Constructing Simplicial Branched Covers

0.58 MB·English

by  Nikolaus Witte

CONSTRUCTING SIMPLICIAL BRANCHED COVERS
NIKOLAUSWITTE
Abstract. Izmestiev and Joswig described how to obtain a simplicial covering space (the
partialunfolding)ofagivensimplicialcomplex,thusobtainingasimplicialbranchedcover[Adv.
Geom.3(2):191-255,2003]. Wepresentalargeclassofbranchedcoverswhichcanbeconstructed
via the partial unfolding. In particular, for d ≤ 4 every closed oriented PL d-manifold is the
partial unfolding of some polytopal d-sphere.
8
0
0
2 1. Introduction

n
Branchedcoversareappliedfrequentlyintopology–mostprominentlyinthestudy,construction
a
J and classification of closed oriented PL d-manifolds. First results are by Alexander [1] in 1920,
3 who observed that any closed oriented PL d-manifold M is a branched cover of the d-sphere.
2 Unfortunately Alexander’s proof does not allow for any (reasonable) control over the number

of sheets of the branched cover, nor over the topology of the branching set: The number of
]
O sheets depends on the size of some triangulation of M and the branching set is the co-dimension
C 2-skeleton of the d-simplex.
. However, in dimension d ≤ 4, the situation is fairly well understood. By results of Hilden [8]
h
and Montesinos [17] any closed oriented 3-manifold M arises as 3-fold simple branched cover
t
a of the 3-sphere branched over a link. In dimension four the situation becomes increasingly
m
difficult. First Piergallini [21] showed how to obtain any closed oriented PL 4-manifold as a
[
4-fold branched cover of the 4-sphere branched over a transversally immersed PL-surface [21].

2 Iori & Piergallini [11] then improved the standing result showing that the branching set may
v
be realized locally flat if one allows for a fifth sheet for the branched cover, thus proving a
1
long-standing conjecture by Montesinos [18]. The question as to whether any closed oriented
1
4 PL 4-manifold can be obtained as 4-fold cover of the 4-sphere branched over a locally flat PL-
1 surface is still open.
.
7 For thepartialunfoldingandtheconstructionofclosed oriented combinatorial 3-manifolds we
0
recommendIzmestiev&Joswig[14]. Theirconstructionhasrecentlybeensimplifiedsignificantly
7
by Hilden, Montesinos-Amilibia, Tejada & Toro [9]. For those able to read German additional
0
: analysisandexamples canbefoundin[24]. Thepartialunfoldingisimplemented inthesoftware
v
package polymake [6].
i
X
This work has been greatly inspired by a paper of Hilden, Montesinos-Amilibia, Tejada &
r Toro [9] and their bold approach. However, the techniques developed in the following turn out
a
to differ substantially from the ideas in [9], allowing for stronger results in dimension three and
generalization to arbitrary dimensions.
Outline of the paper. After some basic definitions and notations the partial unfolding K of a
simplicial complex K is introduced. The partial unfolding defines a projection p : K → K
which is a simplicial branched cover if K meets certain connectivity assumptions. Webdefine
combinatorial models of key features of a branched cover, namely the branching set band the
monodromy homomorphism.
Sections 2 and 3 are related, yet self contained. The main result of this paper is presented
in Theorem 2.1 and we give an explicit construction of a combinatorial d-sphere S, such that
Date: February 1, 2008.
2000 Mathematics Subject Classification. 57M12, 57Q99, 05C15, 57M25.
Key words and phrases. geometric topology, construction of combinatorial manifolds, branched covers.
The authors is supported by DeutscheForschungsgemeinschaft, DFG Research Group “Polyhedral Surfaces”.
1
2 WITTE
p : S → S is equivalent to a given simple, (d+1)-fold branched cover r : X → Sd (with some
additionalrestrictionforthebranchingsetofr). Theorem2.1isthenappliedtotheconstruction
of clbosed oriented PL d-manifolds as branched covers for d ≤ 4. The construction of S and the
proof of its correctness take up the entire Section 2.
Finally, in Section 3 we discuss how to extend k-coloring of a subcomplex L ⊂ K of a
simpliciald-complexK toamax{k,d+1}-coloring ofarefinementK′ ofK,suchthatLisagaina
subcomplexofK′. SinceK′ isconstructedfromK viafinitelymanystellarsubdivisionsofedges,
all properties invariant under these subdivisions are preserved, e.g. polytopality, regularity,
shellability, and others. This improves an earlier result by Izmestiev [12].
