Publ. Ma . 45 (2001), 3–67
SOME TOPICS CONCERNING HOMEOMORPHIC
PARAMETERIZATIONS
S ephen Semmes
Abs ac
In his su ey, we conside se e al ques ions pe aining o home-
omo phisms, including c i e ia o hei exis ence in ce ain ci -
cums ances, and obs uc ions o hei exis ence.
Con en s
1. Wildness and ameness phenomena 4
2. Con ac able open se s 8
2.1. Some posi i e esul s 11
2.2. Ends o mani olds 16
3. In e lude: looking a infini y, o looking nea a poin 16
4. Decomposi ion spaces, 1 18
4.1. Cellula i y, and he cellula i y c i e ion 24
5. Mani old ac o s 26
6. Decomposi ion spaces, 2 29
7. Geome ic s uc u es o decomposi ion spaces 32
7.1. A basic class o cons uc ions 32
7.2. Quo ien spaces can be opologically s anda d, bu
geome ically icky 36
7.3. Examples ha a e e en simple opologically, bu
s ill non i ial geome ically 43
8. Geome ic and analy ic esul s abou he exis ence o good
coo dina es 47
8.1. Special coo dina es ha one migh conside in o he
dimensions 50
9. Nonlinea simila i y: Ano he class o examples 54
10. Doing p e y well wi h spaces which may no ha e nice
coo dina es 55
Re e ences 58
The au ho was pa ially suppo ed by he Na ional Science Founda ion.
4 S. Semmes
1. Wildness and ameness phenomena
Conside he ollowing ques ion. Le nbe a posi i e in ege , and
le Kbe a compac subse o Rn.I Kis homeomo phic o he uni
in e al [0,1], is he e
a global homeomo phism om Rnon o i sel
which maps K o a s aigh line segmen ?
(1.1)
I n= 1, hen Ki sel is a closed line segmen , and he answe is “yes”.
When n= 2, he answe is also “yes”, bu his is mo e complica ed, and
is mo e in he spi i o he Sch¨onflies heo em in he plane. See [Moi],
especially Chap e 10.
When n≥3, he answe o he ques ion abo e can be “no”. An a c K
is said o be “ ame” (o fla ) when a homeomo phism does exis as in
(1.1), and “wild” when i does no exis . See [Moi] o some examples
o wild a cs in R3.
Smoo h a cs a e always ame, as a e polygonal a cs, i.e., a cs made
up o fini ely many s aigh line segmen s. Fo hese one can ake he
co esponding homeomo phism o be smoo h o piecewise-linea as well.
(Compa e wi h Theo em 1 on p. 134 o [Moi], o ins ance.) In o de
o an a c o be wild, some amoun o infini e p ocesses a e needed.
A simple closed cu e in R3migh be smoo h o polygonal and s ill
kno ed, so ha he e does no exis a homeomo phism o R3on o i sel
which maps he cu e on o a s anda d ci cle (inside a s anda d 2-di-
mensional plane in R3). The e a e many well-known examples o his,
like he e oil kno . Thus, o a closed cu e, one defines “wildness” in
a sligh ly diffe en ly way, in e ms o he exis ence o local fla enings,
o ins ance. This u ns ou o be compa ible wi h he case o a cs ( o
which he e is no issue o kno edness), and he e a e some o he na u al
a ian s o his.
He e is ano he basic example, o se s o highe dimension. Suppose
ha γis a simple closed cu e in R3, which is a polygonal cu e, and
which ep esen s he e oil kno . Conside he cone o e γ, which gi es
a 2-dimensional polyhed on in R4, and which is in ac piecewise-linea ly
equi alen o a s anda d 2-dimensional cell. One can show ha his em-
bedding o he 2-cell is no locally fla a he cone poin , i.e., i canno
be s aigh ened ou o ag ee wi h a s anda d (geome ically fla ) embed-
ding by a homeomo phism defined on a neighbo hood in R4o he cone
poin . Simila phenomena occu o codimension-2 embeddings in Rn
o all n≥4, as in Example 2.3.2 on p. 59–60 o [Rus1].
Some Topics Conce ning Homeomo phisms 5
This phenomenon is special o codimension 2, howe e . Specifically, a
piecewise-linea embedding o a k-dimensional piecewise-linea mani old
in o Rnis locally opologically fla i n−k= 2 (o i k= 1 and n=3,
as be o e). See Theo em 1.7.2 on p. 34 o [Rus1].
In he con ex o piecewise-linea embeddings, one can also look o
local fla enings which a e piecewise-linea . A simila ema k applies o
o he ca ego ies o mappings. We shall no pu sue his he e.
Wild embeddings o cells and sphe es (and o he mani olds) exis in
Rn o all n≥3, and o all dimensions o he cells and sphe es ( om
1 on−1). This includes embeddings o cells and sphe es which a e
no equi alen o piecewise-linea embeddings in codimension 2. We
shall mos ly conside he e issues o exis ence o opological fla enings
o local fla enings, and embeddings which a e no no mally gi en as
piecewise-linea .
See [Bin6], [Bu ], [Bu C], [Can1], [Da e1], [Da e2], [Edw1],
[Moi], [Rus1], [Rus2] o mo e in o ma ion, and o u he e e ences.
Le us also men ion ha embeddings, al hough wild, may s ill enjoy sub-
s an ial good beha io . Fo ins ance, hey may be bilipschi z, so ha
dis ances a e no inc eased o dec eased by mo e han a bounded ac o ,
o quasisymme ic, in he sense o [TukV]. Roughly speaking, he la e
means ha ela i e dis ances a e no dis o ed oo much, a he han
dis ances hemsel es, as o a bilipschi z mapping. See [Geh1], [LuuV],
[V¨ai2] o some basic esul s in hese di ec ions.
As ano he e sion o wildness o embeddings, imagine ha one has
a compac se Cin some Rn, and ha Cis homeomo phic o he usual
middle- hi ds Can o se . Can one mo e C o a subse o a s aigh line
in Rn, h ough a homeomo phism om Rnon o i sel ?
When n= 1 his is au oma ically ue. I is also ue when n=2;
see [Moi], especially Chap e 13. In highe dimensions i is no ue in
gene al, as is shown by a amous cons uc ion o An oine (“An oine’s
necklaces”). See Chap e 18 o [Moi] and [Bla].
How can one ell when a se is embedded wildly o no ? As a simple
case, le us conside Can o se s. I Cis a compac subse o Rnwhich
lies in a line and is homeomo phic o he Can o se , and i nis a leas 3,
hen he complemen o Cin Rnis simply-connec ed. This is no ha d
o see. Basically, i one akes a loop in he complemen o Cand fills i
wi h a disk in Rn, and i ha disk happens o un in o C, hen one can
make small pe u ba ions o he disk o a oid in e sec ing C.
The complemen o Cis also simply-connec ed i he e is a global
homeomo phism om Rnon o i sel which maps Cin o a line. This is
6 S. Semmes
me ely because he homeomo phism i sel pe mi s one o educe o he
p e ious case.
Howe e , An oine’s necklaces ha e he p ope y ha hei comple-
men s a e no simply-connec ed. See [Moi], [Bla]. No e ha he ho-
mology o he complemen o a compac se in Rnis con olled h ough
he in insic opology o he se i sel , as in Alexande duali y [Spa]. In
pa icula , while he complemen o an An oine’s necklace may no be
simply-connec ed, i s 1-dimensional homology does anish.
Ve sions o he undamen al g oup play an impo an ole o wildness
and aming in gene al, and no jus o Can o se s. Fo his one may
no ake (o wan o ake) he undamen al g oup o he whole comple-
men a y se , bu look a mo e localized o ms o he undamen al g oup.
A specific and basic e sion o his is he ollowing. Suppose ha Fis
a closed se inside o some Rn. Gi en a poin p∈F,andaloopγin
Rn Fwhich lies close o p, one would like o know whe he i is possible
o con ac γ oapoin inRn Fwhile s aying in a small neighbo hood
o p. This second neighbo hood o pmigh no be qui e as small he fi s
one; a p ecise s a emen would say ha o e e y >0 he e is a δ>0
so ha i γlies in B(p, δ)∩(Rn F), hen γcan be con ac ed o a poin
in B(p, )∩(Rn F).
This ype o condi ion is sa isfied by s anda d embeddings o se s in o
Rn, like Can o se s, cells, and sphe es, a leas when he dimension o
he se is diffe en om n−2. Fo a poin in R2, o a line segmen in
R3, e c., one would ge Z o he co esponding localized undamen al
g oup o he complemen o he se . (In he case o a line segmen in
R3, one should es ic one’s a en ion o poin s pin he in e io o he
segmen o his.)
Con e sely, he e a e esul s which pe mi one o go backwa ds, and
say ha localized undamen al g oup condi ions o he complemen like
hese (localized simple-connec edness condi ions in pa icula ) lead o
ameness o a gi en se , o o he kind o “s anda d” (non-wild) beha io .
See [Bin5], [Bin6], [Bin8], [Bu ], [Bu C], [Can1], [Can2], [Da e1],
[Da e2], [Edw1], [Moi], [Qui1], [Qui2], [Rus1], [Rus2] o mo e in-
o ma ion abou localized undamen al g oups and hei ole in wildness
phenomena and aming heo ems (and o ela ed ma e s and u he
e e ences).
One migh wonde why π1and localized e sions o i play such an
impo an ole. Some basic poin s behind his a e as ollows. Fo ho-
mology (o cohomology), one o en has good in o ma ion om da a in
he gi en si ua ion h ough s anda d esul s in algeb aic opology, like
duali y heo ems. In ci cums ances wi h sui able simple-connec i i y,
Some Topics Conce ning Homeomo phisms 7
one can pass om in o ma ion abou homology o in o ma ion abou
homo opy (in gene al dimensions), as in he Hu ewicz and Whi ehead
heo ems. See [B e1], [Spa].
Ano he basic poin conce ns he effec o s abiliza ion. A wild em-
bedding o a se in o some Rncan become ame when iewed as an
embedding in o an Rmwi h m>n(m=n+1 in pa icula ). The same
is ue o kno ing. Fo ins ance, a smoo h loop may be kno ed in R3,
bu when iewed as a subse o R4, i is always unkno ed. This is easy
o see in explici examples (like a e oil kno ).
Fo some simple and gene al esul s abou wild embeddings in Rn
becoming ame in a la ge Rm, see P oposi ion 4 on p. 84 o [Da e2],
and he co olla ies on p. 85 o [Da e2]. These in ol e a amous de ice
o Klee. In conc e e examples, one can o en see he aming in a la ge -
dimensional space di ec ly, and explici ly. Examples o wild se s a e
o en made wi h he help o a ious linkings, o some hing like ha , and
in a highe -dimensional space one can disen angle he linked pa s. This
can be accomplished by aking indi idual pieces and pulling hem in o
a new dimension, mo ing hem a ound eely he e, and hen pu ing
hem back in o he o iginal Rnin a diffe en way.
This simpli ying effec o s abiliza ion also fi s wi h he ole o local-
ized e sions o undamen al g oups indica ed be o e. Le Fbe a closed
se inside o some Rn, and imagine ha one has a loop γin Rn Fwhich
lies in a small ball cen e ed a a poin in F. In he condi ion ha was
discussed ea lie , one would like o con ac γ o a poin in he comple-
men o F, while emaining in a small ball. I one hinks o γand Fas
being also inside Rn+1, hen i is easy o con ac γ o a poin in he
complemen o Fin Rn+1, while emaining in a small ball. Specifically,
one can fi s ansla e γin o a pa allel copy o Rninside o Rn+1, i.e.,
in o Rn×{a} o some a= 0 a he han Rn×{0}in Rn+1 (using he
ob ious iden ifica ions). This pa allel copy is hen disjoin om F, and
one can con ac he loop in a s anda d way. No e ha his a gumen
wo ks independen ly o he beha io o F.
Using aming heo ems based on localized undamen al g oup condi-
ions, and conside a ions like hose in he p e ious pa ag aph, one can
ge s onge esul s on ameness ha occu s om s abiliza ion han he
ones on p. 84–85 in [Da e2] men ioned abo e. Mo e p ecisely, ins ead
o needing kex a dimensions in some cases, i is enough o go om
Rn o Rn+1. Compa e wi h he bo om o p. 390 and he op o p. 391
in [Da e1], and he e e ences indica ed he e. (Compa e also wi h he
ema ks on p. 452 o [Can1].)
8 S. Semmes
2. Con ac able open se s
Fix a posi i e in ege n.
I Uis a nonemp y con ac able open subse o Rn,is
Unecessa ily homeomo phic o he open uni ball in Rn?
