Differential Geometry II by Chin-Lung Wang

Caution: The manuscripts contains only part of the material given in the class

Chapter 6 Minimal submanifolds

Weierstrass representations of minimal surfaces in R3Kaehler/Calibrated geometry - Algebraic construction of minimal submanifoldsDouglas’ soluton to the Plateau problemMinimal hypersurfaces vs Plateau solutions

Chapter 7 Bundles

Chern classes for complex bundles via Chern-Weil theoryThe case of holomorphic bundles over complex manifoldsEuler sequence and the Leray—Hirsch formulaPontryagin classes and the Euler class (not included)

Chapter 8 Atiyah-Singer index theorem

Heat kernels for harmonic oscillator in Euclidean spaceLocal index formulaApplication 1: Gauss—Bonnet—Chern theorem (not included)Application 2: Hirzebruch signature theorem (not included)Application 3: Hirzebruch—Riemann—Roch theorem (not included)

Chapter 9: On Milnor’s exotic 7 spheres

The LaTeXed pdf is provided

Chapter 10: On exotic R4

Introduction to Donaldson and Freedman’s theorems Construction of exotic R4Introduction to Donaldson’s thesis - ASD moduliADHM instanton and introduction to Taubes’ glueing theorem

Appendix: (not included) - Non-linear perspectives on modern geometry

1. General idea on classification of manifolds. Thurston’s geometrization conjecture. 2. Hamilton’s Ricci flow and Perelman’s theorem.3. Seiberg—Witten theory.

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cw,^ 0n, Lr1 V J-f, -o. *rt1" ds &,Y.*u,ut. I

t I I re

ur)^;G -Fl -'t5,,g Jwdndif5G".- ^^+ X?s:= l*F Jw*ntv'tf "5

<

0,.,u,-d nr'\ pr, ( _r wf s a I' )tf

L4

e y.

( n;nt : 5' -1 5"r*' )tr;

En ft i5 t'et'nYacf '

la*

t4

t e Cl+oi..t ?,,

fu*fu, f r,5

V(t) r- f [+,] Q.

C f; u*t ctw,f

] \u u o1e,r*./W Fg*, ,

IV Of q ,^^. . *t.o r^ o r,1

f; t f(")€o 3 ,.r

t1 hs,aro,

r;weLt *t ,l^"

(b

?q's frfu1

^ve

Ca.ltP 4 q.

[t n s i f t*i,^"u V c,*-;, tq

**: Pet$nillo"- a( l.,s(. ^c-f.Fw (rl ,ll-;+ (*,7 ) (* ^T 4

f is(r;aL'uo) ["f r,.-,ryh:c i( art"ao$F 1o^ 'n&'

tf "7 es J.c ffUyL-,* lW / A/ d^-e- Op f fif A .

Xo "f rJ 1,./otnwylv+" . (

fh;t i5 e1t-ivtar. )

*rec* cf

ga ; M 4r^/ \o* J,.t6',^^14 r a h". thwrprtfofr^#4'l u^ A/ K U\ W =) (\A V -i\U4

f. tf 3,* K,*L,,Lq s*r o\.1 Nl

#-try l-t'q vna"l:ric b [* := dn ] l;',ro

fl*,.J. a Fq,-- ub = *[* q ,[t,^r

etr,:rk.u^A-t, qo Jffe Jttr-l 4 ioJ- E r, lrl,isK,al^Lu n5o Ar.1 r/fix *f d k^$r& Cu, l(^ ^te olq" 6a^t, Igr.

kgj : r' P'" t' d - f "',"- + *; ? Vo. ,-.*$o ;l fw is LtT2nuy vitrW , tt-,-*-,-,

j*, , cw'), [,.r"1 ,",,-4_ + o h^ huqf M;K" )

2 u?/^ '*f4 rru'iFt br--o b.r^^rr wusl 6e !a6!1 -ywtrer

12'1

z*(",Y)L,

z ) = x *[Y,z\4 ^(r.rd ,. )

-Y-(x,z\ +W rytrPa li' r"

J * ( x, Yr

=LIiJ ;.i "i.^tr^^r;

r

" f<'nn s *

e X o ( r:v,z) - Y 3(r*,7) + z l$:l!,) .. .,

a J [(vr; )v ";ffi: ,'['I's 1ri, l:Ei 'i ]

Yr'IYt 1L')

) + X( ?y) :1y,) - b (try7 r t(' )b (1t..y r ); I + 3 (r sz"t,t.)

; X(eryz, K) + ,Lr,ryiL,x\i X (VrzYt X )* , (: vrtlf ,x)

+ ) (N [v,z),x \IFnr) Y,z )

r ,'(rq'l ,v) -w UV+: ,*)vx"t b'-J\ X, Y, L+b h'^-(. d"" Sf

(o,t),**? b ((olr)y,e) + 3((vr,7)zr X) * b((vz.t) f, X) 6&4*Q oe*

e#c"

fol-io l<v<', b((vot)L,c)= -r(lvoT)',b I fN'V lP*' )o, 0T-o i dd-owr^ \^t^v+\, '|t('nr)rr, Xl

J,^,Ix, ryttz) : I livrf]fv,JL) +'((v'vl)t z' X)

JL* - i4 "* T(pnT)v4 l7^'f J'T +Jo97'J-o

@ vv-T ,+l a*i crtt T" .

,l-[x,Y, LJ Aw ia z 3([?xT) y,z

. N(y ,L\|- J [rY, JA1 -J CY'L7 + lJY',V7 f Lv',TZ1

tfu Nu' 1<^^h,.^.is

t +.',tsc,l/

,t^.9a,..r.f. VJ=0 ff Jul:Q a'utL' N=D

r is clzor, rtu r'A u?x ^tA, N=o t'h t '-ol.ta Y''

a.^ dr f / 1 eo lq ^-a-!' utt'rr''1"'\ 7aU''=-)lt'

. l'h t*Ata.& [ M,^' l,,^-'*'t - [Virr'r" L*\ ) : PDC -tv''w

:,,."J ,v?otzt- tJ c o €4 I it ^^ ilat/[Q*' cpY slructrn'-Q*

1r .r-t L lP,^*,

\ltt--ltu.fu,liOw U.I( so u^r, rn. .(i^itt^. A 4+fe_^n+ ) r,8u\rud; o\g/w L lV, Z l = (ty,z I - TCr,z)

+ ry4-zlv:

(i is tr"\

N (Yt z

r/z'tr

6,a* \

)

+ rzYIa

t\is nork%S +n s-<

^ lcnior. ( chcta ) ; y ,

)= L(Y,z) + JL(y,tZLo

I

(lfv,zJ = ,,:!-zfv= rty,zJ-(zf)v )\ [Y,,92) = ylz-irv = s(y,z]+(yg)z /so L(fY,z) - rTfytzT - Tt fyrz J

= f t(v,Z)L(Yflz)= (ryr\ZJ -Jr v,?ZJ

= t crytzJ +[(rv)+Jz _, ( ttyz] r (ys)z=, L(Y4 + [Fvlt:= --(yf)F4

L (ar; r)xJ') e D dec. u._!_ I r J aYt' =' - aX l

,t v 64q' u'* [rf 4 L^rLM, vgcfr'vs = o

6hr (.synb i na li o n

a" Lut-,' qQ.

u"iLta o^yI'vn^-o K*'f K,at^l<n + L? y't 5 ih b Nq" I q^,, drl-v - A1irc^^ g+-rb J-qh.2

lv\ ib.,It*tal itV L . . YI ur; lnor^onr.cr*]l^-f MY+ \J tp'* 'irtt^tA tdlG\Ltyy-fya dqsis v TFM C TrN

€, ,7et /'" €^rT(r- ) (^*r r)(-n, ---" €n,Tgn

9 (ei ,ei ) + B(Te;, te,.) = ( or, (t + v7p, Te; ) "bw 8(,^,fu). (v^tr)t= (tv,.'r)'o J(v^u)M3 JB(u'v) \r'(. g r'5 j-L; Iirwav (via \r1t+rn-++rt2 ) ,rrrtL^h

5o F(",") + B(l\tT,a ): a ie. I ;5 a

(.1

Q;:u.rgiotn: w w^+C rvtin;*r'J )

Wc -- r^",f *"at\ w-eJ N +b ( e ?^ff:*Fls a^".r1 tt' t

:t

K(*,X,v1 w) e R[",?rtrw)+ ( $x,?),8;,w)) - (F(t,w),

Tyl = R(*trX,y,Ty)

t(0 (r,v), 6[rx,,TY)> -(n(x,TY ), s(zx,y))

r[*,rX,Y,]f ) r lt K(r,y)ll )

B(z,t))(1. R(*,TX,yt

t5 ' F (', u

I= F (, .,

+

tf B(q,v) > ( erv J . E l,a,e ) )

(g(J?,v),8(r,Je))

&uras,h I"' 4/,r " ,, %

,e,V ) + & (]r,u, T?-tv )t,e,v) + n(f €,v t7ltv )

c K(e,1,€,v) -r Rbcrn,-Ierv ) -z (B(eru),6(erv)U> ta ,-!t-, S i o ut ,.

P [,nn' l.uva rgq fl *p lr rn S c."wrzJr^,q,Iin q suila(fu t4^",r< - bwf no!

I(ry2tl

( F(,

3 ,h eis:"!

