C OMPOSITIO M ATHEMATICA
R. J. D AVERMAN
Decompositions of manifolds into codimension one submanifolds
Compositio Mathematica, tome 55, n
o2 (1985), p. 185-207
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DECOMPOSITIONS OF MANIFOLDS INTO CODIMENSION ONE SUBMANIFOLDS
R.J. Daverman *
@ 1985 Martinus Nijhoff Publishers, Dordrecht. Printed in the Netherlands.
1. Introduction
This paper
investigates decompositions
ofboundaryless
manifolds intoclosed, connected,
codimension one submanifolds. It wasinspired by
recent work of V.T. Liem
[22],
with a related but distinctemphasis:
typically
thehypotheses
in Liem’s resultsrequire
that thedecomposition
elements have the
shape
of a codimension onesphere.
It can be construedas an
outgrowth
of the samespirit leading
to the studiesby
D.S. Coram[11]
andby
Coram and P.F. Duvall[13]
ofdecompositions
ofS3
intosimple
closed curves. Forexample,
in thatspirit,
because of the severelimitations on the
decomposition
elements within the source manifoldM,
one
expects
to discover agood
deal of information about thedecomposi-
tion space,
M/G.
We prove thatM/G
is a1-manifold, possible
withboundary.
A
unifying
theme is thequestion asking:
which manifolds M admit suchdecompositions?
The obviousexamples
that come to mind are thelocally
trivial fiber bundles overE1
orS’.
Twisted line bundles overclosed
(n - 1)-manifolds provide
otherexamples;
twisted I-bundles over two(n - l)-manifolds Ni
andN2
can beglued together along
theirboundaries,
whenhomeomorphic,
toproduce
still others. L.S. Husch[20]
has described
yet another, exhibiting
adecomposition
G of an n-mani-fold M into closed
(n - l)-manifolds,
all embedded in M as bicollaredsubsets,
such that thedecomposition
space isequivalent
toSI
but wherethe
decomposition
map 03C0: M ~M/G
is nothomotopic
to alocally
trivial fiber map; this manifold M is built from a non-trivial h-cobor- dism W
having homeomorphic boundary components by identifying
these two ends.
At first
glance
thedecompositions
under consideration may appear toprovide partitions
of M similar to those associated with foliations.(Recall
that a k-dimensional foliation of an n-manifold M is apartition
* Research supported in part by NSF Grant MCS 81-20741.
!F of M into k-dimensional submanifolds such that each x E M has a
neighborhood
Uequivalent to En = Ek En-k, where
every level corre-sponding
toEk
X(point} belongs
to some element ofF.)
Such ap- pearances are ratherephemeral,
for the differences are morepronounced
than the similarities.
First,
uppersemicontinuity
necessitates that all thedecomposition
elements becompact;
the moreinteresting foliations,
forthe most
part,
havenon-compact
leaves.Second,
uppersemicontinuity
forces setwise convergence
(in)
togo of a
sequence ofdecomposition
elements
( gi 1
for which some sequence ofpoints xi ~
g, converges to apoint xo e go E G;
foliations have nocomparable
feature - infact,
individual leaves can be dense in M.Third, given
a foliation F of Minto
k-manifolds,
one knows thatlocally
the leaves arealigned
with thelevels
Ek
X(point) ~ Ek En-k in
some atlas of E n-chartscovering M;
the elements of the
decompositions
studied here need not belocally flatly
embedded in
M,
and even whenthey happen
tobe, they
need not line up in any wayparallel
to other elementsnearby.
-The theorems here
frequently supply
structural data about the sourcemanifold M
involving
the alliedconcepts
of "standard structures" and’standard
decompositions".
The standard( n-dimensional)
structures areof two
types: (i)
twisted I-bundlesWT
over(arbitrary) closed,
connected(n - l)-manifolds,
and(ii) near-product
h-cobordisms W(the
term"near-product"
meansthat,
ifBo
andB,
denote thecomponents
ofaW,
thenW - Bi ~ Bj
X[0, 1)
for i~ j).
Eachtype
of standard structure Wadmits certain natural
(non-unique)
standarddecompositions 9
intoclosed
(n - l)-manifolds,
eachlocally flatly
embedded inW,
such thatWlg
istopologically [0, 1].
An n-manifold M is said to have a standardformation {Wi} provided
it has alocally
finite cover{Wi},
where eachW
is a standard structure and
where W ~ Wj~ (i~j) implies W
~W
is aboundary component
of each. Given any standard formation{Wi}
forM,
andgiven
some standarddecomposition Yi
for eachW ,
one obtains astandard
decomposition Y for
Mby setting y = U iyi.
