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LIBRARY

Michigai State
Univer sity

  
     

[XXL-6.;

This is to certify that the

thesis entitled

EMBEDDINGS IN SOME SINGULAR MANIFOLDS

presented by

Lyle Leonard Welch

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Mathematics

 

627%

Major professor

Date $—~ // ‘7/

07639

 

ABSTRACT
EMBEDDINGS IN SOME SINGULAR MANIFOLDS
BY

Lyle Leonard Welch

The singular 2—manifolds in which we will be embedding various
1-, and 2—dimensional spaces are 3-books, a space which is the one
point compactification of an open 3-book, and finite 2-complexes. The
question of the effects of the singularities on the classical topology
of manifolds is investigated in a logical-historical manner for
dimensions 2 and 3.

Beginning with A. Schoenflies it was found that Cantor sets
were equivalently embedded in the plane, and hence they lie on an arc.
It is shown in Chapter II that a Cantor set in a 3—book lies on an
arc in the book.

Somewhat later R. L. Moore investigated uncountable mutually
exclusive collections and found no such collections of triods in E2.
This type of question has been carried over to E3 by C. S. Young and
others. Chapter III deals with this question for an n—book where it
is shown that a flat n—book does not contain an uncountable collection
of mutually exclusive flat (n+l)-frames for n 2 2, but it does contain
an uncountable collection of mutually exclusive (flat) n-frames.

From the notion of connected im kleinen,1 R. L. Moore started
a study of uniform local connectedness (O-ULC) to characterize a
curve in E2. Ultimately a k-LC property was introduced. The O—LC and
l-LC properties of the complement of a polygonal arc in a flat 3—book

are studied in Chapter IV. The results that are obtained are

Lyle Leonard Welch

intermediate between those results already established in E2 and E3.

In Chapter V Eilenberg's thesis is used to study cut sets of
the one point compactification of an open 3-book, g3. An example is
given of a locally connected, G6 set that cuts 33 between two points

2'

and 2", but it contains no G—curve that cuts g3, nor does it contain
an irreducible cut of £3. Furthermore, with modifications this example
becomes a locally connected irreducible cut of 33 between two points

z' and 2" that is not a 9-curve.

With Antoine the study of wild embeddings began, and one
condition that has led in this field is cellularity. Homologically
trivial 2-complexes and cellularity are studied in Chapter VI. It
is shown that a homologically trivial finite 2-complex in a flat
3-book B3 is cellular in E3. An attempt is made to justify defining
a cellular arc in B3 as one that has a homologically trivial finite
2—comp1ex neighborhood in the book contained in each neighborhood of
the arc. An example is given of a wild arc that has such a neighborhood,
and a proof that any wild arc that is bad at just one endpoint doesn't
have such a neighborhood. It is also shown that any "cellular" arc
in B3 is cellular in E3.

The work of C. Persinger and G. Atneosen allows a study of wild
sets in 2-complexes. Chapter VII contains a characterization theorem
of those finite 2-complexes in E3 that contain a wild arc. Finally,
the existence of knots in a 3-book naturally leads to questions about

the Schoenflies Theorem in a 2—complex. A characterization theorem

of those finite 2-complexes in which polygonal simple closed curves

\I N Fk ll.\ \
‘1 1! ml, III [(‘II .III“! In]. llltlllllll I l."
(II-I. ’ l I I (

. I'ulldllll“ [I Olll It I, '

)1 I ll» . ‘ Iii-ll

Lyle Leonard Welch

bound a disk is proved in Chapter VIII.

 

1For a definition of connected im kleinen see page 233 of R. L.
Moore, "Concerning connectedness im kleinen and a related property,"
Fund. Math., Vol. 3 (1922) pp. 232—237.

lull. I flifltlllllltlgx‘l‘iilkf [I [I'llflullilllltllilclnill‘II I‘lllllt.llll.ll|"lll [v ..II .1 Ill '1... .II II

EMBEDDINGS IN SOME SINGULAR MANIFOLDS

By

Lyle Leonard Welch

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1971

|||| I "l [[(IIIIIIII‘III‘r‘Il’lI’I .. l' ‘(1 4. Il'[‘ ‘lllll Ill f.ll1‘l|ll{ lI-lllx

g 7.5) 307

ACKNOWLEDGMENTS

I would like to thank Professor P. H. Doyle for leading me
through a great learning experience during the preparation of this
thesis. I would also like to thank Mr. Hillesland for leading me to
an appreciation of mathematics and Dr. Rebassoo for teaching me my
first mathematics. Most of all, I want to thank my wife, Judy, for
sharing with me during this struggle, and for typing this confusing

mess - to her it was.

ii

TABLE OF CONTENTS

CHAPTER PAGE
I INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . 1
II CANTOR SETS IN 33 LIE ON ARCS . . . . . . . . . . . . . 8
III UNCOUNTABLE MUTUALLY EXCLUSIVE COLLECTIONS IN Bn. . . . 12
IV THE l-LC PROPERTY OF ARCS IN B3 . . . . . . . . . . . . 19
V CUTS OF A 3-BOOK. . . . . . . . . . . . . . . . . . . . 26
VI HOMOLOGICALLY TRIVIAL Z-COMPLEXES IN B3 . . . . . . . . 35
VII A CHARACTERIZATION OF 2-COMPLEXES IN E3 THAT

CONTAIN WILD ARCS O O O O O I O O O O O C O O O O O O O 44

VIII SIMPLE CLOSED CURVES THAT BOUND DISKS IN A 2-COMPLEX. . 54

BIBLIOGRAPIIY. O O O O O O O O O O O O O O O O O O O O I 59

iii

 

f \ bllll l‘fllllnllll.[l ..A'll

LIST OF FIGURES

FIGURE PAGE

3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

iv

CHAPTER I

INTRODUCTION

The singular manifolds in which we will be embedding various
l-, and 2-dimensional spaces are 3-books, and more particularly flat
3-books, as well as a space which is the one point compactification of
an open 3-book, and finite 2-complexes.

The embeddings will be primarily embeddings in a flat 3-book
but since previous results are in a more general setting we need the
following more general definition of an n-book. An 272223.3n is the
union of n-closed disks in E3 such that each pair of disks meets
precisely on a single arc B on the boundary of each. The disks are
called the leaves of Bn, and are denoted by Di’ 1 = l,2,...,n and the
arc is called its back,

This investigation of n-books was initiated by P. H. Doyle in
[12] when he extended an earlier result [11], and showed that if each
of the leaves of an n—book topologically embedded in E3 is tame then
the n-book is tame. C. A. Persinger continued the investigation of
extrinsic properties of subsets of n-books in [28, 29, 30], and finally
G. Atneosen in her thesis [2] investigated the embeddability of compacta
in n-books from two different vieWpoints: the intrinsic properties of
nfbooks, and the extrinsic properties of n—books in E3. In this paper
in Chapters II, III, IV and VI we will investigate some embeddings in

flat 3-books to answer questions of embeddability but more importantly

to obtain some results in B3 that are intermediate between the results
already established in E2 and E3. In Chapter V we will investigate
embeddings in the one point compactification of an open 3-book in an
attempt to extend some of the results of Eilenberg's [16] to this space,
and, finally in Chapters VII and VIII we will characterize those finite
2-complexes that contain wild arcs, and those finite 2—complexes for
which any polygonal simple closed curve bounds a disk in the 2—complex.

The following comments on notation and definitions will be
included for the reading of this paper. The notation will be given in
a general setting although we will be using only the particular cases

for n = 1,2, and 3 in this paper.

En = {xlx = (x ,x

l 2’°"’Xn) an n-tuple of real numbers}.

En is assumed to have the topology determined by the Euclidean metric

d .
n
En = {xlx = (x x x ) X 2 0}
+ 1, 2,..., n n .
Sn—1 = {x E Enldn(x,0) = 1 where O = (0,0,...,O)}.

A homeomorphic image of S1 is called a simple closed curve, and the

 

homeomorphic image of Slkj [-1,1] is called a B-curve.
Closed n-cell = {x E Enldn(x,0) S 1}.
Open n-cell = {x e Enldn(x,0) < l}.

A disk is the homeomorphic image of a closed 2-cell, and an arc is the
homeomorphic image of a closed l-cell. If A and B are topological
spaces, a homeomorphism of A into B is called an embedding.

By a Efdimensional manifold M is meant a separable metric space

 

such that each point has a neighborhood whose closure is homeomorphic

l ll. [- ‘l I I. [..I.

to a closed n-cell. The interior of M, Int M, consists of those points
which have neighborhoods homeomorphic to an open n-cell; the boundary of

M, Bd M, is defined to be M — Int M. By the interior of an n-book is

n

meant the set éga Int Di LIInt B, which will be called an open nfbook.

 

The interior g£_§3, Int Sn, will be the bounded complementary domain of

Sn in En+1

and the exterior gf_§3, Ext Sfi.will be the other component

We also need some terminology from combinatorial topology. The
definitions are essentially those of Zeeman [37].

By a Efsimplex A, 0 S n, is meant the convex hull of n+1 linearly
independent points (the vertices) {lej = 0,1,...,n} in Ep, n15 p, where

the convex hull of a set R is the intersection of all convex sets

 

containing R. A will denote the boundary of A. By a £f£§£g_B of A,
denoted by B < A, is meant the convex hull of r+l distinct points of
{lej = o,1,2,...,n}. Asimplicial complex, K of Ep, p 2 1, is a finite
collection of simplexes of ED such that:

(1) if A e K, then all the faces of A are in K,

(2) if A,B e K, then A,n B is a common face of A and B.
The underlying point set of asimplicial complex K, denoted by |K| , is

called a Euclidean polyhedron or polyhedron.