M ihi,^^^q'..1 It, C"l;bnn*,'or-r1,. to

\ t\d'soA ,4,fnva tu M

J,LV.

lE fr*c ft1 ExLter , TL'e-u t I _

L^,€. oi t'* e4 Tt "P'

I 'uf

i.otd< €+ TpM Cr?fr; ^ upr Sqttyau..-

vx,y(Trfr, - t \r'=

Pl '.

u,[x,Y)'* ( Tf,r Y )'( [zx l'lYl'* (*l'1 Y l'"=' h.td.4. 49 7X c tY , ;e XrY SF* ?x t-L"-r

., t g,r'W l a,J 4 Tf fr

M.r urht;Jtv^'o-lrr*, )r\rr.-.Q il is J(e,.,,.,, slr^ z-ftww-,

h*nr ,* uA ^a, + 3 oNr B €, /.- .r e. z,,u* 4 t M bf .

frlo

\

I

-FlTl& a- f*^

ol"at<

\h = ," ( e"4-, ,(rk )

ie, ;f Fr,

'|1.'9il ,)' .\k Ailr< A0 zt1

..-I-, Ai n.---

Jr^, D, tla 6VL

L[-l)')r

k'-1, "'t tnl

&u ltrtla t

,( f<i t ^TrM ,

,t 0 e.m

"6(Jeat,e*)

+i€.

lvft ttJ = (rt

# € ,\1 ...I,* tUf

-(lus, lh lt r'f , "=" l"'(J< (+ l);l=l V*'

by nlotL, € J€rk-, = t^ €rh , ,'4,TtA ; [-;u,tzt,.;orf"

fil9o r"-" hotd4 witna, I I ' I ttgr ocrr*J wt^e^^

ovt'- hh ar,t 5 xVQ- #^9 $qtr,rl" E?*4 ,t Tl,e.-rsnn TL : nA' "F, + f'A u'\i^ 'K ^ trvr I G t"'-ttr-Z)

v" n',-- r,a, vot(r^)= I, + - I^,# € I*, Jv^t = vot(tw'),; ,, 4+ M, also ?X A

iin.^gia.|"

rCAC

t:

!o ,u,t i-ioa.r J* f' lo**a-,,^ i f,oOlc^^^

K' J *do"^ Lta.*-e.

f" uvr,i l. 4'5 h- | .1.1* J )

A-+K* Pietlwite C'

?' t/r

;r. y is Co

r 6u-,t!ilx- aA tr^.a\ f"',"^e ftS ortna\

Ll uvvp,4 . .( iS C t

For 11

I :aA =S( -)

f ,'s wtulofon-e- ;F Iu

b-l (f ) is uol'\-oureA V P€ f t)

Lowr y et f n5 ctag s t lit,'', J,'lk tu< W' ?rti; 'rhl",ffiXf r= 1 ,f '. A + lR.n st. \ ir pic,auri* Ct t

c^*d' Ylra ' Sl-l f iJ '/va

;n,.1-t;.l 1

fVca t,""^-r-t;,"".r.d F'' z1t z d (z't t3 )

A , Xf \ tR.* urrc \ h1

ntt) ,: Jo [** ^ e,1 I ,*&l

( | dlu,", ? wL l*^ '.

Asr,^^r +L* ol io *?, (KrAQ) <to

fi,"J T e Xp st. A(t) - 6ppiruvf 1,4<+4t-A ; t'U"t Lal"^l,r,u, 4 Ve'a*ialtort-a '.

T^t* !^ e Xp er. iooA(v^) = 67

7 ?. 5 *t" *l*u u6u \-^e.^ T/t ,> Xf

L;C. uo1aLf QLA )

I-^^ 1r'.^t^^!. ho F posl;tk ir"/ . d-fru^4^ryTb-rv.,.

T tt) A J'u.! a/i E; r( '6 -.,1 +4",-oa,.. " \rl

Al .". \ ; P^." 'l- 4lrh carl '. 1* JPa; oa. :

?)?

l'L/rD ir,'c.l^LLf tr' feX""i l€ne*la lLosT il J€4.

o(.r) ,= \^ (l*r[r*ty,,l, ) JxJ-., (y-tve,^,aur u*^J-e.t'

"JA.t /\' [ ) , I

tq*,,^r**t Frae,l.r.

tr"., [vn- (' .-. [vt'(wl'- [v.w I' s ti(1u,r* lwl') J'yr At'r) <ip(v) ,,,^), - ;(f[f*l=tVll a^^A yx.yj :. o

i e , ? : A K" is ^y\ rlr,,,r or r a^ f,r*.,^J ^a,

fScTL: Gf c t Jf , ar:= ;"tJo lor *,1 q € xr t

uF *',.*.=,. ;: Y e xr P(r )

D(k) - Jf € A(g):6f ov_,A y ;r^,A;fu",INoL'o 1,0.'"I ; +e.t ?ln*ta-,,, p,'.^ a!rr^^ ( ta) scL^*Hor-,tvri*,""^1-l- t

** Let^ L or,'. a, b ;fa0'.1 gat *tn-a 14, :ca.r-t nnur' tv

-, l(*r-- / rt D;ll(a)

^ <- o/r

F^, Y a, L'. i* yna,.ysi a t4 |?l.az:)f gt. fl':.r(,f ,t, *t*,^"-1 irr"*r v-etw,*j

AtJ ,"^r"(r*4 pq/\-r^.,^tfn'u^-,'* 4 pe., Su,u,rL S,^(",^+MU?fr 61 J+^. uy4a,-,*"-0- W un (a)

ix. s (,tt'iLiw tvtt,-"f*"^ e,;u. e =n (- &"u*t* ,u*/ll-l L,-( p

So ;i is "I-voWy h eof,tl=e y whicL., r,niniwizq+L\t Pi r^tct^ (.s l- i r^fr3 r a-Q-

?AcT?" (,,t*y btc(sA,,Rn) , Xt i=f V:+t*.n f-c, ,*lSo=L)t( db'=r,)',f p(v)(ro , M Jt. yr txb vit; . D(yr).dbLt k !t,,t-t,^ rb lov. (^r, w;A )*ur^1,-. b

+t^f, l; Jrm.lnr +- Vl"AJe +tu-*l"rt q.^^.u, afi€1,.[_qjrrg,svlP.6+.*d.i--.

G ul fi',, t'v.(*v- a/-

i. D(g) stD(f^,e

To "S%j^" 4 L+4ow't^^-l 4f 'L1,lu, L3 ( AA c^^oA t^oz?-t;de(

li e If*xFlvt?h)=la,We S+rll havz )7 : ).,+ D

c( e xr.,

-( h o. Yc/vn ( P^p e ; Lot*o^ f 67 )

Ler 1"4)Jf , *1.-u" y:= lVis e1*Wntihvt^61/14- h dL gJl.

Gr. Avztla- Astotr' I 1 k .71

+ D.*blanttl

(rt) r+( f^(*,,1 )

Jet f o t tninmeYJion,all ta\

= A(fn,r.) = A(f^,t) <

PPL

h *Y a ! t vL.) lt'"^ l- f,

J+.

kt** h->b leF

67,^ i"f A(VI t xr

Jr,=,hf D(t (xr

f iS ^.r'oqnnv^-( . ,'(

[?" 'l vl I Y2<. 'f 1 z o

),

")

r,3/r)

?)

A(u) = J^l<e*av, (J* \p (,c) - I o (fr-l'+ r Yt l)

^ EA(Y\ < D(? ) ';" fcL

z

6p (

?0, )7

y' itt

f @[

A(f" )

To u 4 la c'f

Le.r tP :tntf,40

4 P^* L (La.*'t rv

* Jf ry ,[ea(.

: tJl- l(9")\ 6P

y"Ftu,,v^rln-t* h -fD

f$ ) * i >""0(tr//a

Fact 2

Y"

),

( c'(t'wht

I

J

7r

+.,

A-J

tl

,VJd inm er dio n i

,(n*')

o

*Jh)

I t . (, , f l6.rSU't^ 61'

z, 3 1

, rx/rJ+"( r+

^(n )

er" -uvlud po,:'< u'

A

k =1,

(q)

lro "(tt| , D(v)<M j

; "r4l , covtt/ - +zr1 ,G 0 ySolwfior+

('vlr

fir )a :

k *f %'t',.'tnn -te

(f Pn,=JA

P,?'l ,^.riX Ct. -----Ll

V o<[<, , v* (ri5t. [(vtcatle s zq

('t,r,. r"* l*r1' ) t xn&rtl

pJviJg

not^A,P(?)gM'

, ?f 'et'f I '

e Ql . ttJ):=J I rO

Jr'JY'o

)e f *-6-'?,M

1,1 (t/t )

-+ v.d-o

'(( ? ?x ' 'i, + *'l '*? tx.(-5,,(+)) -r Y7 (to5 (i ))

*v

t [rt lt ' t('i r'v' (41 -k ('h]L '^5" (l )

L-, -L qr.ctr 5,r"fif 'qs t?)qx nr

r rr.r- *l,lw o I \I z- [(r |

':' I lt'' o'lct> - '

)

I ltnt'1^"'u +('tl*r'\2o I

Yon^'(-

i(. i...= If !.,,*, lu, l')sdr s e(11 < r\4

.thit i, r-snfr' fraln t t , ,, \ dVev.u^i( Yi:c''^LY !u[ (ll ,rl'ttt'lt )fo

-l '- )t r t ' '\ a AheY

J,\ur !,..-v ?. b1 fii1-€a'aal^!-'-t 4 fit,,^'^ -sticJ'on '"h-'

r€ P ,rrltr'rr'Jt J: 'u'o' t t'4

r 4ry)U n^l t^/7-< 5 u\,',t"^n4 z *

" 1. p

'1-l€ -tAW C *, 4/" qt

J qo

.,?"., t" t v a.t6 s!\+ 'a)l il ) J Jr '(x3l '1*,

fl Ir*r* vry

I tz) h - (,>) L I

) t 2l z-'?lf

t trlr\o h1 2 <l 'td- ''4 I

yvw) ll + U c ^g *vvq

.( +., Ja ' .,a C ,,V '2 ?1 ,,! +o u )

2>

l-1, ' t,,r1'., S!')r.,t

7

t)>

h.lwlv,7l,J-d1 /

'l> i'"._q,

I '1"1

wt"+ 'D:of

t+

^/t?

a,+

g ,,1?

Vc )t

v+

)" L-*

trve

.fl- .'t I

7>P>

J u.tu I sr,l

su'1

J ?,J+J

Y*l )

5 ,l 'J\A 'I!

ol.7

-Ja 1Y t

-,4,n l?

