Pacific Journal of
Mathematics
MONOTONE MAPPINGS OF A TWO-DISK ONTO ITSELF
WHICH FIX THE DISK’S BOUNDARY CAN BE CANONICALLY
APPROXIMATED BY HOMEOMORPHISMS
W ILLIAM E MERY H AVER
Vol. 50, No. 2
October 1974
PACIFIC JOURNAL OF MATHEMATICS
Vol. 50, No. 2, 1974
MONOTONE MAPPINGS OF A TWO-DISK ONTO ITSELF
WHICH FIX THE DISK'S BOUNDARY CAN BE
CANONICALLY APPROXIMATED BY
HOMEOMORPHISMS
WILLIAM
E.
HAVER
The theorem stated in the title is proven. As a corollary
it is shown that the space of all such monotone mappings is
an absolute retract.
1. Introduction. Let Dn denote the standard %-ball of radius
one in En and H(Dn) the space of all homeomorphisms of Dn onto itself
which equal the identity on the boundary of Dn. Let H(Dn) denote
n
the space of all mappings of D onto itself which can be approximated
arbitrarily closely by elements of H(Dn). Under the supremum topology,
H{Dn) and H(Dn) are separable metric spaces; H(Dn) is complete under
the supremum metric. It is known that H(Dn) is locally contractible
[7], H(Dn) x l2 & H{Dn) [4], #(IF) jis homogeneous [7], and that
H{Dι)f^ l2 [3]. In this paper we shall be concerned with the case
n = 2, and to simplify notation we shall write D for D2. It is wellknown (cf. [8]) that H(D) is the space of all monotone mappings of
D onto itself which equal the identity when restricted to the boundary
of D.
We shall show [Theorem 1] that the elements of H(D) can be
"canonically approximated" by elements of H(D) and [Theorem 2] that
H(D) is an absolute retract. The work of this paper depends heavily
on W. K. Mason's paper, "The space of, all self-homeomorphisms of
a two-cell which fix the celΓs boundary is an absolute retract", [9].
The crux of Mason's paper is the definition of a basis for H(D) which
can be shown to possess some particularly nice properties. We shall
review the definition of this basis in the following section and then
define a basis for H(D). Familiarity will be assumed with the notation
and basic definitions of [9].
2* Mason's basis for H(D). Consider D to be a rectangle in R2
with horizontal and vertical sides. A grating, P, on D consists of a
finite number of spanning segments (crosscuts) across D, parallel to
its sides, with the same number of horizontal and vertical crosscuts.
Let Pl9 P2j
be a sequence of gratings on D such that (a) the mesh
of P. approaches 0 as i increases and (b) if I is a crosscut of Pi and
j ^ i, then I is a crosscut of Pβ.
Let έ%f be the collection of all polyhedral disks H contained in D
477
478
WILLIAM E. HAVER
such that Bd (H) is the union of a vertical segment in the left side
of Bd (D), a vertical segment in the right side of Bd (JD), a polygonal
spanning arc of D, Hτ, t h a t is contained in the closure of the same
component of H(D) — H as the top of Bd (D), and a polygonal spanning
arc of D, HB, that is contained in the closure of the same component
of H(D) - H as the bottom of Bd (D). Let T be the collection of all
polyhedral disks V contained in D such t h a t Bd (V) is the union of
a horizontal segment in the top of Bd (J5), a horizontal segment in
the bottom of Bd (D), a polygonal spanning arc of D, VL, that is
contained in the closure of the same component of H(D) — V as the
left side of Bd (D) and a polygonal spanning arc of Ό, VR, that is
contained in the closure of the same component of H(D) - F a s the
right side of Bd (D).
Let P3 be a grating from the sequence Pu P2,
. Let {llf
,
ln) be the set of horizontal crosscuts of P3 and {ml9 •••, mn) the set
of vertical crosscuts. Let {Hu
, Hn} c έ%f satisfy Hi Π H3 = 0 if
i Φ j and {Vu
, Vn] c T satisfy Vi n Ty = 0 if i ^ i
Then define
O(P if flίf . . . , f r . ; 7 l f •••, F.)
= {/ e J ϊ φ ) I /(ί 4 ) c H< - {Cl (D - fli)} and
/(m4) c Vi - {Cl (Z> - Vi)} for 1 ^ i ^ n} .
Then the basis for H(D), which Mason denotes HVT, is the collection
of all such open sets.
3. A Basis for H(D). In this section we define a basis, β, for
H(D) and demonstrate that it possesses some nice properties. Let Pjy
{Hlf •••, Hn} and {Vu •••, F J be as in the definition of HVT.
