WELLER, GLENN P.

Weller, Glenn P.

Osaka University Knowledge Archive

Equivalent sizes of Lipschitz manifolds and the smoothing problem

Osaka City University Repository

Weller, G. P.
Osaka J. Math.
10 (1973), 507-510
EQUIVALENT SIZES OF LIPSCHITZ MANIFOLDS
AND THE SMOOTHING PROBLEM
GLENN P. WELLER*
(Received September 2, 1972)
1. Introduction
A criterion for smoothing a closed topological manifold M was given by Y.
Shikata in [3], His criterion was based on the notion of the size of My which will
be denoted by \M\S.
Shikata's Theorem. A closed topological manifold M is smoothable if and
only if | M |
s
= 0 .
In this paper we introduce another "size" of M, denoted ! |M||, which is con-
ceptually somewhat simpler and which is definable for all open or closed Lipschitz
manifolds. We then can prove the following result.
Theorem. A closed Lipschitz manifold M is smoothable if and only if\\M\\ =
0.
This theorem follows directly from Shikata's theorem and the following
proposition, which shows that the two "sizes" are equivalent.
Proposition. For each positive integer n there are positive numbers a(n) and
β(ή) such that if M is a closed n-dimensional Lipschitz manifold then
REMARK. Neither of these sizes is trivial, for L. Siebenmann has given ex-
amples in [4, pp. 135-137] of Lipschitz manifolds Xn, n>6, having no piecewise
linear manifold structure. In particular, Xn is not smoothable, and so
2. Lipschitz manifolds and their sizes
We recall from [2] that if (Xly dλ) and (X2, d2) are metric spaces 2indf:X1-^X2
is a map then the Lipschitz size off (relative to d
x
 and d2) is
* This research is partially supported by NSF Grant GP 28487.
508 G. P. WELLER
Inf {tellk-Vfay^dtfXtfy^kdfay) for all x,
oo, if the above set is empty.
If /(/)<°°,/is a regular Lipschitz map. It is clearly seen that relative to the
obvious distances Z(/)=/(/"1) and l{gf)<l(f)-l{g)
We further recall from [6, p. 165] that a topological w-manifold M is a Lip-
schitz manifold if there is a coordinate cover C={(Uiy A, )}f e / such that C/f is an
open subset of My At is a homeomorphism of an open subset of Euclidean w-space
onto Uiy and
hixh.\ hγ(UiV[ Uj)-+hj\UiΓ\ Uj) is a regular Lipschitz homeomorphism for all
iy j . We may always assume C is finite when M is compact.
For any metric d on M denote
ld(C)— Sup/€Ξ/ /'(Af ) relative to Euclidean metric and </,
/(C)=Sup,
 tye//(Al~1Ay) relative to Euclidean metric.
Then we may define two "sizes" for M:
\M\ = Inf log /
rf(C) and
| |M| | = Inflog/(C),
where both infima are taken over all coordinate covers C and all metrics d on M.
Lemma 1. For all closed Lipschitz manifolds My
Proof. For the first inequality let £*>0 be arbitrary and pick C as above
so that log /(C)<||M||-}-£*. Denote L=l(C). Let {/,-} be a partition of unity
subordinate to {[/,-}, V£=Carrier (/,) and W= U t F t X F t . For (x,y)^ Wy the
functions φi(Xy y)==zfi(x)fi(y)\Σjfj(x)fj'(y)]~1 define a partition of unity on W so
that <pi(x, y)=<Pi(y, x). For (χ>y)<=W define p(x, y)=Jli<Pi(x9 y)\\h7λx-hγλy\\.
We can choose sequences S= {x=x0, x19" 9 xt=y) so that for all /, (Xi-19 Xi)
^W and define [S]=*Σiiρ(xi_1, x{) and d(x,y)=\ni S[*S]. It is easily verified
that d is a pseudo-metric on M giving the original topology on M. By [1, Theorem