(2.1)
Fo he eco d, o say ha Uis con ac able means ha he iden i y
mapping on Uis homo opic o a cons an , h ough (con inuous) map-
pings om Uin o i sel . In pa icula , he homo opy and homology
g oups o U(o posi i e dimension) would hen be i ial, jus as o an
n-dimensional ball.
When n= 1, he answe o he ques ion in (2.1) is “yes”. In his
case, Uis ei he he whole eal line, an open segmen in he eal line,
o an open ay. Each o hese is easily seen o be homeomo phic o he
in e al (−1,1), which is he uni ball in his case.
I n= 2, hen he answe o he ques ion in (2.1) is “yes” again. This
is a well-known ac , and we shall e u n o i la e , in Sec ion 8.
S a ing in dimension 3, he answe o he ques ion in (2.1) is “no”.
We shall say some hing abou examples o his in a momen , bu le us
fi s ask ou sel es he ollowing: how migh one be able o ell ha a
gi en con ac able open se in Rnis no homeomo phic o an n-dimen-
sional ball?
He e again a localized e sion o he undamen al g oup is impo an .
I n≥3, hen a necessa y condi ion o a se U o be homeomo phic
o an n-dimensional ball is ha Ube “simply connec ed a infini y”.
Roughly speaking, his means ha i one akes a closed loop γou nea
infini y in U, hen i should be possible o con ac γ o a poin , while
s aying ou nea infini y oo (al hough pe haps no as much as γi sel
is).
He e is a mo e o mal defini ion. Fo his we also include “connec -
edness a infini y” as a fi s pa .
Defini ion 2.2. Le Ube an open se in Rn, o a opological space mo e
gene ally. (No mally one migh a leas ask ha Ube locally compac .)
Uis connec ed a infini y i o each compac se K0⊆U he e is
a la ge compac se L0⊆Usuch ha e e y pai o poin s in U L0is
con ained in a connec ed se which is i sel con ained in U K0. (One can
define “a cwise connec edness a infini y” in a simila manne . The wo
no ions a e equi alen unde assump ions o local a cwise connec edness,
and o opological mani olds in pa icula .)
Some Topics Conce ning Homeomo phisms 9
Uis simply-connec ed a infini y i i is connec ed a infini y, and i
o e e y compac se K1⊆U he e is a la ge compac se L1⊆Uso
ha i γis an a bi a y closed loop in U L1(i.e., an a bi a y con inuous
mapping om he uni ci cle S1in o U L1), hen γis homo opic o a
cons an h ough con inuous mappings om he ci cle in o U K1.
I Uis he uni ball in Rn,n≥3, hen Uis simply-connec ed a
infini y. Indeed, le B(0, ) deno e he open ball in Rnwi h cen e 0 and
adius , and le B(0, ) deno e he co esponding closed ball. Then e e y
compac subse o B(0,1) is con ained in B(0, ) o some <1. Fo
each <1, B(0, ) is a compac subse o B(0,1), and B(0,1) B(0, )
is connec ed when n≥2, and simply-connec ed when n≥3. This
is because B(0,1) B(0, ) is homeomo phic o Sn−1×( , 1), and Sjis
connec ed when j≥1, and simply-connec ed when j≥2.
The p ope y o being simply-connec ed a infini y is clea ly p ese ed
by homeomo phisms. Thus o ge a con ac able open se Uin Rnwhich
is no homeomo phic o an n-dimensional ball, i suffices o choose Uso
ha i is no simply-connec ed a infini y.
I Uis con ac able, hen i is simply-connec ed i sel in pa icula . I
Uis simply-connec ed, connec ed a infini y, and no simply-connec ed
a infini y, hen i means ha he e is a compac se K⊆Uand loops
γin Uwhich lie as a owa ds infini y as one would like (i.e., in he
complemen o any gi en compac subse o U) such ha (a) γcan be
con ac ed o a poin in U, and (b) γcanno be con ac ed o a poin in
U K. To pu i ano he way, hese loops γcan be con ac ed o poin s in
U, bu in doing his one always has o pass h ough a leas one elemen
o he compac se K.
A mechanism o ha ing his happen o a se Ucon ained in R3is
gi en by he cons uc ion o “ he Whi ehead con inuum” [Whi ]. (See
also [Da e2], [Ki ].) He e is an ou line o he p ocedu e.
S a wi h a s anda d smoo h “ ound” solid o us Tin R3. He e T
should be a compac se , i.e., i should con ain i s bounda y.
Nex one chooses ano he smoo h solid o us T1inside T. Mo e p e-
cisely, T1should lie in he in e io o T. One chooses T1in a pa icula
way, which can be imagined as ollows. (Pic u es can be ound on p. 68
o [Da e2] and p. 82 o [Ki ].) Fi s ake a “small” solid o us in T,
small enough o be con ained in a opological ball in T. One can hink
o g abbing hold o his small solid o us a wo ends, and hen s e ch-
ing hem a ound he “hole” in he la ge o us T. One s e ches hem
a ound he wo diffe en sides o he hole in T. To ge T1, hese wo ends
should hook a ound each o he on he o he side o he hole. In o he
10 S. Semmes
wo ds, one migh imagine ha ing he wo ends o he small solid o us
om be o e, s e ched a ound opposi e sides o T, and hen passing one
ac oss he o he , un il hey do no ouch any mo e, bu a e clasped o-
ge he , like wo hooks, o wo links in a chain. The configu a ion looks
locally like wo hooks o links clasped oge he , bu in ac one has wo
ends o he single solid o us T1, w apped a ound he hole in T.
I T1is chosen in his way, hen i has he ollowing wo basic p op-
e ies. The fi s is ha i is homo opically i ial in T. Tha is, he
iden i y mapping on T1is homo opic o a cons an mapping h ough
(con inuous) mappings om T1in o T. This ollows exac ly he desc ip-
ion abo e; in making he homo opy, one is allowed o s e ch o mo e
T1a ound as much as one like, and one is allowed o ha e diffe en pa s
o (images o ) T1c oss each o he in T. To pu i a bi diffe en ly, he
mappings being de o med a e no equi ed o be injec i e.
The second p ope y is ha T1is no “iso opically i ial”. This
means ha one canno con inuously de o m T1 h ough an iso opy o T
in o a se which lies in a ball con ained in T. In effec , his means ha
one canno con inuously de o m T1inside Tin such a way ha T1ends
up in a ball in T, and so ha he de o ma ions do no e e c oss each
o he (unlike he homo opy in he p e ious pa ag aph). I one could ge
T1inside a ball in T, hen one could con inue he de o ma ion o ge an
iso opy in o an a bi a ily small ball. One would no ask o sh inking
T1 o a poin he e, because his is au oma ically p e en ed by injec i i y
(independen o clasping o no ).
This explains how Tand T1should be chosen. Since T1is a 3-dimen-
sional smoo h solid o us in i s own igh , one can epea he p ocess o
ge ano he smoo h solid o us T2con ained in i , and in ac con ained
in he in e io o T1. In o he wo ds, since Tand T1a e bo h smoo h
solid o i, hey a e diffeomo phic o each o he in pa icula , and his can
be used o make p ecise he idea o “ epea ing he p ocess”. Specifically,
i φ:T→T1is such a diffeomo phism, hen one can ake T2 o be φ(T1).
One hen epea s he p ocess indefini ely, ge ing smoo h solid o i Tj
o j=1,2,... such ha Tj+1 is con ained in he in e io o Tj o each
j, and so ha Tj+1 is a anged in Tjin he same way as T1is a anged
in T.
Now le Wbe he in e sec ion o all hese solid o i Tj. This gi es a
nonemp y compac se in R3. We can hink o Was lying inside o S3,
and hen ake U=S3 W. One can also o a e his a ound so ha U
ac ually lies in R3. One can show ha Uis con ac able, bu no simply-
connec ed a infini y. See [Da e2], [Ki ], [Whi ] o mo e in o ma ion.
(Fo he pu poses o looking a U, he complemen o W, i can be
Some Topics Conce ning Homeomo phisms 11
con enien o use a modes ly diffe en desc ip ion o he cons uc ion, in
which one builds Uup om smalle pieces in an “inc easing” manne ,
analogous o he “dec easing” cons uc ion o Wabo e.)
Al hough Uis no homeomo phic o a 3-dimensional ball in his case,
he Ca esian p oduc o Uwi h a nonemp y open in e al is homeo-
mo phic o a 4-dimensional ball. This is a ibu ed o A nold Shapi o
in [Bin3]; see also Sec ion 10 o [Bin4] and [Ki ]. This is analogous
o he effec o s abiliza ion be o e, in Sec ion 1. In pa icula , one can
check di ec ly ha aking he Ca esian p oduc wi h he in e al ge s
id o he p oblem ha Ui sel has wi h simple-connec i i y a infini y.
Beginning in dimension 4, he e a e con ac able open se s in Rn
which a e no opological n-balls, and which ha e he addi ional ea-
u e ha hei closu es a e compac mani olds wi h bounda y. This las
does no wo k in dimension 3, and, o ha ma e , he complemen o
he Whi ehead con inuum in S3canno be ealized as he in e io o a
compac mani old wi h bounda y, whe he o no his compac mani old
should occu as he closu e o he se in S3. The eason is ha i such
a compac mani old did exis , i s bounda y would be a 2-dimensional
su ace wi h he homology o he 2-sphe e. We shall say mo e abou his
in a momen . In his case he bounda y would ha e o be homeomo phic
o he 2-sphe e. This would con adic he ailu e o simple-connec i i y
a infini y o he o iginal space, since S2is simply-connec ed.
The diffe ence wi h n≥4 is ha he bounda y can be a homology
(n−1)-sphe e (i.e., a mani old wi h he same homology as Sn−1) which
is no simply-connec ed. The in e io hen ails o be simply-connec ed
a infini y again, and is no homeomo phic o an n-ball in pa icula .
Fo some ela ed in o ma ion and e e ences conce ning hese exam-
ples in dimensions g ea e han o equal o 4, see [Da e2], including he
op o p. 94, and he discussion on p. 103–104.
2.1. Some posi i e esul s.
Fo dimensions n≥4, i is known ha e e y con ac able opologi-
cal mani old Mwhich is simply-connec ed a infini y is homeomo phic
o Rn. See [S a] o n≥5, and Co olla y 1.2 on p. 366 o [F e] o
n= 4. A ela ed e e ence is [McMZ]. Ac ually, [S a] is s a ed o
he piecewise-linea ca ego y; one can go om he e o he opologi-
cal ca ego y ia [Ki S]. The ou -dimensional esul does no wo k in
he smoo h o piecewise-linea ca ego ies (which a e equi alen in di-
mension 4), because o he exis ence o “ ake R4’s” (smoo h mani olds
homeomo phic o R4, bu no diffeomo phic o i ). Conce ning he la -
e , see [F eQ] (p. 122 in pa icula ) and [Ki ] (Chap e XIV).
18 S. Semmes
As a special case, imagine now ha
Mis a fini e polyhed on o di-
mension n. Le Ldeno e he codimension-1 link o qin
M.ThusLis
an (n−1)-dimensional fini e polyhed on, and
Mlooks locally a qlike
a cone o e L.
In o de o
M o be an n-dimensional opological mani old in a
neighbo hood o q, he link Lshould be ai ly close o a s anda d (n−
1)-sphe e. In pa icula , i is no ha d o see ha Lshould be homo opy-
equi alen o Sn−1. This implies ha Lshould be connec ed and simply-
connec ed, unde ou assump ion ha nis a leas 3.
In ac , in he case whe e
Mis a fini e polyhed on, he connec edness
and simple-connec edness o he link La ound qa e equi alen o he
local connec i i y and simple-connec i i y condi ions o
M {q}nea q
indica ed abo e, wi h (3.3) and (3.4). This is no ha d o see, and i is
also a he nice. To pu i a bi diffe en ly, imagine ha one s a s wi h
he class o fini e polyhed a, and hen ies o go o mo e gene al con ex s
o opological spaces. The local connec i i y and simple-connec i i y
condi ions o
M {q}a qas desc ibed abo e p o ide a way o cap u e
he in o ma ion in he connec edness and simple-connec edness o he
codimension-1 link a qin he case whe e
Mis a polyhed on, in a man-
ne ha makes sense o a bi a y opological spaces, wi hou special
s uc u e as one has o fini e polyhed a.
4. Decomposi ion spaces, 1
Le nbe a posi i e in ege , and le Kbe a nonemp y compac subse
o Rn. One could also conside gene al mani olds ins ead o Rnhe e,
bu we shall gene ally s ick o Euclidean spaces o simplici y. The main
ideas come up in his case anyway.