1.1. Basic definitions and notations. A simplicial complex K is a combinatorial d-sphere
or combinatorial d-ball if it is piecewise linear homeomorphic to the boundary of the (d+1)-
simplex, respectively to the d-simplex. Equivalently, K is a combinatorial d-sphere or d-ball if
there is a common refinement of K and the boundary of the (d+1)-simplex, respectively the
d-simplex. A simplicial complex K is a combinatorial manifold if the vertex link of each vertex
of K is a combinatorial sphereor a combinatorial ball. A manifold M is PL if and only if M has
a triangulation as a combinatorial manifold. For an introduction to PL-topology see Bjo¨rner [2,
Part II], Hudson [10], and Rourke & Sanderson [22].
A finite simplicial complex is pure if all the inclusion maximal faces, called the facets, have
the same dimension. We call a co-dimension 1-face of a pure simplicial complex Ka ridge, and
the dual graph Γ∗(K) of K has the facets as its node set, and two nodes are adjacent if they
share a ridge. We denote the 1-skeleton of K by Γ(K), its graph.
Further it is often necessary to restrict ourselves to simplicial complexes with certain connec-
tivity properties: A pure simplicial complex K is strongly connected if its dual graph Γ∗(K) is
connected, and locally strongly connected if the star st (f) of f is strongly connected for each
K
face f ∈ K. If K is locally strongly connected, then connected and strongly connected coincide.
Further we call K locally strongly simply connected if for each face f ∈ K with co-dimension ≥ 2
the link lk (f) of f is simply connected, and finally, K is nice if it is locally strongly connected
K
and locally strongly simply connected. Observe that combinatorial manifolds are always nice.
Let (σ ,σ ,...,σ ) bean orderingof thefacets of apuresimplicial d-complex K, and let D =
0 1 l i
σ denote the union of the first i facets. We call the ordering (σ ,σ ,...,σ ) a shelling
0≤j≤i j 0 1 l
of K if D ∩ σ is a pure simplicial (d − 1)-complex for 1 ≤ i ≤ l. If K is the boundary
i−1 i
S
complex of a simplicial (d+1)-polytope, then K admits a shelling order which can be computed
efficiently; see Ziegler [27, Chapter 8].
A simplicial complex obtained from a shellable complex by stellar subdivision of a face is
again shellable, a shellable sphere or ball is a combinatorial sphere or ball, and for 1 ≤ i ≤ l the
intersection D ∩σ isacombinatorial(d−1)-ball(orsphere). AshellablesimplicialcomplexK
i−1 i
is a wedge of balls or spheres in general. If K is a manifold, then D is a combinatorial d-ball
i
(or sphere) for 0 ≤ i ≤ l, and in particular we have that D ∩σ , D , and hence K are nice.
i−1 i i
We call a face f ⊂ σ free if f 6∈ D . In particular the (inclusion) minimal free faces describe
i i−1
all free faces, and they are also called restriction sets in the theory of h-vectors of simplicial
polytopes.
1.2. The branched cover. The concept of a covering of a space Y by another space X is
generalized by Fox [4] to the notion of the branched cover. Here a certain subset Y ⊂ Y may
sing
violate the conditions of a covering map. This allows for a wider application in the construction
of topological spaces. It is essential for a satisfactory theory of (branched) coverings to make
certain connectivity assumptionfor X and Y. Thespaces mostly considered areHausdorff, path
connected, and locally path connected; see Bredon [3, III.3.1]. Throughout we will restrict our
attention to coverings of manifolds hence they meet the connectivity assumptions in [3].
Consider a continuous map h : Z → Y, and assume the restriction h : Z → h(Z) to be a
covering. If h(Z) is densein Y (and meets certain additional connectivity conditions) then there
is a surjective map p : X → Y with Z ⊂ X and p| = h. The map p is called a completion
Z
of h, and any two completions p : X → Y and p′ :X′ → Y are equivalent in the sense that there
CONSTRUCTING SIMPLICIAL BRANCHED COVERS 3
exists a homeomorphism ϕ : X → X′ satisfying p′◦ϕ = p and ϕ| = Id. The map p : X → Y
Z
obtained this way is a branched cover, and we call the unique minimal subset Y ⊂ Y such
sing
that the restriction of p to the preimage of Y \Y is a covering, the branching set of p. The
sing
restriction of p to p−1(Y \Y ) is called the associated covering of p. If h: Z → Y is a covering,
sing
then X = Z, and p = h is a branched cover with empty branching set.
