
13
From a computational point of view, it may make more sense to enumerate the equivalence classes of subgroups of G for which H1 and H2
are equivalent if related by a composite of conjugation and an automorphism of ? (in its action on G as a fundamental group), or (equivalently)
if related by an automorphism of ?~ which normalises G (in its action
on G by conjugation as a group of deck transformations). We could
require that automorphism to be in a specied subgroup of Aut(?~ ).
From a computational point of view, such a denition of equivalence
could well be very useful.
The discussion above suggests that it would be worthwhile to attempt
to design a version of the low-index subgroup algorithm which would
classify the subgroups of index at most m in a nitely presented group
H , up to action by a composite of conjugation and an element of an
explicit group A of automorphisms of H , where A is given by specifying
the images of the generators of H under the generators of A.
An algorithmic approach to combinatorial cell complexes
5. The Algorithms
We describe our algorithms for an arbitrary nite, simple, connected 2complex ?. However, the current GAP/GRAPE implementations of our
algorithms are only for the special case of a clique complex of a nite
simple graph (in which the 2-cells are precisely the triangles). Work
remains to be done on ecient implementation for more general classes
of complexes. We have used our current implementation successfully
on complexes with over 1,000 vertices and over 100,000 edges, but the
range of applicability depends heavily on the nature of the fundamental
group.
5.1. Building a Fundamental Record of ?
Let ? be a nite, simple, connected 2-complex, and for each edge fv; wg
of ?, associate the pair of directed edges (v; w), (w; v). A fundamental
record (P; f ) of ?, produced by the algorithm of this section, consists
of a nite presentation P = (X ; R) (with generators X and relators R)
for the fundamental group G of ?, and a labelling (mapping) f from
the directed edges of ? to the free group on X , such that the G-image
of the label fv;w of (v; w) is the edge-label gv;w encountered in section 4.
The labelling f will be used to construct covers of ? as described in
section 4.2.
The presentation (X ; R) for the fundamental group is constructed
in such a way as to attempt to minimize the number of generators.
This approach is appropriate if the fundamental group is cyclic or if
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14
SARAH REES and LEONARD H. SOICHER
we are abelianizing the fundamental group as we proceed (to compute
the rst homology group). Otherwise, the length of relators in R may
explode.
Roughly speaking, we build a copy of ? one edge at a time, labelling
each directed edge (if necessary with a new generator) as we do so. At
any stage, if a label can be found for a directed edge which is a word in
existing generators, then this label is used. If two distinct such labels
can be found, then a relation has been discovered.
More precisely, the algorithm runs as follows. The input is a nite,
simple, connected 2-complex ?, and the output is a fundamental record
for ?.
We initialize the complex to be a spanning tree T of ?, set P to
be the presentation with no generators and no relators, and set f to
map each directed edge of to the trivial word. If at this stage = ?,
we are done, and output (P; f ). Otherwise, we shall extend edge by
edge until it is equal to ?. At each stage that we modify we shall
update (P; f ) to remain a fundamental record for .
Suppose that 6= ?. Our basic step is to search for an edge fv; wg 2
E (?) n E (), such that, for some v1 ; : : : ; vn 2 V (?),
? v; v1 ; : : : ; vn; w is a path in , and
? c := v; v1 ; : : : ; vn ; w; v is a 2-cell of ?.
Suppose this search is successful. Then we add the edge fv; wg to
, together with all 2-cells of ? involving fv; wg, such that these 2-cells
induce circuits in . We then dene
fv;w := fv;v1 fv1;v2 fvn ;w ;
and
fw;v := (fv;w )?1 :
If P is not obviously a presentation for the trivial group, then we must
check if we need to add more relators to P . To do this, for each 2-cell
c0 6= c in of the form c0 = v; w1 ; : : : ; wm ; w; v, we add to P the relator
fv;w1 fw1;w2 fwm ;w fw;v :
After we have done this, (P; f ) is a fundamental record for . We may
choose to try to simplify the relators of P at this stage, and perhaps to
change the edge-labels to shorter words (if possible) representing the
same elements of the fundamental group of .
If our search of is not successful, a new generator g is now added
to P . We then choose an edge fv; wg 2 E (?) n E (), add this edge to
E (), dene
fv;w := g and fw;v := g?1 :
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15
If at this stage we still have 6= ?, we go back and repeat our basic
search step. Otherwise we are done, and output (P; f ).