The first Structure
Theorem,
Theorem4.4,
attests that if M admits a(upper semicontinuous) decomposition
G into closed(n - 1)-manifolds locally flatly
embedded inM,
then it also admits a standard formation{Wi}
and associated standarddecomposition
9.Furthermore,
in some sense Y serves as a reasonableapproximation
to G.However,
when elements of G are allowed to bewildly
embedded inM,
then M can take ondrastically
different forms. Forinstance,
Exam-ple
5.5depicts
adecomposition
G of an n-manifold M(n 6)
for whichM/G
isEl
and the fundamental group of M atinfinity
isinfinitely generated.
In contrast to what occurs whendecompositions
elements arelocally flat,
thisexample
has an infinite number of(pairwise) homotopi- cally inequivalent
elements.Nevertheless,
thedecomposition
G ofExample
5.5 has theimportant
property
that each inclusion g ~ M induceshomology isomorphisms.
This holds whenever
M/G
isEl (Lemma 6.2).
Itsuggests
thegeneral
structure theorem
(Theorem 6.6):
any manifold Madmitting
a decom-position
into codimension one closed submanifolds has a formation{i},
where theW ’s
arecompact
manifolds withboundary
thatsatisfy homological analogues
of thehomotopy
restrictionsimposed
upon theobjects
of a standard formation.As a consequence, we can
specify (Corollary 7.4)
the 3-manifoldsM3
that admit
decompositions
into closed2-manifolds, by showing
first thatevery such
M3
also has a standard formation and associated standarddecomposition
9.Finally,
weinvestigate
somespecial
cases in whichdecomposition
elements are known to possess some
homotopy-theoretic compatibility,
which is
significant
to view of the aforementionedExample
5.5. Accord-ing
toCorollary 8.4,
if~(Mn/GA) = ~
and each g EGA
has Abelian fundamental group, then all the elements ofGA
arehomotopy equiv- alent ;
as aresult, for n 5, (whether
or not~(Mn/GA) = ~)
M n has astandard
decomposition 9 approximating GA (Theorem 8.7).
The sameconclusion is valid if G is a
decomposition
of Mn(n > 5)
such that eachinclusion g ~ M induces an
isomorphism
of fundamental groups(Theo-
rem
8.8).
2.
Définitions, conventions,
andpreliminary
resultsBy
ann-manifold,
we mean aseparable
metric space modelled onEuclidean n-space En.
Accordingly,
an n-manifold has noboundary.
Ifwe wish to consider the
possibility
ofboundary,
we shall say soexplicitly, referring
to ann-manifold
withboundary Q (a separable
metric space modelled on the n-cellIn),
and we denote itsboundary
asa Q.
One should
always regard
themanifolds,
with or withoutboundary,
discussed herein as connected spaces. In
particular,
the source manifoldM
(or, Mn ) appearing densely throughout
ispresumed
to be connected.The
symbol
G is reserved to denote adecomposition
of the sourcen-manifold M" into closed
(i.e., compact), connected, (n - 1)-manifolds.
Usually
we presume G to be upper semicontinuous(abbreviated
asusc), except
in Section3,
where in some instances we assumeonly
that G is apartition (but
then prove it to beusc).
We use thesymbol
03C0(and, sometimes, p )
to denote the naturaldecomposition
map of M to thedecomposition
space,M/G.
Of course, G is usc iff 03C0 is a closedmapping.
By
an( n-dimensional)
h-cobordism W we mean acompact
n-manifold Whaving
twoboundary components, Bo
andBl,
such that each inclusionBi ~
W( i
=0, 1)
is ahomotopy equivalence. (Note
that W isnot
required
to besimply connected.)
Such a W is called anear-product
h-cobordism
provided
W -Bo = B,
X(0, 1]
and W -B1 ~ Bo X [0, 1).
(The symbol =
means "ishomeomorphic to.")
Eachnear-product
h-cobordism W has a standard
decomposition yW
associated with anyhomeomorphism 03C8
ofBi [0, 1)
ontoW - Bj (i, j ~ {0, 1}, i~j);
namely
It should be clear that
W/yW ~ [0, 1].
The true purpose of the
"near-product" phraseology
isapparent only
for n =
3,
4.For n >
5 thefollowing
holds:PROPOSITION 2.1: For
n 5,
each n-dimensional h-cobordism W is anear-product.
This was established
by
E.H. Connell[10].
For n = 3 this
language helps
circumventpotential ambiguities
thatwould stem from failure of the 3-dimensional Poincaré
Conjecture.