Example 1.1. For k ≥ 2 consider the map
p :C → C : z 7→ zk.
k
The restriction p | is a k-fold branched cover 2 → 2 with the single branch point {0}.
k 2
D D D
We define the monodromy homomorphism
m : π (Y \Y ,y )→ Sym(p−1(y ))
p 1 sing 0 0
ofabranchedcover forapointy ∈ Y \Y asthemonodromyhomomorphismoftheassociated
0 sing
covering: If[α] ∈ π (Y \Y ,y )isrepresentedbyaclosed pathαbasedaty ,thenm maps[α]
1 sing 0 0 p
to the permutation (x 7→ α (1)), where {x ,x ,...,x } = p−1(y ) is the preimage of y and
i i 1 2 k 0 0
α : [0,1] → X is the unique lifting of α with p ◦ α = α and α (0) = x ; see Munkres [19,
i i i i
Lemma 79.1] and Seifert & Threlfall [23, § 58]. The monodromy group M is defined as the
p
image of m .
p
Twobranchedcovers p :X → Y andp′ : X′ → Y′ areequivalent iftherearehomeomorphisms
ϕ : X → X′ and ψ : Y → Y′ with ψ(Y ) = Y′ , such that p′ ◦ϕ = ψ ◦p holds. The well
sing sing
known Theorem 1.2 is dueto theuniqueness of Y , and hencethe uniqueness of the associated
sing
covering; see Piergallini [20, p. 2].
Theorem1.2. Letp : X → Y beabranchedcoverofaconnected manifoldY. Thenpisuniquely
determined up to equivalence bythe branching setY , and the monodromy homeomorphism m .
sing p
In particular, the covering space X is determined up to homeomorphy.
Let Y be a connected manifold and Y a co-dimension 2 submanifold, possibly with a finite
sing
number of singularities. We call a branched cover p simple if the image m (m) of any meridial
p
loop m around a non-singular point of the branching set is a transposition in M . Note that
p
the k-fold branched cover p | : 2 → 2 presented in Example 1.1 is not simple for k ≥ 3.
k 2
D D D
1.3. The partial unfolding. Thepartial unfoldingK of a simplicial complex K firstappeared
in a paper by Izmestiev & Joswig [14], with some of the basic notions already developed in
Joswig[15]. Thepartialunfoldingiscloselyrelatedtotbhecompleteunfolding,alsodefinedin[14],
but we will not discuss the latter. The partial unfolding is a geometric object defined entirely
by the combinatorial structure of K, and comes along with a canonical projection p :K → K.
However, the partial unfolding K may not be a simplicial complex. In general K is a pseudo-
simplicial complex: Let Σ be a collection of pairwise disjoint geometric simplices with sbimplicial
attaching maps for some pairs (σ,bτ) ∈ Σ×Σ, mapping a subcomplex of σ isomborphically to
a subcomplex of τ. Identifying the subcomplexes accordingly yields the quotient space Σ/∼,
which is called a pseudo-simplicial complex if the quotient map Σ → Σ/∼ restricted to any
σ ∈ Σ is bijective. The last condition ensures that there are no self-identifications within each
simplex σ ∈ Σ.
The group of projectivities. Let σ and τ be neighboring facets of a finite, pure simplicial com-
plex K, that is, σ∩τ is a ridge. Then there is exactly one vertex in σ which is not a vertex of τ
and vice versa, hence a natural bijection hσ,τi between the vertex sets of σ and τ is given by
v if v ∈ σ∩τ
hσ,τi : V(σ) → V(τ) :v 7→
(τ \σ if v = σ\τ.
The bijection hσ,τi is called the perspectivity from σ to τ.
A facet path in K is a sequence γ = (σ ,σ ,...,σ ) of facets, such that the corresponding
0 1 k
nodes in the dual graph Γ∗(K) form a path, that is, σ ∩ σ is a ridge for all 0 ≤ i < k;
i i+1
4 WITTE
hγi(v )
v0 v2 1
hγi(v )
0
σ τ
v
1 hγi(v )
2
γ
Figure 1. A projectivity from σ to τ along the facet path γ.
see Figure 1. Now the projectivity hγi along γ is defined as the composition of perspectivities
hσ ,σ i, thus hγi maps V(σ ) to V(σ ) bijectively via
i i+1 0 k
hγi = hσ ,σ i◦···◦hσ ,σ i◦hσ ,σ i.
k−1 k 1 2 0 1
We write γδ = (σ ,σ ,...,σ ,σ ,...,σ ) for the concatenation of two facet paths γ =
0 1 k k+1 k+l
(σ ,σ ,...,σ ) and δ =(σ ,σ ,...,σ ), denote by γ− = (σ ,σ ,...,σ ) the inverse path
0 1 k k k+1 k+l k k−1 0
of γ, and we call γ a closed facet path based at σ if σ = σ . The set of closed facet paths
0 0 k
based at σ together with the concatenation form a group, and a closed facet path γ based at σ
0 0
acts on the set V(σ ) via γ ·v = hγi(v) for v ∈ V(σ ). Via this action we obtain the group of
0 0
projectivities Π(K,σ ) given by all permutations hγi of V(σ ). The group of projectivities is a
0 0
subgroup of the symmetric group Sym(V(σ )) on the vertices of σ .