Remark In our implementation for clique complexes we do no actual
searching in the \basic step". Whenever an edge fx; yg is added to ,
the new further edges (in triangles of ? containg fx; yg) which can be
added to to complete a 2-claw to a triangle are determined, and these
new edges are queued for future addition to . (A bit-array (boolean
list in GAP) is used for fast membership testing in this queue, and no
edge is queued more than once.) Thus, the \search" simply consists of
taking an edge o that queue, or determining that the queue is empty
(in which case a generator must be added to P ).
An algorithmic approach to combinatorial cell complexes
5.2. Computing the First Homology Group of ?
A presentation for the rst homology group of ? can of course be found
by rst computing a presentation for the fundamental group of ?, and
then abelianizing this presentation. In order to keep the size of our
computation down from the start, however, we choose instead to follow the algorithm described above for the fundamental record, except
that the presentation P is always abelianized, and the labelling f is a
mapping from the directed edges of to the free abelian group on the
generators X of P . Doing this, it is often possible to compute quickly
the rst homology group when computing the fundamental group with
the algorithm above would not be practical. We call the output (P; f )
of this abelianized form of the algorithm the abelianized fundamental
record of ?.
Algorithms to determine the structure of a nitely presented abelian
group are discussed in [25, Chapter 8] and [8]. We also mention that a
dierent approach from ours to computing rst homology groups has
been developed by Steve Linton (unpublished).
5.3. Computing the First Homology Group mod p
Suppose that p is a prime and that we wish to compute the rst homology group mod p of ?, that is, H1 (?; F p ). Then we follow an algorithm
similar to that above for the rst homology group, except that instead
of a presentation, we (implicitly) maintain the homology group mod
p of the subcomplex as a vector space V of all d-tuples over F p ,
where d = dim(H1 (; F p )). The labelling f is then a mapping from
the directed edges of to V . When a non-trivial relator is found, the
dimension d of V is decreased by one, and the edge-labels are appropriately rewritten to lie in V .
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SARAH REES and LEONARD H. SOICHER
5.4. Computing Covers
Suppose that ? is a nite, simple, connected 2-complex with fundamental group G. Then, as described in section 4.2, every cover ? is of
the form ? , where is a permutation representation of G.
Suppose that we are given a fundamental record (P; f ) of ?, and a
nite permutation representation : G ! Sym(I ). For example, may
have been obtained from the presentation P by using coset enumeration
or the low-index subgroup algorithm (see [16] or [25]). We can construct
the cover ? of ? as described in section 4.2, after dening, for each
directed edge (v; w) of ?,
;
v;w := gv;w
where gv;w is the natural image of fv;w in G. We remark that if G is
abelian then we may use an abelianized fundamental record instead of
a fundamental record.
5.5. Computing the Deck Group of a Cover
Suppose that we are given a permutation representation : G ! Sym(I )
of the fundamental group G of the nite, simple, connected 2-complex
?. Then a typical bre of the cover (? ; ) is
v?1 = f(v; i) : i 2 I g;
where v 2 V (?). The deck group D Aut(? ) of ? acts on each such
bre.
Let 2 D, v 2 V (?), i 2 I , and suppose
(v; i) = (v; j ):
Then if (v; w) is a directed edge of ?, we must have
(w; iv;w ) = (w; jv;w ):
(5)
If (v; w) is a directed edge of the xed spanning tree T of ? on which the
edge-labels are trivial then v;w = 1 and we have that (w; i) = (w; j ).
Indeed, if w is any vertex of ?, then by induction on the length of a
path in T from v to w, we still must have (w; i) = (w; j ). This gives
us a natural faithful representation
: D ! Sym(I )
dened by i = j if and only if (v; i) = (v; j ), and this denition does
not depend on the choice of v 2 V (?). Equation (5) further tells us
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17
that D must centralize G . Conversely, each element in the Sym(I )centralizer of G denes a deck transformation of ? .
Note that the above gives an alternate way of seeing that the deck
group D of the universal cover ?~ of ? is isomorphic to the fundamental
group G of ?, since the Sym(G)-centralizer of the right-regular representation of G is the left-regular representation of G.