Imagine sh inking K o a single poin , while lea ing he es o Rn
alone, and looking a he opological space ha esul s. This can be
defined mo e o mally as ollows. Le us w i e Rn/K o he se which
consis s o he poin s in Rnwhich do no lie in K, oge he wi h a single
poin which co esponds o Ki sel . In o he wo ds, his is whe e we
sh ink K o a single poin . This se can be gi en a opology in a s anda d
way, so ha a subse Uo Rn/K is open i and only i i s in e se image
back in Rnis open. He e “in e se image” uses he au oma ic quo ien
mapping Rn o Rn/K. (In conc e e e ms, he in e se image o Uin
Rnmeans he se o poin s in Rnwhich co espond o elemen s o U,
whe e one includes all poin s in Ki he elemen o Rn/K associa ed o
Klies in U.)
Some Topics Conce ning Homeomo phisms 19
This ype o quo ien Rn/K is a special case o a “decomposi ion
space”. We shall discuss he gene al si ua ion u he in Sec ion 6, bu
his special case al eady includes a lo o in e es ing examples and phe-
nomena.
Now le us conside he ollowing ques ion:
Gi en Kas abo e, when is Rn/K a opological mani old?(4.1)
This is eally a special case o he si ua ion in Sec ion 3. Fo his i
is be e o use Snins ead o Rn, so ha Sn/K —defined in he same
manne as abo e— is equi alen o he one-poin compac ifica ion o
Sn K.
Le us conside some basic examples. I Kconsis s o only a single
poin , hen Rn/K is au oma ically he same as Rni sel , and he e is
no hing o do. I Kis a fini e se wi h mo e han one elemen , hen i
is easy o see ha Rn/K is no a mani old. I we le qdeno e he poin
in Rn/K which co esponds o K, hen (Rn/K) {q}does no enjoy he
local connec edness p ope y ha i should i Rn/K we e a mani old a q,
as in (3.3) in Sec ion 3. Mo e p ecisely, his local connec edness p ope y
o he complemen o {q}would be necessa y only when n≥2. When
n= 1, one does no ha e o ha e his local connec edness condi ion,
bu hen (Rn/K) {q}would ha e oo many local componen s nea q o
Rn/K o be a mani old a q. (Tha is, he e would be mo e han 2 such
local componen s.)
Now suppose ha Kis a s aigh line segmen in Rn. In his e en ,
Rn Kis homeomo phic o Rnagain. This is no ha d o check. This
would also wo k i Kwe e a s anda d ec angula cell o highe dimension
in Rn.
Mo e gene ally, his wo ks i Kis a ame cell in Rn, meaning he
image o a s anda d ec angula cell unde a homeomo phism o Rnon o
i sel . This ollows au oma ically om he case o s anda d ec angula
cells.
Howe e , i one me ely assumes ha Kis homeomo phic o a s anda d
ec angula cell, hen i is no necessa ily ue ha Rn/K is a mani old!
This is ano he aspec o wild embeddings, om Sec ion 1. We shall say
mo e abou his as his sec ion goes on. A conc e e example is gi en
by aking K o be a copy o he Fox-A in wild a c in R3. (Compa e
wi h [Fe ].)
No e ha we a e no saying ha Rn/K is always no a mani old
when Kis wildly embedded. The con e se is ue, ha Kmus be
wildly embedded when Rn/K is no a mani old (and Kis a opological
20 S. Semmes
cell). This is jus a eph asal o he ema k abo e, ha Rn/K is a
mani old when Kis a amely embedded cell.
He e is a sligh ly mo e oolish example, which one migh iew as a
gene aliza ion o he ea lie commen s abou he case whe e Kis a fini e
se wi h mo e han a single poin . Imagine now ha Kis a copy o he
j-dimensional sphe e Sj,1≤j≤n−1. Fo his le us use a s anda d,
smoo h, ound sphe e; i is no a ma e o wildness ha we wan o
conside .
In his case Rn/K is ne e a opological mani old. I j=n−1, hen
Rn/K is homeomo phic o he union o Rnand an n-sphe e, wi h he
wo mee ing a a single poin . This poin is he one ha co esponds o
Kin Rn/K. Le us deno e his poin by qagain, as abo e. In his case
(Rn/K) {q}does no ha e he igh local-connec edness p ope y a q
in o de o Rn/K o be a mani old, as in (3.3) in Sec ion 3.
I j=n−2, hen one uns in o ouble wi h local simple-connec i i y
o (Rn/K) {q}a q, as in (3.4) in Sec ion 3. Fo his one migh hink
abou he special case whe e n= 3, so ha Kis a s anda d ci cle in
R3. I is easy o ake small loops in R3 K, lying close o K, which
a e none heless linked wi h K. These loops hen p ojec down in o
(R3/K) {q}, whe e hey can be as close o he poin qas one likes,
bu hey a e ne e con ac able in (R3/K) {q}a all, le alone in small
neighbo hoods o q(as in (3.4) in Sec ion 3). This is he same as saying
ha hese loops a e no con ac able inside o Rn K, which is equi alen
o (Rn/K) {q}.
When j<n−2, hen one has simila obs uc ions o Rn/K being a
mani old, bu in e ms o he ailu e o highe -dimensional o ms o local
connec edness o (Rn/K) {q}(using homology o homo opy). This is
analogous o he cases al eady desc ibed, when j=n−1o n−2. We
shall say mo e abou his soon, bu o he momen le us go on o some
o he ma e s.
Fo his example, whe e Kis aken o be a s anda d j-dimensional
sphe e, no e ha Rn/K i sel is locally con ac able a q. This is as
opposed o connec edness p ope ies o (Rn/K) {q}, and i is analogous
o wha happens in he case o fini e polyhed a. Specifically, o fini e
polyhed a one always has local con ac abili y, bu he beha io nea
a gi en poin o punc u ed neighbo hoods a ound ha poin is ano he
ma e . The la e is connec ed o he beha io o he codimension-1
link o he polyhed on a ound he gi en poin , as in Sec ion 3.
In he p esen case, whe e we ha e Rn/K wi h Ka s anda d ound
j-dimensional sphe e, one can see he local con ac abili y o Rn/K a
Some Topics Conce ning Homeomo phisms 21
he poin q(co esponding o K) as ollows. In Rn, one can ake a ubu-
la neighbo hood o K, which is homeomo phic o he Ca esian p oduc
o he j-sphe e Kand an (n−j)-dimensional ball. This neighbo hood
can be con ac ed on o Kin a simple way, and his leads o he local
con ac abili y o Rn/K a q.
Now le us conside he case o he Whi ehead con inuum, om Sec-
ion 2. We should no eally say he Whi ehead con inuum he e, as he e
is some flexibili y in he cons uc ion, which can lead o he esul ing
se Wno being pinned down comple ely. This ambigui y will no eally
cause ouble o us he e, and we can wo k wi h any compac se Win
R3which is ob ained as in he p ocedu e desc ibed in Sec ion 2.
The se Whas he ea u e o being cell-like, as in he ollowing defi-
ni ion.
Defini ion 4.2 (Cell-like se s).A compac se Kin Rnis said o be
cell-like i Kcan be con ac ed o a poin inside o any neighbo hood U
o i sel in Rn.
Compa e wi h [Da e2], especially p. 120. Tha he Whi ehead con-
inuum Wis cell-like is no ha d o see om he cons uc ion o W,as
he in e sec ion o a dec easing sequence o solid o i wi h ce ain p op-
e ies. Specifically, o his he key poin is ha he & h solid o us can
be con ac ed o a poin inside he p e ious one.
I Kis a opological cell, hen Kis con ac able o a poin inside o
i sel , wi hou using he ex a bi o oom p o ided by a small neigh-
bo hood o i sel . This is also independen o he way ha Kmigh be
embedded in o some Rn, i.e., wildly o amely.
Fo W,i isno ue ha R3/W is a opological mani old. I we le q
deno e he poin in R3/W co esponding o W, hen (R3/W) {q}is no
locally simply-connec ed a q(in he sense o he condi ion in Sec ion 3,
a ound (3.4)). In conc e e e ms, his means ha he e a e loops in
R3 W(which is he same as (R3/W) {q}) which lie as close o Was
one likes (in hei en i e y), bu which canno be con ac ed o a poin
in R3 Wwhile emaining easonably close o W.
These loops can be desc ibed conc e ely, as me idians in he solid
o i whose in e sec ion gi es W. The loop om he solid o us Tjcan
be filled wi h a disk inside Tj, bu no wi hou c ossing he smalle
o us Tj+1, o any o i s successo s. This comes back o he way ha each
T+1 is “clasped” inside o T. See [Da e2], [Ki ] o mo e in o ma ion
(including P oposi ion 9 on p. 76 o [Da e2]).
In any e en , he ailu e o he local simple-connec i i y o (R3
/W) {q}
a qis equi alen o S3 Wno being simply-connec ed a infini y, as in
22 S. Semmes
Sec ion 2. This also ollows he discussion in Sec ion 3, and he commen
jus a e (4.1).
This case is qui e diffe en om he one o embedding ound sphe es
in Rn, as discussed be o e. Mo e p ecisely, le us compa e he si ua ion
wi h Wand he example be o e whe e Kis a s anda d ci cle inside o
R3. Fo he la e , he e a e loops in R3 Kwhich lie as close o K
as one wan s, and which a e no con ac able o a poin in R3 Ka
all, le alone in a neighbo hood o K.Fo W, one has ha S3 Wis
con ac able (as men ioned in Sec ion 2), and his implies ha R3 Wis
simply-connec ed. (This is a s aigh o wa d exe cise.) Thus hese loops
nea Wcan be con ac ed o a poin in R3 W, i one allows onesel o
go away om W o he con ac ion.
He e is ano he aspec o his. Al hough one has hese loops in R3 W
which lie nea Wbu canno be con ac ed o a poin in R3 Wwhile
s aying nea W, hese loops can be made homologically i ial in R3 W
while s aying nea W. Tha is, one can fill he loops wi h su aces
inside R3 Wwhile s aying close o W, i one allows he su aces o ha e
handles ( a he han simply being a disk, as in he case o homo opic
i iali y). This is some hing ha one can easily see om he pic u es
(as in [Da e2], [Ki ]). The basic idea is ha one can fill he loops wi h
disks, whe e he disks s ay close o W, bu also pass h ough W(and so
a e no in R3 W). Howe e , one can a oid he in e sec ion wi h Wby
cu ing ou a couple o small holes in he disk, and a aching a handle
o hem which goes along he bounda y o he solid o us in he nex
gene a ion o he cons uc ion. Then Wwill s ay inside his nex solid
o us, h oughou he es o he cons uc ion, and his su ace gi es a
way o filling he loop wi hou in e sec ing W(o being o ced o go a
away om i ).
This kind o filling by su aces does no wo k in he case whe e we
ake K o be a s anda d ci cle in R3. In his si ua ion, we ha e loops
in R3 Kwhich lie close o K, and which a e linked homologically wi h
he ci cle K. In o he wo ds, he linking numbe o he loop wi h Kis
nonze o, and his linking numbe is a homological in a ian which would
anish i he loop could be filled wi h a su ace wi hou in e sec ing K.
(Fo mo e abou “linking numbe s”, see [Bo T], [B e1], [Fla], [Spa].)
He e is ano he ea u e o W, which dis inguishes i om o dina y
ci cles in R3(o sphe es in Rnmo e gene ally). Le us hink o Wnow
as lying in R4 a he han R3, h ough he inclusion o R3in R4by
aking he ou h coo dina e o be 0.
Fo R4, we ha e ha R4/W is a opological mani old (homeomo phic
o R4). The basic poin behind his is he ollowing. In he ealiza ion
Some Topics Conce ning Homeomo phisms 23
o Was he in e sec ion o a dec easing sequence o solid o i in R3, he
& h solid o us was always “clasped” in he p e ious one (as in Sec ion 2,
and [Da e2], [Ki ]). In R4, he ex a dimension p o ides a lo o ex a
oom, in such a way ha his “clasping” is no eally p esen any mo e.
I Tis a solid o us which is embedded and clasped inside o ano he
solid o us Tin R3, one can “unclasp” Tin R4by li ing one end up,
b inging i a ound he hole in T, and lea ing he o he end alone. This
is a s anda d obse a ion, and i is analogous o he way ha kno s in
R3become unkno ed in R4.