0 0
The projectivities along null-homotopic closed facet paths based at σ generate the sub-
0
group Π (K,σ ) of Π(K,σ ), which is called the reduced group of projectivities. Finally, if K
0 0 0
is strongly connected then Π(K,σ ) and Π(K,σ′), respectively Π (K,σ ) and Π (K,σ′), are
0 0 0 0 0 0
isomorphic for any two facets σ ,σ′ ∈ K. In this case we usually omit the base facet in the
0 0
notation of the (reduced) group of projectivities, and write Π(K) = Π(K,σ ), respectively
0
Π (K) = Π (K,σ ).
0 0 0
The odd subcomplex. Let K be locally strongly connected; in particular, K is pure. The link
of a co-dimension 2-face f is a graph which is connected since K is locally strongly connected,
and f is called even if the link lk (f) of f is 2-colorable (i.e. bipartite as a graph), and odd
K
otherwise. We define the odd subcomplex of K as all odd co-dimension 2-faces (together with
their proper faces), and denote it by K (or sometimes odd(K)).
odd
Assume that K is pureand admits a (d+1)-coloring of its graph Γ(K), that is, we assign one
color of a set of d+1 colors to each vertex of Γ(K) such that the two vertices of any edge carry
different colors. Observe that the (d+1)-coloring of K is minimal with respect to the number of
colors, and is unique up to renaming the colors if K is strongly connected. Simplicial complexes
that are (d+1)-colorable are called foldable, since a (d+1)-coloring defines a non-degenerated
simplicial map of K to the (d+1)-simplex. In other places in the literature foldable simplicial
complexes are sometimes called balanced.
Lemma 1.3. The odd subcomplex of a foldable simplicial complex K is empty, and the group
of projectivities Π(K,σ ) is trivial. In particular we have hγi = hδi for any two facet paths γ
0
and δ from σ to τ for any two facets σ,τ ∈ K.
We leave the proof to the reader. As we will see in Theorem 1.4 the odd subcomplex is of
interest in particular for its relation to Π (K,σ ) of a nice simplicial complex K.
0 0
Consider a geometric realization |K| of K. To a given facet path γ = (σ ,σ ,...,σ ) in K we
0 1 k
associatea(piecewiselinear)path|γ|in|K|byconnectingthebarycenterofσ tothebarycenters
i
of σ ∩σ and σ ∩σ by a straight line for 1 ≤ i< k, and connecting the barycenters of σ
i i−1 i i+1 0
CONSTRUCTING SIMPLICIAL BRANCHED COVERS 5
and σ ∩σ , respectively σ and σ ∩σ . A projectivity around a co-dimension 2-face f is a
0 1 k k k−1
projectivity along a facet path γδγ−, where δ is a closed facet path in st (f) (based at some
K
facet σ ∈ st (f)) such that |γ| is homotopy equivalent to the boundary of a transversal disc
K
around |f| ⊂ |st (f)|, and γ is a facet path from σ to σ. The path γδγ− is null-homotopic
K 0
since K is locally strongly simply connected.
Theorem 1.4 (Izmestiev & Joswig [14, Theorem 3.2.2]). The reduced group of projectivi-
ties Π (K,σ ) of a nice simplicial complex K is generated by projectivities around the odd
0 0
co-dimension 2-faces. In particular, Π (K,σ ) is generated by transpositions.
0 0
The fundamental group π (|K|\|K |,y ) of a nice simplicial complex K is generated by
1 odd 0
paths |γ|, where γ is a closed facet path based at σ , and y is the barycenter of σ ; see [14,
0 0 0
Proposition A.2.1]. Furthermore, due to Theorem 1.4 we have the group homomorphism
h : π (|K|\|K |,y ) → Π(K,σ ) :[|γ|] 7→ hγi,
K 1 odd 0 0
where [|γ|] is the homotopy class of the path |γ| corresponding to a facet path γ.
The partial unfolding. Let K be a pure simplicial d-complex and set Σ as the set of all pairs
(|σ|,v), where σ ∈ K is a facet and v ∈ σ is a vertex. Thus each pair (|σ|,v) ∈ Σ is a copy of
the geometric simplex |σ| labeled by one of its vertices. For neighboring facets σ and τ of K
we define the equivalence relation ∼ by attaching (|σ|,v) ∈ Σ and (|τ|,w) ∈ Σ along their
common ridge |σ ∩ τ| if hσ,τi(v) = w holds. Now the partial unfolding K is defined as the
quotient space K = Σ/∼. The projection p : K → K is given by the factorization of the map
Σ → K :(|σ|,v) 7→ σ; see Figure 2. b
b b
v
3
v v
2 2
v v
1 3
v
0
v v
1 3
v v
1 2
Figure 2. The starred triangle and its partial unfolding. The complex on the
right is the non-trivial connected component of the partial unfolding, indicated
by the labeling of the facets by the vertices v , v , and v . The second connected
1 2 3
component is a copy of the starred triangle with all facets labeled v ; see also
0
Example 1.1 for k = 2.