PROPOSITION 2.2.
If
W is a 3-dimensionalnear-product h-cobordism,
withboundary components Bo
andBl,
then W =Bo
X I.PROOF:
Let 03C8: BI
X(0, 1] -
W -Bo
be ahomeomorphism.
Construct anembedding À
ofBo X [0, 1] ]
in Wgiving
a collar onBo
there. Then03C8-1(03BB(B0
X{1/2}))
islocally
flat andincompressible (cf. [18,
pp.42-43])
in
BI
X(0, 1]; according
to[18, Appendix],
theregion
boundedby
03C8-1(03BB(B0 {1/2})]
andB1 {1}
ishomeomorphic
toBo X [1/2, 1].
The result follows.
Let N denote a closed
(n - 1)-manifold,
on which is defined a 2 - 1covering
map p: N - X. A twisted I-bundle over X is a spaceWT equivalent
to( N
XI)/R,
where Rrepresents
the(usc) partition
of N X Iwhose
only nondegenerate
sets are those of the formThen
WT
has a standarddecomposition yW
into the(n - 1)-manifolds corresponding
to theimages
ofN {t},
t ~ I. Whent 1,
these arehomeomorphic
toN,
but theimage
of N X{1}
ishomeomorphic
to X.As
before, WT/yW ~ [0, 1].
Repeating
a definitiongiven previously,
we say that{Wi}
is a stan-dard
formation for
ann-manifold
Mprovided {Wi}
is alocally
finitecover of
M,
eachW
is either anear-product
h-cobordism or a twisted I-bundle over some(n - 1)-manifold, and W
nWj ~ (i ~ j) implies
W
~W
is aboundary component
of each. Associated with any standard formation{Wi}
for M is a standarddecomposition 9
of Mgiven by y = ~iyi, where gi represents
some standarddecompostion
ofWi.
Here is an obvious result:
PROPOSITION 2.3 :
If {Wi}
is a standardformation for
M such that eachW
is
homeomorphic
toNi
X[0, 1],
then M is alocally
trivialfiber
bundle overSI
orEl (according
to whether or not M iscompact)
withfibers
homeo-morphic
toN.
Recall the
example
of Husch[20,
p.912] demonstrating
thenecessity,
for our purposes, of
considering near-product
h-cobordisms that are notproducts.
3.
Analysis
of thedecomposition
space.The aim in this section is to prove that
MIG
is a 1-manifold withboundary. Boundary
must be tolerated because03C0(g)
E~(M/G)
iff g is 1-sided in M(equivalently,
gseparates
no connectedneighborhood
ofitself).
In
addition,
it is shownthat,
forcompact
manifoldsM,
theobjects
ofstudy
can beregarded
asarbitrary partitions
G of M intoclosed,
connected codimension onemanifolds;
every such G is usc.First,
a technical result.LEMMA 3.1: Each
closed,
connected( n-1)-manifold g in
M has a connectedneighborhood Vg
such that every closed( n - 1)-manifold
inVg -
g separatesvg.
PROOF: Select a connected
neighborhood Vg
of g that deformationretracts to g in M.
Suppose
to thecontrary
that some closed(n - 1)-manifold
N inTlg -
g fails toseparate Vg.
Then an arc A inVg meeting
N atjust
onepoint
andpiercing
N at thatpoint
can becompleted
to asimple
closedcurve J in
Vg,
with J r1 N = A ~ N.By
construction ofVg,
J is homo-topic
in M to aloop
J’ in g. Thisyields:
J n N ={point}
whileJ’ r1 N
= ~, violating
the invariance of(mod 2)
intersection number.COROLLARY 3.2:
If
G is a uscdecomposition of
M intoclosed,
connected(n - l)-manifolds,
then the setQ of
allg*
E G that are 1-sided in M islocally finite.
Next,
the central result of this section.THEOREM 3.3:
If
G is a uscdecomposition of
M intoclosed,
connected( n - l)-manifolds,
thenMIG
isa 1-manifold
withboundary.
PROOF : Focus first on some go E G
having
a connectedneighborhood Uo separated by
go.Apply
Lemma 3.1 to obtain a G-saturated connectedneighborhood Vo,
with go cVo
cUo,
such that every g E G inVo
sep-arates
Va.
Then03C0(V0)
is anon-compact, locally compact, connected, locally connected, separable
metric space, eachpoint
of whichseparates 03C0(V0)
into twocomponents.
Thisimplies 03C0(V0) ~ E1.