In o he wo ds, his p ocedu e gi es a way o make a de o ma ion o
R4, in which he solid o us Tis mapped o a se o small diame e ,
while no mo ing poin s some dis ance away a all. By con as , back
in R3, i is no possible o make an iso opy which sh inks T o a se o
small diame e , while lea ing he poin s in he complemen o he la ge
solid o us fixed. This is exac ly because o he way ha Tis “clasped”
in T, so ha i canno be “unclasped” by an iso opy in T. When one
has he ex a dimension in R4, one can “undo” he clasping, by li ing
one end up and mo ing i a ound, as indica ed abo e.
Once one has his kind o “sh inking” in R4, one can use his o
show ha R4/W is homeomo phic o R4. One can do his di ec ly,
using sh inking homeomo phisms like his, and combina ions o hem,
o make a mapping om R4 o i sel which sh inks W o a poin while
emaining injec i e (and con inuous) e e ywhe e else. One pu s homeo-
mo phisms like his on op o each o he , and deepe and deepe in he
cons uc ion o W, un il Wi sel is sh unk all he way o a poin . The
a ious solid o i Tjin he cons uc ion, o which Wis he in e sec ion,
a e made smalle and smalle in his p ocess. The ick is o do his
wi hou sh inking e e y hing, so ha he mapping ha esul s emains
a homeomo phism on he complemen o W.
This idea o sh inking can be gi en a gene al o m, and is discussed
in de ail in [Da e2]. See also [Edw2], [Ki ].
By con as , le us conside he case o a ci cle Kin R3. I one
iews Kas a subse o R4in he same way, hen R4/K is s ill no
a opological mani old. This ollows om ou ea lie discussion abou
ci cles and sphe es o highe dimensions inside o Rnin gene al. One
also does no ge a mani old by eplacing R4wi h Rm o la ge m’s.
No ice, howe e , ha he e is a kind o “imp o emen ” ha occu s
in adding dimensions in his way. I Kis a ci cle in R3, and i qdeno es
he poin in R3/K which co esponds o K, hen (R3/K) {q}is no
locally simply-connec ed a q. Fo ha ma e , (R3/K) {q}∼
=R3 K
is no simply-connec ed a all. When one conside s Kas a subse o
24 S. Semmes
R4, and asks analogous ques ions o R4/K (o R4 K), hen he e is
no longe any ouble wi h simple-connec i i y. The basic unde lying
p oblem con inues, hough, in he o m o 2-dimensional connec i i y.
This is no ha d o see.
Simila ly, i one iews Kas a subse o Rn o la ge n, hen he
ouble wi h connec i i y in lowe dimensions goes away, bu (n−2)-di-
mensional connec i i y s ill does no wo k.
Wi h he Whi ehead con inuum we a e mo e o una e. The p ob-
lem wi h local simple-connec i i y goes away when we p oceed om R3
o R4, and difficul ies wi h highe -dimensional connec i i y do no hen
a ise in hei place. One should no be oo su p ised abou his, since
he Whi ehead con inuum is cell-like, while ci cles o sphe es o highe
dimension a e no a all cell-like. In o he wo ds, wi h ci cles o sphe es
(and hei complemen s in Rn), he e is some clea and simple non i -
ial opology a ound, while he Whi ehead con inuum is much close o
some hing like a s anda d cell, which causes less ouble.
4.1. Cellula i y, and he cellula i y c i e ion.
Now le us look a some gene al no ions and esul s, conce ning he
possibili y ha Rn/K be a opological mani old (and, in ac , homeo-
mo phic o Rn).
Defini ion 4.3 (Cellula i y).A compac se Kin Rn(o , mo e gen-
e ally, an n-dimensional opological mani old) is said o be cellula i
i can be ealized as he in e sec ion o a coun able amily o se s Bi,
whe e each Biis a opological n-cell (o , equi alen ly, homeomo phic o
he closed uni ball in Rn), and i each Bi+1 is con ained in he in e io
o he p eceding Bi.
Compa e wi h [Da e2], especially p. 35, [Edw2], and p. 44 o [Rus1].
Al e na i ely, a compac se Kis cellula i and only i any neighbo hood
o Kcon ains an open se which con ains Kand is homeomo phic o he
s anda d n-dimensional ball.
Theo em 4.4. Le Kbe a compac subse o Rn. Then Rn/K is a
opological mani old i and only i Kis cellula in Rn. In his case,
Rn/K is homeomo phic o Rn.
See Exe cise 7 on p. 41 o [Da e2] o he fi s asse ion, and P oposi-
ion 2 on p. 36 o [Da e2] o he second one. (Conce ning he la e , see
Sec ion 5 in [Da e2] oo. No e ha some o he no a ion in Exe cise 7
on p. 41 in [Da e2] is explained in he s a emen o P oposi ion 2 on
Some Topics Conce ning Homeomo phisms 25
p. 36 o [Da e2].) See also [Edw2], especially he heo em on p. 114,
and p. 44ff o [Rus1].
Fo he eco d, le us men ion he ollowing.
P oposi ion 4.5. Le Kbe a compac subse o Rn.I Kis cellula ,
hen Kis cell-like. Con e sely, i nis equal o 1o 2, hen Kis cellula
i i is cell-like.
The ac ha cellula i y implies cell-likeness ollows easily om he
defini ions. When n= 1, he con e se is e y simple, since connec edness
implies ha a se is an in e al, and hence cellula . In R2, he a gumen
uses special ea u es o plane opology. See Co olla y 4C on p. 122
o [Da e2].
In highe dimensions, cell-like se s need no be cellula . Examples
a e gi en by Whi ehead con inua, and some wild embeddings o cells.
Howe e , he e is an exac cha ac e iza ion o cellula se s among cell-like
se s, which is he ollowing. Basically, he poin is o include he same
kind o localized simple-connec i i y o Rn Ka ound Kas discussed
be o e.
Theo em 4.6. Le Kbe a compac se in Rn, wi h n≥3. Then Kis
cellula inside o Rni and only i (a) i is cell-like, and (b) o e e y
open neighbo hood Uo Kin Rn he e is ano he open neighbo hood Vo
Kso ha e e y con inuous mapping om S1in o V Kcan be con ac ed
o a poin inside o U K.
This cha ac e iza ion o cellula i y is s a ed in Theo em 5 on p. 145
o [Da e2]. This uses also he defini ion o he cellula i y c i e ion gi en
on p. 143 o [Da e2]. When n≥4, his esul wo ks o subse s o gen-
e al n-dimensional opological mani olds, and no jus Rn. When n=3,
he e is ouble wi h he gene al case o mani olds, ela ed o he 3-di-
mensional Poinca ´e conjec u e being unse led; i he cellula i y c i e ion
holds o gene al mani olds, hen he 3-dimensional Poinca ´e conjec u e
would ollow, as discussed on p. 145 o [Da e2]. See Theo em 1.11
on p. 373 o [F e] conce ning he 4-dimensional case, and [McM] and
Sec ion 4.8 o [Rus1] o dimensions 5 and highe .
Co olla y 4.7. Le Kbe a compac subse o Rn,n≥3.I Kis cell-
like in Rn, hen K×{0}is cellula in Rn+1.
See Co olla y 5A on p. 145 o [Da e2]. The main poin behind he
de i a ion o Co olla y 4.7 om Theo em 4.6 is ha by passing o a
Euclidean space o one highe dimension, po en ial ouble wi h local
simple-connec edness o he complemen o Kgoes away. This fi s wi h
basic examples, and he Whi ehead con inuum in pa icula .
26 S. Semmes
Le us no e he ollowing simple con e se o Co olla y 4.7.
Lemma 4.8. Suppose ha Kis a compac subse o Rn.I K×{0}is
cellula in Rn+1, hen Kis cell-like in Rn.
Indeed, i K×{0}is cellula in Rn+1, hen i is also cell-like in Rn+1,
as in P oposi ion 4.5. I is easy o check ha cell-likeness o K×{0}
in Rn+1 implies cell-likeness o Kinside Rn, jus by he defini ions.
(Thus cell-likeness, unlike cellula i y, is no made mo e easible by he
ex a oom o ex a dimensions.) This implies Lemma 4.8.
Fo conc e e examples o cell-like se s, o en he cellula i y in highe -
dimensional spaces, as in Co olla y 4.7, can be seen in ai ly di ec and
simple e ms. The oom om he ex a dimensions makes i easy o mo e
pieces o he se apa , wi hou he claspings, kno ings, e c., which
occu ed o iginally. Some aspec s o his came up ea lie , conce ning
Whi ehead con inua.
No e ha he localized simple-connec i i y condi ions ha a e used
he e a e a bi diffe en om hose employed in he con ex o aming
heo ems, as in Sec ion 1. To make his p ecise, le Kbe a compac
subse o some Rn. The condi ions ha come up in he p esen sec ion
in ol e he beha io o Rn K, localized a ound K( he whole o K).
Tha is, one looks a he beha io o Rn Kwi hin a bi a ily-small
neighbo hoods o Kin Rn. In he con ex o Sec ion 1, one would look
a he beha io o Rn Knea indi idual poin s in K.
To pu i ano he way, he e one seeks o con ac loops in Rn K
ha a e close o K o poin s, while s aying close o K. In he con ex
o Sec ion 1, one looks a small loops in Rn Knea K, and ies o
con ac hem o poin s in he complemen o Kwhile s aying in small
balls, and no jus s aying nea K.
5. Mani old ac o s
Le Wbe a Whi ehead con inuum, cons uc ed h ough a dec easing
sequence o solid o i in R3, as in Sec ion 2.
Theo em 5.1. I R3/W is defined as in Sec ion 4, hen (R3/W)×R
is homeomo phic o R4.
In pa icula , (R3/W)×Ris a opological mani old, e en hough
R3/W i sel is no . Thus R3/W is a mani old ac o .
The ac ha (R3/W)×Ris homeomo phic o R4is gi en as Co ol-
la y 3B on p. 84 o [Da e2]. See also [AndR], [Ki ].
No e ha he exis ence o a homeomo phism om (R3/W)×Ron o
R4is no he same as he obse a ion men ioned in Sec ion 4, ha
Some Topics Conce ning Homeomo phisms 27
R4/(W×{0}) is homeomo phic o R4. In conside ing (R3/W)×R, one
is in effec aking R4, and hen sh inking each copy W×{u}o W o a
poin , whe e u uns h ough all eal numbe s. Fo R4/(W×{0}), one
sh inks only a single copy o W o a poin .
Al hough he cons uc ion is mo e complica ed o (R3/W)×R han
o R4/(W×{0}), he e a e some common aspec s. As be o e, one o
he main poin s is ha he solid o i in R3which a e “clasped” (inside
o o he solid o i) become unclasped in R4. Wi h he ex a dimension
in R4, one can pick up one end o one o hese o i, b ing i a ound, and
hen lay i down again, so ha he clasping is undone. Fo he p esen
si ua ion wi h (R3/W)×R, one pe o ms his kind o ac ion o all o
he copies W×{u}o Wa once, u∈R, a he han jus a single copy.
(Compa e also wi h Sec ion 6, and he gene al no ion o decomposi ion
spaces men ioned he e.)
In Sec ion 2, i was men ioned ha S3 Wis a con ac able open
se which is no homeomo phic o a 3-ball (because i is no simply-
connec ed a infini y), and ha (S3 W)×Ris homeomo phic o a
4-dimensional open ball. (See [Bin3], [Bin4], [Ki ].) This esul is
simila in some ways o Theo em 5.1, bu he conclusions a e no qui e
he same ei he .
In his ein, le us make he ollowing obse a ion. As usual, deno e
by q he (singula ) poin in R3/W ha co esponds o W. Le us w i e L
o he subse o (R3/W)×Rgi en by {q}×R.ThusLis homeomo phic
o a line.
Using a homeomo phism om (R3/W)×R o R4, one ge s an em-
bedding o Lin o R4. I is no ha d o see ha any such embedding o
Lin o R4has o be wild. Jus as R3 Wis no locally simply-connec ed
nea W,i Ldeno es he image o Lin R4by an embedding as abo e,
hen R4 L is no locally simply-connec ed nea L. (No e ha R4 L is
homeomo phic o (R3 W)×R, by cons uc ion.) This ensu es ha L
is wild in R4, no ma e wha homeomo phism om (R3/W)×Ron o
R4one migh use, since o dina y s aigh lines in R4do no beha e in
his way.
One can also make local e sions o his a gumen , o show ha Lis
locally wild in he same manne .
I one we e o wan o pass om a homeomo phism om (R3/W)×R
on o R4in Theo em 5.1 o a homeomo phism om (S3 W)×Ron o R4,
hen in pa icula one could be lead o y o figu e ou some hing abou
wha happens when one dele es L om R4. Con e sely, i one wan ed o
go in he o he di ec ion, one migh ha e o figu e ou some hing abou
34 S. Semmes
o some hing like dila ions, ansla ions, o a ions, and eflec ions. In
o he wo ds, excep o a uni o m scale ac o , one migh hope ha he
geome y does no ha e o change.