The partial unfolding of a strongly connected simplicial complex is not strongly connected in
general. We denote by K the connected component containing the labeled facet (|σ|,v).
(|σ|,v)
Clearly, K = K holds if and only if there is a facet path γ from σ to τ in K with
(|σ|,v) (|τ|,w)
b
hγi(v) = w. Itfollows thattheconnectedcomponentsofK correspondtotheorbitsoftheaction
of Π(K,σb) on V(σb). Note that each connected component of the partial unfolding is strongly
0 0
connected and locally strongly connected [24, Satz 3.2b.2]. Therefore we do not distinguish
between connected and strongly connected components of the partial unfolding.
The problem that the partial unfolding K may not be a simplicial complex can be addressed
in several ways. Izmestiev & Joswig [14] suggest barycentric subdivision of K, or anti-prismatic
subdivision of K. A more efficient solutionb(with respect to the size of the resulting triangula-
tions) is given in [24]. b
6 WITTE
1.4. The partial unfolding as a branched cover. As preliminaries to this section we state
two theoremsbyFox [4]andIzmestiev &Joswig[14]. Together theyimplythatunderthe“usual
connectivity assumptions” the partial unfolding of a simplicial complex is indeed a branched
cover as suggested in the heading of this section.
Theorem 1.5 (Izmestiev & Joswig [14, Theorem 3.3.2]). Let K be a nice simplicial complex.
Then the restriction of p : K → K to the preimage of the complement of the odd subcomplex is
a simple covering.
b
Theorem 1.6 (Fox [4, p. 251]; Izmestiev & Joswig [14, Proposition 4.1.2]). Let J and K be
nice simplicial complexes and let f :J → K be a simplicial map. Then the map f is a simplicial
branched cover if and only if
codimK ≥ 2.
sing
Since the partial unfolding of a nice simplicial complex is nice Corollary 1.7 follows immedi-
ately.
Corollary 1.7. Let K be a nice simplicial complex. The projection p : K → K is a simple
branched cover with the odd subcomplex K as its branching set.
odd
b
For the rest of this section let K be a nice simplicial complex and let y be the barycenter
0
of a fixed facet σ ∈ K. The projection p : K → K is a branched cover with K = K by
0 sing odd
Corollary 1.7, and Izmestiev & Joswig [14] proved that there is a bijection ı : p−1(y ) → V(σ )
0 0
that induces a group isomorphism ı : Symb(p−1(y )) → Sym(V(σ )) such that the following
∗ 0 0
Diagram (1) commutes.
(1) π (|K|\|K |,y )
1 (cid:15)(cid:15) mopddQQQQ0QQQhQKQQQQQ((
M // Π(K,σ )
p ı∗ 0
Let r : X → Y be a branched cover and assume that there is a homomorphism of pairs
ϕ : (Y,Y ) → (|K|,|K |), that is, ϕ : Y → |K| is a homomorphism with ϕ(Y ) = |K |.
sing odd sing odd
Then Theorem 1.8 gives sufficient conditions for p : K → K and r : X → Y to be equivalent
branched covers. It is the key tool in the proof of the main Theorem 2.1 in Section 2.
b
Theorem 1.8. Let K be a nice simplicial complex and let r : X → Y be a (simple) branched
cover. Further assume that there is a homomorphism of pairs ϕ :(Y,Y )→ (|K|,|K |), and
sing odd
lety ∈ Y beapointsuchthatϕ(y )isthebarycenterof|σ |forsomefacetσ ∈ K. Thebranched
0 0 0 0
covers p : K → K and r : X → Y are equivalent if there is a bijection ι : r−1(y ) → V(σ ) that
0 0
induces a group isomorphism ι : M → Π(K,σ ) such that the diagram
∗ r 0
b
(2) π (Y \Y ,y ) ϕ∗// π (|K|\|K |,ϕ(y ))
1 sing 0 1 odd 0
mr hK
(cid:15)(cid:15) (cid:15)(cid:15)
M ι∗ // Π(K,σ )
r 0
commutes. In particular, we have K ∼= X.