4s'a

, (v)

,i*.. i *,' *'J|- u^-*t I r; ' ! fl "'F* ('r-*r-t 5 *[ *-o *-r't L1

( - (tt..: 'F )

*1 f^1 "o \

+ tl,. l"*It*t'-t(?rl'

t?"'YY 1 - \ + "r'e- . ?w ?l afea^ 1$-^*l-"'.^,-"

I

o(

.* &.1,

ttrffuieh^-r*

or^ (t;W"'-Q-=t.;

J

Z It-+[tfl--+

x+A

X-+

'Lh

h,[ at'n(Ju" f )\\

o

2

\l

('rL I

fl "ol +lt l"|2

is

r+lvfJ'

i :- t-t g-

\*-/- ' J

L-+?tf "u-':' :'i.!d&Y jt-

i -".*.:*-

I

Ll

r+[ "rl'(, lv

e h aj* u/<- o-t','- .> tI rq.^^o.^ Lv/?-. , "'^f "l!t.,,' q'Fr -* fta ,.* i'* *-L W yf rv tn ,;'1-a*q

)'^[*1* *I

tiir i'Lt :

"i\bvoop.

t/

2;-a/, o "LtenftrlT q,rh'tr'.',.

( o(.0 1"\ -_-----\

-

Jtu 1*[ol lt

= I l^,rch) -f)n u )

\ k ;)t")A-*t ..'-*

t--u-/ l^ ,' 1 LJ/v uar a\-a-! | et^t'^-^

t2/\."' + wv'eu"'Y [? l" ] '

yi-;,:,rl',tof lL l€:'

;(

, ln ) = , (U''f lh f *| * *. M r**i '"^'"*

Vw,*f*-**

*G- r),:^,1

if t'(

n,ttf t

r^J\ft(ut?,;qo"r*o utq7-uL A'( t )v l,wi a f-t on.

a A (F* t,h ) a t+ I r*tvll',ht'

\JtJL-t' tIvof ,oh )2

\* Jr@zA"(o) = ( ["4 l' A

-

-//v I\ (r (vf I,

(of,ohL

So Wy V..'^I"** is ,*n'" % sh^b I'4 7 o

L l*C nQ J'*"'vl i u- uw )

"Loto[ 9to6;1if,, " 6 A'1 ,''ait^ac^l gvLcvati6/t; lt c.y , !-o; *4 " a 6 S sW/<, wtt'u ,' r.r4.vt.A ',

,

JmJ" 3

;, ut,^*.:y'-a-'r,i

fn

:o

s0

"', f n )

))(*,'l

(,' ,

,! n y * "-o.'a-+,,,i- i;,, li i^.1,., ; F-,1 ,"on i uo^**frg

h= 6t I

r, sic.nrt,,;9^:':[u1 o^"1 T )

c..4. 'cv t: A j ( grz_7t^, t+t_Q-

(*.J ) r--r (*,1 , fj(x,y), --.,

( .5r, c Y*.Y, e t+ [Fr lt)( 5rz 3 Yx'?.1 c Fx.Frll\ 5r' = ?y,'tJ =. l+ tFA .|'

6ir' : -! t t+ tfY l' -f-'lJ IJ i -f-'ll + lf*l' )

.1 = C,*lf,t ) (t*tfal')- (fr.f;

F=

)'

A? c 3'lt +

$:,_r_ !)iF lilfr ?,:t ) t _r,(' _r',_ ).', .

r (,*tf")'J fra I

,i't4 ! .t',1

l'r lirj-

?5

)

J;J.il

-

5\A''tl i

Irl,I

Ji J,. .p + -:/J fiJ

('* u' t' )rl

Y, - Y ,

aii ) : o

Iq

L(

:-;,- i-

5')

+

,1. f;i

)

J.'d.

(

\ crkt s,' jpl-Jfi = *hr

A |ik e;3trq 3t,r",i(l (e olto e1h/r.v. n

= &','v(vl, )= ah

*t ti:1lil+-r,F o,r

J

4a

S ,\ru tt-

=O

_i_q);(q

hr

:f =2,

):l

ri ("m t;i tih

)'j L ) = on)t

€icEd

vt

Cl4 r',tt '. t\;< )1St g@ t+ uI, u

t,itt gh",6r +4n^F

?'t'e,)i [ = o i^f i = o

t&r,d.

( * *\ tr's.*l

5o D1'"

a; I

is ?.

1=lrL

1;, 1

f* z(r,r, ) trl

]L

rJ'

I . );?al

ef u^^t V. +

'f "1 tt: ;J, rn -) -r 1.1 ^nC V-r t q:

( ! [ r] ":r rE

sl o e

h3 =b

s'b 1

tle ('b

=A )z)

r.J +'r., ) t

?l

Q-]

"?i +. lhtl*, 1'J)+rl Y--, ylvl

\ .t.l )'-tl

+l

fut'Jtn Ji r'?v,) ITJT; +

nrJ.x3 +

r. ll!

ta *,^.- nr,. l; rr. \': '

^(

,'J1'J

tlr r)

n,)

I Je

+

+

('d )

r) )\.

) t .(+.t Jl

lbl+l)

1+.t Jl

,a ("tJlr-r)s.b +(.( .t t'J ) -.r L t? tJl +-.lr

Jl 'r) s'la + (.llt.r-r )

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.(,r") i= (,*j' )^*l J-tPh

A REPORT ON

HIRZEBRUCH SIGNATURE FORMULA

AND

MILNOR’S EXOTIC SEVEN SPHERES

CHIN-LUNG WANGDecember 1993

Introduction: In 1956 J. Milnor constructed a non-standard smooth structureson S7 which startled the whole mathematical society because it is the first knownexample that a manifold can admit more than one smooth structures! This article isan attemption to understand this construction.

Start from the sphere bundle S(E) of an oriented four plane bundle E overS4, can show the total space M of S(E) is a topological S7 if the Euler numbere(E) = 1. In this case, if M is diffeomorphic to the standard S7, we can attach an8-disk to the disk bundle D(E) along the boundary M via this diffeomorphism to geta smooth closed 8-manifold W . By applying the Hirzebruch signature formula to W ,we will obtain some divisibility condition on its Pontryagin numbers. By a detailedcomputation of the characteristic classes, this can not be true for some E, and weget the “exotic spheres”.

The main construction is in §4, the computation of characteristic classes is donein §1 (1.2) (1.3). Other part of this paper is devoted to a proof of Hirzebruch signa-ture formula. Since I adopt the topological approach, Thom’s cobordism theorem isdiscussed in §2.

1

§1 Topological preliminaries.

(1.0) Cohomology group of grassmanian manifolds. Let Gn(Cm) be thegrassemannian of all n dimensional complex linear subspaces of Cm. It is knownas the classfying space of Cm bundles in the sense that any Cm bundle E → M

arises from f∗γn for some f : M → Gn(Cm) with m large enough, where γn is the“universal bundle” with fiber γn

[X] the vector space X. In order to attach a “naturalcharacteristic class” c(E) to E, it must satisfies c(f∗γ) = f∗c(γ). So it is necessaryto study cohomology of Gn(Cm). To begin with, we construct a cell decompositionas follows. Let

C0 ⊂ C1 ⊂ C2 ⊂ · · · ⊂ Cm

be a fixed filtration. For any X ∈ Gn(Cm), the sequence of m numbers

0 ≤ dimC(X ∩C1) ≤ · · · ≤ dimC(X ∩Cm) = n

has n jumps, and denote the sequence of jumps by j(X). We call a sequence σ =(σ1, . . . , σn) with 1 ≤ σ1 < σ2 < · · · < σ ≤ m a Schubert symbol. For each Schubertsymbol σ, we associate a subset e(σ) in Gn(Cm) as the collection of all X withj(X) = σ. e(σ) is topologically a cell of complex dimension

(σ1 − 1) + (σ2 − 2) + · · ·+ (σn − n)

This dimension formula is easy to see by deform X a little bit, but it needs somework to show it is really an open cell (cf. [MS] p.76), anyway it is elementary. Thereare totally

(mn

)such cells, and the whole collection of these cells form a CW complex

structure of Gn(Cm).

In particular, there is no cell of (real) odd dimension, and for cells of evendimension 2r, they corresponds to those σ such that (let τi = σi − 1):

0 ≤ τ1 ≤ τ2 ≤ · · · ≤ τn ≤ m− n

τ1 + τ2 + · · ·+ τn = r.

When m is large, say m − n ≥ r, and n ≥ r, the number of all such σ is exactlythe partition number p(r) of r. Since no cells are of odd dimension, the cohomologygroups are then clear:

H2r+1(Gn(Cm);Z) = 0

H2r(Gn(Cm);Z) ' Zp(i).

In the case m, n both large, one will suspect that this cohomology ring is a polynomialring with each even dimension 2i a generator ci (the universal chern class) for thefollowing reason: If this is true, than for cs1

1 · · · csrr (= ci1 · · · ci`

with i1 ≤ · · · ≤ i`)to be of degree 2r, there are exactly p(r) terms of such monomials! (corresponds to

2

the partions I = i1, . . . , i`.) This is maybe the origine how chern classes invented.Since the construction of characteristic are not the goal of this article, I will just listthe formal axioms of chern classes and define the Pontryagin classes from it. Andthen compute them in some special cases that will be used later.

(1.1) Characteristic classes. Start from the following (without proof)

Theorem. For any complex vector bundle E over a manifold M , we can uniquelyassociate an element c(E) =

∑i≥0 ci(E) ∈ H2∗(M ;Z), the (total) chern class of E,

such that

(1) c0(E) = 1, ci(E) ∈ H2i(M ;Z) and ci(E) = 0 for i > rank(E).(2) Naturality: c(f∗E) = f∗c(E) for any f : M ′ → M .(3) Whitney sum formula: c(E ⊕ F ) = c(E) · c(F ).(4) Normalization: c(γ1) = 1 − g, where γ1 is the universal line bundle over Pn(C)and g is the Poincare dual of the hyperplane class.