 

 

The phrase finite Euclidean polyhedron is used to emphasize
the fact that we are only considering,simplicialcomplexes consisting

of finitely many simplexes. L is called a subdivision of K if ILI = IKI

 

and every simplex of L is contained in some simplex of K. L is a

subcomplex of K if L is asimplicial complex and LC K. The subcomplex

 

of K consisting of all q-simplexes of K, where q S m, is called the

(m) .

mfskeleton of K, K If P is a Euclidean polyhedron then a simplicial

complex K such that [K] = P is called a triangulation of P, and P is

 

said to be the carrier of K. The mg§h_of a triangulation, K, is the
supremum of the diameters of all the simplexes of K. The dimension of
zisimplicialcnmplex K is the largest interger n such that K contains
an n-simplex, and we call K an nfcomplex. The carrier of a l-dimensional
complex is called a graph,

If A and B are simplexes in Ep such that the union of their
vertices forms a linearly independent set of points in Ep, then A and
B are joinable, and the join of A and B, denoted by A*B, is defined to
be the simplex spanned by the union of their vertices.

The s_t_a£ and gig of a simplex A of a simplicial complex K are

defined:
St(A,K) = {BIA < B}, lk(A,K) = {BIB*A e K}.

A subdivision L of K is said to be obtained from K by starring A_at a,

if-a 6 Int A and L = {K - St(A,K)} \I{a*A*lk(A,K)}. A first derived

 

subdivision of K is obtained by starring all the simplexes of K in

An rth derived

 

some order such that if 01 > 02 then 01 precedes O

2.
subdivision of K is defined inductively as the first derived of an
(r-l)th derived.

If K 3 L, we say there is an elementary simplicial collapse from

 

K to L if I(- L consists of a principal simplex A of K, i.e. A is not
the proper face of any simplex in K, together with a free face. We say

Ksimplicially collapses to L, written K‘SAL, if there is a sequence of

 

elementary simplicial collapses from K to L. If L is a point we say

that K is collapsible.

 

A topological polyhedron P in En is tamely embedded in En if

 

there is a space homeomorphism that carries P onto a finite Euclidean

polyhedron. Otherwise P is wildly embedded. A set x in En is to be

 

locally tame §t_§_point p_ of X if there is a neighborhood N of p and

 

a homeomorphism h of N’(the closure of N) onto a polyhedron in En such
that h(N'n X) is a finite Euclidean polyhedron. A set X is said to be
.Zil§.é£.§.2212£. 2_ if it is not locally tame at p. These definitions
of tame and locally tame are due to Fox and Artin [18] and Bing [5],

respectively. A set P in En is locally polyhedral at-a point x. of P

 

if there is a neighborhood of x whose closure meets P in a finite
Euclidean polyhedron.
The notions of wild and tame can also be applied to spaces that

are not polyhedrons. By a Cantor set is meant any homeomorphic image

 

of the classical Cantor ternary set, i.e., any compact, perfect,
O-dimensional, non—empty metric space is a Cantor set. A Cantor set
A.c En is said to be tame if it lies on a tame arc in E“; otherwise
A is said to be wild.

Examples of wild arcs in E3 were known as early as 1921 when
Antoine [l] constructed a wild Cantor set in E3, and thus an are through
this Cantor set is wild - the arc is called an Antoine's Necklace. In
1948 Fox and Artin [18] gave a number of examples of wild arcs and
spheres in E3 with one or two wild points.

In connection with n-books, C. A. Persinger proved in [30] the

following two theorems.

Theorem 1.1 No wild Cantor sets lie in a tame n-book in E3.

 

Theorem 1.2 There exists wild arcs and disks in tame n-books in

E3, n-> 2.

 

A set C in En is said to be cellular if there exists a sequence

of topological closed n-cells {Ci} such that C1+1 C Int C1 and

(D
C = inl C1. This notion was defined by M. Brown [8]. An are A is

pfshrinkable if A has an end point q and in each open set U containing

 

q in B“, there is a closed n-cell V45 U such that q lies in Int V with
Ed V meeting A in exactly one point. This notion was defined by P. H.
Doyle in [13] where he shows that all arcs that are bad at just one end
point are cellular.

A k-cell in E“, k S n, is said to be flatly embedded or flat,

 

if there is a space homeomorphism of En onto itself mapping it onto a
k-simplex. A f1§£_nfbggk_is a n-book such that each of its leaves is
a Euclidean 2—simplex.

A topological space is separated if it is the union of two
disjoint, non-empty open sets. Let S be a connected space then a closed
subset C separates S if S - C is a separated set. A point p of S is
called a £g£.pgin£_of S if S - p is separated. These definitions are
in Hocking and Young [21].

Finally, in Chapter IV, we need the following definitions from
homotopy theory which are in Dugundji [15]. If X and Y are two topological
spaces and I is the unit interval, then two maps f,g:X + Y are called
homotopic (written f = g) if there exists a continuous function
H:X X I + Y such that H(x,0) = f(x) and H(x,1) = g(x) for each x e X.

An f:X + Y homotopic to a constant map is called nullhomotopic, written

 

f = O. A space Y is contractible if the identity map on Y is nullhomotopic.

 

The image of H is called a homotopy path.

 

One theorem that should be mentioned because we will continually

refer to it is due to Schoenflies [32]. We will refer to it as the

plane Schoenflies Theorem.

Theorem 1.3 If J is a simple closed curve in E2, and h is a
homeomorphism of J onto the unit circle S1 in E2, then h can be

extended to a homeomorphism of E2 onto itself.

CHAPTER II

CANTOR SETS IN B3 LIE ON ARCS

One of the original proofs that was given to show that a Cantor
set in the plane lies on an (tame) arc was by Antoine [1]. However,
the proof of the generalization of that result to a flat 3-book follows
the outline of a proof by A. S. Besicovitch [4].

We need the following two lemmas in order to prove the theorem.
The first is a theorem1 of Dugundji [15], and the second is a generali-
zation of Lemma 1 of A. S. Besicovitch [4] to a 3-book, where the proof

follows the proof of Lemma 1 found there.

Lemma 2.1 If A is a compact subset, and B is a closed subset of a

metric space X, and if A n B 5‘ ¢ then d(A,B) = e > 0.

Lemma 2.2 Given a perfect set F in a 3-book, all of whose components
are points and a positive number d, there exists a finite set of
disjoint closed disks and "closed" 3-books of diameter less than d
containing F in their interior and each of them contain a perfect subset

of F.

Proof: As in the proof of Lemma 1 of [4], if we let {FIW B,6} be
the set of points at most 6 distance from F'f\B in the back of the

3-book, B, then there exists a 6' > 0 such that all the components of

 

1Theorem 4.4, Chapter XI, page 234.

8

{F11 B,5'} are of diameter less than d. Clearly these components are
intervals of B.

Let I be one of the intervals of B and construct a closed
"spherical" neighborhood U of the midpoint of I of diameter the length
of I in the book. The Bd U might contain points of F off the back of
the book but inside U there exists a "closed" 3-book neighborhood whose
boundary misses F and intersects B in 1, since the perfect set F is a
Cantor set and Cantor sets in the plane are tame. Do this for each
interval of {F n 13,6'}.

Finally, by Lemma 1 of [4], in the interior of each leaf of the
book the remainder of F can be covered by open disks whose closures are
disjoint, of diameter less than d, and disjoint from the "closed"
3-books constructed above.

Clearly each of these closed disks and "closed" 3-books contain

a perfect subset of F since their boundaries miss F.

Theorem 2.3 Let C be a Cantor set in the interior of a 3-book B3

 

then there exists an arc in B3 such that C is contained in the arc.

Proof: Let 61 = d(C, Bd B3) then El

By Lemma 2.2, there exists a finite collection of disjoint

> O by Lemma 2.1.

closed disks and "closed" 3-books of diameter less than min{€1,1}

containing C in their interior and each of them contains a Cantor set.

'
Denote this collection by {D11} (1 S i S n1). Now let x11, x11 8 Bd Dli
x11 f xii and connect xii to xli+1 (l S i S nl-l) by an arc in
n
Int 33 - fianli' This can be done because the Dli's which are closed,
are contained in the interior of B3. For convenience let a =' x1 and
b = x' .

10

Now let 62 = min{d(CI\ D Bd Dli)}’ then 82 > 0 since

Bd D11) > O by Lemma 2.1 for each i and there are only a

li’

d(C n D11,

finite number of 1's.

Since each D11 contains a Cantor set of points of C, by applying

Lemma 2.2 to each D11, there exists a finite collection of disjoint

closed disks and "closed" 3-books of diameter less than min{€2,k}
containing C in their interior and each of them contains a Cantor set.

Denote this collection by {DZj} (l S j S n2), where D ,¢: D11 (n; S j < n' )

23 i+l
' =3 ' = ' y
with n1 1 and nn1 n2. Now let x2j’ x2j 6 Bd D2j’ x2j ¥ x2j and
I ' < < V _ V '
connect x2j to x2j+1 if ni _ j ni+1 1, x11 to xZni and x2“i+1'1 to xli
ni+l'1
by arcs in D - I.) D . This can be done as above.
11 jzni Zj

If we continue in this manner we want to show that the limit of
this process, call it K, is an are which clearly contains C.

By a lemma2 of [21], K is a continuum containing a and b so
by a theorem3 of [21], we can show that K is an arc if we can show K
has only two non-cut points, namely, a and b.

To this end let x e K - {a,b}. If at some stage of the construc-

tion of K, x is a point on an arc that connects a D to D for some
ni ni+l

n then it is clear that x is a cut point of K. Hence we can suppose

that x e D for all n. Now let P by the union of all D with

nin n ni

i < in and all arcs up to and including the are connecting Dni _1 to
n

D except the point x Similarly let Fn be the union of all Dn

nin nin' i

with i > i and all arcs starting with the arc from D to D .
n nin nln+l

 

2Lemma 2-8, page 43.

3Theorem 2-27, page 54.

11

(I) 00
except the point x' . Now let P = UP n K and F = UF n K. Then
n1n n=1 n n=1 n
P and F are disjoint relatively open non-empty subsets of K, and each
point of K - x lies in one or the other. Therefore x is a cut point

of K, and hence K is an are from a to b.