Proof. Corollary 1.7 ensures that p : K → K is indeed a branched cover, and commutativity of
b
Diagram (1) and Diagram (2) proves commutativity of their composition:
b
π (Y \Y ,y ) ϕ∗// π (|K|\|K |,ϕ(y ))
1 M(cid:15)(cid:15) sminrg 0 ι∗ 1 // Π(K(cid:15)(cid:15),ohdσKd)PPoo PPP0Pım∗PPpPPPPPP(( M
r 0 p
Theorem 1.2 completes the proof. (cid:3)
CONSTRUCTING SIMPLICIAL BRANCHED COVERS 7
2. Constructing Branched Covers
Throughout this section let r : X → Sd be a branched cover of the d-sphere with branching
set F. The main objective is to give a large class of branched covers r, such that there is a
combinatorial sphere S with p : S → S equivalent to r as a branched cover. In particular this
implies the existence of a homeomorphism of pairs ϕ :(Sd,F) → (|S|,|S |). Note that by the
odd
natureofthepartialunfoldingandbtheprojectionp : S → S anybranchedcover r equivalenttop
has to be simple and (d+1)-fold. A theorem similar to Theorem 2.1 may easily be formulated
for branched covers of d-balls. b
Recall that we associate to a facet path γ in S the (realized) path |γ| in |S|, and that the
square brackets denote the homotopy class of a closed path. Thus we write m ([ϕ−1(|γ|)]) for
r
the image of an element in π (Sd\F,y ) represented by the closed path ϕ−1(|γ|), which in turn
1 0
is obtained from a closed facet path γ based at some facet σ ∈ S with barycenter ϕ(y ) by first
0 0
considering its realization |γ| and then its preimage under ϕ.
Theorem 2.1. For d ≥ 2 let r : X → Sd be a (d+1)-fold, simple branched cover of the d-sphere,
and assume that the branching set F of r can be embedded via a homeomorphism ϕ : Sd → |S′|
into the co-dimension 2-skeleton of a shellable simplicial d-sphere S′. Then there is a shellable
simplicial d-sphere S, such that p : S → S is a branched cover equivalent to r. Further more,
the d-sphere S can be obtained from S′ by a finite series of stellar subdivision of edges. If S′
is the boundary of a simplicial (d +b1)-polytope then also S is the boundary of a simplicial
(d+1)-polytope.
To make the proof of Theorem 2.1 more digestible we first give the (algorithmical) back-bone
of the proof and defer some of the more technical aspects to the Lemmas 2.2, 2.3, and 2.4. Fix
a point y ∈ Sd \F and we may assume ϕ(y ) to be the barycenter of some facet σ ∈ S′ and
0 0 0
|σ |∩ϕ(F) = ∅tohold. Furtherfixabijection ıbetweenthepreimage{x ,x ,...,x } = r−1(y )
0 0 1 d 0
of y and the vertices of σ , and color the vertices of σ via ı by the elements in r−1(y ).
0 0 0 0
ϕ
Figure 3. The base space of the branched cover r : X → S2 (left) and a
polytopal 2-sphere S with marked beginning (σ ) of a shelling (right). On
i j 0≤j≤li
theleftthepreimageofD = σ underthehomomorphismϕ : S2 → |S |is
i 0≤j≤li j i
shadedandthebranchingsetismarked. TheoddsubcomplexofD is markedon
i
S
theright. Thebranchedcoversr : X → S2 (restrictedtoϕ−1(|D |))andD → D
i i i
are equivalent.
c
Thed-sphereS is constructed inafiniteseries (S′ = S ,S ,...,S = S)of shellabled-spheres,
0 1 l
andeachd-sphereS comeswithashellingofitsfacetwithmarkedbeginning(σ ,σ ,...,σ ).
i i,0 i,1 i,li
Thecomplex S is obtained from S by (possibly) subdividingσ in a finite series of stellar
i+1 i i,li+1
subdivisions of edges not contained in any σ for 0 ≤ j ≤ l . Thus we may choose the shelling
i,j i
of S such that it extends (σ ,σ ,...,σ ) and we denote the marked beginning of the
i+1 i,0 i,1 i,li
shelling of S simply by (σ ,σ ,...,σ ).
i 0 1 li
8 WITTE
w
γ x0
v
x1 σv x1 γ v
σ x0
f
σ
σ x0f 0
x2
x0 x1 δ
σ δ
w
x1 Di
w
Figure 4. Case (i): The 2-ball D with the facet σ colored via ı by the preim-
i 0
age {x ,x ,x } of y and induced coloring of the ridge f on the right hand side
0 1 2 0
of the figure. The vertex v is colored x if any element of M corresponding
0 r
via m ◦ϕ−1 to a facet path of the form γδγ−1 maps x to itself. Case (ii): The
r 0
induced coloring of the co-dimension 2-face f and the vertices v and w on the
left. Theedge{v,w}issubdividedifthefacetpathγδ(σ ,σ,σ )γ−1 corresponds
w v
via m ◦ϕ−1 to the identity in M .
r r
Let D = σ then the main idea of the proof of Theorem 2.1 is to construct S such
i 0≤j≤li j i
that the branched covers r : X → Sd (restricted to ϕ−1(|D |)) and D → D are equivalent. To
S i i i
this end we prove that ϕ restricted to ϕ−1(|D |) is a homomorphism of pairs (ϕ−1(|D |),F ∩
i i
ϕ−1(|Di|)) → (|Di|,|odd(Di)|) and that the following Diagram (3) ccommutes; see Figure 3.
