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T R I A N G U L A T I O N O F M A N I F O L D S . IB Y R. LASHOF A N D M . ROTHENBERG1

Communicated by William Browder, December 26, 1968.I. Introduction. The purpose of th is note is to out l ine a proof thata lm os t e ve ry c om pa c t m a n i fo ld M of dimension greater or equal tos ix , (or dimension greater or equal to f ive i f dM = 0) can be t r i an gula ted as a com bina tor ia l mani fo ld . T h e word a lm os t re fe r s to two

types o f res t r i c t ions we impose on M. F i r s t t h a t TTI(M) be sufficientlyn i c e . Se c ond ly , a nd m ore unp l e a sa n t ly we r e qu i r e H*(M; Z2) = 0 a ndIP(dM; Z2 )=0 . S ince the ex is tence of t r i angula t ion impl ies un iqueness , a consequence of our resu l t i s a fo rm of the Hauptvermutungwhich covers some cases p rev ious ly no t done . For exampleW h O n ( J l f ) ) = 0 .The proofs o f the announced theorems make use o f a number o fr e c e n t r e su l t s , a nd t hus r e p r e se n t a c o l l e c t i ve m a the m a t i c a l a c h i e ve m e n t . T he funda m e n ta l b r e a k th ro ugh is due t o K i rby i n [ 3 ] , whof i r s t e s tab l i shed wha t we re fe r to as Lemma 2 , con jec tured Propos i t ion 3 , an d recognized i t s im po r tan ce fo r t r i an gu la t in g mani fo lds a ndfo r t he a nnu lus c on j e c tu r e . O the r im por t a n t ge om e t r i c i ng re d i e n t sa re due to Lees [4 ] an d Lashof [5 ] . A se l f -con ta ined a nd s t ra igh t forward geometr ic proof of the theorem based on these resul ts ispossible if one is will ing to assume M i s 4 -connec ted . The more general form sta ted above depends on del icate and dif f icul t a lgebraicresu l t s of H s iang and Shan eson [2] , Shan eson [7] , and W al l [8 ] . W eunde r s t a nd t ha t K i rby a nd S i e be nm a nn a l so ha ve ob t a ine d s im i l a rresu l t s .In this note , I , we out l ine a proof of our main technical resul t ,Theorem 3. In I I , we show how this implies the resul ts of the f i rs tp a r a g r a p h .The fol lowing is a useful t r ick, f i rs t observed by Kervaire .

L E M M A 1.Let: XXR >YXR be a homeomorphism where X, Y arecompa ct spaces and R the real numb ers. T hen there exist a homeom or-phism \f/\ XXS l-^YXS l such that f or e > 0 and sufficiently small thefollowing diagram commutesXX [-e,e]^>YXR

I 1 X exp i 1 X expXXS1 > YXS1

1 Partially supported by an NSF Grant.75

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TRIANGULATION OF MANIFOLDS. I 751Further if X and Y are PL manifolds and \dXX R is PL (piecew iselinear) y then we can choose \p such that ^ | dXXS1 is PL.

The fo l lowing lemma i s the fundamenta l geomet r ic observa t ion ofK i rby [2 ] .L E M M A 2 . Let X:BkXtn^> B kXtn be a homeomorphism where Bk is ahomeomorph of the unit k ball in R k and tn is the n torus, the Cartesianproduct of S1 with itself n times. Suppose \\dBkXtn = identity. LetX: Bk XRn >B k XRn be a covering map of X, determined u niquely incase k^l by the condition that \\dBkXRn = identity. T hen X is isotopicto the identity by an isotopy that is the identity on the boundary , dBk XRn-P R O P O S I T I O N 3 . Suppose k^3 and h: BkXtn-*B kXtn is a homeomorphism such that h restricted to a neighborhood of dB kXtn is PL.T hen some finite cover of hy h, is homo topic to a PL homeo morphism h'with h = h' on a neighborhood of dB kXtn.If k^5 the n an easy app l ica t ion of the resu l t s of Far re l l [ l l ] an dinduc t ion on n s h o w s t h a t h i t se l f i s homotopic to an appropr ia te PLh o m e o m o r p h i s m h'. F o r k B kXtn. Th i s is pos s ib l e by Le m m a 3 . T he n by Le m m a 2 t hea pp rop r i a t e un ive r sa l c ove r ing o f orW, r1 ^ 1 , is isotopic via yt to aP L h o m e o m o r p h i s m w h e r e y t = ide n t i t y on a ne ighb orhood ofdB kXRn. T h e n yt i s an i so top y of X to a PL hom eom orph ism wi thyt =X on a ne ighborhood of dBkXRn.We now come to our major theorem, which has a l so been proveni n d e p e n d e n t l y b y K i r b y .T H E O R E M 1. Let : BkXRn-*B kXRn be an imbedding of the pair(B kXRn, dB kXRn)->(B kXRn, dB kXRn) such that f maps a neighborhood of dB kXRn piecewise linearly onto a neighborhood of dBkXRn.Suppose ky^3 f k+n^5. T hen is isotopic to a PL imbedding fthrough an isotopy f\ (fo= ) such that f t= on a neighborhood ofdBk XRn.P R O O F . We wil l prove in deta i l the weaker resul t which is suff ic ientfor our purposes , which says th a t fo r an y e > 0 , res t r i c ted to BkXD^,