CHAPTER III

UNCOUNTABLE MUTUALLY EXCLUSIVE COLLECTIONS IN B11

In this chapter will will use a result of R. L. Moore [27] to
point out some differences between a closed disk, the Euclidean plane,
and flat n-books for different n's. In particular we will show that
a flat n-book does not contain an uncountable collection of mutually
exclusive flat (n+1)-frames for n 2 2, but does contain an uncountable
collection of n-frames.

An ETEEEES is the homeomorphic image of n-line segments which
are joined at a common end point. The homeomorphic image of each line
segment is called a branch of the n-frame, and the vertex of index n

will be called the branch point. If in addition the branches of an

 

n-frame are line segments we will say that the n-frame is flat. In
the particular case when n = 3 we will call the 3-frame a trigd, and if
the branches of the triod are line segments we will call it a flag
312d.-

Because we will be referring to the result of R. L. Moore [27]

we will state it now for easy reference.

Lemma 3.1 There does not exist an uncountable set of mutually

exclusive triods in the Euclidean plane.

Using this lemma we can prove a result, which points out some

differences between a closed disk and the Euclidean plane, and that

12

13
will help shorten some of the proofs that follow.

Lemma 3.2 A closed disk does not contain an uncountable collection
of mutually exclusive arcs that intersect the boundary of the disk in

an interior point of the arc.

Proof: Suppose we have such an uncountable collection of arcs.
Since there exists a homeomorphism of the closed disk onto the unit
disk in E2 we can construct an uncountable collection of mutually
exclusive triods in the Euclidean plane in the following manner.
Construct a line segment normal to the unit circle and in the comple—
ment of the unit disk in E2 from each point of intersection of one of
these arcs with the boundary of the disk. But this contradicts Lemma

3.1.

However, there exists an uncountable collection of mutually
exclusive triods in a 3-book, as we can clearly see by the following
construction.

Let B: be the open 3—book which is the union of the xy—plane

and one-half of the yz-plane, and let B3 be the 3—book contained in

l
B: that has square leaves that intersect in the interval [-1,1] on the
y-axis. We can obtain uncountably many mutually exclusive triods in Bi
by joining three line segments perpendicular to the back of the book,
one in each leaf of Bi, at each irrational number in [-1,1]. Since

any 3-book is a homeomorphic image of B3 it contains uncountably

1)
many mutually exclusive triods.
Furthermore this is essentially the only way we can get

uncountably many mutually exclusive flat triods in a flat 3-book.

14

Lemma 3.3 If there exists uncountably many mutually exclusive flat
triods in a flat 3-book, then there exists an uncountable subcollection
of these triods with the branch point on the back of the book and with
one line segment in each leaf of the book. Moreover, all the branches

of the triod have length at least 8, for some 6 > 0.

2322:: The branch point has to lie on the back of the book for
uncountably many of the triods for otherwise one of the leaves of the
book would contain uncountably many mutually exclusive triods, which
would contradict Lemma 3.1.

There exists an E > 0 such that there are uncountably many of
these triods with all their line segments of length greater than or
equal to E, for if not there then would only be a countable number of
these triods with all their line segments of length at least fi-for
each n. But since the countable union of countable sets is countable
we have a contradiction of the fact that there are uncountably many
triods.

In this uncountable subcollection of the triods let us suppose
that there are only a countable number of these triods with one line
segment in each leaf of the book. Now since each line segment of these
triods has length at least 6, there would exist at most a finite number
of these triods with one or two line segments in the back of the book.
Thus there would exist an uncountable subcollection of these triods
with one leaf of the book containing at least two line segments of the

triods. But this would contradict Lemma 3.2.

We can also use a similar argument to show that there exist

uncountably many mutually exclusive O—curves in a 3-book. To Obtain

15

uncountably many mutually exclusive 6-curves in Bi

spheres centered at the origin with radius an irrational number between

just intersect the

O and l with Bi. Again since each 3—book is a homeomorphic image of
Bi, each contains an uncountable collection of mutually exclusive
B—curves.

In general we can embed uncountably many mutually exclusive
polygonal knots and links in a flat book.

To construct this uncountable collection, we will first embed
an arbitrary polygonal knot in a flat 3-book by the method of C. A.
Persinger [30]. At one of the points where the knot crosses the back
of the book we will alter the knot as in Figure 3.1, by replacing the
line segment(s) of the knot by the three dotted line segments. In the
third leaf construct a polygonal are connecting any two of the three

points on the back of the book, and meeting the new knot and the back

of the book in only these two points.

 

 

 

Figure 3.1

Now construct a "

tubular" neighborhood of this polygonal knot
and link in the book, and let do be the distance from the boundary of
the neighborhood to the knot and link.

Let ro be an irrational number between 0 and do’ and construct

a set of line segments in the book parallel to the line segments of the

16

knot and link and at a distance ro from them. By shortening or
lengthening these line segments, if necessary, we obtain two polygonal
knots and links in the "tubular" neighborhood of the original knot and
link. If we do this for each irrational number between 0 and d0 we
have an uncountable collection of mutually exclusive polygonal knots

and links in the book.

Remark: It is clear that the two points of index 3 can lie off the
back of the book for only countable many of any uncountable collection

of G-curves in a 3-book.

We need the following definitions that are found in Whyburn's

book Analytic Topology, to complete the proof of the next theorem.

Definition 3.1 A point P is said to be a condensation point of a

 

 

set M provided every neighborhood of P contains uncountably many points

of M.

Definition 3.2 A collection G of disjoint subsets of M will be

 

called non—separated provided that no element of G separates in M

 

two points belonging to any other single element of G.

Definition 3.3 A non-separating collection [G] of subsets of M
will be said to be saturated provided that if G e [G] and P is any point
of M - G, there exists at least one element 6' of [G] which separates

P and G in M.

Theorem 3.4 A flat 3-book doesn't contain an uncountable collection

 

of disjoint 4-frames if the branches of the 4-frames are polygonal or

at most one of the branches is an arbitrary arc.

l7

Prggf: Suppose we have an uncountable collection of mutually
exclusive 4-frames in a flat 3-book. Let us consider the 4-frames
consisting of the first line segment in each of the three polygonal
branches of the 4—frame starting from the branch point and the
arbitrary arc. Now by Lemma 3.3 there exists an uncountable subcollection
of these 4-frames with the branch point on the back of the book and with
one line segment in each leaf of the book. Moreover all the branches
of the flat triod contained in the 4-frame have length at least e, for
some 6 > 0. Furthermore, there exists an uncountable subcollection of
these 4-frames with the distance from the branch point to the other
end point of the arbitrary arc at least 8' > 0, where 8' exists for the
same reason that 6 exists in Lemma 3.3.

Since there exists an uncountable number of branch points in
the back of the book, there exists a branch point, call it v0, that
is a point of condensation of the set of all branch points. Let U be
the e-neighborhood of V0 in the book. Now the collection of all flat
triods that consist of the three line segments of the 4-frame which
intersect U, is an uncountable non-separated collection, and each member
separates U. By a theorem1 of [35], this collection contains an
uncountable saturated subcollection. But if we take an element of the
saturated subcollection there must be an arc joining a point not on it
to the branch point (since this element is a 4-frame and 6' > O) which

is a contradiction. Hence we have the desired conclusion.

Remark: The above proof cannot be generalized to an uncountable

collection of 4-frames because a triod with the branch point on the

 

1Theorem 2.2, Chapter 3, page 45.

. u I n . Ilu‘ I‘ll lllll. I

II III II: II III ll lilo-Ill III III.
1"11 l1!
1" l [I

18

back of a 3-book does not necessarily locally separate the 3-book. Thus
the question, is there an uncountable number of 4-frames in a 3—book, is

still an open question.

This theorem, and Lemmas 3.1 and 3.2 lead us to the following

theorem which points out differences between n-books for different n.

Theorem 3.5 A flat n-book does not contain an uncountable collection

 

of mutually exclusive flat (n+l)-frames for n 2 2, but does contain an

uncountable collection of n-frames.

3592:; If n = 2, we have the desired conclusion by Lemma 3.1.
If n = 3, the previous theorem gives us the desired conclusion.
Let n > 3, and suppose we have an uncountable collection of
mutually exclusive flat (n+l)-frames. As before there exists an E > 0
such that there is an uncountable subcollection of theseflat
(n+l)-frames with the length of each branch at least E, and with the
branch point on the back of the book. Now there are clearly only a
finite number of these (n+l)-frames with one of the branches along the
back of the book, so let us assume that all the branches have only
one point in common with the back of the book. Then there must exist
an uncountable subcollection with at least two branches in one leaf
of Bu, which contradicts Lemma 3.2. Hence we have the desired conclusion.
We can clearly construct an uncountable collection of (flat)

n-frames in Bn as we did in the case n = 3.

CHAPTER IV

THE l-LC PROPERTY OF ARCS IN B3

In this chapter we will investigate the effects that the
singularities of a flat 3-book have on the l-LC property of the comple—
ment of a polygonal arc in a flat 3-book. In this regard we will prove
some intermediate results between the l-LC property of the complement
of a polygonal arc in E2 and the complement of a polygonal arc in E3.

The k-LC prOperty was first defined by S. Eilenberg and R. L.

Wilder [17], and is included here for completeness.

Definition 4.1 Let Y be a set in a metric space M. If fzx + Y is

 

a map, let B(f) be the infimum of the diameters of all homotopy paths
in Y shrinking f to a constant. If f is not nullhomotopic B(f) = W.
Y is said to be k-LC at a point x in M if given any sequence {fn} of

maps of the k-sphere Sk into Y, fn(Sk) + x, then B(fn) + 0.

The following definition is a modification of a definition
due to E. E. Moise [26], and will be used to facilitate the statement

and proof of the following theorem.