(3) π (ϕ−1(|D |)\F,y ) ϕ∗// π (|D |\|odd(D )|,ϕ(y ))
1 i 0 1 i i 0
mr hDi
(cid:15)(cid:15) (cid:15)(cid:15)
M ı∗ // Π(D ,σ )
r i 0
Commutativity of Diagram (3) is obtained by ensuring that for each closed facet path γ in D
i
(which is not a facet path in D ) the projectivity hγi acts on V(σ ) as m ([ϕ−1(|γ|)]) acts
i−1 0 r
on r−1(y ).
0
The pair (S ,(σ ) ) is constructed from the pair (S ,(σ ) ) as follows. Let σ =
i+1 j 0≤j≤li+1 i j 0≤j≤li
σ be the first facet in the shelling of S not contained in D , let γ be a facet path in D ∪σ
li+1 i i i
from σ to σ, and let f ⊂ σ be a face. Further let H be the subgroup of M which is induced
0 f,γ r
via m by all elements of π (Sd\F,y ) of the form [ϕ−1(|γδγ−1|)], where δ is any closed facet
r 1 0
path in st (f) based at σ. The subgroup H has at least dim(f)+1 trivial orbits, namely,
Si f,γ
the orbits corresponding to the vertices of f, and for g ⊂ f we have that the set of trivial orbits
of H contains the trivial orbits of H . We consider the following three case:
f,γ g,γ
(i) The intersection σ ∩D is a ridge f. Let γ be a facet path in D ∪σ from σ to σ, and
i i 0
color σ (and hence f) by the coloring induced along γ by the fixed coloring of σ . Now
0
keep the coloring of f, butchange the color of the remaining vertex v = σ\f to any trivial
orbit of H ; see Figure 4 (right).
v,γ
(ii) The intersection σ ∩D equals two ridges f ∪v and f ∪w with a common co-dimension
i
2-face f. Let σ ∈ D be the facet intersecting σ in f ∪ v, let σ ∈ D be the facet
v i w i
intersecting σ in f ∪w, and choose facet paths γ from σ to σ in D and δ from σ to σ
0 v i v w
in st (f). The fixed coloring of σ induces along γ, respectively γδ, colorings on f ∪v
Di 0
and f ∪w, and the colorings coincide on f. Now we change the color of w according to
m ([ϕ−1(|γδ(σ ,σ,σ )γ−1|)]), which is either a transposition (changing the color of w) or
r w v
the identity; see Figure 4 (left).
CONSTRUCTING SIMPLICIAL BRANCHED COVERS 9
(iii) Otherwise set S = S and let (σ ,σ ,...,σ ,σ) be the marked beginning of a shelling
i+1 i 0 1 li
of S .
i+1
We obtained a (possibly inconsistent) coloring of the vertices of σ in the cases (i) and (ii).
Note that the coloring of σ induces a consistent coloring on D ∩σ, and that there is at most one
i
conflicting edge {v,w}, that is, v and w arecolored the same. A consistently colored subdivision
of σ is constructed in at most d−1 subdivisions of σ with exactly one conflicting edge e each,
where each subdivision is obtained from the previous one by stellar subdividing e: Let f ⊂ σ
e
be the unique minimal face such that |e| ⊂ |f| holds and denote by C the set of trivial orbits
e
of H . Now color the new vertex v with an element of C which is not the color of any
fe,γ e e
vertex ve′ subdividing an edge e′ with fe′ ⊂ f. Note that Ce is the entire preimage r−1(y0) if fe
is a co-dimension 1-face, and that Ce has at least one element distinct from the colors of all ve′
for fe′ ⊂ fe. If Ce contains the one color x∈ r−1(y0) not used in the coloring of σ, color ve by x
and terminate the subdivision process.
{x }v
2
{x }v
2
{x ,x ,x ,x }
0 1 2 3
{x ,x }
0 2
{x ,x ,x ,x }
0 1 2 3 {x }w
2
{x }
1
{x } {x }
0 1
{x }
{x } 0
2
Figure 5. Coloring of the vertices of the refinement of σ in case (i) (on the
left) and case (ii) (on the right). The minimal free face v, respectively {v,w}, is
marked. Each vertex v is labeled by the trivial orbits of H and the vertex
e fe,γ
color is printed bold.
This completes the construction of S in the cases (i) and (ii), and we define the marked
i+1
beginning of a shelling of S by (σ ,σ ,...,σ ) followed by the facets of the refinement of σ
i+1 0 1 li
in an appropriate order.
It remains to prove that the algorithm described above terminates and that p : S → S is a
branched cover equivalent to r : X → Sd. Since S is shellable and hence nice, p is a branched
cover by see Corollary 1.7. The following Lemmas 2.2 and 2.3 prove the equivalencebof p and r,
while termination of the construction above is provided by Lemma 2.4.