Remark. In fact cn(E) = e(ER), the Euler class of the underlying real orientedvector bundle. This is the first step to define the chern class.

For a complex manofold M , we denote c(TM) by c(M), for example let’s com-pute c(Pn(C)). Let ε be the trivial line bundle, it can be show that TPn(C) ⊕ε ' ⊕n+1

γ1 and we have ci(E) = (−1)ici(E), so apply the sum formula, we getc(Pn(C)) = c(TPn(C)⊕ ε) = c(γ1)n+1 = (1 + g)n+1.

Now we define the Pontryagin classes pi(E) of a real vector bundle E by

pi(E) = (−1)ic2i(E ⊗C) ∈ H4i(M ;Z).

and p(E) =∑

pi(E) the total Pontryagin class. It has similar properties as (1) to(4), with (3′) p(E⊕F ) = p(E) ·p(F ) (mod 2-torsion). and (4′) p(γ1

R) = 1+g2. Theseare all easy consequences of E ⊗C 'C E ⊗C if E is real, and ER ⊗C ' E ⊕ E ifE is complex. For example, c(γ1

R ⊗C) = (1− g)(1 + g) = 1− g2, so p(γ1R) = 1 + g2.

Also for the tangent bundle, p(Pn(C)) = (1 + g2)n+1.

Consider the carnonical quaternion line bundle γ over Pm(H) = (Hm+1−0)/H×,by the right action of H×. It has the sphere bundle S(γ) ' S4m+3 ⊂ Hm+1. (Butthe total space E(γ) is not Hm+1 − 0!). Notice that S(γ) has the fiber S3 = Sp1 =the unit group of H, hence S(γ) is in fact a S3 principal bundle. This is the Hopffibration in the quaternion case. Since γ has a natural underlying real and complexbundle structure, we will compute c(γC) and p(γR).

The principal bundle S(γ):

S3 i−→ S4m+3

yp0

Pm(H)

3

has a homotopy long exact sequence end at π0(S3) term,

πk(S3) i∗−→πk(S4m+3)p0∗−→πk(Pm(H)) ∂−→πk−1(S3)

When k =1,2, and 3 we get πk(Pm(H)) = trivial. Now from the cohmology Gysinsequence:

Hi−1+4(S4m+3) −→ Hi(Pm(H)) ∪e−→Hi+4(Pm(H))p∗0−→Hi+4(S4m+3)

where e = e(γR) = c2(γC) ∈ H4(Pm(H)). We have for i + 3, i + 4 6= 4m + 3,ie. i 6= 4m, 4m − 1, ∪e gives an isomorphism Hi(Pm(H)) ' Hi+4(Pm(H)). Inparticular, H4i(Pm(H)) = ei · Z. Combine with the above result on π1, π2, and π3,using Hurwicz theorem, we conclude Hi(Pm(H)) = 0 for i = 1,2, and 3. This thenimplies Hi(Pm(H)) = 0 for i 6 |4. That is, H∗(Pm(H)) = Z[e]/em+1 as a truncatedpolynomial ring generated by e. (Note. If one feel a cell decomposition of Pm(H) isvisible, then this result is in hand.) So c(γC) = 1 + e, use the eailier technique, have

(1− p1 + p2 − · · ·) = (1− c1 + c2 − · · ·)(1 + c1 + c2 + · · ·)= (1 + c2)(1 + c2)

= 1 + 2e + e2.

so p(γR) = 1− 2e + e2.

Now we specialize to m=1, then P1(H) ≡ S4. Denote e by u (for we will computethe Euler class of other bundles later). We summarize what we have done: The Hopfbundle γ has e(γ) = u, p1 = −2u. Its sphere bundle S(γ) is a S3 bundle over S4 withtotal space S7.

There are still many other S3 bundles over S4. We will classify them and computeall their characteristic classes.

(1.2) SO(4) bundles over S4. To begin with, those vector bundles are classifiedby the homotopy group π3(SO(4)) by glueing two trivial bundles along the equatorS3. So let’s compute it and find its generators.

Recall a well known result ([S] p.115), SO(3) ' P3(R), this is proved by considerthe map ρ:S3 → SO(3) defined by using quaternion multiplication

(∗) ρ(u)v = uvu−1

here v ∈ S2 is considered as the unit sphere of the space spanned by i, j, and k in thequaternion H (this map ρ in fact says S3 = Sp1 = Spin(3)). So π1(SO(3)) = Z/2Zand πi(SO(3)) = πi(S3) for i ≥ 2. Now consider the principal bundle structure:

SO(3)j−→ SO(4)

yp

S3

4

where p is defined by p(g) = g · 1, and define a map σ: S3 → SO(4) by

(∗∗) σ(u)v = uv

also use the quaternion multiplication. Since p(σ(u)) = σ(u) · 1 = u · 1 = u, σ is asection, and by standard result on principal bundle we conclude SO(4) ' S3×SO(3).So we have

π3(SO(4)) ' π3(S3)× π3(SO(3)) = Z⊕ Z

and its two generators are [σ] and [j ρ], it is reasonable to still denote the later oneby ρ because the behavior of j is compatible with the quaternion structure underconsideration. In view of (*) and (**) we can now describe the general form of allsuch bundles, they all come from the form fhj : S3 → SO(4), where

(∗ ∗ ∗) fhj(u)v = uhvuj

This is just a corollary of the following

Lemma. Let G be a topological group, then for k ≥ 2 the (pointwise) multiplicationof two homotopy classes in πk(G) corresponds to the composition law of homotopyclasses.

Proof. Let φ1, φ2 : (Ik, ∂Ik) → (G, e) and let φ0 be the constant map e, then clearlyhave homotopies

φ1 + φ0 ∼ φ1, φ0 + φ2 ∼ φ2

Multiply these two homotopies, we get

(φ1 + φ0) · (φ0 + φ1) ∼ φ1 · φ2

By the definition of composition law, the left hand side is exactly φ1+φ2. This provesthe lemma. Qed.

In fact, it is very clear that σ corresponds to the Hopf bundle discussed beforebecause we use right action there, and as a S3 = Sp1 bundle its coordinate transformwill be the left transformation. So we have an even simpler set of generators, namelythe right Hopf bundle γ and the left Hopf bundle γ defined by left action. It corresponsto the homotopy class σ := σρ−1, that is

(∗∗)′ σ(u)v = vu.

This complete the description of the SO(4) bundles over S4.

Since γ, γ are isomorphic as real bundles, they have the same Euler class e = u,but since the “quaternion orientation” is changed, we will have p1 = 2u. Now we willuse a general argument to compute the characteristic classes of all these bundles.

5

(1.3) Characteristic classes of SO(4) bundles over S4. In general, SO(n)bundles over a space X are classified by the set of homotopy classes [X, Gn(R∞)].Where Gn(Rm) is the grassmanian of all oriented n planes in Rm, or more explicitly,it is SO(m)/SO(n)× SO(m− n). When X happens to be a sphere Sm, it is just aswhat we have done in (1.2) that this set ' πm(Gn(R∞)) ' πm−1(SO(n)), which isalso easily deduced from the following fibration structure:

SO(n) i−→ SO(N + n)/SO(N)yp

Gn(RN+n)

Now we will show both maps e and p1: π4(G4) → H4(S4) are group homomorphisms.For example, let [f ] ∈ π4(G4), p1 is the map

[f ] 7→ p1(f∗γ4)([S4])

so p1(f∗γ4)([S4]) = f∗(p1γ4)([S4]) = p1γ

4(f∗[S4]) by the definition of f∗ and f∗, andthe last map [f ] 7→ f∗([S4]) is exactly the Hurwicz homomorphism

π4(G4) → H4(G4).

Combine with the isomorphism π4(G4) ' π3(SO(4)), we furnish the computations:

Proposition: The SO(4) bundle Ehj defined by fhj has e(Ehj) = (h + j)u andp1(Ehj) = 2(h− j)u. In another word, for k ≡ l (mod 2), there is an unique SO(4)bundle E such that p1(E) = 2ku and e(E) = lu.

Proof. Obviously by looking at the characteristic classes of the right and left hopfbundles γ and γ. Another way to see this is to see the tangent bundle TS4, which hase(TS4) = 2u (the Euler number) and p1(TS4) = 0 since TS4⊕ε = TS4⊕NS4 =

⊕5ε

and by the sum formula. (TS4 corresponds to f11, the “sum” of γ and γ.) Qed.

(1.4) Characteristic numbers. For a complex manifold M of dimension n, it iseasy to see (described below) there are exactly p(n) terms of products of ci’s to be ofthe top degree 2n (the real dimension of M). Also for a 4n dimensional real manifoldM , there are exactly p(n) terms of products of pi’s to of the top degree 4n. Here allcharacteristic class are understood to be of the tangent bundle TM .

Let I = i1, . . . , ir be any partition of n with i1 ≥ · · · ≥ ir. Define cI =ci1 · · · cir and pI = pi1 · · · pir they are all of top degree classes. By evaluating thesetop degree classes on [M ], we get some intergers and called them the “characteristicnumbers”. (chern numbers and Pontryagin numbers) We will see in the next sectionthat the Pontryagin numbers have strong relation to the cobordism problem. Herewe compute some examples that will be used later.

6

Again let I = i1, . . . , ir be any partition of n with i1 ≥ · · · ≥ ir. Define

M IC = Pi1(C)× · · · ×Pir (C)

M IR = P2i1(C)× · · · ×P2ir (C)

so dim(M IC) = 2n, dim(M I

R) = 4n as real manifolds. We want to show, the char-acteristic numbers are good enough invariants to distinguish these manifolds, thatis

Lemma. The p(n) × p(n) matrix [cI(MJC)]IJ of characteristic numbers is non-

singular. And the same statement holds for [pI(MJR)]IJ .