Definition 4.2 Let B be the back of a flat 3-book BB, and let A be

 

an arc in Int B3 such that B P!A.is a point p. If for each sufficiently
small open neighborhood U of p in B3, U — B is the union of three
disjoint open sets two of which intersects the component of Ulfi A that

contains p, then A pierces B at p. If the component of U n A that

19

20

contains p intersects only one of the three disjoint open sets of

U-B, then A will be said to be tangent to B at p.

Theorem 4.1 Let A be a polygonal arc in the interior of a flat

3-book B3. B3- A is not l—LC at x C A if and only if there exists a

 

subarc of A that is tangent to B at x.

2523:: If there exists a subarc of A that is tangent to B at x,
then Figures 4.1 a and b indicate that if x is an end point or an
interior point of A that there exists a sequence of functions

fn:S1 + B3- A that are not nullhomotopic in B3- A. Hence B3- A is not

l-LC at these points.

 

 

(a) (b)

Figure 4.1

Conversely, if x e A is not a point where any subarc of A is
tangent to B, then x belongs to one of the following three cases:

Case A: x e A and x ¢ B. Since x is in one of the leaves of
B3, for n 2 N, for some N, fn(Sl) will be in the complement of A in one
leaf of B3, and thus B3- A is clearly l-LC at x.

Case B: x is a point of an interval of A.O B. If the diameter
of fn(Sl) is less than the length of the interval in A r\B, then we can

clearly see that fn:S1 + B3- A is nullhomotopic. See Figure 4.2 3. Hence

B3— A is clearly 1-LC at x.

21

Case C: There exists a subarc of A that pierces B at x.

Then locally 83- A looks like Figure 4.2 b, since A is polygonal.

Z 24/ / A /
Z, fn(sl)+ fl x

L L
(a) (b)

 

 

 

 

Figure 4.2

It is clear that by collapsing the two leaves that contain a portion of
A first and then the other leaf that this neighborhood of x is

contractible. Hence B3- A is l-LC at x.

Now continuing with the discussion of differences between the
plane, a flat 3-book, and 3-space, the previous theorem provides a little
additional information. It is clear that the complement of a polygonal
arc in the plane is l-LC at every point on the arc. In the interior of
a flat 3-book, B3, the complement of a polygonal arc in B3 is not l—LC
only at points where some subarc of the arc is tangent to the back of
the book, and is l-LC at all other points of A. Finally, in 3-space the
complement of a polygonal arc is not l-LC at every interior point of
the arc, and is l—LC only at the end points of the arc.

Furthermore, if we allow the arc A to be wild and locally
polygonal except at one end point then D. R. McMillian [25]1 has shown
that E3- A is not l-LC at the bad point. A similar result can be

proved for the complement of such an arc in a flat 3-book.

 

1Lemma 4, page 524.

{IiIIIIIII l
III-I I‘ll

 

22

Theorem 4.2 Let A be a wild arc in the interior of a flat 3-book

B3. If A is locally polygonal except at one end point of A, then B3- A

 

is not l-LC at this end point.

ngpf: Let P be the bad end point of A, then P is clearly in the
back of the book, and let Un be a neighborhood of P in B3 of diameter
%- starting with n sufficiently large so that Uh«c B3. In one leaf
of 33 we can obtain a sequence of disjoint arcs, {An}, that converge to
P by picking An in Un' This sequence exists for if there were only a
finite number of these arcs near P in this leaf then the arc would be
locally tame at P since locally it would lie in a plane.

Case A: There exists an infinite subset of these arcs, call
them {Ah.} where at one end point, an,, of these arcs in the back of the
book a subarc of A is tangent to B at an, or at an, the are A intersects
the back of the book in an interval and then A goes back into the same
leaf of B3. Now since A is locally polygonal at an, we can construct a
simple closed curve about an, in (B3- A) r\Uh. that isn't contractible
in 83- A. See Figure 4.3. Clearly these simple closed curves converge

to P, and hence the arc is not l—LC at P.

/m / /
L/

/ \ L l l

 

 

 

 

 

 

Figure 4.3

Case B: There doesn't exist an infinite subset of {An} as in

Case A. Then there exists an infinite subset of {An} call them {An"}

23

where the arc pierces the back of the book at the end point(s) in the
back of the book or intersects the back of the book in an interval and
then goes into one of the other two leaves of BB. Now as in Figure 4.4
we can construct a simple closed curve in (33— A)¢fl Un"’ since A is
locally polygonal at the end points of Ad,,that isn't contractible in

3

B - A. These simple closed curves clearly converse to P, and hence

83— A is not l—LC at P.

 

 

rm 4
WW
I

 

 

A (possibly)

 

—.

Figure 4.4

Note: P could lie on the interval in the back of the book between
the end points of Ah" but that won't make any difference as indicated in

Figure 4.4 by staying close to An" with the simple closed curve.

Remark: Tangent points and other non-piercing points, can always
be removed without changing the E3-embedding of an arc. Thus for an
arc that is locally polygonal except at an end point P there is an
embedding in which all points on the back except P are piercing points,

and the arc has the same E3-embedding.

This study could just as well have been made with the O-LC
property of the complement of the arc rather than the l—LC property.
The only reason for doing the latter is that it was what was initially

looked at.

 

 

llllll‘fllll‘lll'llll

24

For the case of the complement of a polygonal arc A in the
plane it is clear that it is O-LC only at the end points, and E3- A
is clearly O—LC at all points of A. The result for a flat 3—book would

be the following.

Theorem 4.3 Let A be a polygonal arc in the interior of a flat

3-book B3. B3— A is O-LC at x e A if and only if x satisfies one of the

 

following two cases:
(1) x is an end point of A, or

(2) there exists a subarc of A that pierces B at x.

Ppppf: If x is an end point of A then clearly B3 - A is O-LC at
x, so let us suppose there exists a subarc of A that pierces B at x.
By looking at Figure 4.5 it is clear that the are connecting the two
points of fn(S°) is a homotopy path which can be made arbitrarily small

for n sufficiently large.

 

 

 

Figure 4.5

Conversely, there are three cases to consider:
Case A: x 6 Int A and x ¢ B. Then locally the arc lies in a
plane, and hence B3- A is not O-LC at x.

Case B: x 6 Int A and there exists a subarc of A that is tangent

25

to B at x. By mapping S0 as in Figure 4.6 a it is clear that the map
is not nullhomotopic in this neighborhood, and there exists a sequence

of these maps {fn} such that fn(S°) + x. Hence B3- A is not O-LC at x.

/fn(s°).§\\:/ ///£'fn(s°)/
/l ' /I V

L l J
(a) (b)

 

 

 

 

 

Figure 4.6

Case C: x 6 Int A and x is a point of an interval of A11 B.
If A is locally as in Figure 4.6 b by mapping So as in the figure, it
is clear that the map is nullhomotopic, but the length of the homotopy
path is at least twice the distance from x to the farthest end point of
this interval of Aln B. Furthermore there exists a sequence of these
maps {fn} such that fn(S°) + x. Hence B3- A is not O—LC at x.

If A is locally as in Figure 4.2 a then there exists a sequence
of maps of So that are not nullhomotopic in this neighborhood as in

Case B.

CHAPTER V

CUTS OF A 3-BOOK

During the translation of Eilenberg's thesis [16], it became

apparent that some of the theorems on unicohérence could be extended

to the one point compactification of an open 3-book B3, but that there

are counterexamples for many of the generalizations of the theorems on
A3

cuts of 82 to cuts of B .

A connected space X is called unicohérent, when for any

 

decomposition of X into two closed and connected sets X1 and X2, the

set X1!) X2 is connected. A set Y contained in a space S cuts S

between two points 3 s e S - Y if there doesn't exist a continuum

1’ 2
KC S - Y such that 31,32 e K.
Note: This definition of a out due to Eilenberg is slightly less

restrictive than the definition of a cut point and of cuttings found
in Whyburn [35], and slightly more restrictive than the definition of

cuttings found in Hocking & Young [21].

Let us define E3, and a restriction of a G-curve in BB, which

we will call a 6'-curve.

Definition 5.1 Let C be a great circle on 82, and let D1 be the

closed disk contained in the plane of C with boundary C. Then we will
3

 

'denote SZtJ D1 by B

the two components of 33— D

, and furthermore we will denote the closure of

by D and D .

l 2 3

26

27

Note: The Di's were also used in the definition of a 3—book, but

no confusion should arise if we use the same notation here.

 

Definition 5.2 Z C'B3 is called a 6'-curve if Z is a G-curve,
Z lei = Z1 is an arc, and 21/) C = {a,b} a f b for each i = 1,2,3.
We will denote the 6'-curve that is obtained by intersecting the plane

3

that is perpendicular to the plane of C with E by 6;, and call it the

standard 8'-curve.

 

Theorem 5.1 E3 is unicohérent.
Proof: By a theorem1 of Eilenberg's [16] both 32, and D1 are uni-
x , 2
cohérent. Hence B3 = SZIJ D1 is unicoherent, since S ’1 D1 . C is
connected, and both S2 and D1 are closed and locally connectedz.
Remark: This is a generalization of Corollary 1.6 of [16] to B3,
n+1

which states in part that S is unicohérent for n = 1,2,...

The following theorem is a generalization of the plane Schoenflies

A

Theorem to B .

Theorem 5.2 If Z C B3 is a 6'-curve, then the homeomorphism

 

h:Z + 6; can be extended to a homeomorphism of B3 onto itself.

Proof: Clearly h exists, so suppose that a and b lie on a common
diameter — if not we can easily find a homeomorphism £43 B3 such that

f(a) and f(b) are on a common diameter. To that end let f<£ C such

 

1Corollary I.6, page 73 of [16].

2Theorem 1.4, page 72 of [16].

D I .III I I‘Illll- II- I

28

that f(a) and f(b) are on a common diameter, and extend f to D by
mapping the radius OE linearly onto CITE) where 0 is the center of D
and c e C. Extend f to all of E3 by mapping each circle on 82 that is
obtained by intersecting $2 with a plane parallel to D, as f maps C
(map the "north" and "south" pole onto themselves).