The
basis, β, will consist of all sets of the following form:
B ( P 3 ; Hlf 11Ll_Hn;
Vu •••, F . )
= {/ e H(D) I f-\k) c J3i - {Cl (D - fli)} and
f-'im,)
c Vi - {Cl (D - V4)}, for 1 ^ i ^ ^} .
We note that feB(P3; HL, - , ff,; F x ,
, Vn) Π Jϊ(2)) if and only
1
if f- e 0{Ps\ fli, --,g»; Vl9
, Vw). To see that the elements of /S
are open subsets of H(D), let / be an arbitrary element of B(P3; H19
- ,Hn; Vlf •••, Vn). Then let
ε = min {d(f(Hl
l^i^
U H?)f (j hi d(f(VΪ
l
U Ff), U my)} .
il
Let βf be an arbitrary element of H(D) satisfying d(f, g) < ε. Suppose
that g $ B{P5] Hu
, Hn; Vl9
, VΛ). Then, without loss of generality, we can assume that there exists an integer ί such that g~\li)
MONOTONE MAPPINGS OF A TWO-DISK
479
is not contained in Hi - Cl (D - if,). Since (i) g-\U) Π (Bd D) =
f-l{U) n (BdZ>) c ί/i - Cl (D - Hi), (ii) flΓ1^) is a connected set and
(iii) HI U -Hi5 separates H, - Cl φ - ίZ,) from D - Hif there is an
α? G i?7 U H? such that gr(x) e l{. But this implies that d(f(x), g(x)) ^
U Hf), U) ^ ε and hence d(f, g) ^ ε.
LEMMA
1.
β is a basis for
H(D).
Proof. Let / e iJ(D) and ε > 0 be given. We wish to find Beβ
such that / e B and eZ(/, #) < ε for all xeB.
Pick a grating, P, , such
that diam | st(x, P3) | < ε for every xeD.
Let l{ be the ίth crosscut
from the top of D. Choose H[ to be a polygonal spanning arc of D
with one endpoint in each side of D that separates the top of D from
f~ι{lι).
Choose H* to be a polygonal spanning arc of D that separates
f^ik)
from f^ik).
The polyhedral disk Hx is thus uniquely defined.
Assume inductively that disks Hu
, i Z ^ have been defined in such
a manner that HjΠHk — 0 if 1 ^ i ^ i — 1, 1 ^ & ^ i — 1, and i =^ ^
and that for each j,l <^j <Zί — 1, HJ separates Hf^ from f~\l3) and
JS/ separates f"\lj)
from f~\lj+ι).
Choose ί ί f to be a polygonal
spanning arc of D that separates ί f ^ from f~l{U).
Finally choose
Hi to be a pylygonal spanning arc of D that separates f~ι{li) from
f^ih+i) ( o r from the bottom of D if i = %). We have thus uniquely
defined iϊ; in such a way that the inductive hypothesis is satisfied.
Define Vi, •••, V« in a similar manner. Now, by construction
feB(P3;Hu
••.,#„; Vlf ••-, K ) and if geB(P3;Hly
••-,#„; Vi, ••.,
F%) then d(f, g) < ε.
2. Lβί J5,,
is an element of β.
LEMMA
, B3 be elements of β.
Then B = Γ\ί=1Bk
Proof. Assume B Φ 0 and t h a t Pk is the grating associated with
Bk,l^k£j.
Hence for any U,l<,k<j,
every crosscut of Pk is a
crosscut of P3. Let lx be the first horizontal crosscut of P3. For each
kfl^k£j,
let jfflfjfe be the element of 3ίf associated with lx and Bk
(if there is one). Let Hγ be the component of D - (ULi H^ U U*=i -ff^)
that contains f~\l^).
Define in an analogous manner H2,
, Hn and
Vlf
, K . It is clear that B(P3; Hlf
, Hn; Vlf
, Vn) = Π ύ ^ .
The elements of {ίZΊ,
, Hn) are pairwise disjoint since each Hi is
contained in Hi>n and the elements of {Huj, •• ,ίZ Λ ) J } are pairwise
disjoint.
Lemma 3 will follow as a corollary to the following theorem of
Mason. (The proof of this theorem constitutes the bulk of [9].)
THEOREM
(Mason). Let U be an element of HVT and K a finite
480
WILLIAM E. HAVER
dimensional compact subset of U. Then there is an embedding ψ of
the cone over K into U such that ψ(f, 0) = /, for all f e K.
LEMMA 3. Let B = B(Pf, Hu
, Hn; Vu
, Vn) be an element
of β and K a finite dimensional compact subset of B Π H{D). Then
there is an embedding λ of the cone over K into B Π H(D) such that
Ί K / , 0 ) = / , for
all f e K.