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752 R. LASHOF AND M. ROTHENBERG [JulyD* is the n d isk of rad ius e , i s homotopic th rough immers ions ft toa P L im m e r s i on / i whe re *= on a ne ighbo rhood of dBkXRn- W ewi l l ind ica te how the same method wi th suf f ic ien t a t t en t ion to thede ta i l s wil l imp ly Th eo rem 1 as s ta te d .By the engul f ing theorem, the re ex is t s an i so topy of the iden t i tyh t: BkXR n-> B kXRn, A*= ide n t i ty on a ne ighb orhoo d of dBkXRn\Jf(BkXD?) s u c h t h a t hf(BkXR n)DB kXD? for a n y finite r.F i r s t c h o o s e r ' w i t h f( BkXD?)CB kXD?>. W e can choose a sm oothtn-xCRn w i th O G ^ 1 s u c h t h a t Dnr'C tn-xXRCRn. Fin al ly we choosea n BnT wi th D^Ct^XRCD^CR - W e now use the i so topy ht a sa bove t o a l t e r / , o r wha t i s t he s a m e th ing , we c a n a s sum e BkXD?Cf(B kXRn) .W e t h e n h a v e BkXD^Cf-1(B kXD^)Cf-1(B kXtn-1XR)CBkXRn.Now as in (6 ) the re ex is t s a PL homeomorphism i: BkXtn~1XR+f-^BtXt^XR) a n d t h u s f0=fi: BkXtn-1XR->BkXtn~ lXR is ahom e om orph i sm wh ic h i s PL on a ne ighbo rhood o f t he bounda ry .F u r t h e r i~l: B kXD ->B kXtn~1XR is P L a nd by s t r e t c h ing R, one cana s s u m e t h a t i~l(B kXD 7)CB kXtn-1X [-a, a], w h e r e a > 0 i s sm al l .W e now a pp ly Le m m a 1 t o ge t a hom e om orph i sm \[/, PL ne a r t heb o u n d a r y ,\f :B kXtn-^B kXtn a n d a P L e m b e d d i n g j B kXD -+B kXtns u c h t h a t xp'j' fai 1 =f\ BkXD nt. W h er e is a l ift ofj to BkXtn~ l XRa n d ipr is a corresponding cover of ip .W e now h av e th e followingc o m m u t a t i v e d i a g r a m

Bk X Rn > Bk X Rn exp i exp

Bk X tn~ l XR-*BkX tn~ l XRCBk X Rui tB kXD:^BkXtn^,BkXtn

whe re is a P L e m be dd ing a nd f\BkXD = exp \f/J. By Coro l la ry 4 ,$ i s i so top ic re la t ive to a ne ighborhood of the boundary to a PLh o m e o m o r p h i s m ^ i , t h r o u g h i s o t o p y \j/t. B u t t he n e xp \f/tJ prov idesthe r e qu i r e d hom otopy th rough im m e r s ions . Th i s p rove s t he we a ke rresu l t .To prove the theorem as s ta ted i t i s suff ic ient to show \p jis iso top icto \f/'j' th rough a n a m b ie n t PL i so topy o f t he i de n t i t y on BkXRn.T hi s can be do ne a long the l ines of [ ] . Q .E .D .T H E O R E M 2 . Let n>k, h^Z and w ^ S . Then irk(PLn) -*Trk{Topn) issurjective.P R O O F . L e t (Dny Dk) be t he s t a nda rd d i sk pa i r . Le t p : U-^Dn be

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1969] TRIANGULATION OF MANIFOLDS. I 753a topolog ica l immers ion , where U is a ne ighbo rhood of Dk in Dn,s u c h t h a tpr e s t r i c t e d to a ne ighbo rhood ofdDk=i d e n t i t y .We ident i fyp , p' if the y a g re e on a ne ighbo rhood of Dk. Iso top ies of such (i.e.r e gu l a r hom otop i e s ) are defined in the obv ious way, and we let7T'OP be the set of isotopy c lasses .By Le e s im m e r s ion t he o re m [4], IrZP=Wk(Topn) . If we rep lacetopolog ica l by piecewise l inear in the a b o v e , we define ip'*, and byt he r e su l t of Haef l ige r -Poenaru [l] , ipL=7TA;(PLn) and we h a v e ac o m m u t a t i v e d i a g r a m i n d u c e d by the map PLn T o p n :

/ P L S 7T* PLn)I

/To*pS 7T* T0pn)T h e n the theorem will follow if we can show tha t e ve ry ge rm of atopo log ica l immers ion p: U-*Dn, p =i d e n t i t y in a ne ighbo rhood ofdD n, is isotopic to a PL im m e rs ion .