Now, since Z is a 6'—curve, by extending hIZ n D by the

1
identity on C, we have a homeomorphism h' : (Z n D1) U C + (6") n D1) U C,

where 9; is the standard 6'-curve that contains the points a and b.

h' can be extended to h"<$ D by a generalization of the plane

3

l

Schoenflies Theorem for a B—curve in the plane.

Since we can do the same thing on D2 and D3 and since the three
3

homeomorphisms agree on C we have an extension of h to B .

A

Corollary 5.3 Each 6'-curve in B3 separates E3 into two open 3-books

 

and it is their common boundary.

Remark: The following three examples show that the restrictions on

a 6'-curve are necessary for the above theorem and corollary.

Example 5.4 Z f\C = {a,b,xo} , Figure 5.1 a. The complement of Z

 

on the right is not an open 3-book. It is also clear that the homeomor-
phism of the theorem can't be extended to B3 since h(xo) would be an in

interior point of D.

Example 5.5 Z.“ C = {a,b,xo}, Figure 5.1 b. Z doesn't cut B3. It

 

is also clear that the homeomorphism of the theorem can't be extended

to B3 since h(xo) would be in 82 — C.

 

3The proof involves using Lemma 4.4, page 15 of [14].

29

Example 5.6 ZfiDi is a simple closed curve for i = 2 or 3 and {a,b}

 

for the other, Figure 5.1 c. The complement of Z has just two components
but it is clear that neither is an open 3-book and that the homeomorphism

of the theorem can't be extended to B3.

 

(a) (b) (C)
Figure 5.1
Example 5.7 A locally connected, Gd set that cuts B3 between two

 

points z' and 2" on C. However, it contains no 6~curve that cuts B3
nor does it contain an irreducible cut of B3. This then is a counter-

example of a generalization of Theorem 11.17 of [16] for B3.

Let xi + x1 be a sequence of points on C such that the lengths

A A
of the arcs x1x1+1 in C is k the length of the arc x1_1x1

in D2 join x1 to x1+1 by a "semicircle" of diameter the length of the

chord xix1+1 (i > 1). In D2 also join the x1

pole" of D by one quarter of great circles, and in D1 join the center

(i > 1), and

's (i > 1) to the "north

2

of D1, call it 0, to the x1

x1 to x2 by a great circle in D

's (i > 1) by radii of D Finally connect

1.
3 and call the union of all these arcs X,
Figure 5.2. Let 2' and 2" be as in the figure.

Clearly X is locally connected since the only point to check

is x1 and any neighborhood of it in X contains a path connected

30

neighborhood.

 

 

 

 

Figure 5.2

X is a Gd set. If we st0p the construction of.X at X.1 then

that portion of X, call it X1, is closed and by taking smaller and

smaller "tubular" neighborhoods of X.j it is clear that Xj is a G6 set.

Thus X =- jgl X3 = jgfgcaj} -- 912191 Gaj} and hence X is a 65 set since

00

I} C is open for each n.
j=1 0:]

To show that X cuts 33 between 2' and 2" first let

r:33— X‘+ (Sz-X)‘J xl be the map defined by mapping the radii of D

onto their end points.

1

Suppose X doesn't cut B3 between 2' and 2"; then there exists
a continuum K4: B3 — X such that z' and z" e K. Further if we suppose

2-.X, then we get a continuum in 82- X containing 2' and 2"

that r(K) C 8
so that 82/) X doesn't cut S2 between 2' and 2". However, 821‘ X contains

a simple closed curve that cuts 82 between 2' and 2". Hence r(K):D {x}

. l l

31

or equivalently KflOxl 3‘ fl.

Now suppose every neighborhood of some point y e K (\Ox1 contains

points of K that are also in some sector of D determined by 0,xj,xj+l

for some j 2 3, then each sector of D determined by 0,xi contains

’xi+l
points of K for i 2 N.

Further, if £§ik+14w K = O for some k 2 N, then it is clear that
K is not connected since the sector 0,xk,xk+l and its complement in B3

intersected with K separates K into two open sets. Thus izki+14~ K ¥ ¢

v ' A
for i 2 N. Let x1 be a sequence of points such that x1 6 xixi+1,

xi + x1 so x1 6 K since K is closed. This is a contradiction.

then

Therefore for each y e K FIOxl there exists a neighborhood small
enough so that it contains no points of K in these sectors of D1, and

a finite number of these cover KIT Ox since this set is compact. We

1
can move the points of KIN 0x1 away from Ox1 by the following geometrical
argument - we don't move 0 or x1 since the distance from K to O and x1

is positive.

First map the diameter through y onto the broken line through

I

y - see Figure 5.3 - and extend the map to the boundary by the identity

map. Then extend the map by the plane Schoenflies Theorem for a B-curve.

 

Figure 5.3

.———

If we let K' be the continuum that is moved away from Ox1 by

the above argument then r(K')<: 32- X and we obtain a contradiction of

32

the fact that 821% X cuts 82 between 2' and 2". Hence X cuts B3 between

2' and 2".

It is clear no G-curve in X cuts B3 by looking at Example 5.5.
To show that X contains no irreducible cut suppose that X' is
an irreducible cut. Then X' contains a subsequence of {xi}, call it

{xi} converging to x e X'; otherwise X' wouldn't cut B3. But now

1

(X' —‘O;;) LI{O,x;}, i.e., just remove one of the radii of D leaving

1
the end points, cuts E3 between 2' and 2" if n 2 3, which is a contra-

diction.

Example 5.8 A locally connected irreducible cut of B3 between two

 

points z' and 2" on C that is not a G-curve. This then is a counter-

example of a generalization of Theorem 11.19 of [16] for B3.

 

 

 

 

Figure 5.4

Let xi + x1 be a sequence of points on C as in Example 5.7,

and connect x to x by a "semicircle" of diameter the length of the

1 2

33

chord x x in D as in Example 5.7. Then connect x to x by a "semi-

1 2 3 2 3

circle" in D2, and continue alternating these "semicircles" between

D2 and D3. Now connect the center of D1 to each xi (1 > 1) by a radius

of D1 and let X' be the resulting set, Figure 5.4. Let 2' and 2" be as

in the figure.
We can show that X' cuts B3 between 2' and 2" by the same

argument as in the previous example. Clearly X' is locally connected.

Now x'rw $2 is an irreducible cut of S2 so if X' is not an

irreducible cut of B3 we could take an open interval out of one of

the radii of X'I‘ D1 and what would be left would still cut E3. But it

is clear how one could construct an arc, like A in Figure 5.4, in

B3- X" containing 2' and 2", which is a contradiction.

Remark: X' cuts B3 but it does not separate B3, for if EB- X' were
the union of two disjoint closed sets then it is clear than one of the
closed sets would contain most of the upper hemisphere of 52, every other

sector of D determined by O,x

1 and the line segment from 0 to

Zi-l’x2i

x1 not including the end points. The other closed set would contain

part of the lower hemisphere and every other sector of D determined by

1

O,x21,x21+1. But there exists a sequence of points in the sectors

determined by O,x converging to an interior point of 0x1, which

2i’x2i+l

is a contradiction.

From the above constructions we can prove the following result

which is the only theorem that we could prove for cuts of B3.

Theorem 5.9 If X¢= S2 - {z',z"} is closed and cuts 82 between 2' and

 

2 then there exists a component Y of X such that Y LIA, where A is the

Set of line segments from 0 (the center of D) to C n Y, cuts B3 between

34
the two points z' and 2" if z', z" 6 C.

Prppr: Since X cuts 82 between 2' and 2", then there exists a
component Y of X that cuts 52 between 2' and 2".4
Suppose Y U'A doesn't cut B3 between 2' and 2" then there
exists a continuum K C E3- (Y U A) containing 2' and 2". Define
r":B3- (Y U A) + 82— Y by mapping the radii of D - A onto their end
points — r can be defined since C n Y must contain at least one point
in the two components of C - {z',z”} and thus A is nonvoid. Since r

is continuous r(K) is a continuum in 82— Y containing 2' and z". This

2
contradicts the fact that Y cuts S between 2' and 2".

If we generalize the concept of local cuts to B3 we will also
easily find counterexamples to the generalizations of the results of
S. Eilenberg found on page 87 of [16].

A set XCLS2 is said to cut 82 locally at a point x e 82, when

X cuts each sufficiently small neighborhood U of x.

Example 5.10 A continuum X.c,E3 that doesn't locally cut B3 at a

 

point x0 8 X that cuts X. This is a counterexample of a generalization

of Theorem 11.22 of [16] for B3.

Let X be an arc in B3 that pierces C in the sense of Definition
3.2 at a point xo 8 C. X doesn't locally cut B3 at x0, see Figure 4.5,

but xo clearly cuts X.

 

4Theorem II.3, page 76, of [16].

i II III All ||Ill I. III: III! '1' l t 0'] 11 l llu Ill: !!l Ill ‘4 I." I I I'll It I [III II III. I

CHAPTER VI

HOMOLOGICALLY TRIVIAL 2-COMPLEXES IN B3

This chapter will deal with homologically trivial finite
2-complexes in a flat 3—book, and with homologically trivial finite
2-complex neighborhoods of arcs in a flat 3—book where the arcs are
cellular when considered in E3. The following remark indicates why
we might want to look at homologically trivial finite 2-complexes in

a 3—book, and their relation to a cellular arc.

Remark: If A is a cellular set in 3-space then by the definition
of cellularity inside any neighborhood, U, of A there exists a
topological 2—sphere, 82, such that A c Int 82. Hence by the Bing

Approximation Theorem [6] there exists a polyhedral 2-sphere Si in U

such that A c Int Si.

To first of all prove that homologically trivial finite

2-complex in a flat 3—book are collapsible, we need the followirug lemma.