Proof. Since H(D) is a topological group, the function G: H(D) —•
1
H(D) defined by G(f) = f" is a homeomorphism. Therefore, by the
note following the definition of β, G{K) is a finite dimensional compact
subset of U{Pό;Hly -- ,Hn; Vl9 •••, Vn). Hence by Mason's theorem
there is an embedding ψ of the cone over G{K) into U(Pό; Hu
,
Hn; Vlf
, Vn) such that ψ(f, 0) = /, for all / e G(K). Define λ: K x
I~>Bf) H(D) by λ(ft, t) = G~\f{G{k), t)).
4* The main results* The following theorem shows that the
elements of H(D) can be canonically approximated by elements of H(D).
THEOREM 1. Let a be an open cover of H(D). Then there exists
a locally finite polyhedron, &, and maps b: H(D) —*&, ψ: έ^
and Θ:~Ή(D) x I-^~H(D) such that
(a) for each feH(D), there is an element, Uf, of a such that
θ{f, t) 6 Uf, for each t e I,
(b) θ(f, 1) - /, for each f e H{D),
(c ) θ(f, 0) = ψb(f), for each feH{D),
(d) θ(f, t) e H(D) for each f e H(D) and t e [0, 1).
Proof. Let α' be an open barycentric refinement a (i.e., if / € H(D),
I t(f, ®r) I is contained in some element of a). For each positive integer,
k, let ak be an open cover of H(D) such that
( i ) ak is a refinement of a!,
(ii) if Veak, diam V<l/k,
(iii) if Feα f c , then Veβ.
s
We next define an open cover, Ύ], of H(D) x [0, 1).
Let η = {Vx [0,l/2)| F e α J U (
Let 7 be a countable refinement of rj such that
(a) if h e y, st(h, r) is a finite set,
(b) iί hey, then there is an element of η, V x J, such that
I st(hf y) I c V x J.
Let & be the nerve of 7 and B: H{D) x [0, 1 ) - * ^ be the standard
MONOTONE MAPPINGS OF A TWO-DISK
481
barycentric map. Order the element of 7, and for each ^ e γ , let
Vi x Ji be an element of ΎJ such that | st(hi9 7) | c V{ x J*. Note that
We shall define a map τ/τ: & —> £Γ(D) by induction on the skeletons
of ^ . For each vertex (ht) of ^ * let ψ°((hi)) be an arbitrary element
of iϊ(#) intersected with the projection of /&< onto H(D).
Now assume that for m = 1, 2,
, w we have defined α/τ
w
1
£Γ(D) such that ^ extends ψ™" and for each simplex crm = (hiQ,
( a ) ψm(σm) is finite dimensional,
( b ) ψ « ( O c fl(JD) Π {F« | Λ, c Π?=o βί(λ,if 7)}.
Now let σnJrl = (h0,
, hn+1} be any simplex of &*n+1. Let
n { i i
l
n (
y
, ) }
Since each Vi is an element of β, by Lemma 2, Ueβ.
By the inductive hypothesis the image under ψn of the boundary of σn+1 is a
finite dimensional compact subset of U Γ) H(D)y denoted K. By Lemma
3 there is an embedding
λ: c ( i Q —
UnH(D)
such that λ(/, 0) = / for all f e K. We consider σn+1 to be the cone
over its boundary, and so for (x9 t) e σn+1, let ψn+1(x, t) = λ(ψ%(a?), ί).
Extending over each n + 1 simplex in this manner gives
φ*+1: &*n+1—> H(D) and completes the induction. Hence limΛ_>oβψw =
α/r: ^ —> H(D) is continuous by the continuity of each τ/r% and the local
finiteness of ^ *
Let δ:Ή(5) — ^ be defined by δ(/) = B((f, 0)).
We next define the homotopy θ: H(D) x J—> fl(-D) in the following
manner:
Conditions (b), (c), and (d) are obviously satisfied. We show simultaneously that θ is continuous and that for each / e H(D) there is an
element Uf of a such that for each t e I, θ(f, t) e Uf.
Suppose that (f,t)eH(D)x
[0,1) and that (2k- 3)/2* <t<(2k-l)/2k.
Let ho be any element of 7 which contains (/, ί). By the definition
of α/r, ψB(f, t) e Vo. But Vo e ak^ U ak U α:fc+1 and therefore the diameter of Fo is less than l/(k — 1) which implies that d(ψB(f, t), f) <
l/(Jc — 1) and thereby that θ is continuous. Since each ak is a refinement of α', there exists an element of a!, U{ftt), such that
482
WILLIAM E. HAVER
{/} U {fB(f, ί ) } c F o c Uif,t). Since a! is a barycentric refinement of
a, there is some element, Uf, of α such that Uίβ[o,i) U{f,t) c Uf and
hence #(/, ί) e Uf9 for each £ e I.