No mally his will no be he case. Some amoun o bending o wis -
ing, e c., will (in gene al) be in ol ed, and needed, o accommoda e
he kind o opological beha io ha is p esen . This includes linking,
clasping, o hings like ha .
Fo he pu pose o choosing a geome y ha migh fi wi h a gi en
decomposi ion space, howe e , one can modi y he usual Euclidean me -
ic so ha he embeddings in ol ed in he basic “ ule” do ha e he kind
o beha io indica ed abo e, i.e., a cons an scale ac o oge he wi h an
isome y. The scale ac o s should be less ha 1, o eflec he sh inking
ha is supposed o ake place o he decomposi ion spaces (e en a a
pu ely opological le el).
I is no ha d o see ha one can make de o ma ions o geome y like
his. One can do his in a kind o di ec and “in insic” way, defining
me ics on Rnwi h sui able p ope ies. One can also do his h ough em-
beddings o he decomposi ion spaces in o highe -dimensional Euclidean
spaces. In hese highe -dimensional Euclidean spaces, he sel -simila i y
ha one wan s, in ypical si ua ions, can be ealized in e ms o s anda d
linea sel -simila i y, h ough dila ions and ansla ions.
Mo e p ecisely, in hese ci cums ances, he quo ien o Rnby he de-
composi ion can be ealized opologically as an n-dimensional subse X
o some RN(wi h N=n+ 1, o ins ance), in such a way ha X
is a smoo h submani old away om he na u al singula i ies, and Xis
sel -simila a ound hese singula i ies.
To build such a se X, one can s a wi h he complemen o he
o iginal domain Din Rn. One would iew Rn Das an n-dimensional
submani old o RN. In place o he i e a ion o he basic ule o he
decomposi ion om be o e, one now s acks some “basic building blocks”
in RNon op o Rn D(along he bounda y o D), and hen on op o
he o he building blocks, o e and o e again.
These basic building blocks a e gi en by n-dimensional smoo h mani-
olds in RN(wi h bounda y). They a e diffeomo phic o a single “model”
in Rn, which is he o iginal domain Din Rn, minus he in e io s o he
mcopies o Dembedded inside D, as gi en by he basic “ ule” ha
gene a es he decomposi ion. The building blocks a e all diffeomo phic
o each o he , since hey a e all diffeomo phic o his same model, bu
hey a e also cons uc ed in such a way as o be “simila ” o each o he .
Tha is, hey can all be gi en by ansla ions and dila ions o each o he .
Some Topics Conce ning Homeomo phisms 35
This is a key diffe ence be ween his cons uc ion and he o iginal de-
composi ion in Rn.
Fu he , he building blocks a e cons uc ed in such a way ha hei
ends a e all simila o each o he (i.e., e en diffe en ends on he same
building block). Specifically, he building blocks a e chosen so ha when
one goes o s ack hem on op o each o he , hei “ends” fi oge he
p ope ly, wi h smoo hness ac oss he in e aces.
These hings a e no difficul o a ange. Roughly speaking, one uses
he ex a dimensions in RN o s aigh en he “ends” in his way, so ha
he diffe en building blocks can be s acked p ope ly. Typically, his
would in ol e some hing like he ollowing. One s a s wi h he basic
model in Rn, gi en by Dminus he in e io s o he membedded copies
o Din D. One hen makes some ansla ions o he membedded sub-
domains in D, up in o he ex a dimension o dimensions in RN.Up
he e, hese subdomains can be mo ed o ben a ound, un il hey a e
simila o Di sel (i.e., being he same modulo ansla ions and dila-
ions). This can be done one a a ime, and wi hou changing any hing
nea he bounda y o he o iginal domain D. In his manne , he o igi-
nal model egion in Rnbecomes ealized as an n-dimensional compac
smoo h submani old (wi h bounda y) in RN, wi h he ends ma ching up
p ope ly unde simila i ies.
To pu i ano he way, he main “ ade-off” he e is ha one gi es
up he “fla ness” o he o iginal model, as a egion in Rn, o ge basic
building blocks in RN ha a e n-dimensional cu ed submani olds whose
ends a e simila o each o he . The cu ing o he in e io s o hese
building blocks compensa es o he s aigh ening o hei ends.
As abo e, one hen s acks hese building blocks on op o each o he ,
one a e ano he , o ge a ealiza ion o he decomposi ion space by an
n-dimensional subse Xo RN. (One also pu s in some limi ing poin s,
a he ends o he owe s o he building blocks ha a ise. In o he
wo ds, his makes Xbe a closed subse o RN. These ex a poin s a e
he singula i ies o X.) This subse is smoo h away om he singula -
i ies, and sel -simila a he singula i ies, because o he co esponding
p ope ies o he basic building blocks.
By choosing he scale- ac o s associa ed o he ends o he basic build-
ing blocks o be less han 1, he diame e s o he ends end o 0 (and in
a good way) as one s acks he building blocks on op o each o he many
imes. This co esponds o he ac ha he se s in he decomposi ion
a e supposed o be sh unk o single poin s in he quo ien space. This is
also pa o he s o y o he “limi ing poin s” in he p e ious pa ag aph.
The limi ing poin s a e exac ly he ones associa ed (in he end) o he
36 S. Semmes
nondegene a e se s in he o iginal decomposi ion in Rn, which a e being
sh unk o single poin s.
The ac ual homeomo phic equi alence be ween he se Xin RNp o-
duced h ough his me hod and he decomposi ion space Rn/G wi h
which one s a s is ob ained using he diffeomo phic equi alence be ween
he building blocks in RNand he o iginal model in Rn(Dminus he
in e io s o he membedded copies o i sel , as abo e). In ough e ms,
a he le el o he opology, he same kind o cons uc ion is occu ing in
bo h places, Xand he decomposi ion space, and one can ma ch hem
up, by ma ching up he indi idual building blocks. This is no ha d o
ack.
Ins ead o s acking building blocks on op o each o he infini ely
many imes, one can s op a e fini ely many s eps o he cons uc ion
(and add in sui able plugs o fill in he holes). This gi es a se which
is s ill smoo h, and diffeomo phic o Rn, and which app oxima es he
non-smoo h e sion ha ep esen s he decomposi ion space.
When one makes cons uc ions like hese —ei he fini e app oxima-
ions o infini e limi s— he sel -simila i y helps o ensu e ha he spaces
beha e geome ically abou as well as hey could. See [Sem3] o mo e
in o ma ion, and some sligh ly diffe en e sions o hese basic hemes.
7.2. Quo ien spaces can be opologically s anda d, bu geo-
me ically icky.
We ha e seen be o e how decomposi ions o Rnmigh lead o Rnagain
opologically in he quo ien , bu do so in a manne ha is s ill somehow
non i ial. Fo ins ance, he decomposi ion migh a ise om a non i ial
mani old ac o , o lead o wild embeddings in he quo ien which seem
e y simple (like a s aigh line) a he le el o he decomposi ion. In
hese si ua ions, one can s ill ha e highly non i ial geome ies om
he p ocedu es desc ibed in Subsec ion 7.1, e en hough he unde lying
space is opologically equi alen o Rn.
As a special case, wild embeddings in he quo ien can ha e nice me ic
p ope ies in he kind o geome ic ealiza ions discussed he e, while he
same p ope ies would no be possible in Rnwi h he s anda d Euclidean
me ic.
He e is a conc e e ins ance o his. Le Wbe a Whi ehead con inuum
in R3, as in Sec ion 2. Conside he co esponding quo ien space R3/W,
as in Sec ion 4. One can ealize R3/W opologically as a subse o R4,
whe e his subse is smoo h away om he singula poin , and has a sim-
ple sel -simila i y a he singula poin , as in Subsec ion 7.1. Simila ly,
Some Topics Conce ning Homeomo phisms 37
one can hink o (R3/W)×Ras being gi en as a subse o R5, namely,
as he p oduc o he one in R4wi h R.
Le us w i e q o he singula poin in R3/W, i.e., he poin in he
quo ien which co esponds o W, and se L={q}×R. Wi h espec o
he embedding o (R3/W)×Rin o R5,Lhas Hausdo ff dimension 1,
and bounded subse s o i ha e fini e 1-dimensional Hausdo ff measu e.
Howe e , he image o Linside o R4unde a homeomo phism om
(R3/W)×Ron o R4will be wild. As in Sec ion 5, he image o Lin R4
unde such a homeomo phism has Hausdo ff dimension a leas 2, wi h
espec o he usual Euclidean me ic in R4. This uses Theo em 5.2.
This shows ha he geome y ha we ha e o (R3/W)×Rhas o be
subs an ially diffe en om he usual Euclidean geome y on R4,e en
hough he wo spaces a e opologically equi alen . Specifically, e en
hough he e a e homeomo phisms om (R3/W)×Ron o R4, no such
homeomo phism can be Lipschi z, o e en H¨olde con inuous o o de
la ge han 1/2. (Recall ha a mapping is Lipschi z i o each pai o
poin s in he domain, he dis ance be ween hei images is bounded by a
cons an imes he dis ance be ween he poin s hemsel es. A mapping
is H¨olde con inuous o o de αi he dis ance be ween he images o
wo poin s is bounded by a cons an imes he dis ance be ween he wo
o iginal poin s aised o he powe α. Fo his condi ion, i is o en
na u al o es ic one’s a en ion o pai s o poin s which a e no mo e
han dis ance 1 apa , o o poin s in bounded egions.)
Al hough (R3/W)×R—wi h he kind o geome y desc ibed abo e—
is qui e diffe en om R4wi h he usual Euclidean me ic, he e is a
s ong and nice ea u e ha i has, in common wi h R4. We shall call
his p ope y “uni o m local coo dina es”.
Since (R3/W)×Ris homeomo phic o R4, i has homeomo phic
local coo dina es om R4a e e y poin . “Uni o m local coo dina es”
asks o a s onge e sion o his, and is mo e quan i a i e. Specifically,
a ound each me ic ball Bin (R3/W)×R(wi h espec o he kind o
geome y ha we ha e), he e a e homeomo phic local coo dina es om
a s anda d Euclidean ball βo he same adius in R4, such ha
he image o βunde he coo dina e mapping
co e s he gi en ball Bin (R3/W)×R,
(7.1)
and
he modulus o con inui y o he coo dina e mapping and i s
in e se can be con olled, uni o mly o e all choices o me ic
balls Bin (R3/W)×R, and in a scale-in a ian manne .
(7.2)
38 S. Semmes
He e “modulus o con inui y” means a unc ion ω( ) so ha when
wo poin s in he domain (o a gi en mapping) a e a dis ance ≤ , hei
images a e a dis ance ≤ω( ). Also, would ange h ough posi i e
numbe s, and ω( ) would be nonnega i e and sa is y
lim
→0ω( )=0.(7.3)
This las cap u es he con inui y in ol ed, and, in ac , gi es uni o m
con inui y.
Fo a mapping om a compac me ic space o ano he me ic space,
con inui y au oma ically implies uni o m con inui y, and ha implies
he exis ence o some modulus o con inui y. This is no o say ha one
knows much abou he modulus o con inui y, a p io i. (One can always
choose i o be mono one, o ins ance, bu one canno in gene al say
how as i ends o 0 as →0.)
Conc e e examples o moduli o con inui y would include ω( )=C
o some cons an C, which co esponds o a mapping being Lipschi z
wi h cons an C,o ω( )=C
α,α>0, which co esponds o H¨olde
con inui y o o de α. One can ha e much slowe a es o anishing, such
as ω( ) = (log log log(1/ ))−1.
In ou case, wi h he p ope y o “uni o m local coo dina es”, we wan
o ha e a single modulus o con inui y ω( ) which wo ks simul aneously
o all o he local coo dina e mappings (and hei in e ses). Ac ually,
we do no look a moduli o con inui y o he mappings hemsel es,
bu eno malized e sions o hem. The eno maliza ions a e gi en by
di iding dis ances in he domain and ange by he (common) adius
o Band β. In his way, Band βa e iewed as hough hey ha e
adius 1, independen ly o wha he adius was o iginally. This gi es a
kind o uni o m basis o making compa isons be ween he beha io o
he indi idual local coo dina e mappings and hei moduli o con inui y.