3
PEEK—ii Let K be a finite 2-complex in a flat 3-book B , the“ K

has a free face.

eatlt
M: Suppose that K has no free face, and consider a compo“

3 (:11¢3
1’ call it M: Where Di is one of the leaves of 13 . Then

free faces of this simplicial complex are 311 on the back of the

of K44 D
book'

To show that M is a 2-manifold with boundary, fixrst let

35

36

x €‘M 0 Int Di' There are three cases to consider: if x is a point
of the interior of a 2—simplex then it clearly has an open 2-ce11
neighborhood; if x is a point of a l—simplex and not a vertex of the
2-complex, then x is contained in exactly two 2—simp1exes because
there are no free faces in the interior of Di’ and hence x has an open
2-cell neighborhood; finally, if x is a vertex of the 2—complex then
x is contained in at least three l~simplexes each of which is contained
in exactly two 2—simplexes because again there are no free faces in
Int Di and hence x has an open 2-cell neighborhood.

Now if we let x e M n B, where B is the back of the book, there
are two cases to consider: if x is not a vertex of the 2-complex then
it clearly has a closed 2-cell neighborhood; if it is a vertex of K,

by again using the fact that there are no free faces in Int D x is

19

contained in at least three l-simplexes and two of these l—simplexes

are on the back of the book so x has a closed 2-cell neighborhood.
Thus M is a 2—manifold with nonvoid boundary. But the boundary

of all compact 2-manifolds are disjoint simple closed curvesl. Hence

we have a contradiction since Bd M c B.

Theorem 6.2 Any homologically trivial finite 2-complex K in a

 

flat 3-book B3 collapses to a point.2

Proof: By Lemma 6.1, K has a 2-simplex with a free face, and thus

we can collapse this simplex to get a homologically trivial 2-complex

 

1Chapter 1 of Massey [24].

2This theorem follows directly from an unpublished result of
P. Dierker [10] : If K is a collapsible 2—complex and L is a homologically
trivial subcomplex, then L is collapsible.

.!‘ Ill ll Ill Ill

 

III“ II 1 Ila" [It'll I‘lll' '- '-

37

with one less 2—simp1ex. By a finite number of these elementary
simplicialcollapses we have a homologically trivial l-complex, which

is a connected tree. This tree can clearly be collapsed to a point.

Remark: To prove this theorem all we needed was the fact that K

was simply connected.

By exactly the same arguments as in Lemma 6.1 and Theorem 6.2
we can show that a homological trivial finite 2—complex in the plane
is collapsible. In fact, the complex could be thought of as lying in
two of the leaves of a flat 3-book, and hence collapses to a point.
However, the dunce cap provides a counterexample to the generalization
of this result to E3, because a triangulated dunce cap has no free

faces but is homologically trivial.

Corollary 6.3 A dunce cap cannot be embedded in a flat 3-book.

 

Theorem 6.4 A homologically trivial finite 2-complex, K, contained

 

,.
in a flat 3—book B3 is cellular in E3.

Prppf: First triangulate E3 so that K or a subdivision of it is

a subcomplex of the triangulation of E3. To this end first of all
triangulate E3, and then take three copies of E: and join them along
the x-axis so that they make an angle of 1200 with each other in E3 -
call the complex Bo' Since B3 has planar leaves it can be embedded in
B0’ thus IKI<Z IBOI, and therefore there exists a subdivision of K and
of Bo’ call them K and 30’ such that K is a subcomplex of Bo' Now each

i, D' of B0 separates E3 into two cepies of E2. Let us

:1

consider the one that doesn't contain the other leaf of Bo. Since this

1
domain is E+

pair of leaves D

X (DitJ D3) it can be triangulated so that the triangulation

38

iLJ D3. If we do this for

each pair of leaves of B0 we have a triangulation of E3 with K a sub-

is compatible with the triangulation of D

complex of it.

Let U be an open neighborhood of K in E3, and take the rth
derived subdivision (r 2 2) so that the mesh of the triangulation of
E3 is less than %d(Bd U,IK|).

The derived neighborhood N of K in E3 is clearly contained in
U, and since K is collapsible by Theorem 6.2, then N is a 3-ce113.

An arc in a flat 3-book will be said to have a pfppprp (pseudo-
basis) if for each open neighborhood U of the arc there exists a

homologically trivial finite 2—complex neighborhood of the arc in the

book contained in U.

Lemma 6.5 Any are that is bad at just one end point doesn't have a

p—basis even though it is cellular in E3.

grppr: Persinger [30] has shown that we can embed the arc in a flat
3—book B3, and since 83 has planar leaves we can assume it is contained
in B2, where B3 is the open 3-book which is the union of the xy-plane
and one-half of the yz—plane.

Now we clearly can find a plane y = y0 such that the plane
doesn't intersect the arc, and since the arc doesn't separate Bi, we
can extend the arc by a polygonal arc to get an arc A' that meets the
plane y = y0 in only one point, call it a, which is an end point of A'.

If we reflect A' in the plane y = y0 to get A" and let

A = A'\J A", Martin [23] has shown that A is not cellular. Therefore

 

3Theorem 2.11, page 57 of [22].

39

there exists an open neighborhood U of A in E3 such that no 3—cell
that contains A in its interior is contained in U.

If we now suppose the original arc has a p-basis, then there
exists a homologically trivial finite 2-comp1ex neighborhood of the arc,
call it K1, contained in B3 such that IKIIC: U. If A'led K1 is a vertex
of K1 move A' slightly so that the intersection is an interior point of
a l—simplex, again call this arc A'. we can construct a simplicial

tubular neighborhood K of A'- (IKII Uta) in B: contained in U such that

2

K1!) K2 is a l-simplex and K intersects the plane y = y0 in a l—simplex.

2
If we look at the Mayer—Vietoris sequence,

_)
. .. Hq(K1r\ K2) '* Hq(Kl) e Hq(K2) '*' Hq(K1U K2) "* Hq_l(Kln 1(2) + ...
we can see that Hq(K1(J K2) = O for all q > 0 since
2
= a: = > ...-z
Hq(KlIW K2) H2(K1) Hq(K2) O for all q 0. Let K1 K1(J K2.
Similarly A" has a homologically trivial finite 2-complex

neighborhood |K§|<3 U and K: intersects the plane y = y0 is the same

l-simplex as Ki, and, therefore Ki1J K: is homologically trivial by the
same argument as above. However KI'J K3 is then cellular by Theorem 6.3,

and hence there exists a 3-cell inside U containing IKitj Kil in its
interior, which contradicts the choice of U.

The fact that the original arc is cellular follows from the
fact that every subarc of it is p—shrinkable and hence by a result of

3
P. H. Doyle [13] the arc is cellular in E .

Remark: From this lemma it is clear that the p-shrinkable condition
of [13] is not a sufficient condition for an arc in 83 to have a p-basis.
Hence a p-shrinkable argument of the type of [13] will not work in a flat

3"b00k o

40

At this point it seems we could define a cellular arc in a
flat 3—book as one that has a p—basis. The polygonal arcs would have
a p—basis by triangulating tubular neighborhoods of the arc in each
leaf of the book and for points on the back of the book triangulate a
small 3-book neighborhood of the point being sure that the triangulations
are compatible. However, the following example of an embedding of a
wild arc with a p-basis seems to indicate that how the arc is embedded
is important. Since no general results could be obtained except in the
converse sense, no definition of cellularity in a flat 3-book is put

forward.

Example 6.6 An embedding of a wild arc which is locally polygonal

 

except at one point in a flat 3-book 33 such that the arc has a p-basis.

Let p be a point on the back of 33 and let B: be a sequence

of flat 3—book neighborhoods of p in B3 such that 33 <2 Int B3 with

i+l i
B3 = B3 Let E = Int 33 — B3 Now embed a 01 onal trefoil knot
l ' i i i+1' p yg

in I; (the closure of E1) for each i by the method of C. A. Persinger

[30] so that the arc meets the Ed 33

1+1 in exactly one point which is in

only one of the leaves of BB, and such that for i > 1 the arc intersects
the Bd B3 in the same point that the polygonal trefoil knot in'Ei_1
intersects Bd Bi. Thus by induction we have an embedding of a sequence
of trefoil knots that converge to p and the resulting arc is locally
polygonal except at p.

Now if U is any open neighborhood of the arc in B3 there exists
a B: for some j such that B: C U. Furthermore in B3- 83 the arc is

polygonal, and thus by a previous argument there exists a homologically

trivial finite 2—complex neighborhood, K1, of it in U, and since the

41

are intersects B3 in one point that is in only one leaf of B3, K can be

j 1

made to intersect B3 in a segment. To make IK1I\J [Bil a 2—complex just

add the two vertices of the l—simplex of K in the l-simplex of B3 and

1 J
extend the new triangulation of this l-simplex to all of B?. As in
Lemma 6.5, KltJ B: is homologically trivial.

Now since the back of the book has an orientation - its just a
line segment - if we had embedded the arc so that all the points of the
arc that are also on the back of the book were all on one side of p
then we could extend the arc by a line segment along the back of the
book, to get a wild arc that is locally polygonal except at p. If
this line segment is not contained in B3

J
to the one above to get the whole are in a homologically trivial

we can use a similar argument

finite 2-comp1ex inside U.

This example seems to depend quite strongly on the fact that
the are as constructed is L.P.U. at p. Thus in an attempt to obtain
a general result that an arc in a flat 3-book that is locally polygonal
except at one point has a p—basis, the restriction that it be L.P.U. at
each point was added. Even with this additional restriction, the
intersection of a polyhedral 2—sphere with a flat 3—book can be
sufficiently bad to make it seem impossible to use this line of proof.

However the following converse theorems can be proved.

Theorem 6.7 Let A be an arc embedded in a flat 3—book B3 such that

 

it is locally polygonal except at one point p and at p there exists
an element of the p—basis of p in each neighborhood of p whose
boundary meets the arc in one point if p is an end point of A, and

two points if p is an interior point of A. Then if we consider A as

III!) III 1! I'll" ll 1| Ill. 4
i l

42
being embedded in E3, A is L.P.U. at p.