The following result is an immediate corollary of Theorem 1 and
a theorem of Hanner [5] which states that a metric space X is an
ANR if given an arbitrary cover, α, of I there exists a locally finite
polyhedron M maps b: X—> ^ , <f: ^ —> X, and Θ:X x I-^X such that
0(8, 0) = <f &(&) for all ί c e l , 0(a?, 1) = α; for all xeX and for each α; e X
there is an element U of α such that θ(x, t) e U for all t e [0, 1].
THEOREM
2.
£Γ(JD)
is an absolute retract.
Proof. By the preceding comments, H(D) is an ANR.
But i?(£>)
is contractible by the Alexander isotopy [1] applied to H{D). The
theorem follows since every contractible absolute neighborhood retract
is an absolute retract.
5. Applications, ( a ) The author has shown [6] that H{M),
the space of all mappings of a compact manifold onto itself which can
be approximated arbitrarily closely by homeomorphisms, is weakly
locally contractible. Theorem 1 can be used [7] to show that for any
compact 2-manifold, M\ H(M2) is locally contractible.
( b ) A problem of current interest is whether H{D) is homeomorphic to l2; it can easily by shown using a result of Anderson [2]
that if H(D) is homeomorphic to l2f then H(D) is homeomorphic to l2.
Perhaps the results of this paper and the fact that H(D) is complete
under the usual metric will be helpful in showing that H(D) is homeomorphic to l2.
( c ) L. C. Siebenmann [10] has asked whether the inclusion map
i: H(M) —> H(M) is a homotopy equivalence. Theorem 1 provides an
affirmative answer to the question for the special case ί: H(D) —> H(D).
Added in proof. Recent work of Torunczyk ("Absolute retracts
as factors of normed linear spaces," Fund. Math., to appear) implies
that since H(D) is an AR and H{D) x ^ ΊΪ(D), WO) is in fact
homeomorphic to l2.
REFERENCES
1. J. W. Alexander, On the deformation of an n-cell, Proc. Nat. Acad. Sci., U.S.A., 9
(1923), 406-407.
2. R. D. Anderson, Strongly negligible sets in Frechet manifolds, Bull. Amer. Math.
Soc, 75 (1969), 64-67.
3. R. Geoghegan, On spaces of homeomorphisms, embeddings, and functions I,
Topology, 11 (1972), 159-177.
MONOTONE MAPPINGS OF A TWO-DISK
483
4. R. Geoghegan and D. W. Henderson, Stable function spaces, to appear.
5. 0. Hanner, Some Theorems on Absolute Neighborhood Retracts, Arkiv Fur Matematik,
I (1951), 389-408.
6. W. Haver, Homeomorphisms and UV°° maps, Proc. of SUNY Bingh. Conf. on
Monotone Mappings and Open Mappings, (1970), 112-121.
7.
, The Closure of the Space of Homeomorphisms on a Manifold, Trans.
Amer. Math. Soc, to appear.
8. J. Hocking, Approximations to monotone mappings on noncompact two-dimensional
manifolds, Duke Math. J., 2 1 (1954), 639-651.
9. W. K. Mason, The space of all self-homeomorphisms of a 2-cell which fix the celΓs
boundary is an absolute retract, Trans. Amer. Math. Soc, 1 6 1 (1971), 185-206.
10. L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology,
II (1972), 271-294.
Received September 6, 1972. Research partially supported by NSF Grant GP. 33872.
UNIVERSITY OF TENNESSEE
PACIFIC JOURNAL OF MATHEMATICS
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University of California
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Department of Mathematics
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University of Washington
Seattle, Washington 98105
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Pacific Journal of Mathematics
Vol. 50, No. 2
October, 1974
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example to the Blum Hanson theorem in general spaces . . . . . . . . . . . . . .
Huzihiro Araki, Some properties of modular conjugation operator of von
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E. F. Beckenbach, Fook H. Eng and Richard Edward Tafel, Global
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David W. Boyd, A new class of infinite sphere packings . . . . . . . . . . . . . . . . . . .
K. G. Choo, Whitehead Groups of twisted free associative algebras . . . . . . . .
Charles Kam-Tai Chui and Milton N. Parnes, Limit sets of power series
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Allan Clark and John Harwood Ewing, The realization of polynomial
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Dennis Garbanati, Classes of circulants over the p-adic and rational
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Arjun K. Gupta, On a “square” functional equation . . . . . . . . . . . . . . . . . . . . . .
David James Hallenbeck and Thomas Harold MacGregor, Subordination
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William Emery Haver, Monotone mappings of a two-disk onto itself which
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