Let Q
denote the set of allg* ~
Ghaving
no connectedneighborhood separated by g*. By Corollary 3.2, Q
islocally
finite.In case
Q
=0, clearly 03C0(M) = M/G
must be a 1-manifold.In case
Q ~ ~,
the setMQ
= M -~ {g*|g* E Q 1
is aG-saturated, connected,
openmanifold,
and03C0(MQ)
must betopologically (0, 1),
theonly boundaryless non-compact
1-manifold. For distinct elementsg*, g**
ofQ, w(g*)
andi7(g**) compactify
distinct ends of03C0(MQ). Hence, Q
consists of either two or oneelements,
andM/G
ishomeomorphic
to[0, 1]
or[0, 1), according
to whether M iscompact
or not.THEOREM 3.4:
Suppose
M is a closedn-manifold
and G is apartition of
Minto
closed,
connected(n - l)-manifolds.
Then G is usc.PROOF: There exists a finite subset F of G such that every
g E
G - Fseparates M
= M - U{g| g E F}.
To seethis,
suppose otherwise. Choosesuccessively
gl, g2, ... from G such thatM - ( gl U g2 U - - - U gk ) is
connected. The
impossibility
of such choices becomes clear for klarger
than the rank
(over Z2 )
ofH1(M; Z2 ).
For i ~{1, 2,...,k}
one canconstruct a
simple
closedcurve J
in M that meets andpierces
gi atprecisely
onepoint
and that intersects no otherg,
selected.By
thehomological
invariance of(mod 2)
intersectionnumber, no J
is null-ho-mologous (over Z2 )
in M. The size of k ensures that some curve,say Jl,
is
homologous
in M to some union of the otherJ’s (i > 1),
anotherviolation of intersection number invariance.
Fix go e G - F and a
neighborhood Uo
of go in M. Determine a smallerneighborhood ho
of go withg0 ~ V0 ~ C1 V0 ~ U0 ~
and Cl% compact.
LetM+ represent
the closure(in M )
of one of thecomponents
ofM -
go. DefineG+
as the set of all g EG, g ~
go, inM+.
For g EG+,
let
Xg
denote the closure of thecomponent
ofM+ -
gcontaining
go. The collection{Xg|g ~ G+}
istotally
ordered and its intersection is go. Thecompactness
ofFr % guarantees
the existence of someg+ ~ G+
forwhich
Xg+ c %.
Ifg-
is theanalogous
element of G inM - M+,
thenevery
g E
G in thecomponent of M - (g+ ~ g-) containing
go satisfies:g c
%
cUo, revealing
that G satisfies the usual definition of uppersemicontinuity
atgo E
G - F.One can argue, in similar
fashion,
that G is upper semicontinuous ateach g E F. The
only
extra technical matter to account for is that someg E F may
separate
no connectedneighborhood
of itself.This argument establishes the
stronger
result below.COROLLARY 3.5:
Suppose
M is ann-manifold
withoutboundary
such thatH1(M; Z2 )
isfinite,
and suppose G is apartition of
M intoclosed,
connected
(n - l)-manifolds.
Then G is usc.COROLLARY 3.6:
If
M is aclosed,
connectedn-manifold
such thatHl ( M; Z)
isfinite
andif
G is apartition of
M into closed connected( n - l)-manifolds,
then there existexactly
two elementsof
G that are1-sided in M.
PROOF:
By
Theorem3.4,
G is usc;thus, by
Theorem3.3, M/G
is either[0, 1] or SI.
It cannot beSI
because then one couldproduce
aloop
J inM such that 03C0
1
J : J -SI
hasdegree 1, by finding
J thatpierces
someg E G
exactly
once,indicating
that 03C0*:H1(M; Z) ~ HI(MjG; Z)
is anepimorphism (monotonicity implies
thatautomatically).
COROLLARY 3.7:
If
M is a closedn-manifold
such thatH1(M; Z2 )
=0,
then M admits no
partition
intoclosed,
connected( n - l)-manifolds.
PROOF:
By
the Universal CoefficientTheorem, Hl ( M; Z)
is finite.Moreover,
sinceH1(M; Z2 )
=0,
each closed(n - 1)-manifold
in Mseparates.
This contradicts thepreceding Corollary.
We close
by stating,
withoutproof, improvements
to the main results that can be establishedby
similararguments.
THEOREM 3.3’:
If
K is a uscdecomposition of
ann-manifold
M intocompacta having
theshapes of closed, connected, ( n - l)-manifolds,
thenMIK
isa 1-manifold
withboundary.