Prppr: Suppose that p is an end point of A, let U be an open
neighborhood of p in E3, and let K be a homologically trivial finite
2—complex neighborhood of p in 83 whose boundary meets A in one point,
and K1: U n B3. Since A is locally polygonal at A O Bd K = b, we can
retriangulate A so that b is a vertex of A.

We want to construct a derived neighborhood N of K in E3 whose
boundary meets A in just one point and chU. Then Bd N will be a
polyhedral 2-sphere that meets A in one point, since N is a 3—ce114,
and hence A is L.P.U. at p.

To this end let K1 be the l-simplex in A - K with vertex b,
and then triangulate E3 so that K.b’K1 is a subcomplex of the triangu-
lation as we did in Theorem 6.4. To make sure that the boundary of the
derived neighborhood we construct misses the arc except on K1 let
B'= A - (K.U K1), and take the rth derived subdivision (r 2 2) of the
triangulation of E3 so that the mesh is less than or equal to kd(K,BO> 0.
Also take the rth derived subdivision of this triangulation of B3 so
that the mesh is less than or equal to %d(Bd U, IKI).

Now it is clear by the definition of the rth derived neighborhood
N of K in E3 that it would miss B'and hence its boundary would meet A
in only one vertex of N which would be a point of K1. Further Nit U,
and since r 2 2 the rth derived neighborhood of K in E3 is a derived
neighborhood of K in E3.

It is clear that the same type of argument would work if p is

an interior point of A, and if A n‘Bd K is two points.

 

4Theorem 2.11, page 57 of [22].

43

The following theorem is proved using Theorem 6.4.

Theorem 6.8 If an arc A in a flat 3-book has a p—basis, then A is

 

cellular in E3.

CHAPTER VII

A CHARACTERIZATION OF 2—COMPLEXES IN E3 THAT CONTAIN WILD ARCS

In this chapter a characterization theorem is obtained for
those finite 2-complexes in E3 that contain wild arcs, where the restric-
tion that is placed on the 2-complex is that it contains a subcomplex
whose carrier is a 3—book. In order to facilitate this proof we need
the following definitions of Whittlesey [34].

Let us assume initially that we are given a connected finite
simplicialZ-complex, K', and in K' every edge is a face of some 2-simplex.
Let P be an arbitrary point of K', and let St(P) be the star of the

open simplex containing P.

Definition 7.1 If St(P) is homeomorphic to the Euclidean plane,

 

then P is called regular. The points of K' which are not regular are

called singular.

Definition 7.2 If St(P) is an open flat n-book (n#2) then P is

 

called line singular. If St(P) is topologically equivalent to the

 

space obtained by identifying the origins of a certain m(>l) copies of

the Euclidean plane, then P is called a conical point - St(P) is called

 

a cone, and the components of its regular part are called the leaves
of the cone. If St(P) is singular and neither conical nor line

singular then P is called a node.

Definition 7.3 Let K be an arbitrary finite simplicial 2-complex.

 

44

45

If a point P of K is not in the closure of some 2—simplex, then we

say P lies in the lfdimensional part of K.

 

Remark: If P is a node of K', then P is the vertex of at least

one line singular l-simplex which is either the back of a flat n—book
(n 2 3), or a free face of a 2-simplex. Furthermore, the components
of the regular part of St(P) are either cone leaves or are topologically
an open triangle with P as a vertex. If the component of the regular
part of St(P) that is topologically an open triangle has two distinct
singular edges with P as a vertex then that component is called a £22)
and if there is only one singular edge with P as a vertex it is called
a cornet. See Figure 7.1. In addition, from the figure it is clear
that if the 2—complex contains no subcomplex whose carrier is a 3-book
then the regular part of St(P), has no components that are cornets.
Furthermore, the singular edges of a fan with P as a vertex must be

free faces of some 2—simplexes.

 

 

 

Figure 7.1

46

Theorem 7.1 An arbitrary finite 2-complex K in E3 contains a wild arc

 

if and only if K has a subcomplex whose carrier is a 3-book.

Prppr: If K has a subcomplex whose carrier is a 3-book, then C. A.
Persinger [30] has shown that K contains a wild arc.

Conversely, suppose K has no subcomplex whose carrier is a
3-book, but K does contain a wild arc, A. By Binge result that locally
tame sets are tame [5], A is not tame at some point P.

To show that the arc is actually tame at P, first notice that
P can't be an end point of A for otherwise near P the arc would lie in
a plane and hence would be locally tame at P.

Now let us further suppose that every l-simplex of K is the
face of some 2-simplex, then by the previous definitions P clearly
can't be a regular point or a line singular point of K, since then
the arc would be locally planar at P. Thus by the previous remark P
is either a conical point of K or P is a node of K where the components
of the regular part of St(P) are either leaves of a cone or fans (the
singular edges are free faces of some 2—simplex(es)).

Case A: P is a conical point of K. The St(P) must have at
least two leaves, and furthermore the arc must intersect two of these
leaves near P for otherwise the arc would be locally planar at P. Call
these leaves L1 and L2. Since K C E3, there clearly exists a
homeomorphism of E3 onto itself so that there exists a plane, H,

through P that doesn't meet St(P) - P and L is in one component of

1
E3- H and all the other leaves of St(P) — P are in the other component

of E3- H. Furthermore by a relative general position argument we can

pick H so that it is not perpendicular to any 2-simplex of LII) P.

 

..‘I‘ ...-I «I'll [III

 

47

Let U" be a closed convex neighborhood of P in L111 P

contained in an e-neighborhood of P, and sufficiently small so that
Bd U"IW A # ¢. In fact, U" can be chosen so that Ed U" is n line
segments, one on each 2-simplex of L111 P. Now let H' be a plane

parallel to H and at a distance of e from H in the same component of
E3- H as L1. Consider the volume generated by lines perpendicular to
H going through points of U" and bounded by H and H', call this volume

U'.

 

 

 

Figure 7.2

Let P' be the first point of A starting from P that is in Ed U"
and let the subarc of A from P' to P by A'. Now in U"- A' connect P
to another point P" 6 Bd U" by an arc A" (U"- A' is clearly path
connected since A' is in fact tame). A't/ A" is homeomorphic to the
line segments PPTEJ PP“ and this homeomorphism can be extended to the
Ed U" by the identity, call it b". By applying the Schoenflies
Theorem to the B—curve, h" can be extended to U", call this homeomorphism
h'.

To extend h' to the U', let U be the volume determined by U"

48

between H' and H" where H" is a plane parallel to H and at a distance

6 from H with H' i H". Further let B 6 H"11 Bd U, and B' E H'IW Bd U.
Now there exists a function f:U' + U such that f is a homeomorphism on
U'- P and f(P) = PE, and h' can be extended to hcz U by first extending
it to the Bd U by the identity and then by mapping E; linearly onto
'EHTTE) and ET; linearly onto EThsz) for all x e U". Extend by the
identity to the remainder of U. Since h'(P) = P, hIPE is the identity
on PE; and therefore f-lhf is a homeomorphism on U' and an extension of
h'. Furthermore, hIBd U' is the identity, and thus none of the other
2-simplexes in the other component of E3- H are moved, so if we do the

same thing in L (J P, and extend by the identity to all of E3 we have

2
that A is in fact locally tame at P, which is a contradiction of our
original assumption.

Case B: P is a node of K. Then the regular part of St(P)
contains at least one fan, and another fan and/or a leaf of a cone.
Again the arc must intersect two of the components of St(P) — P near P.
Furthermore if the regular part of both of these components are fans
then the arc is locally planar at P, and if both components are leaves
of a cone we arrive at the same contradiction as in Case A. Thus the
only case left to consider is if one of the components is a leaf of a
cone, and the regular part of the other component is a fan. Call the
fan plus the two singular l-simplexes F1.

As in Case A, there exists a homeomorphism of E3 onto itself
so that there exists a plane H such that H f)St(P) = P and F1 is the

only component of St(P) - P in one component of E3— H. Furthermore,

the homeomorphism can be chosen so that F is planar. Let V' be a

1

closed neighborhood of P in F contained in an E—neighborhood of P,

1

49

and such that Bd V'I) A f ¢. Now if we pick two points one on each
side of F' in this component of E3— H and take the suspension of V'
over these two points we obtain a closed neighborhood V in E3. See
Figure 7.3. As for the closed neighborhood U in Case A there exists a
homeomorphism h d V such that h maps the component of V' I) A that
contains P onto a line segment starting from P and hIBd V is the identity.
For that portion of the arc in a leaf of a cone we can use Case A, and
hence as in Case A we reach a contradiction of the assumption that A
was not locally tame at P.

If K is an arbitrary 2—complex, we need only remark that the
only additional case is if P is a node of K, where locally the arc lies

in a leaf of a cone and a l—simplex that is not a face of any 2—simplex,

and then we reach a contradiction by Case A.

 

 

 

Figure 7.3

Remark: The complex K may or may not contain a node, as Figures 7.4

a and b illustrate. P is a node of a, but b has no nodes.

50

 

 

P P
(a) (b)
Figure 7.4
Corollary 7.2 An arbitrary finite 2-complex K in E3 contains a wild

 

are if and only if for some vertex v of K the Lk(v) contains a

subcomplex which is a triod.

Prppr: If the Lk(v) contains a triod then K contains a schomplex
whose carrier is a 3-book, which can be seen by joining v to Lk(v).
Hence K contains a wild are by the previous theorem.

Conversely, if K contains a wild arc then it contains a subcomplex
whose carrier is a 3-book. Hence if v is a vertex on the back of the

book the Lk(v) contains a subcomplex which is a triod.

Corollary 7.3 Let S be the l-dimensional part of an arbitrary finite

 

2-complex K in E3. K contains a wild arc if and only if some component

of IR] - S contains a line singular point P where St(P) is an n-book

with n 2 3.
Proof: This is just a restatement of the previous theorem.
Corollary 7.4 A necessary and sufficient condition that a finite

 

2-complex in E3 contains no wild arcs is that each homologically trivial

subcomplex contains no wild arcs.