Le us e u n now o he special case o (R3/W)×R, wi h he geom-
e y as be o e. In his case one can ge he condi ion o uni o m local co-
o dina es om he exis ence o opological coo dina es (wi hou uni o m
bounds), oge he wi h he sel -simila i y and smoo hness p ope ies o
he se . He e is an ou line o he a gumen . (A de ailed e sion o his,
o a modes ly diffe en si ua ion, is gi en in [Sem3]. Specifically, see
Theo em 6.3 on p. 241 in [Sem3]. No e ha he p ope y o uni o m lo-
cal coo dina es is called “Condi ion (∗∗)” in [Sem3], as in Defini ion 1.7
on p. 192 o [Sem3].)
Le Bbe a me ic ball in (R3/W)×R, o which one wan s o find
sui able coo dina es. Assume fi s ha Bdoes no ge oo close o he
singula line {q}×Rin (R3/W)×R, and in ac ha he adius o
Some Topics Conce ning Homeomo phisms 39
Bis easonably small compa ed o he dis ance om B o he singula
line. In his case (R3/W)×Ris p e y smoo h and fla in B, by con-
s uc ion ( h ough he me hod o Subsec ion 7.1). This pe mi s one o
ge local coo dina es a ound Bqui e easily, and wi h sui able uni o m
bounds o he moduli o con inui y o he coo dina e mappings and
hei in e ses. The bounds ha one ge s a e scale-in a ian , because o
he sel -simila i y in he geome ic cons uc ion ( om Subsec ion 7.1).
In ac , one can ha e Lipschi z bounds in his case, as well as s onge
o ms o smoo hness.
I he ball Bis easonably close o he singula line {q}×R, hen one
can educe o he case whe e i is ac ually cen e ed on {q}×R. Tha is,
one could eplace Bwi h a ball which is cen e ed on {q}×R, and which
is no oo much la ge . (The adius o he new ball would be bounded
by a cons an imes he adius o B.) This subs i u ion does no cause
ouble o he kind o bounds which a e sough he e.
Thus we suppose ha Bis cen e ed on he line {q}×R.Wemay
as well assume ha he cen e o Bis he poin (q,0). This is because
(R3/W)×Rand he geome y ha we ha e on i a e in a ian unde
ansla ions in he Rdi ec ion, so ha one can mo e he cen e o (q,0)
wi hou difficul y, i necessa y.
Using he sel -simila i y o (R3/W)×R, one can educe u he o
he case whe e he adius o Bis app oxima ely 1. Fo ha ma e , one
can educe o he case whe e i is equal o 1, by simply inc easing he
adius by a bounded ac o (which again does no cause p oblems o
he uni o m bounds ha a e being conside ed he e). (To be hones , i
one akes he geome y o R3/W o be fla ou side a compac se , as in
Subsec ion 7.1, hen his educ ion is no ully co e ed by sel -simila i y.
Tha is, one should handle la ge scales a bi diffe en ly. This can be
done, and a simila poin is discussed in [Sem3] in a sligh ly diffe en
si ua ion ( o examples based on “Bing doubling”).)
Once one makes hese educ ions, one ge s down o he case o he
single ball Bin (R3/W)×R, cen e ed a (q,0) and wi h adius 1. Fo his
single choice o scale and loca ion, one can use he ac ha (R3/W)×R
is homeomo phic o R4 o ge sui able local coo dina es.
Fo his single ball B, he e is no issue o “uni o mi y” in he moduli
o con inui y o he coo dina e mappings. One simply needs amodulus
o con inui y.
A key poin , howe e , is ha when one wo ks backwa ds in he e-
duc ions jus made, o go o a bi a y balls in (R3/W)×Rwhich a e
ela i ely close o he singula line {q}×R, one does ge local coo di-
na es wi h uni o m con ol on he (no malized) moduli o con inui y o
40 S. Semmes
he coo dina e mappings and hei in e ses. This is because o he way
ha he educ ions coope a e wi h he scaling and he geome y.
A any a e, his comple es he ou line o he a gumen o showing
ha (R3/W)×Rhas uni o m local coo dina es, in he sense desc ibed
be o e, and wi h he kind o geome y o (R3/W)×Ras in Subsec-
ion 7.1.
I is easy o see ha a me ic space which is bilipschi z equi alen
o some Rnhas uni o m local coo dina es ( ela i e o Rn a he han
R4, as abo e). (Recall ha wo spaces a e bilipschi z equi alen i he e
is a homeomo phism om one on o he o he which is bo h Lipschi z
and has Lipschi z in e se.) The equi ed local coo dina es can simply
be ob ained om es ic ions o he global bilipschi z pa ame e iza ion.
The con e se is no ue in gene al, i.e., a space can ha e uni o m local
coo dina es and no be bilipschi z equi alen o he co esponding Rn.
An example o his is gi en by (R3/W )×Rwi h he kind o geome y
ha we ha e been conside ing.
One can also ge much simple examples, by aking snowflake spaces.
Tha is, one can ake Rnequipped wi h he me ic |x−y|α, whe e |x−y|
is he usual me ic, and αis a posi i e eal numbe s ic ly less han 1.
I is no ha d o check ha his has he p ope y o uni o m local coo -
dina es. I is no bilipschi z equi alen o Rn, because i has Hausdo ff
dimension n/α ins ead o Hausdo ff dimension n. The p e ious example
based on (R3/W)×Rbeha es much be e han his, hough, wi h he
co ec Hausdo ff dimension (namely, 4) in pa icula .
No ice ha he uni o m local coo dina es p ope y would imply a
bilipschi z condi ion i he local coo dina es all came om es ic ions
o a single global pa ame e iza ion. This is because o he way ha he
scaling wo ks. In gene al, he uni o m local coo dina es p ope y allows
he local coo dina e mappings o change as one changes loca ions and
scales, and his is why i allows o he e o be no global bilipschi z
pa ame e iza ion. The case o (R3/W)×Rp o ides a good example o
his.
Ano he way o hink abou he uni o mi y o e all loca ions and
scales in he uni o m local coo dina es p ope y is ha i is a condi ion
which implies he exis ence o homeomo phic coo dina es e en a e one
“blows up” he space (in he Hausdo ff o G omo -Hausdo ff senses)
along any sequence o loca ions and scales in he space. Wi h he uni-
o m local coo dina es p ope y, he local coo dina es could be “blown
up” along wi h he space, wi h he uni o m bounds o he moduli o
con inui y p o iding he equicon inui y needed o ake limi s o he co-
o dina e mappings (a e passing o sui able subsequences).
Some Topics Conce ning Homeomo phisms 41
Ins ead o looking a uni o m local coo dina es in connec ion wi h
bilipschi z equi alence wi h Rn, one can conside quasisymme ic equi a-
lence. Roughly speaking, a quasisymme ic mapping be ween wo me ic
spaces is one ha app oxima ely p ese es ela i e dis ances, in he same
way ha bilipschi z mappings app oxima ely p ese e ac ual dis ances.
In o he wo ds, i one has h ee poin s x,y, and zin he domain o such
a mapping, and i xis much close o y han zis, hen his should also be
ue o hei images unde a quasisymme ic mapping, e en i he ac ual
dis ances be ween he poin s migh be changed a lo . See [TukV] o
mo e in o ma ion abou quasisymme ic mappings. Two me ic spaces
a e quasisymme ically equi alen i he e is a quasisymme ic mapping
om one on o he o he . As wi h bilipschi z mappings, composi ions
and in e ses o quasisymme ic mappings emain quasisymme ic.
I a me ic space admi s a quasisymme ic pa ame e iza ion om Rn,
hen i also sa isfies he condi ion o uni o m local coo dina es. This is
ue o nea ly he same eason as o bilipschi z mappings; gi en a quasi-
symme ic mapping om Rnon o he me ic space, one can ge sui able
local coo dina es o he space om es ic ions o he global mapping
o indi idual balls. The e is a diffe ence be ween his case and ha o
bilipschi z mappings, which is ha one should allow some ex a escal-
ings o compensa e o he ac ha dis ances a e no app oxima ely
p ese ed. Specifically, a ball Bin he me ic space may be co e ed in
a nice way by he image unde a quasisymme ic mapping o a ball β
in Rn, bu he adii o Band βneed no ma ch up. Fo a bilipschi z
mapping, one would be able o choose βso ha i has adius which is
compa able o ha o B. In he quasisymme ic case, one may no ha e
ha , bu i one adds an ex a escaling on Rn(depending on he choices
o balls), hen one can s ill ge local coo dina es wi h he kind o uni o m
con ol on he (no malized) moduli o con inui y as in he uni o m local
coo dina es condi ion.
I a me ic space has uni o m local coo dina es wi h espec o Rn,
and i hese coo dina es come om es ic ions and hen escalings (on
Rn) o a single global pa ame e iza ion, as in he p eceding pa ag aph,
hen ha pa ame e iza ion does ha e o be quasisymme ic. This is an
easy consequence o he defini ions, and i is analogous o wha happens
in he bilipschi z case.
I a me ic space admi s uni o m local coo dina es om some Rn,i
s ill may no be ue ha i admi s a quasisymme ic pa ame e iza ion.
This is ickie han be o e, and in pa icula one does no ge examples
42 S. Semmes
simply by using snowflake me ics |x−y|αon Rn. Indeed, he iden-
i y mapping on Rnis quasisymme ic as a mapping om Rnwi h he
s anda d me ic o Rnwi h he snowflake me ic |x−y|α,0<α<1.
Howe e , he e a e coun e examples, going back o esul s o Rickman
and V¨ais¨al¨a. Tha is, hese a e spaces which ha e uni o m local coo di-
na es (and a e e en somewha nice han ha ), bu which do no admi
quasisymme ic pa ame e iza ions. Basically, hese spaces a e Ca esian
p oduc s, whe e he indi idual ac o s can beha e nicely in hei own
igh , and whe e he combina ion mixes diffe en ypes o geome y. A
basic example (which was he o iginal one) is o ake a p oduc o a
snowflake wi h a s aigh line. Quasisymme ic mappings y o ea
diffe en di ec ions in a uni o m manne , and in he end his does no
wo k o pa ame e iza ions o hese examples. See Lemma 4 in [Tuk],
and also [V¨ai3] and [AleV].
These examples occu al eady in dimension 2. They do no beha e
well in e ms o measu e, hough. This is a basic pa o he s o y;
compa e wi h [AleV], [Tuk], [V¨ai3].
As usual, dimension 1 is special. The e a e posi i e esul s s a -
ing om mo e p imi i e condi ions, and good cha ac e iza ions o he
exis ence o quasisymme ic pa ame e iza ions, in ac . See Sec ion 4
o [TukV].
In dimension 2, he e a e posi i e esul s abou ha ing global quasi-
symme ic pa ame e iza ions o a gi en space, and wi h bounds, unde
addi ional assump ions o good beha io in e ms o 2-dimensional mea-
su e, which include ha ing Hausdo ff dimension 2. Ins ead o “uni o m
local coo dina es”, one can assume a p io i weake condi ions abou he
geome y and opology. See [Da iS1], [HeiKo1], [Sem1]. Fo hese
esul s, i is impo an ha he dimension be 2, and no la ge , because
o he way ha hey ely on he exis ence o con o mal mappings. We
shall say a bi mo e abou his in Sec ion 8.
In dimension 3, he e a e coun e examples, e en in he con ex o good
beha io in e ms o measu e. These examples a e based on decompo-
si ions o R3, using geome ic ealiza ions as in Subsec ion 7.1. This is
discussed in [Sem3]. The absence o a quasisymme ic pa ame e iza ion
in his case is close o a esul in [F eS], al hough he se ing in [F eS]
is diffe en .
Conce ning he space (R3/W)×Rconside ed be o e, equipped wi h a
nice geome y as om Subsec ion 7.1, i is no clea ( o my knowledge)
whe he quasisymme ic pa ame e iza ions om R4exis o no . We
saw ea lie ha bilipschi z mappings do no exis , because o he line in
(R3/W)×Rwhich has o ha e Hausdo ff dimension a leas 2 a e any
Some Topics Conce ning Homeomo phisms 43
homeomo phism om (R3/W)×Ron o R4. These conside a ions o
Hausdo ff dimension o measu e do no by hemsel es ule ou he exis-
ence o a quasisymme ic mapping, as hey do o bilipschi z mappings.
(Compa e wi h [V¨ai2], o ins ance.)
Simila ema ks apply o double-suspensions o homology sphe es. In
pa icula , i is no known ( o my knowledge) whe he o no quasisym-
me ic pa ame e iza ions exis o hem.
7.3. Examples ha a e e en simple opologically, bu s ill non-
i ial geome ically.
Le us men ion ano he class o examples, which one can also hink
o in e ms o decomposi ions (al hough hey a e “ i ial” in his e-
spec ). These examples a e based on “An oine’s necklaces”, which came
up be o e, in Sec ion 1.