Proof. It is convinient (although not strictly necessary) to introduce another set ofcharacteristic classes, the chern character ch =

∑i≥0 chi, which is defined to be

∑n

i=1exi =

∑n

i=1(1 + xi +

12x2

i + · · ·) = n + c1 +c21 − 2c2

2+ · · ·

that is, ch1 = c1, ch2 = (c21 − 2c2)/2, . . .. We don’t need to know the exact formula

between c and ch, we only have to know they are equivelent Q-bases, which is quiteobvious. The advantage to use ch is the following fact, ch(E ⊕ F ) = ch(E) + ch(F ),which is clear by definition. In our case it reads

(∗) ch(M1 ×M2) = ch(M1) + ch(M2),

this linearity allows us to handle product space much easier. We only have to provethe matrix formed by a(I, J) = chI(MJ

C) is nonsingular. Then the first part of thelemma follows. (chI is defined by the same way as cI)

Since chi(M) = 0 if i > dimC(M), by the property (*), we get a(I, J) = 0if |I| < |J |. (At least one is is larger than all jt.) Furthermore, even in the case|I| = |J |, a(I, J) = 0 unless I = J . Introduce a total order on all partitions whichextends the partial order |I|, then the matrix [a(I, J)] is in a triangular form. So toprove it to be nonsingular, only have to prove the diagonal elements (ie. I = J) arenonzero. Let I = J = i1, . . . , ir, then a(I, I) is

r∏

`=1

chi`(Pi1(C)× · · · ×Pir (C))[M I

C]

=r∏

`=1

(chi`(Pi1(C)) + · · ·+ chi`

(Pi`(C)))[M IC]

where the obvious zero terms are omitted. By evaluating on the [M IC] from ` =

1, · · · , r gradually, again use chi(M) = 0 if i > dimC(M), we find

a(I, I) =r∏

`=1

chi`(Pi`(C))[Pi`(C)].

7

so we only have to compute chn(Pn(C)). By the formula TPn(C)⊕ε ' ⊕n+1γ whichis already used in (1.1), get chn(Pn(C)) = (n + 1)chn(γ). Since γ is a line bundle,by def chn(γ) = c1(γ)n/n! and by the definition of chern class the later term is gn,which is the generator of H2n(Pn(C)) and gn([Pn(C)]) is not zero. This proves the(first part) of the lemma.

For the Pontryagin case, we define for a real bundle E, phi(E) = ch2i(E ⊗C),then for M a complex manifold, phi(TM) = ch2i((TM)R ⊗C) = ch2i(TM ⊕ TM)= ch2i(TM) + ch2i(TM) = 2ch2i(TM). Then by the same manner we also get thematrix with entries phI(MJ

R) is nonsingular. Since these classes phi are an equivalentbasis of pi over Q, the result follows. Qed.

(1.5) Cohomology ring of Gn(R∞).

For later use, we state the following theorem and refer its proof to the literature([MS] p.179).

Theorem. Let R be an integral domain with 2 to be invertible, and denote pi, e thecharacteristic classes of the universal bundle ξ, then

H∗(G2n+1(R∞); R) = R[p1, . . . , pn]

H∗(G2n(R∞); R) = R[p1, . . . , pn−1, e].

8

§2 Thom’s Cobordism Theorem

(2.0) Introduction. We will consider the oriented case only. Two closed n-manifolds M1,M2 are said to be cobordant if there is an (n+1)-dimensional compactoriented manifold-with-boundary W such that M1−M2 = ∂W . This equality meansan orientation preserving diffeomorphism. This relation clearly defines an equiva-lence relation on all closed n-manifolds. Using the disjoint union as addition law, theset of cobordism classes form an abelian group, denoted by Ωn. Toghther with thecartisian product as multiplication law, the set Ω =

⊕n≥0 Ωn then becomes a com-

mutative graded ring, with the class of all manifolds that bound as the zero elementand [pt] the multiplication unit. We have to check that the product is well-defined:If M1 −M2 = ∂W1 and M ′

1 −M ′2 = ∂W ′

2, then we have

M1 ×M ′1 −M2 ×M ′

2 = ∂(W1 ×M ′1 −M2 ×W ′

2)

so the map: Ωm × Ωn → Ωm+n is actually well-defined. The commutativity andassociativity are clear.

The goal of this section is to study the structure of the ring Ω, or actuallyΩ⊗Q. We will start from the Thom transversality theorem which will lead us to arepresentation of Ωn as a certain homotopy group of the Thom space of a universalbundle, we then need some results of rational homotopy groups or Serre’s theory of C-ismorphism to transform these homotopy groups into homology groups of classifyingspaces. In this step we have to consider Ω⊗Q or Ω(modC) to get a satisfictoryresult.

An important observation is that by using the deRham cohomology and Stokes’theorem, corbodant manifolds have the same Pontryagin numbers, (say all the pon-tryagin numbers are zero if a manifold bounds). The result in (1.4) about the Pon-tryagin numbers of (products of) complex projective spaces then shows, the variousproducts of P2nj (C)’s with j1 + · · ·+ jr = m are all in distinct cobordism classes inΩ4m, and the total number of such products is p(m).

Since the cohomology rings of classifying spaces are known to be generated bycharacteristic classes (Pontryagin classes in our case), and by counting the dimension,we can finally conclude that the ring Ω⊗Q is freely generated by [P2n(C)]|n ≥ 0.This is exactly the context of Thom’s Cobordism Theorem in the oriented case. Andwe will use this result to prove the Hirzebruch’s Signature Theorem in the nextsection.

The P2k+1(C)’s do not appear because they bound some manifolds in the fol-lowing manner: Consider the indentification of C2k+1 and Hk+1, then taking pro-jectilization (the compatibility of both action by C× and H× is obvious), we get afiber bundle f :P2k+1(C) → Pk(H) with fibre P1(C) ' S2, this bundle has structuregroup SO(3), so this bundle is in fact the sphere bundle of some vector bundle andthen it bounds the disk bundle.

9

Another remark is the determination of cobordant relation by Pontryagin num-bers. In this fashion, Thom’s cobordism theorem can also be stated as “Two man-ifolds are coborbant if and only if they have the same pointryagin numbers”. Theresult of this section can implies this statement to be true when we ignore the torsionpart.

(2.1) Transversality. Let f :X → Y be a smooth map between two smooth mani-folds, A a subset of X, and Z a submanifold of Y . f is said to be transversal to Z onA (denoted by f |∩A Z or write f |∩Z on A), if ∀x ∈ f−1(Z) ∩ A, have the followingsurjectivity condition

Df(TxX) + Tf(x)Z = Tf(x)Y.

We dorp A if A = X. In the case Z reduced to be a point, and A = X, this isjust the usual definition of a regular value. The reason to study the transversality isdue to following observation: If f |∩Z and Z is of codimension k in Y , then by theimplicit function theorem we have f−1(Z) a smooth submanifold of X of codimensionk (empty set is allowed). And the normal bundle of f−1(Z) in X is isomorphic to thepull back bundle f∗(N) of the normal bundle N of Z in Y .

Now we begin to prove the transversality theorem which says that any smoothmap can be approximated by transversal ones.

Theorem (Thom). Let f :X → Y be smooth, and f |∩A Z, where A is a closedset of X and Z is a submanifold of Y , let d be any metric compatible with theunderlying topology of Y and ε > 0, then there is a smooth map g:X → Y such thatg |∩Z, d(f(x), g(x)) < ε and f |A = g|A.

Proof. There are several steps:

(1) Since transversality is an open condition, there is an open set U ⊂ A such thatf |∩U Z. Now we will choose some appropriate coordinate coverings to reduce theproblem to the case of Euclidean sapces. First of all, let Y0 = Y −Z and Yi be chartscover Z with Z ∩Yi coordinate planes. Secondly, choose Vi charts such that Vi is arefinement of both X −A,U and f−1(Yi). Since we can not modify f on A, weneed an even more refine covering of those Vi that do not interesect A. We choose(by paracompactness) Wi a family of locally finite relatively compact charts withWi finer than Vj. Since the index will no longer be preserved, we re-indexing Vi

(so some Vi may equals Vj). Finally we disgard those Wi which are contained in U .Still denote the final family by Wi. It is claerly a covering of X − U .

(2) We will construct fi inductively such that

a) d(fi(x), fi−1(x)) ≤ ε/2i ∀x ∈ X.b) fi ≡ fi−1 outside a compact neighborhood of Wi in Vi.c) fi |∩Z on f−1

i (Z) ∩ (W1 ∪ · · · ∪ Wi).

Once this is done, we get lim fi = g: X → Y the desired smooth map. Notice

10

there is no limit process since the covering is chosen to be locally finite. Actuallywhen X is compact, Wi is a finite family. We remark also that in the whole process,we never change the value of f near A.

(3) For each i ≥ 1, fi−1(Vi) ⊂ Yj(i) for some j(i). Since Vi, Yj are all coordinatecharts, by the induction steps, we only have to treat everything in the Euclideanspases. Namely, K ≡ W ⊂ V ⊂ Rn, Z = Y ∩Rq ⊂ Y ⊂ Rp, and f : V → Y smoothwith f |∩Z on a relatively closed set S (S is to be thought as W1 ∪ · · · ∪ W i−1).

Consider the projection p:Rp → Rp−q, then Z = p|−1Y (0), so p f : V → Rp−q

has 0 as a regular value if and omly if f is transversal to Z, thus it suffices to considerthe case Z = 0, that is, a point.

(4) Since 0 is now a regular value of f on S, what we have to do is to modify it tobe regular on all S ∪K. Using a smooth partition of unity, construct a smooth mapλ: V → [0, 1] which equals 1 on a neighborhood of K and equals 0 outside a compactneighborhood K ′ of K in V . By sard’s theorem, there are always points arbitrarilynear 0 and are still regular values, pick y with |y| < ε and consider

g(x) = f(x)− λ(x)y.

then by the definition of λ, clearly have

1. g has 0 as a regular value (ie. g |∩ 0) on K.2. g ≡ f outside K ′.3. |g(x)− f(x)| < ε.