COROLLARY 3.5’:
If
M is ann-manifold
such thatHl(M; Z2)
isfinite
andK is a
partition of
M intocompacta having
theshapes of closed, connected, ( n - l)-manifolds,
then K is usc.4.
Decompositions
withlocally
f lat submanifoldsLEMMA 4.1:
Suppose M/G ~ El
and each g ~ G islocally flat
in M. Thenfor
each go EG,
M ishomeomorphic
to go X( -1, 1),
with go c M corre-sponding
to go Xtoi-
PROOF: Without loss of
generality, MIG
=El
and03C0(g0)
= 0. It suffices to prove that03C0-1([0, ~))
=M+
ishomeomorphic
to go X[0, 1).
Since go is
locally
flat in M andseparates M,
it is bicollared[5].
Fixan open collar C = go X
[0, 1)
on go inM+.
Let X denote the set of allpoints
t in[0, oc)
for which there exists ahomeomorphism
h ofM+
ontoitself
having compact support, fixing (pointwise)
someneighborhood
ofgo, with the
property
thath(C) ~ 03C0-1[0, t ]. Certainly X,
which containsa
neighborhood
of0,
isnonempty
and connected. It has no upperbound,
because if xo were the least upper
bound,
one could stretch the collar C out very near03C0-1(x0)
and use the bicollar on03C0-1(x0)
toexpand
it stillfarther to encompass
03C0-1[0, x0]. Hence,
X =[0, oo).
By
thetechniques developed by
M. Brown in[6], M+
is homeomor-phic
to go X[0, 1).
The
proof
of Lemma 4.1 also establishes:COROLLARY 4.2:
Suppose
each g E G islocally flat
inM, M/G ~ El,
andA c
MIG
is an arc. Then7r -’A
is anear-product
h-cobordism.Nothing
aboutCorollary
4.2 should be construed assuggesting
anycompatibility
between thelayers 7r -la,
a EA,
and the levels in a stan-dard
decomposition 9 resulting
from thenear-product
features of03C0-1A.
COROLLARY 4.3 :
Suppose
each g E G islocally flat
in M and~(M/G) = .
Then all
pairs {g1, g2} of
elements in G arehomotopy equivalent.
PROOF : In
M/G
choose an open set U =E1
and an arc A c U with aA= {03C0(g1), 03C0(g2)}.
Thenapply Corollary
4.2.At this
point
we areprepared
togive
the first Structure Theorem.THEOREM 4.4:
Suppose
M admits a uscdecomposition
G intoclosed,
connected(n - 1)-manifolds,
eachof which
islocally flat
in M. Then M hasa standard
formation {Wi}
and a standarddecomposition cg,
with decom-position
map p: M -Mlg. Moreover, for
each continuous e:MIG - (0, (0), cg
can be obtained so that there exists ahomeomorphism À of M jcg
onto
MIG satisfying
PROOF: In case
~(M/G) = ,
expressM/G
as the union of a collectionof
arcs {Ai},
whereA; F) Aj ~ (1 ~ j) implies Ai n Aj
is anendpoint
ofeach,
and where the arcs are smallenough
thatdiam Ai 03B5(ai)
for allai ~ Ai. Corollary
4.2 certifies that each03C0-1Ai
is anear-product
h-cohordism.
Hence, {03C0-1Ai}
is a standard formation onM,
asrequired.
The
homeomorphism
À almost constructsitself;
one canreadily
define itso 03BBp(03C0-1Ai) = Ai
for all i.Somewhat more
interesting
is the caseM/G
=[0, 1).
Set go =03C0-1(0).
Since go does not
locally separate M,
it has a nice twisted I-bundleneighborhood B
in M. There exists a map p: M - M withcompact support
whosenondegenerate preimages
are the fiber arcscorresponding
to the I-"factor" in B. Let G’ denote the
decomposition
of M inducedby
03C003BC: M ~[0, 1);
thatis,
Note that
03BC-103C0-1(0)
=B,
and that03BC-103C0-1(t)
z03C0-1(t)
f or t > 0.Modify
G’ to form another
decomposition
G*consisting
of the naturallevels,
from a standard
decomposition
ofB, together
with the elements of G’ -{B}. Clearly
G * is adecomposition
of M withlocally flat, closed,
connected
(n - l)-manifolds. Furthermore,
if ’1T*: M -MjG*
denotesthe new
decomposition
map, we canequate MjG*
with[0, 1)
andpresume that 03C0* is
sufficiently
close to 03C0. Now M = B UC1(M - B)
andC1(M-B)=(’1T*)-1[to,1),
whereto=’1T*(aB»O.