An oine’s necklaces a e compac subse s o R3which a e homeomo -
phic o he usual middle- hi ds Can o se in he eal line, bu o which
he e is no global homeomo phism om R3on o i sel which maps hese
se s in o subse s o a line. In dimension 2 his does no happen, as in
Chap e 13 o [Moi].
The “wildness” o hese se s is mani es ed in a simple undamen al-
g oup p ope y. Namely, he complemen o hese se s in R3ha e non-
i ial undamen al g oup, whe eas his would no be ue i he e we e a
global homeomo phism om R3 o i sel which would ake one o hese
se s o a subse o he line. This las uses he ac ha hese se s a e
o ally disconnec ed (i.e., o ha e simple-connec i i y o he complemen
in R3i he se we e o lie in a line).
An oine’s necklaces a e discussed in Chap e 18 o [Moi]. See also
p. 71ff in [Da e2]. The basic cons uc ion o hem can be desc ibed in
e ms o he same kind o “ ules” as in Subsec ion 7.1.
One s a s wi h a solid o us Tin R3. Inside his o us one em-
beds some mo e o i, which a e disjoin , bu which o m a chain ha is
“linked” a ound he hole in he o iginal o us. (See Figu e 18.1 on p. 127
o [Moi], o Figu e 9.9 on p. 71 o [Da e2].) In each o hese smalle
o i, one can embed ano he collec ion o linking o i, in he same way
as o he fi s solid o us.
One can epea his indefini ely. In he limi , one ge s a Can o se ,
which is a necklace o An oine.
Ac ually, we should be a li le mo e p ecise he e. In saying ha we ge
a Can o se in he limi , we a e implici ly imagining ha he diame e s
o he solid o i a e going o 0 as one p oceeds h ough he gene a ions
o he cons uc ion. This is easy o a ange, i one uses enough o i in
50 S. Semmes
A con o mal de o ma ion o he s anda d me ic is defined by 1 eal-
alued unc ion o n eal a iables, i.e., o he con o mal ac o . A
gene al diffeomo phism in ndimensions is desc ibed by n eal- alued
unc ions o n a iables. Thus, allowing o gene al changes o a iables,
he me ics which a e con o mally-equi alen o he s anda d me ic a e
desc ibed by n+1 eal- alued unc ions o n eal a iables. When n=2,
his is equal o n(n+1)/2, bu o n>2 one has ha n(n+1)/2>n+1.
In ac , one knows ha in dimension 3 he e a e nume ous examples o
spaces which sa is y geome ic condi ions analogous o ones ha wo k in
dimension 2, bu which do no admi quasisymme ic pa ame e iza ions.
The e a e also diffe en le els o s uc u e which occu in dimension 3,
be ween basic geome ic p ope ies and ha ing quasisymme ic pa ame-
e iza ions, and which would come oge he in dimension 2. Pa s o
his a e e iewed o discussed in Sec ion 7, especially Subsec ion 7.2;
see [Sem3] o mo e in o ma ion.
Thus, no only does he me hod based on con o mal and quasicon-
o mal mappings no wo k in highe dimensions, bu some o he basic
esul s ha one migh hope o ge o expec simply a e no ue, by
examples which a e p e y conc e e.
This is all p e y nea ! One has kinds o “pa allel acks”, wi h geo-
me ic opology on one side, and aspec s o geome y and analysis on
he o he . A p io i, hese wo acks can exis independen ly, e en i
he e a e ways in which each can be in ol ed in he o he .
Each o hese wo acks has special ea u es in low dimensions. This
conce ns he exis ence o homeomo phisms wi h ce ain p ope ies, o
ins ance. Each has s a emen s and esul s and machine y which make
sense o he gi en ack, and no o he o he side, e en i he e a e
also some o e laps (as wi h applica ions o Riemann mappings).
Each o hese wo acks also s a s unning in o ouble in highe
dimensions, and a abou he same ime! The kinds o ouble ha hey
encoun e can be a he diffe en a p io i, e en i he e is again significan
o e lap be ween hem.
8.1. Special coo dina es ha one migh conside in o he di-
mensions.
Le us now b iefly conside a couple o hings ha one migh y in
highe dimensions on he side o geome y and analysis, in simila eins
as abo e.
One basic app oach would be o y o find and use mappings which
minimize some kind o “ene gy”. As be o e, one can conside smoo h
me ics on smoo h mani olds (like Sn), and y o ge pa ame e iza ions
Some Topics Conce ning Homeomo phisms 51
wi h uni o m bounds on hei beha io , unde modes condi ions on he
geome y o he spaces. (One can also y o wo k di ec ly wi h spaces
and me ics ha a e no smoo h.)
A e y s anda d ene gy unc ional o conside would be he L2no m o
he diffe en ial, as wi h ha monic mappings. In dimension 2, con o mal
mappings can be placed in his amewo k. One can also conside ene gy
unc ionals based on Lpno ms o diffe en ials o mappings. This is mo e
complica ed in e ms o he diffe en ial equa ions ha come up, bu i
can ha e o he ad an ages. The choice o pas he dimension nhas some
pa icula ly nice ea u es, o ha ing he ene gy unc ional coope a e
wi h he geome y (and analysis). (This is one o he ways ha n=2is
special; o his one can ha e bo h p=nand p= 2 a he same ime!)
In pa icula , he ene gy becomes in a ian unde con o mal changes in
he me ic when pis equal o he dimension.
In elas ici y heo y, one conside s mo e elabo a e ene gy unc ionals
as well. Fo ins ance, hese migh include in eg al no ms o he in e se o
he Jacobian o he mapping, in addi ion o Lpno ms o he diffe en ials.
In o he wo ds, he unc ional can y o limi bo h he way ha he
diffe en ial becomes la ge and small, so ha i akes in o accoun bo h
s e chings and comp essions.
In any case, al hough he e is a lo o wo k conce ning exis ence and
beha io o minimize s o unc ionals like hese, I do no eally know
o esul s in dimensions n≥3 whe e hey can be used o ob ain well-
beha ed pa ame e iza ions o spaces, wi h bounds, unde modes o gen-
e al geome ic condi ions. This is especially ue in compa ison wi h
wha one can ge in dimension 2, as discussed be o e.
In dimension 3, he e is ano he kind o special s uc u e ha one
migh conside . Namely, ins ead o me ics which a e con o mal de o -
ma ions o he s anda d Euclidean me ic, le us conside me ics g=gi,j
o which only he diagonal en ies gi,i a e nonze o.
In his case he diagonal en ies a e allowed o a y independen ly.
Fo con o mal de o ma ions o he s anda d Euclidean me ic, he off-
diagonal en ies a e ze o, and he diagonal en ies a e all equal.
In dimension 3, he p oblem o making a change o a iables o pu
a gi en me ic in o diagonal o m like his is “de e mined”, in he same
way as o con o mal de o ma ions o Euclidean me ics in dimension 2.
Specifically, one can compu e as ollows. A gene al Riemannian me ic is
desc ibed by n(n+1)/2 eal- alued unc ions o n a iables, which means
6 eal- alued unc ions o 3 eal a iables in dimension 3. Me ics wi h
only diagonal nonze o en ies a e defined by 3 eal- alued unc ions o
52 S. Semmes
3- eal a iables, and changes o a iables a e gi en by 3 eal- alued unc-
ions o 3 eal a iables as well. Thus, allowing o changes o a iables,
he me ics ha can be educed o diagonal me ics can be desc ibed
by 6 eal- alued unc ions o 3 eal- a iables, which is he same as o
he o al amily o Riemannian me ics in his dimension. In dimensions
g ea e han o equal o 4, his would no wo k, and he e would again be
oo many Riemannian me ics in gene al compa ed o diagonal me ics
and ways o educing o hem ia changes o a iables.
O cou se his is jus an in o mal “dimension” coun , and no a jus i-
fica ion o being able o pu me ics in o diagonal o m in dimension 3.
(One should also be ca e ul ha he e is no significan o e lap be ween
changes o a iables and diagonal me ics, i.e., so ha he e was no
“o e coun ing” o he combina ion o hem.) Howe e , i does u n ou
ha one can pu me ics in diagonal o m (in dimension 3), a leas lo-
cally. This was es ablished in [DeTY] in he case o smoo h me ics.
The e we e ea lie esul s in he eal-analy ic ca ego y. (See [DeTY] o
mo e in o ma ion.)
Howe e , his ype o “no mal o m” does no seem o be as use ul
o he p esen ype o issue as con o mal pa ame e iza ions a e. As in
he case o con o mal coo dina es, pa o he p oblem is ha e en i
one has such a no mal o m, one does no a p io i know any hing abou
he beha io o he diagonal en ies o he me ic in his no mal o m.
One would need me hods o ge ing es ima es wi hou his in o ma ion,
and only he na u e o he no mal o m. In he con ex o con o mal
mappings, one has ex emal leng hs, con o mal capaci ies, and o he
con o mal and quasicon o mal in a ian s and quasi-in a ian s. Fo diag-
onal me ics, i is no clea wha one migh do.
A ela ed poin is ha he analysis o he pa ial diffe en ial equa ions
which pe mi s one o pu smoo h Riemannian me ics in dimension 3 in o
diagonal o m is oughly “hype bolic”, in he same way ha he co e-
sponding diffe en ial equa ions o con o mal coo dina es in dimension 2
a e ellip ic. See [DeTY]. This is closely connec ed o he kind o s abil-
i y ha one has o con o mal mappings, and he possibili ies o ha ing
es ima es o hem unde mild o p imi i e geome ic condi ions.
In a way his is all “jus ai ”, and nicely so. Wi h diagonal me ics one
does ha e some hing analogous o con o mal coo dina es in dimension 3.
On he o he hand, his analogue beha es diffe en ly in undamen al
ways, including es ima es. This is compa ible wi h o he aspec s o he
s o y as a whole, like he opological and geome ic examples ha one
has in dimension 3 (whe e homeomo phisms may no exis , wi h he
p ope ies ha one migh o he wise hope o ).
Some Topics Conce ning Homeomo phisms 53
In any e en , his illus a es how analy ic and geome ic me hods
seem o beha e a he diffe en ly in dimensions 3 and highe , compa ed
o he special s uc u e and phenomena which occu in dimension 2.
This is somewha ema kable in analogy wi h opological phenomena,
which ha e simila diffe ences be ween dimensions. Wi h he opology
he e a e bo h some c ossings and o e laps wi h geome y and analysis,
and much ha is sepa a e o independen .
On he side o geome y and analysis, le us also no e ha he e a e
some o he special ea u es in low dimensions ha we ha e no men-
ioned. As a basic example, he la ge amoun o flexibili y ha one has
in making con o mal mappings in dimension 2 leads o some possibili ies
in dimension 3 ha a e no a ailable in highe dimensions. Tha is, he
la ge eedom ha one has in dimension 2 can some imes pe mi one o
make mo e limi ed cons uc ions in dimension 3, e.g., by s a ing wi h
submani olds o dimension 2, and wo king om he e (wi h ex ensions,
gluings, e c.) These possibili ies in dimension 3 can be much mo e e-
s ic ed han in dimension 2, bu ha ing hem a all can be significan ly
mo e han wha happens in highe dimensions.
Conce ning a ia ional p oblems, one migh also keep in mind he
app oaches o [Da iS2], [Da iS3] (and some ea lie ideas o Mo el and
Solimini [Mo eS]). Fo hese one does no necessa ily wo k di ec ly wi h
mappings o po en ial pa ame e iza ions o se s, and in pa icula one
may allow se s hemsel es o be a iables in he minimiza ion ( a he
han mappings be ween fixed spaces). This b oade ange can make i
easie o he minimiza ions o lead o use ul conclusions abou geome -
ic s uc u e and complexi y, unde na u al and modes condi ions. In
pa icula , one can ge subs an ial “pa ial pa ame e iza ions”, as wi h
uni o m ec ifiabili y condi ions.
These app oaches a e also nicely compa ible wi h he ouble ha
one knows can occu , ela ed o opology and homeomo phisms (and
in geome ically mode a e si ua ions, as in Subsec ions 7.2 and 7.3,
and [Sem3], [Sem4]).
Finally, while we ha e men ioned a lo abou he special phenomena
ha can occu in dimension 2, and wha happens in highe dimensions,
we should also no o ge abou dimension 1. This is e en mo e special
han dimension 2. This is a amilia heme in geome ic opology, o
he ways ha one can ecognize and pa ame e ize cu es. In geome y
and analysis, one can look o pa ame e iza ions wi h bounds, and hese
a e o en cons uc ible.
