Topoioyp,k.01.9. pp. 119--15I. PcrgamonPres.
COUNTTNG
1970.
Prmrsdin Gre~rBrinin
TOPOLOGICAL
MANJFOLDS?
J. CHEEGER and J. M. KISTER
(Recciced
25 Jme
1969; rec?rcr-l 6 October
1969)
WE CONSIDER
the class % of all compact topological manifolds, boundaries permitted. It is
that there are only a countable number of homotopy types in Y. [l] and [I]. The
number of topologically
subclass f8pr, of piecewise linear manifolds has only a countable
distinct elements, since each could be regarded as a finite simplicial complex and a simple
known
argument shows there are only a countable number of those, up to isomorphism.
Recently,
however, Kirby and Siebenmann
[3] have discovered some examples of topological manifolds admitting no PL structure, so that route for counting homeomorphism
types in % is
rather unpromising
(whether manifolds can be triangulated
without a PL structure is still
open).
We take a direct approach and with the aid of a very useful result of Edwards and
Kirby [4] examine overlapping coordinate neighborhoods
to show that % has only countably
many elements up to homeomorphism.
Our argument is similar to one in a differential
setting in [3].
Siebenmann
has informed LIS that hc and Kirby obtained the same result earlier for
closed manifolds having dimension at least 6, and for bounded manifolds of dimension at
least 7, as a result of their topological handlebody theory 161. This approach will not give the
general result, however, since they also show that there is a closed manifold of dimension
4 or 5 that has no handle decomposition.
Our proof is much more elementary and it is
purely
An interestin g by-product
is a topological
proof of a key lemma in [S]. See the remark
geometric.
after the proof.
THEOREM.
There are precisely
(boundary permittedj,
a countable
submersion
number of compact
theorem
topological
and a
manifolds
up to homeomorphism.
Proof. We first consider those with empty boundaries.
Suppose not. then there are an
uncountable
number of some dimension, say n. Let {I1lZ”)ZE,,be s?!ch a collection. For each
M,” find a collection
of imbeddings
hZj : B(2) --+M,“, _j = I, 2,. . ., k,, such that
{hzj(f3( l))>j21 covers M,“, where B(r) is the closed ball of radius r and center 0 in i?“. By
possibly choosing an uncountable
subcollection,
we can assume without loss of generality
that k, = k for all U. We can also assume, by reparametrizin g, that /lZj 1B(1) can be extended
toanimbeddingofB(k+
l)intoM,,j=
I, . . . . k. Wz shall also regard each ‘M,” as a subset
of R’,e.g. let I = 2n f I. Ifrlis the metric in R’,dsfine F,~,,, = rl(&(B(nzj), M, - /IZj(B(IH i 1)))
iThis research
was partially
supported
by
NSF Grant
119
GP-S-N.
150
J. CHEEGER and J. M. KISTER
lCz,m,.
f? Bv, choosing a subcollection
again we can assume there exists an
C > 0 such that E, > E for ail x in .j.+ Each :Zf,” determines an imbeddins
gz : B(k + 1) -+
and let E, = mini,,
I?’ = R’ x R’ x . . . x R’ bvd .Ic/ (x) = (A,,(x). . . , k,(x)). The set of all such imbeddings
the uniform
metric: dU(g,, gD) = maxy,B Ck+I) d(g,(x), g&.‘c)), is separable
metric.
under
hence some gz,, is a limit point
of a sequence
will show that .Cfzo is homeomorphic
to ?if,‘,
homeomorphism
can be taken to be arbitrarily
of distinct imbeddings
for i sttficiently
large.
close to the identity
g,, , g,, , .
Furthermore,
as measured
. We
this
by the
metric d, which we use in all that fol!ows.
Let V,(m) = I~,~~(B(iiz)), j = 1, 3, . . . , k, nz = 1, 2,
, 1~+ 1. To simplify notation we
denote &I,, by M’, for fixed but arbitrarily large i. and kve denote /~,,~(B(rn)) c M’ by Vj’(m),
j=1,2
,...,
li,nz=1,7
,...,
h-+l.Nowlet
Note that U,(l) = AJ and U,‘(l) = M’. Define fj : L5(X-+ I) --) V,‘(k + 1) as /I,~, 0 /r$
j= l,...,
k and note that each fjcan be taken as close to the identity as we like for Al’
suficiently
far out in the sequence M,, , M,, , . . . . We now proceed to construct a small
homeomorphism
from M to M’ inductively on the sets Ui(m) as follows.
Suppose we can construct an imbedding gj : Uj(m) -+ M’
identity by choosing M’ sufficiently far out in the sequence.
construct
an imbedding gj+l : Uj+l(m - 1) + M’ as close to
Hence, starting off by letting gI =f, and tn = k, in k - 1 steps
as
We
the
we
close as we like to the
will show that we can
identity as we please.
will have an imbedding
gr. : U,(l) + M’, the desired homeomorphism.
First we see that gj(c/j(m) n Vj+,(t7z)) c VJ+l (tn + 1) if M’ is chosen sufficiently far
out and gj is close to the identity
(relative to our previous E). Then fj;: gj is defined
on Uj(rn) n vi+,(m) and close to the identity. Letting IV be an open set in A4 with
Uj(m - 1) n Vj+l(m
-
1) c N c Uj(nz) n Vj+ l(m) we can extend fj; :gj 1N:
to an onto homeomorphism
h : Vj+l(m)
+ Vj+, (m) close to the identity,
N + Vi+,(m)
using the theorem
of [4].
Now define gj+ I : Uj+l(tu
gj+l(,r)
- 1) -+ &I’ by
=
(Sjb)
for
\fj+1/7(x)
.y in
uj(irz
-
1)
for x in Vj,l(nl
- 1).
,
By the definition of /I, gjcL is well-defined. Since gj+l can be extended to IV, as well, using
since it is
either half of the definition, it is easily seen that gj+I is a local homeomorphism,
an imbedding
on the two open sets Uj(m - 1) u (N I~ Vj+l(nz - 1)) and vj+l(m - 1) u
(N n Uj+l(m - 1)). It would fail to be an imbedding
only if gj+I(x) = gj+l(_Y) for some
x and y in Uj(m - 1) - N and vj+,(m - 1) - N respectively, two compact disjoint sets with
jWe are indebted to Mr. A. Shiiepsky
inclusion we need later.
for pointin, 0 out thar this condition
is necessary to obtain an
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