Since y can be chosen arbitrarily near 0 and | ∂∂xi

λ| is globally bounded, we canmake g,Dg near f,Df uniformly. By 2. g |∩ 0 on S − K ′, so we only have to careabout the set (S ∩K ′) ∩ g−1(0). Notice S ∩K ′ is a compact set, so when |y| smallwe have Dfx onto ⇒ Dgx onto. So g |∩ 0 on S ∪K as required. Qed.

(2.2) Thom homorphism: τ : πk+n(T(ξ)) → Ωn.

Let ξ be an oriented k plane bundle over the base manifold B equipped with abundle metric, we define the Thom space T(ξ) of ξ to be D(ξ)/S(ξ), that is, identifyall vectors with length ≥ 1 to a single point, we always denote this point by t0 andrefer to it the base point. We notice that different metrics give the same Thom space.When B is compact, T(ξ) is just the one point compatification of the total space E(ξ)of ξ. Actually, this point of view is used more often, and for simplicity, we assumethat B is compact.

Given a map f : Sk+n → T(ξ) with ∞ 7→ t0, have f−1(B) ⊂ Sk+n − f−1(t0), sof−1(B) can be considered to be in some open set U with U ⊂ Sk+n−f−1(t0) ' Rk+n.We can operate everything inside U , that is we will not change f outside U . Noticethat by definition f is surely transverse to B outside U . By transversality theorem,we may modify f in its homotopy class and keep it unchanged outside U , so we may

11

assume f |∩B, then f−1(B) is an n dimensional closed oriented submanifold of U .That is we get an element of Ωn.

To prove that τ is well defined, suppose F :Sk+n × [0, 1] → T(ξ) be a homotopysuch that F (x, 0) = f0(x), F (x, 1) = f1(x). We may choose F such that F (·, [0, 1

3 ]) =f0, F (·, [ 23 , 1]) = f1. Apply the transversality theorem to X = F−1(E(ξ)) ∩ (Sk+n ×(0, 1)), get a map F coinside with f0 near 0, and coinside with f1 near 1, and F−1(B)is a manifold with boundary f−1

1 (B)− f−10 (B). So τ is well defined as a set map.

To prove τ is a group homomorphism we go back to the definition of the additionrule in homotopy groups, clearly it corresponds to the disjoint union of manifolds inour construction of τ , say, one in the north half sphere and one in the south halfsphere. This complete the proof.

In the following consider in particular B = Gk(Rk+p) and ξ = ξkp the oriented

universal k plane bundle. Denote the bundle projection by π. Then we have

(2.3) Theorem (Thom). τ is an ismorphism for k ≥ n + 2 and p ≥ n.

Proof. Surjectivity: we only need k ≥ n, p ≥ n in this part.

Let [M ] ∈ Ωn. By Whitney embedding theorem, M can be embeded in Rk+n.(Whitney’s theorem is much easier to prove if k ≥ n+1.) Let ν be the normal bundleof M in Rk+n. By the existence of tubular neighborhood of M , denoted by U , wecan construct the generalized Gauss map:

g:U ' E(ν) −→ E(ξkn) → E(ξ)

Complete g to be a map g: Sn+k → T(ξ) by sending all points outside U to t0. Thendo the same as before, can assume g to be transversal to B. It is then clear thatτ([g]) = [M ].

Injectivity: This is much more involved than the surjective part. In fact wedon’t need this part in the remaining sections. Even more, when we finally prove theThom’s cobordism theorem for Ω⊗Q, we obtain the injectivity for those homotopyclasses that are not torsion elements as a corollary. Anyway, for completeness we givethe proof.

So let g: Sk+n → T(ξ) be transversal to B and g−1(B) = M = ∂N , have toshow g ∼ constant map. M is an closed n manifold in Rk+n ⊂ Sk+n.

Claim 1: Can embed N in Rk+n × [0, 12 ] such that N ∩ (Rk+n × [0, 1

4 ]) = M × [0, 14 ]

(it’s true that near ∂N , N has a product structure, the collar theorem, but how oneget such embedding globally?)

Assume this, then let V be a tubular neighborhood of N in Rk+n × [0, 1] withd(V,N) < ε and then U = V ∩ (Rk+n × 0) is the tubular neighborhood of M inRk+n.

12

Claim 2: We can deform the map g such that g|U : U → E(ξ) is a bundle map.

Once this is done, g can be extended to be a bundle map g:V → E(ξ) by theclassification theory of vector bundles (cf. [Hirsh] p.100). Define g on the wholeRk+n × [0, 1] by sending the complement of V to t0. Then g gives the desired homo-topy from g to the constant map t0.

To prove claim 1, let h: M × [0, 1) diffeomorphic onto a neighborhood of ∂N (thecollar). Define β:R → [0, 1] to be a smooth increasing cut-off function such thatβ(x) = 0 for x < 1

2 + ε and β(x) = 1 for x > 1− ε. Then define h1: N → Rk+n× [0, 12 ]

by

h1(y) =

(x, 12s) for y ∈ h(M × [0, 1

2 ])

p for y 6∈ h(M × [0, 1))

(1− β(s))(x, 14 ) + β(s)p for 1

2 ≤ s < 1

where p = (x0,12 ) with x0 ∈ M an arbitrary point. Although h1 is a smooth map, the

resulting image h1(N) is never a manifold. It looks like the roof of an old fashionedhouse. But it still contains the collar M × [0, 1

4 ].

Since k ≥ n+2, k+n+1 ≥ 2(n+1)+1 = 2 dim(N)+1. We can apply the smoothapproximation theorem which says in such a dimension, embeddings are dense in thespace of smooth maps. (cf. []) So we find a map

h2: N → Rk+n+1

and h2 ≡ h1 in h(M × [0, 14 ]). Claim 1 is proved.

To prove claim 2, we first deform the map g such that it sends all points outsideU to t0 and keep g unchanged on a smaller neighborhood of M = g−1(B). Recallthe universal bundle π: E(ξ) → B and denote the bundle projection of U → M by p.Let x ∈ U , by using the linear structures on Up(x), which is induced by the normalbundle ν (it is just a rescaling of each fiber), and the linear structure on ξπ(g(x)), candefine for s ∈ [0, 1],

H(x, s) =g(s~x)

s,

with H(x, 0) = Dgπ(x)(~x), which is surely a linear map on the normal space over p(x)(since Dg is linear on the whole tangent space), and this linear map is an isomorphismby the transversality of g Thus g0 ≡ H(·, 0) defines a bundle map U → E(ξ) andH(·, s), s ∈ [0, 1] defines the homotopy from g ≡ g1 to g0. The construction behaviorwell on points near ∂U , so claim 2 is proved. This complete the proof of this “Thomisomorphism”. Qed.

(2.4) Rational Homotopy Groups.

Although we have established the ismorphism of Thom homomorphism, in gen-eral it is still difficult to compute the higher homotopy groups. The Hurwicz theorem,

13

which established a good relation between the homology group and homotopy group,is such a type of theorem that we need now, but its original form needs strong as-sumptions which are surely not satisfied in our situation. Anyway, since we onlyconcern with the non-torsion part Ω⊗Q, so we only have to compute the so-calledrational homotopy groups. (But it is still not possible to include the theory here.) Inthis case, we have (cf. [DFN] p.129)

Theorem. Let X be a finite CW complex which is r-connected with r ≥ 1, thenafter ⊗Q, the Hurwicz homomorphism

πi(X)⊗Q −→ Hi(X;Q)

is an isomorphism for i ≤ 2r.

In order to apply this to our case, we must show T(ξ) have some higher connec-tivity, but since B = Gk(Rk+p) whose cell decomposition is well known, say el is anopen j cell of it, then the inverse image π−1(el) is clearly an open j + k cell of T(ξ),Together with the zero cell t0, we obtain a CW complex structure of T(ξ) withoutcells of dimension between 0 and k, that is, T(ξ) is (k− 1)-connected. So we get, forn ≤ k − 2:

πk+n(T(ξ))⊗Q ' Hk+n(T(ξ);Q).

Now we have the Thom isomorphism

Hk+n(T(ξ), t0;Z) ' Hk+n(D(ξ), S(ξ);Z) ' Hn(B;Z).

So by connecting the three ismorphisms, we finally obtain for k ≥ n + 2,

Ωn ⊗Q ' Hn(B;Q).

As noted (1.5), by letting k large enough, Hn(B;Q) is freely generated by Pontryaginclasses of the universal bundle ξ, so it is zero when 4 6 |n, and is of dimension p(m)if n = 4m. Toghther with the calculation on products of P2n(C)’s. which shows thevarious products of P2nj (C)’s with j1 + · · · + jr = m are all in distinct cobordismclasses, since the total number of such products is exactly p(m), we finally get the

Thom’s Cobordism Theorem:

Ω⊗Q = Q[P2(C),P4(C),P6(C), . . .]

Qed.

Remark. In the proof we do not need each step to be isomorphism. We only need

p(r) = dimHn(B;Q) = dimHk+n(T(ξ), t0;Q)

≥ dim(πk+n(T(ξ))⊗Q) ≥ dim(Ωn ⊗Q)

≥ p(r)

So we need only the surjective part of the Thom homomorphism τ (which is the easierpart), and the injective part of the rational Hurwicz homomorphism.

14

§3 Hirzebruch’s signature theorem

(3.0) Genera. Let R be a commutative ring over Q. An R-genus is defined to bea ring homomorohism φ : Ω ⊗Q → R. Since as a ring, Ω ⊗Q is generated by theprojective spaces P2i(C), i ≥ 0, we only have to know the value of φ on these spaces.This section is in fact a realization of this simple observation.

(3.1) Signature. Let M be a 2n dimensional oriented closed manifold. By Poincareduality, the intersection form qM on Hn(M,Z) is a nondegenerate pairing, it is alter-nating when n is odd, symmetric when n is even, in the later case, it is unimodular.Since we have some classification theory of integarl quadratic forms and alternatingforms, it is very hopeful that the study of the intersection form will provide someimportant information about the topology of M . Now we define the signature σ(M)of M to be zero when 4 6 | dim(M) and to be the signature of qM (that is, the numberof positive eigenvalues σ+ minus the number of negative eigenvalues σ−). We noticethat we can also define the signature by using cohomology, the intersection pairingis then the cup product and evaluated on the fundamental class [M ]. Using deRhamcohomology, the intersection pairing then can be viewed as the integration of wedgeof closed differential forms over M . This view-point will be very useful in some cases.