54 S. Semmes
A undamen al poin along hese lines is he abili y o make pa ame-
e iza ions by a cleng h, o cu es o locally fini e leng h. Mo e gene -
ally, one can use pa ame e iza ions adap ed o o he measu es ( a he
han leng h), when hey a e a ound.
A cleng h pa ame e iza ions p o ide a e y obus and use ul way
o ob aining pa ame e iza ions in dimension 1 wi h good beha io and
bounds. In dimension 1, simple condi ions in e ms o mass can o en
be immedia ely “in eg a ed” o ge well-beha ed pa ame e iza ions, in
ways ha a e no a ailable (o do no wo k nea ly as well) in highe
dimensions, e en in dimension 2.
To pu he ma e in mo e conc e e e ms, in dimension 1 one can
o en make pa ame e iza ions, o app oxima e pa ame e iza ions, sim-
ply by o de ing poin s in a good way. This does no wo k in highe
dimensions. Once one has he o de ing, one can egula ize he geome y
by pa ame e izing acco ding o a cleng h, o some o he measu e (as
app op ia e).
Fo ano he e sion o his, in connec ion wi h quasisymme ic map-
pings, see Sec ion 4 o [TukV].
In diffe en ial-geome ic language, one migh say ha dimension 1
is special o he way ha one can make isome ies be ween spaces,
h ough a cleng h pa ame e iza ions. This no longe wo ks in dimen-
sion 2, bu one has con o mal coo dina es he e. Nei he o hese a e
gene ally a ailable in highe dimensions. In highe dimensions one has
less special s uc u e o ge ing he exis ence in gene al o well-beha ed
pa ame e iza ions, and hen he kinds and anges o geome ic and opo-
logical phenomena which can exis open up in a la ge way.
9. Nonlinea simila i y: Ano he class o examples
A e y nice and conc e e si ua ion in which issues o exis ence and
beha io o homeomo phisms can come up is ha o “nonlinea simi-
la i y”. Specifically, i is possible o ha e linea mappings A,Bon Rn
which a e conjuga e o each o he by homeomo phisms om Rnon o
i sel —i.e., B=h◦A◦h−1, whe e his a homeomo phism o Rnon o
i sel — and which a e no conjuga e by linea mappings!
Examples o his we e gi en in [CapS1], [CapS2]. Fo ela ed ma -
e s, including o he examples, condi ions unde which one can deduce
linea equi alence, and mo e on he beha io o conjuga ing homeomo -
phisms when a linea equi alence does no exis , see [CapS3], [CapS4],
[CapS5], [CapS+], [CapS∗], [HamP1], [HamP2], [HsiP1], [HsiP2],
Some Topics Conce ning Homeomo phisms 55
[KuiR], [MadR1], [MadR2], [Mio], [Rha1], [Rha2], [Ro W],
[Wei1], [Wei2], [Wilk].
10. Doing p e y well wi h spaces which may no ha e
nice coo dina es
I one has a opological o me ic space (o wha e e ) which has nice
coo dina es, hen ha can be p e y good. Howe e , he e is a lo ha
one can do wi hou ha ing coo dina es.
Le us begin wi h some basic condi ions. Le Mbe a opological
space which is compac , Hausdo ff, and me izable. We shall assume
ha Mhas fini e opological dimension, in he sense o [Hu W]. In
hese ci cums ances, his is equi alen o saying ha Mis homeomo phic
o a subse o some Rn. (See [Hu W].)
Fo simplici y, le us assume ha Mis a compac subse o some Rn.
I is also con enien o ask ha Mbe locally con ac able. This means
ha o each poin p∈M, and each neighbo hood Uo pin M, he e is
a smalle neighbo hood W⊆Uo pin Msuch ha Wcan be con ac ed
o a poin in U.
As a class o examples, fini e polyhed a a e locally con ac able. Fi-
ni e polyhed a make a nice special case o conside h oughou his sec-
ion, and we shall e u n o i se e al imes.
Fo ano he class o examples, one has cell-like quo ien s o opological
mani olds (and o some locally con ac able spaces mo e gene ally), a
leas when he quo ien spaces ha e fini e opological dimension. See
Co olla y 12B on p. 129 o [Da e2].
As a gene al ac abou local con ac abili y, le us no e he ollowing.
P oposi ion 10.1. Le Mbe a compac subse o some Rn. Then Mis
locally con ac able i and only i he e is a se V⊆Rnwhich con ains M
in i s in e io , and a con inuous mapping :V→Mwhich is a e ac ,
i.e., (w)=w o all w∈V.
This is a ai ly s anda d obse a ion. The “i ” pa is an easy con-
sequence o he local con ac abili y o Rn( h ough linea mappings).
Specifically, o ge local con ac ions inside o M, one makes s anda d
linea con ac ions in Rn, which no mally do no s ay inside M, and
hen one applies he e ac ion o keep he con ac ions inside M.
Fo he con e se, one can begin by defining on a disc e e and
easonably- hick se o poin s ou side M, bu nea M. Fo a poin win
such a se , one could choose (w)∈Mso ha i lies as close o was pos-
sible (among poin s in M), o is a leas app oxima ely like his. To fill
56 S. Semmes
in in he a eas a ound hese disc e e poin s, one can make ex ensions
fi s o edges, hen 2-dimensional aces, and so on, up o dimension n.
To make hese ex ensions, one uses local con ac abili y o M. I is also
impo an ha he local ex ensions do no go o a om he selec ions
al eady made, so ha :V→Mwill be con inuous in he end, and his
one can ge om he local con ac abili y.
The no ion o “Whi ney decomposi ions”, as in Chap e VI o [S e], is
help ul o his kind o a gumen . I gi es a way o decomposing Rn M
in o cubes wi h disjoin in e io s, and some o he use ul p ope ies. (In
pa icula , his kind o decomposi ion can be help ul o keeping ack o
bounds, i one should wish o do so.) One can use he e ices o hese
cubes o he disc e e se in he complemen o Mmen ioned abo e.
See also [Da e2] conce ning P oposi ion 10.1, especially p. 117ff.
Le us e u n now o he gene al s o y. Suppose ha Mis a compac
subse o Rn, and ha Mis locally con ac able. Le :V→Mbe a
con inuous e ac ion on M, as in P oposi ion 10.1. Thus Vcon ains M
in i s in e io . By eplacing Vby a sligh ly smalle subse , i necessa y,
we may assume ha Vis compac , and in ac ha i is a fini e union o
dyadic cubes in Rn.(Adyadic cube in Rnis a cube which is a Ca esian
p oduc o in e als o he o m [ji2−k,(ji+1)2
−k], i=1,2,... ,n,
whe e he ji’s and ka e in ege s.)
This ype o choice o Vis con enien o ha ing nice p ope ies in
e ms o homology and cohomology. In pa icula , Vis hen a fini e
complex. The inclusion o Min o V, and he mapping :V→M,
induce mappings be ween he homology and cohomology o Mand V.I
ι:M→Vdeno es he mapping coming om inclusion, hen ◦ι:M→
Mis he iden i y mapping, and hus i induces he iden i y mapping on
he homology and cohomology o M. Using his, one can see ha he
mapping om he homology o Min o he homology o Vinduced by ι
is an injec ion (in addi ion o being a g oup homomo phism, as usual),
and ha he mapping om he homology o V o he homology o M
induced by is a su jec ion. This ollows om s anda d p ope ies o
homology and mappings, as in [Mas]. Simila ly, induces a mapping
om cohomology o M o cohomology o Vwhich is injec i e, and ι
induces a mapping om cohomology o Vin o cohomology o Mwhich
is su jec i e.
This p o ides a simple way in which he algeb aic opology o Mcan
be “bounded”, unde he ype o assump ions on M ha we a e making.
(The e a e mo e efined hings ha one can also do, bu we shall no
wo y abou his he e.) Local con ac abili y, and he exis ence o a
e ac ion as in P oposi ion 10.1, a e also nice o making i clea and
Some Topics Conce ning Homeomo phisms 57
easy o wo k wi h con inuous mappings in o M. In pa icula , one can
ge con inuous mappings in o M om con inuous mappings in o V, when
one has a e ac ion :V→M, as abo e. This is as opposed o s anda d
examples like he closu e o he g aph o sin(1/x), x∈[−1,1] {0}. (This
se is connec ed bu no a cwise connec ed.)
Now le us conside he ollowing s onge condi ions on M.
Defini ion 10.2 (Gene alized k-Mani olds).Le Mbe a compac sub-
se o Rnwhich is locally con ac able. Then Mis a gene alized k-man-
i old i o e e y poin z∈M, he ela i e homology Hj(M,M {z})is
he same (up o isomo phism) as he ela i e homology Hj(Rk,Rk {0})
o each j.
We a e implici ly wo king wi h homology defined o e he in ege s
he e, and he e a e analogous no ions wi h espec o o he coefficien
g oups (like a ional numbe s, o ins ance). One may also wish o
use weake condi ions han local con ac abili y (as in [B e2], [Wild]).
The e a e o he na u al a ia ions o his concep .
I Mis a fini e polyhed on, hen he p ope y o being a gene al-
ized mani old is equi alen o asking ha he links o Mbe homology
sphe es o he igh dimension (i.e., wi h he same homology as a s an-
da d sphe e, up o isomo phism).
Ano he class o examples comes om aking quo ien s o compac
opological mani olds by cell-like decomposi ions (Sec ions 4 and 6), a
leas when he quo ien space has fini e opological dimension. See Co ol-
la y 1A on p. 191 o [Da e2] (and Co olla y 12B on p. 129 he e), and
compa e also wi h Theo em 16.33 on p. 389 o [B e2], and [Fe ].
As usual, dimensions 1 and 2 a e special o gene alized mani olds,
which a e hen opological mani olds. See [Wild], Theo em 16.32 on
p. 388 o [B e2], and he in oduc ion o [Fe ].
Fo mo e on ways ha gene alized mani olds can a ise, see [Bo 2],
[B e2], [B y+], [B y∗], [Da e2], [Fe ], [Wei2] (and he e e ences
he ein). A ela ed opic is he “ ecogni ion p oblem”, o de e mining
when a opological space is a opological mani old. Some e e ences
o his include [B y+], [B y∗], [Can1], [Can2], [Can3], [Da e2],
[Edw2], [Fe ], [Wei2].
Wha a e some p ope ies o gene alized mani olds? In wha ways
migh hey be like mani olds?
A undamen al poin is ha Poinca ´e duali y (and o he duali y he-
o ems o mani olds) also wo k o gene alized mani olds. See [Bo 1],
[Bo 2], [B e2], [Wild] and p. 277–278 o [Spa]. This is p e y good,
58 S. Semmes
since Poinca ´e duali y is such a undamen al aspec o mani olds.
(See [Bo T], [B e1], [Mas], [MilS], [Spa], o ins ance.)
A mo e in ol ed ac is ha a ional Pon jagin classes can be defined
o gene alized mani olds. (See he in oduc ion o [B y+].)
Fo smoo h mani olds, he defini ion o he Pon jagin classes is clas-
sical. (See [Bo T], [MilS].) Mo e p ecisely, one can define Pon jagin
classes o ec o bundles in gene al, and hen apply his o he angen
bundle o a smoo h mani old o ge he Pon jagin classes o a mani-
old. As in eg al cohomology classes, he Pon jagin classes a e p ese ed
by diffeomo phisms be ween smoo h mani olds, bu no , in gene al, by
homeomo phisms. Howe e , a amous heo em o No iko is ha he
Pon jagin classes o smoo h mani olds a e p ese ed as a ional coho-
mology classes by homeomo phisms in gene al. Fu he de elopmen s
lead o he defini ion o a ional cha ac e is ic classes on mo e gene al
spaces.
Fo fini e polyhed a, he e is an ea lie ea men o a ional Pon ja-
gin classes, which goes back o wo k o Thom and Rohlin and Schwa z.
See Sec ion 20 o [MilS]. Mo e p ecisely, his gi es a p ocedu e by
which o define a ional Pon jagin classes o fini e polyhed a which
a e gene alized mani olds, and which is in a ian unde piecewise-linea
equi alence. (Fo his, he gene alized-mani old condi ion can be gi en in
e ms o a ional coefficien s o he homology g oups.) I one s a s wi h
a smoo h mani old, hen he e i can be con e ed o a piecewise-linea
mani old (unique up o equi alence) by ea lie esul s, and he classi-
cal a ional Pon jagin classes o he smoo h mani old a e he same
as he ones ha a e ob ained by he p ocedu e o polyhed al spaces.
See [MilS] o mo e in o ma ion.
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Depa men o Ma hema ics
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