Lemma 2.2. The branched covers p : S → S and r :X → Sd are equivalent.
Proof. In order to show the equivalence of the branched covers p and r we prove by induction
b
that the following holds for 0 ≤ i≤ l:
(I) For any closed facet path γ based at σ in D we have
0 i
hγi = ı ◦m ([ϕ−1(|γ|)]).
∗ r
(II) Let v ∈ D be a vertex, and let γ be a facet path in D from σ to a facet σ containing v.
i i 0
Then the color induced on v along γ by the fixed coloring of σ is a trivial orbit of H .
0 v,γ
We remark that (I) implies that ϕ restricted to ϕ−1(|D |) is a homomorphism of pairs
i
(ϕ−1(|D |),F ∩ϕ−1(|D |)) → (|D |,|odd(D )|) and that the Diagram (3) commutes. Finally, (I)
i i i i
and (II) are met for the pair (S ,D ) = (S′,σ ), and commutativity of Diagram (3) proves the
0 0 0
equivalence of r :X → Sd and p : S → S for i= l; see Theorem 1.8.
b
10 WITTE
x1
γ
δ′
e
f
σ
σ 0 x2
Di x0
δ
f
Figure 6. Case (iii): The paths γ, δ, and δ′ if the corresponding edge e of a
f
co-dimension 2-face f is non-free.
Weshowthat(I)and(II)holdforthepair(S ,D )providedtheyholdforthepair(S ,D ).
i+1 i+1 i i
Recall that we denote the first facet σ of the shelling of S not contained in D by σ. The
li+1 i i
simplicial complex D is contractible and hence Π (D ,σ ) = Π(D ,σ ) is generated by closed
i 0 i 0 i 0
facet paths around (odd) co-dimension 2-faces by Theorem 1.4. Thus it suffices to verify (I) for
closed facet paths around (interior) co-dimension 2-faces by examining the three cases (i), (ii),
and (iii).
(i) The intersection σ∩D is a ridge f. New interior co-dimension 2-faces in D arise only
i i+1
in the refinement of σ, which is foldable by construction. Since ϕ(F) does not intersect
the interior of |σ|, any facet path around a new interior co-dimension 2-face corresponds
to the identity of M and (I) holds by Lemma 1.3.
r
(ii) The intersection σ ∩D equals two ridges f ∪v and f ∪w with a common co-dimension
i
2-face f. By induction hypothesis (II) holds for the vertices of f in D and thus (I) follows
i
for the new interior co-dimension 2-face f of D by construction. As for any new interior
i+1
co-dimension 2-face in the refinement of σ, (I) holds (as in case (i)) since the refinement is
foldable and ϕ(F) does not intersect the interior of |σ|.
(iii) Otherwise there is no co-dimension 2-faces f ⊂ σ with a free correspondingedge e =σ\f
f
and (I) follows from Lemma 2.3.
Having established (I), it suffices to verify (II) for a single facet path γ in D from σ to
i+1 0
any facet containing a given vertex v. Thus (II) holds by choice of color for any vertex added
to D in the construction of the pair (S ,D ). (cid:3)
i i+1 i+1
Lemma 2.3. If f ∈ σ is a co-dimension 2-face with a non-free corresponding edge e = σ\f,
f
then (I) holds for any closed facet path based at σ around f in D .
0 i+1
Proof. Let γδγ−1 be a closed facet path based at σ around f in D , where δ is a closed
0 i+1
path around f in st (f). Since {v,w} = e is a non-free edge, there is a facet path δ′ in D
Di+1 f i
with |δ′| homotopy equivalent to |{f ,f ∪v,f ∪w}| in |D |\|odd(D )|, and we assume δ and δ′
e i i
to have the same orientation; see Figure 6. Note that the complex {f ,f ∪v,f ∪w} itself is
e
homotopy equivalent to S1.
W.l.o.g. let m ([ϕ−1(|γδγ−1|)]) either be the identity or the transposition (x ,x ) ∈ M .
r 0 1 r
Each transposition (x ,x ), for i 6= j, appears at most once in the (unique) reduced represen-
i j
tation of the element a = m ([ϕ−1(|A|)]) ∈ M corresponding to the facet path A = γδ′γ−1,
r r
sinceAiscomposedfromfacetpathsaroundco-dimension2-facesofσ. Letb = m ([ϕ−1(|B|)]) ∈
r
M denote the element corresponding to the facet path B = γδ′δ−1γ−1, then a = (x ,x )◦b
r 0 1
holds if and only if (x ,x ) is in the reduced representation of a, and we have a = b otherwise.
0 1
Since (I) holds for D and hence in particular for the facet path A, and with
i
A = γδ′γ−1 = γδ′δ−1γ−1γδγ−1 = Bγδγ−1,