As an example, we will now show that the signature of manifolds is a genus.

Lemma. The signature is a Z-genus, that is,

(1) σ(V + W ) = σ(V ) + σ(W ), σ(−V ) = −σ(V ).(2) σ(V ×W ) = σ(V )× σ(W ).(3) σ(M) = 0 if M bounds some compact oriented manifolds.

Proof. (1) is clear since the intersection form of disjoint union of manifolds splits asthe direct sum of the individual ones.

(2) We use the cohomology with coefficient in R. Let M4k = V n ×Wm, by theKunneth formula,

H2k(M) =⊕

s+t=2k

Hs(V )⊗Ht(W ).

Let vsi , wt

j be the basis of Hs(V ),Ht(W ), such that vsi v

n−sj = δij , w

tiw

m−tj = δij

for s 6= n2 , t 6= m

2 . (We can not do this in the middle dimension in general) LetA = H

n2 (V ) ⊗H

m2 (W ) when n,m are both even, and let A = 0 in other cases. Let

B = A⊥ in H2k(M), that is, the space spanned by elements not in A (elements inA can not be orthogonal to A because the intersection product on A is the tensorproduct qV ⊗ qW which is also nondegenerate). The set

vsi ⊗ wt

j |s + t = 2k, s 6= n

2(t 6= m

2)

is an orthogonal basis of B. We will show in fact σ(B) = 0, and σ(A) = 0 when4 6 |n(4 6 |m). Then σ(M) = σ(A) + σ(B) = σ(A) = σ(V )× σ(W ) as required.

15

To prove σ(B) = 0, observe (vsi ⊗ wt

j) · (vs′i′ ⊗ vt′

j′) 6= 0 only when i = i′, j = j′

and s + s′ = n, (t + t′ = m), and it equals ±1 in this case. The intersection matrixthus has ±(

01

10

)as its building blocks. It is then clear that σ(B) = 0.

If 4 6 |n (so 4 6 |m), we have to show σ(A) = 0. This amounts to say that thesymmetric bilinear form obtained from the tensor product of two alternating formsmust has zero signature. By the structure theorem of nondegenerate alternating formover R, we know it has a matrix representation with

(0−1

10

)as the building block.

And the tensor product of two such matrix gives

0 1 0 0

1 0 0 0

0 0 0 −1

0 0 −1 0

so the intersection matrix on A is a direct sum of the above matrix, it clearly haszero signature.

(3) Let i : M4k → W 4k+1 be the boundary inclusion. We have the followingcommutative diagram:

· · · → H2k(W ) i∗−→ H2k(M) ∂∗−→ H2k+1(W,M) → · · ·yy

y· · · → H2k+1(W,M) ∂∗−→ H2k(M) i∗−→ H2k(W ) → · · ·

The verticle maps are all isomorphisms by Poincare-Lefschetz duality. Let A =Im i∗, B = Ker i∗. Since i∗, i∗ are dual vector space maps, A is dual to H2k(M)/B

under the duality between H2k(M) and H2k(M). So dim(A) = dimH2k(M) −dim(B). By the exactness of the above diagram, dim(A) = dim(B), so we getdim(A) = dim(B) = 1

2 dimH2k(M). Now by Stokes’ theorem, for any ω ∈ A,

∫

M

(i∗ω)2 =∫

W

d(i∗ω ∧ i∗ω) = 0

so the zero cone of the intersection form contains A. Since A has half the dimension,this can happen only when σ+ = σ−, that is, σ(M) = 0. This complete the proof ofthis lemma. Qed.

(3.2) Multiplicative sequences. Let φ be a genus, it is natural to consider thegenerating power series

Pφ(x) =∞∑

n=0

φ(P2n(C))x2n.

16

In the case of signature, It is easy to see that σ(P2n(C)) = 1, so the generatingpower series is Pσ(x) = 1 + x2 + x4 + · · · = 1/(1 − x2). What we want to do nowis to find a polynomial K(p) = K(p1, p2, . . .) with (formal) Pnotryagin classes as itsvariables, such that K(p(M))[M ] = φ(M), the left hand side means we substitute thepontryagin classes of M into the formal variables p, then evaluate at the fundamentalclass [M ] by using the term of degree n when dim(M) = 4n, it is a linear combinationof Pontryagin numbers. Unfortunately this polynomial can not be obtained verydirectly from Pφ. The method to do this was invented by Hirzebruch (also thedefinition of genera is due to him).

Let’s start from an arbitrary even power series Q(x) = 1+ q1x2 + q2x

4 + · · · withcoefficients in R (we use even power series in order to make some expression clear,see later), and form

∏n

i=1Q(xi) = 1 + q2

∑n

i=1x2

i + · · ·= 1 + K1(p1) + k2(p1, p2) + · · ·

+ Kn(p1, . . . , pn) + Kn+1(p1, . . . , pn, 0) + · · ·

here we regard the variables xi’s as have weight 2, and the weight 4r parts of the firstproduct is a symmetric polynomial of x2

i . Let pi be the i-th elementary symmetricpolynomial of x2

i , then we can transformed the weight 4r part into the an uniguepolynomial of these pi’s, this is the definition of Kr. By the theory of symmetricpolynomials, we know that Kr is independent of the number of variables n if n ≥ r.In the following, we will always assume n is large enough, in fact we can take n = ∞,the resulting series is denated by K(p), here the variable p denates 1 + p1 + p2 + · · ·,the formal total pontryagin classes. In the special case p = 1 + p1 = 1 + x2, we haveK(1 + x2) = Q(x).

Lemma. The function φQ(M) := K(p(M))[M ] is an R genus.

Proof. (1) We first check that if M4n = ∂W 4n+1 then φQ(M) = 0. In this casewe have TW |M = TM ⊕ ε, where ε is the trivial normal bundle of M in W . Thenp(M) = p(W )|M , that is, all pontryagin classes of M are restriction of those of W ,since any Pontryagin number pI of M is the integration of the corresponding closed(wedge of) Pointryagin form ωI on M = ∂W , by Stokes’ theorem,

pI =∫

M

ωI =∫

W

dωI = 0

where ωI is the Pontryagin form on W such that ωI |M = ωI . This proves (1).

(2) We have to show the adivitity and multiplicativity. The aditivity is obvious,the multiplicativity is more subtle. We have to prove: Let p′ = 1 + p′1 + p′2 + · · ·,p′′ = 1 + p′′1 + p′′2 + · · ·, (p′k, p′′k of weight 4k). If p = p′p′′ = 1 + p1 + p2 + · · · by

17

collecting corresponding terms, then

K(p) ≡ K(p′p′′) = K(p′)K(p′′).

(This is why people call Kr the multiplicative sequences.) Let Q(x) =∑∞

i=0 qix2i as

before. For any partition I = i1, . . . , ir of n, let qI = qi1 · · · qir , and let

sI(p1, . . . , pn) =∑

x2j11 · · ·x2jr

r

where the sum is over all distinct permutations (j1, . . . , jr) of I. Since it is symmetricin x2

i ’s, it is uniquely represented by a polynomial in p′is, this is the definition of sI .Then we have

sI(p′p′′) =∑

H,J=IsH(p′)sJ(p′′),

which is just a partition of standard monomials into two parts of fewer variables.Again by comparing corrseponding terms, we have

Kr(p1, . . . , pn) =∑

IqIsI(p1, . . . , pn).

Take summation over all n, we get

K(p′p′′) =∑

IqIsI(p′p′′)

=∑

I

∑H,J=I

qH,JsH(p′)sJ (p′′)

=∑

H,JqHqJsH(p′)sJ(p′′)

=∑

HqHsH(p′)

∑J

qJsJ(p′′)

= K(p′)K(p′′).

This is what we want. Qed.

Remark. Actually we have already proved: Given any even power series Q(x)begins with 1, there exists an unique multiplicative sequence K such that K(1+x2) =Q(x). We have proved the existense, for the uniqueness, use a formal decompositionp =

∏i(1 + x2

i ), then K(p) =∏

i Q(xi). This is exactly our construction.

(3.3) We have the following very important lemma. Let f(x) = x/Q(x) = x + · · · ∈R[[x]], it is clearly invertible. Let g(y) = f−1(y), then:

Lemma. g′(y) =∑∞

n=0 φQ(Pn(C))yn.

Proof. Strat from p(Pn(C)) = (1 + p1)n+1 (cf. (1.1)), have K(p) = K(1 + p1)n+1 =

18

Q(p1)n+1. Thus

φQ(Pn(C)) = Q(p1)n+1[Pn(C)]

= coefficient of xn in Q(x)n+1 =(

x

f(x)

)n+1

= residue at 0 of1

f(x)n+1dx

=1

2πi

∫

C

1f(x)n+1

dx =1

2πi

∫

f(C)

g′(y)yn+1

dy

= coefficient of yn in g′(y)

Since f(x) = x+ · · ·, when the circle C around 0 is small enough, f(C) is also a curvearound 0 with winding number 1. In fact we even don not need f to be convergentsince the above argument is essentially a sequence of substitution of formal powerseris. This complete the proof. Qed.

This lemma says g′(y) is exactly the generating power seris of φQ! So if we startfrom a genus φ, with generating power series Pφ, then we set g =

∫Pφ with constant

term zero, and set f = g−1, then Q = x/f(x) is the fundamental power series whoseassociated multiplicative sequence K defines the genus φQ = φ.

Apply this to the signature σ, since Pσ(y) = 1 + y2 + y4 + · · · = 1/(1 − y2), sog(y) =

∫1

1−y2 = tanh−1(y), and then f(x) = tanh(x), finally we get the fundamentalpower series Q(x) = x/ tanh(x). Hirzebruch gave the corresponding K =

∑Kr a

name, the “L polynomials”: L =∑

Lr, and called signature the “L genus”.