CoverMjG* by
small arcs
{A0, A1,...},
asbefore,
withAo
=[0, t0].
Then{(03C0*)-1Ai}
isa standard formation on
M,
asrequired.
Finally,
the caseMjG z [0, 1]
isjust
a double-barreled variation tothe
preceding
one, obtainedby modifying
G in the same fashion overneighborhoods
of bothpoints
ina(MjG).
COROLLARY 4.5:
Suppose
M admits a uscdecomposition
G intoclosed, connected, locally flat ( n - l)-manifolds.
Then( i ) if MjG z [0, 1),
M ishomeomorphic
to the interiorof
a twistedI-bundle over
03C0-1(0);
(ii) if MjG z [o, 1],
M has a standardformation (Wl, W2, W3}
con-sisting of
two twisted I-bundles and anear-product h-cobordism;
(iii) if MIG (o, 1),
then M = go X( -1, 1),
go EG;
and(iv) if MIG S1,
then M can be obtainedfrom
somenear-product
h-cobordism W
by homeomorphically identifying
the two compo- nentsof
aw.Since every
1-sphere
in a 2-manifold islocally
flatthere,
we have anotherproof
of a resultgiven by
Liem[22, Corollary l’].
COROLLARY 4.6:
Suppose
G is a uscdecomposition of
a2-manifold M2
without
boundary
into1-spheres.
then(i ) if M2 IG [o, 1), M2
is an open Mobiusband;
(ii) if M2 IG ~ [0, 1 J, M2
is a Kleinbottle;
(iii) if M2jG ~ E1, M2
is an openannulus;
and(iv) if M2 IG sI, M2
is a torus(SI
XSl)
or a Klein bottle.Theorem 4.4 also
brings
us to thesubject
ofapproximate fibrations,
introduced andanalyzed by
Coram and Duvall. See[12]
for the defini- tion.COROLLARY 4.7:
Suppose
each g E G islocally flat
in M. Then ’1T:M -
MIG
is anapproximate fibration if
andonly if a(MjG) = 9.
PROOF : When
~(M/G) = 9,
03C0 is anapproximate
fibration because it canbe
approximated,
in a reasonable sense,by
theapproximate
fibrations p : M -Mlg
associated with standarddecompositions
y. WhenA(MIG)
= ,
03C0 fails to be anapproximate
fibration because the inclusion of go into a twisted I-bundle B over go is nothomotopic
in B to a map whoseimage
misses go.5.
Examples
This section is devoted to
displaying
some of the bizarrepossibilities
forthe
decompositions
under consideration.First,
wegive
anexplicit
re-minder that wildness can occur
by incorporating
aclassical,
wild2-sphere
in
E 3
as adecomposition
element in adecomposition
ofE 3-origin
into2-spheres,
and then weproduce
morecomplex examples revealing
unex-pected changes,
due towildness,
among thehomotopy types
of decom-position
elements.EXAMPLE 1: A
decomposition G 1
ofM3 = E 3-origin
into2-spheres,
oneof which is
wildly
embedded inM3.
Let S denote the
2-sphere
inE3
describedby
Fox and Artin[17, Example 3.2],
which bounds a 3-cell C there. CoordinatizeE 3
so that theorigin
isplaced
interior to C. One can fill up Int S ~M3
with2-spheres
that cobound with S 3-dimensional annuli in
M3; by construction,
Ext S ~
S2
X(1, oo),
and Ext S can be filled up with2-spheres
aswell,
but the
regions
boundedby S and
any of the latter are not manifolds withboundary.
Controls on the Fox-Artin construction can be
imposed by noting
thefollowing:
there exists adecomposition
ofS2 [-1, 1]
into2-spheres,
where both
S2
X{
±1}
are members and some other member is(wildly)
embedded in
S2 (-1, 1)
cE 3
like the Fox-Artinsphere.
Withthis,
theFox-Artin wildness can be inserted
repeatedly
anddensely,
to obtain:EXAMPLE 2: A
decomposition G2
ofM 3
=E 3-origin
such that theimage
of those
g E G2 wildly
embedded inM3
is dense inM3 jG2.
By taking products
withSn-3,
one canproduce
similardecomposi-
tions of n-manifolds.
Although examples
like thepreceding
illustrate that wildness must be accountedfor, they
do not exhibit the full range ofpathology
that can bepresent. Among
thedecompositions
encountered thusfar,
aslong
as thedecomposition
space hasempty boundary,
all the elements have beenhomotopy equivalent.
Theexample
below shows that this is notalways
the case.