51

Prppr: If the 2-complex contains no wild arcs then clearly each
subcomplex contains none, and hence each homologically trivial subcomplex
contains no wild arcs.

Conversely, suppose the 2—complex does contain a wild arc, then
by the previous theorem the 2-complex contains a subcomplex whose carrier
is a 3-book. This subcomplex is homologically trivial and contains a

wild arc.

Corollary 7.5 A necessary and sufficient condition that a finite

 

2-complex in E3 contains no wild arcs is that each collapsible subcomplex

contains no wild arcs.

Proof: The proof is obtained from the proof of Corollary 7.4 by

replacing "homological trivial" by "collapsible".

Having answered the questions of what types of finite 2—complexes
contain wild arcs, let us turn to the question of which wild arcs can

be embedded in a finite 2—complex that does contain a wild arc.

Theorem 7.6 Let K be a finite 2-complex in E3 that contains a wild

 

arc. An arbitrary arc A can be embedded in K if and only if it can be

embedded in a flat 3-book.

Prppf: Suppose A can be embedded in a flat 3-book. Since K contains
a wild arc, then by the previous theorem K contains a subcomplex whose
carrier is a 3—book and hence A can be embedded in K.

Conversely, suppose that A can be embedded in K. If the
arbitrary are A is tame, then A can be embedded in a 3-book by the
method of C. A. Persinger [30]. Thus we can suppose that A is wild.

The points of A where A fails to be locally tame clearly lie

52

in the singular set of K. In fact, by the previous theorem, it is
clear these points all lie on the back of some 3-book which is contained

in IK

 

Gail Atneosen [2] has shown that the set of wild points of
A in each 3-book of IR] is a compact, totally disconnected set, and
hence the set of wild points of A is a compact, totally disconnected set
since it is a finite union of such sets, call it C'.

First of all, let us connect up the various components of the
line singular sets of K by line segments so that C' is contained in a
finite connected graph, G —- G need not be contained in the l-skeleton
of K. Next add a Cantor set of points in G around each isolated point
of C' to get a Cantor set, C, in G.

We want to show that C is locally tame and hence is tame by a
result due to Bing [7]. If c e C is an interior point of a l—simplex
of G or a vertex of G of index at most 2 then G is clearly locally tame
at C. Thus we need only check the vertices of G with index greater
than 2. To show that the Cantor set is locally tame at such a vertex
of G, construct a neighborhood, U, of the vertex whose boundary misses
C (see Figure 7.5). Now the Cantor set inside this neighborhood is

planar and hence is tame.

1-

Figure 7.5

53

Thus there exists a tame arc that contains C, and therefore
contains the set of wild points of A. By a theorem of Posey's [31] and
the discussion that follows the statement of that theorem in C. A.

Persinger's paper [30], we have that A can be embedded in a flat 3-book.

Remark: In the proof that an arbitrary arc A in K can be embedded
in a flat 3-book we don't need the additional assumption that K contains

a wild arc. However, the only interesting case is when A is a wild arc.
The proof of the theorem contains the following two corollaries.

Corollary 7.7 The set of wild points of an arc in a finite 2-comp1ex

in E3 is a compact, totally disconnected set.

 

Corollary 7.8 Let T, and E be subsets of an arc A such that

 

(i) A is locally tame at each point of T.
(ii) E = A - T
(iii) each neighborhood of each point of E meets T in a non-
empty set.
If E lies on a tame arc, then A can be embedded in a finite

2-complex in E3 that contains a wild arc.

Remark: This is just a restatement of a result of Posey's [31] which

was referred to in the proof of the theorem.

CHAPTER VIII

SIMPLE CLOSED CURVES THAT BOUND DISKS IN A 2-COMPLEX

Because of the existence of flat 3-books in the carrier of a
2-complex, and the existence of knots in a 3—book this naturally leads
to the question of under what conditions on the 2-complexes does the
plane Schoenflies Theorem hold. Clearly the theorem holds if the
2-complex is planar and has no "holes". The following theorem
generalizes this to get a necessary and sufficient condition on a
2-comp1ex so that each polygonal simple closed curve in it bounds a
disk in the 2-complex. The outline of the proof is from Lemma 3.6

of G. Atneosen's Thesis [2].

Definition 8.1 The order of a l—simplex, O, in a 2-complex is the

 

number of closed 2-simplexes that have 0 as a face.

To this point in this paper it hasn't been necessary to
specify which homology was being used nor what the coefficients were.
However, in this theorem we will need simplicial homology with coefficients
in 22, the group of integers modulo two. For an exposition on simplicial
homology theory and related terminology see Chapter 6 of [21]. By a
m—dimensional chain on a simplicial complex K with coefficients in Z is

2
To

 

meant a function cm on the m—simplexes of Ii with values in 22.
facilitate the notation let cm also denote the subcomplex of K which

is all m—simplexes of K on which cm has non—zero value. The carrier

of this subcomplex will also be denoted by cm rather than Icml. No

54

55

confusion should arise since 22 coefficients are particularly well
suited to geometric interpretations, and this notation will be used

only in relation to chains.

Theorem 8.1 A necessary and sufficient condition that each poly-
gonal simple closed curve in a finite collapsible 2-complex K bounds

a disk in [KI is that each l-simplex in K is of order at most 2.

Proof: If there exists a l-simplex in K of order 3 or more, then
IKI contains a 3-book. It is clear by looking at the following figure,
which is Figure 3.1 a of [2], that there exists a polygonal simple

closed curve in K that doesn't bound a disk.

 

 

 

Figure 8.1

Conversely, suppose that each l-simplex in K is of order at
most 2 and let G be a polygonal simple closed curve in K. Now we can
retriangulate K, which we will again call K, such that C is contained
in the carrier of the l-skeleton of K. Consider the simplicial homology
of K with coefficients in 22’ and let C also denote the l—chain that

has value 1 on all l-simplexes of K that are contained in C and 0 on

56

all other l-simplexes of K. Since K is collapsible, H1(K,ZZ), the first

3

simplicial homology group of B with coefficients in Z is trivial.

2.
Thus there exists a 2-chain MC on K such that BMC = C, since BC = 0. It
will now be shown that Mc is a disk.

(1) MC is compact because it is the point set union of a
finite number of compact 2-simplexes. To show that Mc is connected,
suppose that Mc can be expressed as the disjoint union of two closed sets,

A and A . Since C is connected it may be assumed that C C A Let

1 2 l'
M; be the 2-chain on K that has non-zero value only on those 2-simplexes

 

of K contained in A2. Since aMc = C, and since all the l-simplexes of

K are of order at most 2, each l-simplex in A is of order exactly 2.

2
Thus EN; = 0, but H2(K,Zz) a 0, since K is collapsible, and hence there
must be a 3-chain on K whose boundary is Mé. Moreover, since the only

3-chain in K is the zero 3-chain, this implies that A is empty and

2
hence Mc is connected.

(ii) To show that Me is a manifold, let us consider the
following four cases for a point x in Mc'

Case A: x is contained in the interior of 2-simplex of MC.
Then x clearly has a neighborhood in Mc homeomorphic to an open disk.

Case B: x is contained in the interior of a l-simplex in Mc- C.
Since this l-simplex does not lie in C, and since all l-simplexes of K
are of order at most 2, it is of order exactly 2. Hence x has a
neighborhood in MG homeomorphic to an open disk.

Case C: x is contained in the interior of a l-simplex in C.
Again since each simplex in K has order at most 2, and since l—simplexes

of C have odd order, the l-simplex is of order exactly one. Thus x has

a neighborhood in Mc that is homeomorphic to a closed disk.

0 I‘llllll 1 I'll

57

Case D: x is a vertex of Mc' If in addition x e MC- C then
x can't be a node in MG because each l-simplex that has x as a
vertex in Mc is of order exactly two, and hence x is conical point'
of Mc or a regular point of Mc' Similarly, if x e C, then the regular
part of St(x) has exactly one fan - the two singular simplexes would be
the two simplexes of C that have x as a vertex - and any other
components of the regular part of St(x) would be cones.

We wish to show that the x is regular if x 6 Mc- C and a line
singular point with one fan in the regular part of St(x) if x e C in
order to complete the proof the Mc is a 2-manifold.

To this end let us consider Mc- x. There are two cases to
consider.

Case 1: Mc- x has at least two components. Then let Cl be a
component that doesn't contain C. Now since each l-simplex in
C11] {x} is a face of exactly two 2-simplexes, we can't start in

C1 U {x} when we collapse K, and in fact, we would never be able to

collapse C because it intersects the rest of K in just a vertex of K.

1
This contradicts the fact that K is collapsible.

Case 2: Mc- x is connected. Since there are at most a finite
number of these points like x in MC we have that Mc is topologically
a compact, connected 2-manifold with boundary with a finite number of
interior points identified, not necessarily to the same point, and a
finite number of boundary and interior points identified, again not
necessarily to the same point, and such that no two boundary points are
identified.

Now if in fact there exists a conical point in MC- C or a node

in C, then it is clear from the above discussion of what Mc would be,

58

that there would exist a polygonal simple closed curve in Mc contained
in the l-skeleton of K through the conical point or node which will not
bound a mod two 2-chain in Mc’ but it must bound in K. However, every
l-simplex in Mc— C has all its 2-simplexes that have it as a face and
thus the polygonal simple closed curve doesn't bound in K. Hence we
have shown by contradicting the collapsibility of K that Mc can't have
any nodes or conical points, or in other words that Me is a 2-manifold.
Furthermore we can show by the argument above that Mc is
homologically trivial for if a l—cycle doesn't bound a mod two 2—chain
in MG it can't bound one in K. Thus Mc is a homologically trivial
compact, connected 2—manifold with boundary C, and hence C bounds a

disk in K.

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