W e can a s s u m e by sh r ink ing U, t h a t U=Dkx> ~k. The i m m e r sion p U-*Dn induc e sa P L - s t r u c t u r e on Us u ch t h a t if wed e n o t e Uwi th the i n d u c e d P L - s t r u c t u r e by W, p=hp'; w h e r ep' : W>Dn is aP L - i m m e r s i o n and h: U-^W is a h o m e o m o r p h i s m w i t h h\dU PL.W e c ons ide r two cases .CaseI. k*, aD *) -> ( F, dT0. Since k

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754 R. LAS HOF AND M . ROTHE NBE RGP R O O F . F o r k7Tfc(PL) and ^(PL)>7nfc(Top) are monomorphisms (see[6] an d [ lO ]) .F o r k=n1 , cons i de r t h e co m m ut a t i ve d i ag r am :

7T n(P L n_|_i) 7r n ( P L n + i , P L n ) > 7T n- l( P L w ) 7T n_l(PL n_|_i)4 OJl J, 2 J,OLz I 4

7 T n ( T 0 p n + i ) > 7 T n ( T o p n + l , T o p n ) 7T n - l ( T o p n ) >7 Tn - l ( T o p n + i )By the f i r s t paragraph of th i s proof, a4 is an isomorphism. By (1) of[ lO] , 7Tn - i (PL n + i , PL n ) =0 ; and by ( 12) o f [ l O] , a2 is a m ono mo r -ph i s m, a 3 , # i a re onto by Theorem 2 . I t fo l lows by d iagram chas ingt h a t as i s an i somorphi sm.

R E M A R K . S i nceotz is an i somo rphi sm a ll n 5,a\ i s an i somorphi sm,a n d h e n c e a2: 7 rn (P L n + i , PL n) >7rn(Topn+i , Top n ) i s an i somorphi sma l l 5 .R E M A R K . I f we denote by PLn , T o p n t he co r r e s pond i ng b l ockb u n d l e g r o u p s , T h e o r e m 1 m a y b e i n t e r p r e t e d a s s t a t i n g t h a t

7T^(PLJ =7r*(Top n) all n, fe^3. Also ^ ( T o p ^ l P L J =7TA;(Top| P L ) al lk, n^S.B I B L I O G R A P H Y

1. A. Haef l ige r and V. Poena ru , La classification des immersions co-combinatoires,I n s t . Ha u t e s t u d e s S ci . P u b l . M a t h . No . 2 3 , 1 9 64 , p p . 7 5 - 9 1 .2 . W. C . Hs i a n g a n d J . L . S h a n e s o n , Fa ke tori and the annulus conjecture, M i m e o g r a p h e d No t e s , Ya l e U n i v e r s i t y , Ne w Ha v e n , Co n n . , 1 9 6 8 .3 . R . C . K i r b y , Stable homeomorphisms, M i m e o g r a p h e d N o t e s , I n s t i t u t e f orAd v a n c e d S t u d y , P r i n c e t o n , N . J . , 1 9 6 8 .4 . J . Lees ,Imm ersions and surgeries on topological manifolds. M i m e o g r a p h e d N o t e s ,Rice Unive r s i ty , Hous ton , Tex . , 1968 .5 . R . Lashof, Lees1 immersion theorem and the triangulation of manifolds, M i m e o gra ph ed N ote s , Un ive r s i ty o f Ch icago , C h icago , 111., 1968 .6 . R . La s h o f a n d M . Ro t h e n b e r g , Hauptvermutungfor manifolds, Confe rence ont h e To p o l o g y of M a n i f o l d s, Co m p l e m e n t a r y S e ri e s i n M a t h e m a t i c s , P r i n d l e , W e b e ra n d S c h m i d t , Bo s t o n , M a s s . , 1 96 8 , p p . 8 1 - 1 0 5 .7 . J . Shaneson , Embeddings with codimension two of spheres in spheres, Bu l l . Ame r .M a t h . S o c . 7 4 ( 1 9 6 8) , 9 7 2 - 9 7 4 .8 . C. T. C. Wal l , ve rba l r epor t .9 . L . S i e b e n ma n n , T he obstruction to finding a boundary for an open manifold of

dimension greater than five, Thes i s , P r ince ton Unive r s i ty , P r ince ton , N. J . , 1965 .10. M . H i r s c h , On tubular neighborhoods, Co n f e r e n c e o n t h e To p o l o g y of M a n i fo lds , P r in d le , W eber , an d Schm id t , Bos to n , M ass . , 1968 , pp . 63 - 80 .11 . F . T . F a r r e l l , P h . D . Th e s i s , Ya l e U n i v e r s i t y , Ne w H a v e n , Co n n . , 1 9 67 .12. C . T . C . Wa l l , Surgery on compact manifolds, M i me o g r a p h e d No t e s , L i v e r p o o l ,1967.U N I V E R S I T Y O F C H I C A G O , C H I C A G O , I L L I N O I S 60637