Fix n
6. LetF" - 2
denote anonsimply-connected homology ( n - 2)-
cell
(explicitly,
acompact (n - 2)-manifold having
trivialintegral
ho-mology),
and let X denote aspine
ofF" - 2 (that is,
thecomplement
inF" - 2
of an open collar onFn-2).
Form V =Fn-2 [-1, 1].
LetNn-1
be the
homology (n - l)-sphere
obtainedby doubling V along
itsboundary,
withN"-’ = V+U
V_representing
this double. Set M’ =Nn-1 E1.
Name a Cantor set C in( -1, 1 ).
Let K denote the de-composition
of M’ intosingletons
and the sets X{c}
X{0},
c eC,
inV+ {0} ~ Nn-1
X{0}. Finally,
define M as thedecomposition
spaceM’/K,
withdecomposition
map p : M’ ~M’/K =
M.According
to the Main Lemma of[16],
M is an n-manifold. Obvi-ously,
it admits adecomposition G3,
whereFor t ~
0, p(Nn-1
X{t}) is naturally homeomorphic
to thenonsimply-
connected
homology sphere Nn-1,
whilep(Nn-1
X{0})
is homeomor-phic
toS" -1 (cf.
theproof
ofProposition
1 in[16]). Thus,
we have:EXAMPLE 3:
For n >
6 there exists adecomposition G3
of an n-manifoldM into codimension 1 closed submanifolds such that
MIG3 = El yet
not allpairs
of elements fromG3
arehomotopy equivalent.
EXAMPLE 4:
For n
6 there exists adecomposition G4
of an n-manifoldM into codimension 1 closed submanifolds such that
MIG4 = [0, 1)
andeach
g E G4
thatseparates
M islocally flat,
but M is not a twisted line bundle over any(n - l)-manifold.
For
Example 4,
start with a twisted line bundle M* over some closed(n - l)-manifold
S*. Let S’ denote the connected sum of S* andNn-1,
the
homology sphere
ofExample 3,
and let M’ denote the line bundleproduced by piecing together
relevantparts
of M* and N XE1.
Let Y denote a standard kind ofdecomposition
on M’(where
M’ is consideredas the interior of a twisted I-bundle over
S’).
Name a
decomposition K4
of M’ intosingletons
and the sets X{c}
(c ~ C)
in that copy ofV = Fn-2 [-1, 1 ]
contained in the trivial section S’ of M’. Define M asM’IK4
and p: M’ ~ M as the decom-position
map. Asbefore,
M is ann-manifold,
and it admits adecomposi-
tion
G4 = {p(g)|g~y}.
Then the fundamental group of M is isomor-phic
to03C01(S*),
while03C01(M -
p(S’))
isisomorphic
to the freeproduct
of’1T 1 (M* - S*)
and twocopies
of03C01(Nn-1) ~ 03C01(F). Certainly
S* can bechosen with
sufficiently simple 03C01(S*)
that it has nosubgroup
of index 2isomorphic
toqri ( M* - S*)
*qri ( F )
*’1T1(F) (03C01(S*)
=Z2,
forexample).
This
prevents
M frombeing
anE1-bundle
over any manifold X.Finally,
to indicate how thepathology
canproliferate,
wegive:
EXAMPLE 5:
For n
6 there exists adecomposition G5
of an n-manifoldM into codimension
1,
closed submanifolds such thatMIG5 = El
andthe fundamental group of M at oo is
infinitely generated.
Let D denote the
complement
of astandardly
embedded(n - 1)-cell
in
Nn-1,
thehomology sphere
ofExamples
3 and4, arranged
so Dcontains one of the
homology
cellsV = F [-1, 1] of
whichNn-1 is
the double. InS n -1 pick
out a null sequence ofpairwise disjoint ( n -1 )-cells Bo, B1, B2,... converging
to somepoint
andreplace each Bi
with a copyDi
ofD, thereby forming
anobject
Y(a non-ANR, topologized
so as tobe
compact
andmetric).
Let Z = YE1.
Form adecomposition K5
of Zinto
singletons
andwhere
(X {c})i
denotes thecompactum
inDi corresponding
to X X{c} ~ V ~ D.
It can be
proved
that thedecomposition
space M =ZjK5
is ann-manifold
(by showing
it isgeneralized
n-manifold that satisfies theDisjoint
DiskProperty).
The central reason behind itsbeing
a gener- alized manifold isthat,
for ti, p ( Dl
X( t 1)
is a contractible(n - 1)-
manifold bounded
by
an(n - 2)-sphere (cf. [16]), forcing
it to be an( n - 1)-cell.