5.26, p. 154] there is a Lebesgue number η for the cover {V£} with respect to d.
Assertion, d is a metric, i.e., d(xyy)=0 implies x=y.
Choose £ > 0 with 6<η. Then pick [S]<£. Since d(xiy Xj)<[{xiy xi+ly •••,
Xj}]<[S].<£, diameter (S)<v and Sd Vj for some;. Now
AA^'-i-^VJ^II a n d
EQUIVALENT SIZES OF LIPSCHITS MANIFOLDS 509
and
Thus, x=y since € is arbitrary.
Now let {Wi} be a cover of M so that diameter {W{)<η. Then
for some j. Let kjλ=hγ \ Wi and C*= {(Wi9 kt)}.
Assertion. k{ is regular Lipschitz relative to Euclidean metric and d.
If hjXf hjy^WidVj and d(hjX,hjy)<v, then d(hjX, hjy)<p(hjXy hjy)=
!
Also we may pick 5 = {Ay#=Λ:0, #„ •••, xp=h.y\ with d(hjX, hjy)<[S]<η.
Then for all ί, έ/(AyΛ?, *,) < Σ U i P ( * * - n **) < [S] < ?7 and *,<= F y . Hence
Thus, L^-Wx-yW^dikiX, k£y)<L.\\x-y\\ and /(^)<L=/(C). Hence,
</(C) and |MI < | |M| |+ε*. Since 6* is arbitrary, | M | <| |M| | .
For the second inequality let £>0 and choose C and d so that 2 log ld(C)<
2 |Λf| +£. Then liK'h^KlQiiyiφ^Kl^C)2 and l(C)<ld(C*)2. Thus ||M||
<2 | M | + f and | |M| |<2. | M | .
The above proof also shows an equivalent way of defining Lipschitz mani-
folds. We state this as a corollary.
Corollary. A closed manifold M is Lipschitz manifold if and only if there is a
coordinate cover C= {(£/,-, /*,)} iei such that each homeomorphism hi is regular Lip-
schitz with respect to the usual metric on Euclidean space and some metric d on M.
3. A reformulation of Shikata's criterion
Shikata [3] defined the "size" of a compact topological w-manifold M to be
where γ > 0 depends only on n and m(C) is the maximum number of Uj that
any £/,• can intersect. It is clear that any C can be replaced by a coordinate
cover C" refining it, provided we use the restrictions of the appropriate home-
omorphisms h£ from C in computing ld(C). In [5] we show that there is a
positive integer μ(n) depending only on n such that any C has a refinement C
as above with m(C')<μ(n). Let a'(n)=(8y)μ™ and β\n)=\ if 8γ<l , and
510 G. P. WELLER
vice versa if 8 γ ^ l . Then noting the definitions of \M\ and | M | S , we have
proved the following lemma.
Lemma 2. There are positive numbers a\ri) and β\n) depending only on n
such that for all closed manifolds M
a'(n)\M\<\M\
s
<β'(n)\M\ .
Taken together, Lemmas 1 and 2 clearly yield the proposition. Hence, the
theorem is established.
THE UNIVERSITY OF ILLINOIS AT CHICAGO CIRCLE
References
[1] J.L. Kelley: General Topology, Van Nostrand, Princeton, 1955.
[2] Y. Shikata: On a distance function on the set of differentiable structures, Osaka J.
Math. 3 (1966), 65-79.
[3] : On the smoothing problem and the size of a topological manifold, Osaka J.
Math. 3 (1966), 293-301.
[4] L.C. Siebenmann: Topological manifolds, Proceedings of the International Congress
of Mathematicians, Nice, 1970.
[5] G.P. Weller: The intersection multiplicity of compact n-dimensional metric space,
Proc. Amer. Math. Soc. to appear.
[6] J.H.C. Whitehead: Manifolds with transverse fields in euclidean space, Ann. of Math.
73 (1961), 154-212.


EQUIVALENT SIZES OF LIPSCHITZ MANIFOLDS AND THE SMOOTHING PROBLEM

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EQUIVALENT SIZES OF LIPSCHITZ MANIFOLDS AND THE SMOOTHING PROBLEM

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