We can compute the deck group of an r-fold cover ? of ? by computing
D := CSym(r) (G ):
If G is transitive, which is equivalent to saying that ? is connected,
then this computation is very easy. In this case, for i = 1; : : : r, there is
at most one 2 D mapping 1 to i. It is easy to determine if such an
exists, and if so, to determine this . (Suppose that 2 D, 1 = i,
and j 2 f1; : : : ; rg. Then j = 1g for some g 2 G, and so j = 1g =
1g = ig .)
An algorithmic approach to combinatorial cell complexes
5.6. Lifting an Automorphism to a Cover
We now suppose that we have a fundamental record (P; f ) for our nite,
simple, connected 2-complex ?, with fundamental group G, and that
?^ is a connected r-fold cover of ?, dened by a transitive permutation
representation
: G ! Sym(r):
Hence, for any directed edge (v; w) of ? we can determine the associated
permutation v;w 2 G .
Suppose that is an automorphism of ?. The theory of section 4
tells us exactly when should lift to an automorphism of ?^ . If there is
a lift, its eect is completely dened by the image of one vertex. Hence
we attempt to lift by dening an image for a chosen vertex of ?^ and
then attempting to extend the image outwards from that vertex. If the
image extends, then has such a lift and we have found it; if it does
not, then has no such lift.
The vertices of ?^ are the pairs (v; i), where v is a vertex of ? and
i 2 f1; : : : ; rg.
We want to see if can lift to an automorphism ^ of ?^ which maps
(v1 ; 1) to (v1 ; i).
We form a queue, which initially consists just of (v1 ; 1), containing
vertices of ?^ whose images under ^ have been dened. Then, as long
as this queue is non-empty, we remove a vertex a from the queue, and
do the following for each vertex b adjacent to a:
? If no image of b under ^ has been dened then we dene an image:
if a = (v; x), a^ = (v; y), and b = (w; xv;w ), then b^ must be
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18
SARAH REES and LEONARD H. SOICHER
(and is) dened as the vertex b0 = (w; yv;w ). If the image of a
vertex dierent from b has already been dened to have image b0
under ^ , then ^ cannot dene a permutation of V (?^ ) and so the
required ^ does not exist. Otherwise we insert b in the queue and
continue.
? If, on the other hand, an image of b under ^ has already been
dened, then we check that the image of b is adjacent to the image
of a, and if not, then the required ^ does not exist.
Hence, we shall eventually construct a lift ^ of , such that ^ maps
(v1 ; 1) to (v1 ; i), or show that no such lift exists.
Suppose that D is the deck group for the cover ?^ of ?, and that S
is a set of orbit representatives of D on the bre f(v1 ; j ) : 1 j rg.
Then 2 Aut(?) has a lift to an element of Aut(?^ ) if and only if has such a lift taking (v1 ; 1) to an element in S .
5.6.1. Another Approach
Let ? and ?^ = ? be as in the previous section, such that G is the
fundamental group of ? with respect to a basepoint b, and H G is the
fundamental group of ?^ . Here we describe another approach (which we
have not yet implemented) to determining whether an automorphism
of ? lifts to the cover ?^ , and if so, determining one such lift.
We proceed as follows:
1. Dene the action of on G by nding, for each element g of the
(nite) generating set X of G, a circuit of ? based at b and labelled
by g, and tracing out the label of the image circuit. In order to
dene the tv; (which we need in order to construct the lifts of
, but not actually to prove they exist), for each vertex v of ?,
compute the label of the image under of the path from b to v in
our xed spanning tree T of ?.
2. Compute generators hj for H as words in the generators of G. (We
may already know such generators, or we can compute Schreier
generators.)
3. For each such hj compute hj , and hence the permutation in G corresponding to that element. Provided there are points in f1; 2; : : : ; rg
xed by all such permutations, has lifts to ?^ . In fact there is one
lift for each point k which is xed by all such permutations; where
g 2 G satises 1g = k, then there is a lift ^ with
(v; Hx)^ = (v; Hgx tv; ):
homotopy.tex; 15/05/1998; 17:39; no v.; p.18
19
This approach could be particularly useful if we are interested in studying several covers of ?. The information computed in the rst step can
be used for any one of them, as it is independent of H .