Asusual,
p: Z ~ M denotes thedecomposition
map; as should now beexpected,
thedecomposition GS
is defined tobe {p(Y (til-
The manifold M deformation retracts to
p ( Y X {0}),
which is homeo-morphic
toS’n-1. Moreover,
for eachinteger
i1, p(Y [ i, ~))
defor-mation retracts to
p(Y {i}),
the fundamental group of which has rank i times the rank of[03C01(D) = 03C01(F)]. Explicitly,
fort ~ (i, i + 1], p(Y {t}) is topologically
the connected sum of i + 1copies
ofN n -1.
6.
Décompositions
intoarbitrary
submanifolds.In view of the
pathology
manifested in Section5,
we nowattempt
togain
some
understanding
of the manifolds M that admitdecompositions
Ginto
closed,
codimension 1submanifolds, possible wildly
embedded. Thegoal
is to establish a structure theorem that is thehomology analogue
toTheorem 4.4. In
particular,
we want torepresent
M as a union of somequasi-standard objects {i},
and the initial concern is to findobjects J7vi
that are n-manifolds with
boundary.
LEMMA 6.1: The set e
of all
x GM/G
such that03C0-1(x)
is bicollared in M contains a denseG8
subsetof MIG.
PROOF: Without loss of
generality,
we assumeM/G = El. According
to[9],
there exists(up
tohomeomorphism) just
a countable collection{Ni |
i =1, 2,...}
ofdistinct, closed,
connected(n - 1)-manifolds.
Parti-tion
El
into subsetsT (i = 1, 2,...) by
the rule:x E T
iff03C0-1(x)
ishomeomorphic
toNi.
For each x ET
name aspecific homeomorphism 03BBx
ofN
onto03C0-1(x). Topologize
the various sets{03BBx| x
ETi} by
meansof the sup-norm metric in M.
As
suggested by Bryant [7,
p.478]
or in theunpublished
work ofBing [3],
one can prove thefollowing
claim :For each
integer j
there exists a countable subset0
ofT
such that toeach x e
T - 0
therecorrespond
two sequences{s(i)}
and{u(i)}
ofreal numbers in
T
such thats(i)xu(i)
for all i and each of theassociated
sequences (03BBs(i)}
and{03BBu(i)}
converges toÀ x (in
the sup-normmetric).
Because then
03C0-1(x)
can behomeomorphically approximated
in eachcomponent
ofM - 03C0-1(x),
a classicalargument (cf. [4,
Theorem9])
shows that M -
03C0-1(x)
is 1-LC at eachpoint
of03C0-1(x) (for
x ET - Cj).
As a
result, 03C0-1(x)
is bicollared in M(this
comes from[4,
Theorem6]
incase n
= 3,
from[8]
or[14]
in case n >5,
and from[25,
Theorem2.7.1]
incase n = 4.
Now it is
transparent
that the Lemmaholds,
sinceMjG - P)
iscontained in the countable set U
0.
Our intention is to find
quasi-standard objects {i}
in M such thatfl
ishomologically
like anobject
in a standardformation ( Wi 1.
Thecrux of that matter is
given
in the next lemma.LEMMA 6.2:
Suppose
c EEl
=M/G. Then, for
anycoefficient
moduler,
the inclusion-induced
i *: H,(7r - 1(c); 0393) ~ H*(M; r)
is anisomorphism.
PROOF: For
simplicity
we assume c = 0 and we suppress any further mention of r.First,
we showthat 1 *
is 1 - 1. DefineCertainly S ~ {0},
for’1T-I(O)
is a déformation retract in M of someneighborhood
of itself.Moreover,
ifso E S and 0
s’ so, then obvi-ously
s’ E S.Consequently, S
must be one of:[0, ~), [0, d],
or[0, d) (for
somed > 0). However, S= [0, d ]
isimpossible by
the same rea-soning indicating
S ={0}
isimpossible.
Inaddition,
because the inclu- sion-inducedH*(03C0-1(-d, d)) ~ H*(03C0-1[-d, d])
is anisomorphism, S = [0, d)
is alsoimpossible. Therefore,
S =[0, oc).
The claim thatH*(03C0-1(0)) ~ H*(M)
is 1 - 1 follows.Next we show
that i *
is onto. We compareimages by setting
and we argue, as
above,
that S’ =[0, (0),
whichyields
imi*
=H*(M).
Details are left to the reader.
COROLLARY 6.3 :
Suppose M/G
=E1, [ a, b] ~ E1
and c E[ a, b]. Then, for
everycoefficient
moduler,
the inclusion-induced a*:H *( ’1T -l( c); r)
~