The subgroup of Aut(?) consisting of all automorphisms which lift
^
to ? is just the subgroup of all such which satisfy H = H g for some
g 2 G. It is a subgroup which contains the subgroup of Aut(?) which
xes H (as a subgroup) as a subgroup of index at most jNG (H ) : H j.
An algorithmic approach to combinatorial cell complexes
5.7. A Sample Calculation
We now give an example of the use of our implementation of the
algorithms above. We use GAP/GRAPE together with this implementation to construct explicitly two distance-regular (in fact, distancetransitive) graphs discovered by Thomas Meixner; see [15, Proposition 4.4]. These graphs are respectively 4-fold and 2-fold covers of a
graph having 672 vertices and valency 176, and automorphism group
containing U6 (2) = PSU (6; F 4 ).
This calculation is given in the form of a GAP-logle, and was originally performed for Aleksandar Jurisic, who wished to study the Meixner graphs. The computer used was a 266 Mhz Pentium PC running Linux. The calculation of the fundamental record took about 56 seconds,
and the total CPU-time used was about 103 seconds.
gap> RequirePackage("grape");
Loading GRAPE 2.31 (GRaph Algorithms using PErmutation groups),
by [email protected].
gap> GRAPE_RANDOM:=true;;
gap> # We will use certain randomized methods in GRAPE
gap> # (which do not affect the correctness of results).
gap> SU:=SpecialUnitaryGroup(6,2);;
gap> orb:=Orbit(SU,GF(4).one*[1,1,Z(4),0,Z(4),0],OnLines);;
gap> Length(orb);
672
gap> #
gap> # orb is an orbit of non-isotropic projective points.
gap> #
gap> gamma:=EdgeOrbitsGraph(Operation(SU,orb,OnLines),[1,2]);;
gap> if VertexDegree(gamma,1) >= OrderGraph(gamma)/2 then
>
gamma:=ComplementGraph(gamma);
> fi;
gap> GlobalParameters(gamma);
[ [ 0, 0, 176 ], [ 1, 40, 135 ], [ 48, 128, 0 ] ]
gap> #
gap> # gamma is the primitive quotient of the Meixner graphs.
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20
SARAH REES and LEONARD H. SOICHER
gap> # We have associated the group U_6(2) ( <= Aut(gamma) )
gap> # with gamma.
gap> #
gap> # We now build the universal cover of the clique complex of
gap> # gamma, using the algorithms described in this paper.
gap> #
gap> Read("/home/alg1/leonard/gapprogs/complex.g");
gap> Runtime(); # runtime in milliseconds
5170
gap> F:=FundamentalRecCliqueComplex(gamma);;
gap> Runtime();
61400
gap> G:=F.group;
Group( _x1, _x2 )
gap> Size(G);
4
gap> IsElementaryAbelian(G);
true
gap> #
gap> # G is the fundamental group of the clique complex of
gap> # gamma, and is isomorphic to C2 x C2.
gap> #
gap> H:=TrivialSubgroup(G);
Subgroup( Group( _x1, _x2 ), [ ] )
gap> delta:=CoveringGraph(gamma,G,F.edgeLabels,H);;
gap> #
gap> # delta is the (1-skeleton of the) universal cover of
gap> # the clique complex of gamma.
gap> #
gap> GlobalParameters(delta);
[ [ 0, 0, 176 ], [ 1, 40, 135 ], [ 12, 128, 36 ], [ 135, 40, 1 ],
[ 176, 0, 0 ] ]
gap> #
gap> # delta is the Meixner 4-fold cover.
gap> #
gap> H:=Subgroup(G,[G.generators[1]]);
Subgroup( Group( _x1, _x2 ), [ _x1 ] )
gap> Size(H);
2
gap> epsilon:=CoveringGraph(gamma,G,F.edgeLabels,H);;
gap> GlobalParameters(epsilon);
[ [ 0, 0, 176 ], [ 1, 40, 135 ], [ 24, 128, 24 ], [ 135, 40, 1 ],
[ 176, 0, 0 ] ]
gap> #
gap> # epsilon is the Meixner 2-fold cover.
gap> #
gap> Runtime();
102550
gap>
homotopy.tex; 15/05/1998; 17:39; no v.; p.20
An algorithmic approach to combinatorial cell complexes
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