To actually compute the L polynomials Lr’s, we need the Taylor expansion ofx

tanh(x) . It reads

x

tanh(x)= 1 +

13x2 − 1

45x4 + · · ·+ (−1)k+1 22kBk

(2k)!x2k + · · ·

where Bk’s are the Bernouli numbers ([MS] App.B), the first few terms are 1/6, 1/30,1/42, 1/30, 5/66, 691/2730, · · ·. Then a direct computation gives the L polynomials:

L1(p1) =13P1

L2(p1, p2) =145

(7p1 − p21)

...

Since we will need only L2 in the following sections, we don’t list the higher L

polynomials here.

19

As a conclusion, we have already established the

Hirzebrich Signature Theorem: σ(M) = L(M).

(3.4) Remarks (about the theorem.)

(1) Using other Q(x) we can get other interesting genus (cf. [ ]), for example takeQ(x) = x/2

sinh(x/2) , we get the so-called “A-roof genus”, A(M), which appears inthe Atiyah-Singer Index Theorem.

(2) The Signature Theorem (as well as the Gauss-Bonnet-Chern Theorem and theHirzebruch’s Riemann-Roch Theorem) is in fact a special case of a more generaltheorem, namely Atiyah-Singer Index Theorem, but historically the Index The-orem was first proved by a “twisted version” of Hirzebruch Signature Theorem.Now there are several new methods to prove the Index Theorem without involvethe Signature Theorem, so it is actually a corollary of the Index Theorem.

20

§4 An Exotic Seven Sphere.

(4.0) As showed in §1, the Hopf fibrations are typical examples of spheres whichcan be realized as a “sphere fibered by sphere”. Milnor in his 1956’ paper ([M1])described a lot of S3 bundles over S4, with total space the seven sphere, but withdifferent smooth structures from standard S7. In this section we will describe these“exotic spheres”. We will see the power of the Signature Theorem. Although we havealready discussed some topological properties of such sphere fibration, we will startwith the most naive way in (4.1) and put the results of §1 into consideration after(4.6).

(4.1) Let D4+, D4

− denote the upper and lower hemi-sphere of S4, then any (oriented)vector bundle on S4 can be described as identifying two trivial vector bundles overD4

+, D4− (contractible space!) along their common boundary, the equator S3, this

identification is given by a map f : S3 → SO(4), and this map is unique up tohomotopy.

Now consider fhj : S3 → SO(4) by the rule: (identify R4 = H and usingquaternion multiplication)

fhj(u) · v = uhvuj .

This defines an SO(4) bundle Ehj on S4 with fiber R4, Let ξhj be the sphere bundleof Ehj , that is, ∂D(Ehj), we will show when h+j = 1, (so h−j = k is odd), the totalspase of ξhj , denoted by M7

k , is a topological seven sphere. (In (1.3) we have alreadyproved these two numbers h + j, 2(h − j) correspond to e, p1.) Since h + j = 1, k

determines the pair (h, j) uniquely, so in the following we write the lower indices ask instead of hj.

(4.2) The idea is to construct a “Morse function” f on M7k with exactly two critical

points. (Recall that a Morse function is a real valued smooth function with discretecritical points and the hessian of each critical point is a nondegenerate quadraticform.) Once this is done, since our manifold is compact, this implies the two criticalpoints, say y0, y1, are exactly the minimal and maximal points of f . We may assumethat f(y0) = 0, f(y1) = 1. By considering the gradient flow:

dx

dt= ∇f(x).

we know that f−1([0, a]) are all diffeomorphic for 0 < a < 1. When a is small,by Morse lemma, there exists a coordinate system (x1, · · · , x7) about y0, such thatf(x1, . . . , x7) = x2

1 + · · ·+ x27, so f−1([0, a]) is clearly diffeomorphic to the seven disk

D7. By the flow, we conclude that M7k − y1 = f−1([0, 1)) is diffeomorphic to D7, so

M7k is a smooth “topological sphere”. (Remark: the above argument surely works

for all dimensions.)

21

(4.3) Now we will show M7k can be realized as an identification of two R4×S3 along

(R4 − 0)× S3 via the diffeorphism g of (R4 − 0)× S3 :

g: (u, v) 7→ (u′, v′) =(

u

|u|2 ,uhvuj

|u|)

this makes sense because (we should verify that v′ ∈ S3)∣∣∣∣uhvuj

|u|

∣∣∣∣ =|u|h|v||u|j

|u| =|u|h+j

|h| = 1.

Here h + j = 1 is essentially used. The formula u′ = u/|u|2 is nothing but thecoordinate change of the two stereographic projections: S4 → R4, one from thesouth and one from the north. It suffices to check this in the 1 dimensional case, andit is trivially done by the similar triangle rule.

To check the glueing really gives M7k , we consider the equator S3, that is, |u| =

|u′| = 1, in fact u = u′. The map g restrict on this equator then defines a mapg : S3 → SO(4) by g(u)v = uhvuj . which is exactly the map fhj , since any bundleover S4 is classified by this map as mentioned before, this space is exactly M7

k .

(4.4) To construct the desired function f on M7k , consider the following two coordi-

nate charts, (u, v) and (u′′, v′), where u′′ = u′(v′)−1. Define f : M7k → R by

f(x) =Re(v)√1 + |u|2 =

Re(u′′)√1 + |u′′|2

Check equality: later term =

Re(u′(v′)−1)√1 + |u′|2 =

Re(|u|u′(v′)−1)√1 + |u|2

(· · ·) = u(|u|v′)−1 = u(uhvuj)−1 = u · u1−jv−1u−h = uhv−1u−h. When we repre-sent H as 4 × 4 matrix over R, we have Re(x) = 1

4 trace(x), so Re(uhv−1u−h) =14 trace(uhv−1(uh)−1) = 1

4 trace(v−1) = Re(v−1). Since |v| = 1, v−1 = v, we haveRe(v−1) = Re(v) = Re(v). So left = right.

Now we consider the critical points. From the right expression of f we easilysee that no critical points exists in the chart (u′′, v′): the function x1/

√1 + |x|2

in the direction x1, so ∂1f(x) > 0. Hence all critical points lie in the (u, v) chart,and in fact lie in the set (0, v). But in this set f(x) reduces to be Re(v) (the heightfunction) on S3, the unit sphere of H. So the critical points are clearly the two pointsv = ±1, that is, (0,±1). By (5.2), M7

k is a topological sphere.

(4.5) Now we will show some M7k are not diffeomorphic to the standard S7. Suppose

M7k is diffeomorphic to S7, then we can attach an standard 8 dimensional disk D8

22

to the boundary of the total space of the disk bundle D(Ek) along M7k ' S7 via the

assummed diffeomorphism. Denote the resulting closed 8 dimensional manifold byW 8

k . We compute the signature σ(W 8k ) as follows:

We notice W 8k is nothing but the Thom space T(Ek), by the Thom isomorphism

theorem, we get (by excision and ∪e(Ek)):

Hi(S4) ' H4+i(D(Ek), S(Ek)) ' H4+i(T(Ek), t0).

The integral cohomology groups of W 8k therefore equal Z in dimension 0, 4, and 8,

and zero in other dimensions. Actually, it is Z ⊕ Ze(Ek) ⊕ Ze(Ek)2. This impliesσ(W 8

k ) = ±1. Choosing an orientation, may assume σ(W 8k ) = 1. Now apply the

Hirzebruch signature theorem, we have

1 = σ =7p2 − p2

1

45.

Thus all we have to do now is to compute the pontryagin classes of W 8k .

(4.6) Recall the result of (1.3), which says

e(Ehj) = (h + j)u, p1(Ehj) = 2(h− j)u.

In the present case, e(Ek) = u and p1(Ek) = 2ku.

To pass this result to W 8k , denote by π:Ek → S4 the bundle projection, we always

have TEk ' π∗(TS4) ⊕ π∗(Ek), so apply the Whitney sum formula and naturalityas in (1.1), and p(TS4) = 1, we get p(TEk) = π∗p(Ek) and p1(TEk) = π∗p1(Ek) =π∗(2ku) = 2ku = 2ke(EK). So p2

1(TW 8k ) = p2

1(TEk) = 4k2. This is true because ofnaturality, the pontryagin classes of Ek are the restriction of Pontryagin classes ofW 8

k which is a smooth closed manifold and only have one more point than Ek, andso have the same value when evaluted on the fundamental class.

Put it into the signature formula, we get 4k2+45 = 7p2 ≡ 0 (mod 7), (Pontryaginnumbers are integers!) this implies 4(k2 − 1) ≡ 0 (mod 7) and so k ≡ ±1 (mod 7).But k can be any odd integers! We get a contradiction for those Ek with k 6≡ ±1(mod 7), that is, the hypothesis in (4.5) is wrong: M7

k is not diffeomorphic to S7!

(4.7) Final remarks.

(1) There is a quick way to prove: E(ξhj) is a topological sphere if and only ifh + j = 1. First, calculate its cohomology by using Gysin sequence as in (1.1),but then we need Smale’s throrem on the generalized Poincare conjecture toconclude the result.

(2) There is still something not so good: although there are a lot of exotic sevenspheres, they may be diffeomorphic! For example, Are M7

3 and M75 diffeomor-

phic? In Milnor’s original approach, he put everything in the category of man-ifolds with boundary, and from this he constructed a diffeomorphism invariant

23

which is exactly k − 1 (mod 7), in this way he can distinguish some of theseexotic spheres.

(3) But there is still another even more sophesticated question: How many smoothstructures can a topological sphere have? The following section is a summary ofKervaire and Milnor’s result to this question. I do not include the proofs here.Instead, I will describe Brieskorn and Hirzebruch’s construction of these exoticspheres (including higher dimensional exotic spheres) in later sections.

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§5 Summary of Kervaire/Milnor’s results ([ ])

All manifolds are smooth oriented with dimension ≥ 5 and all bundles are smoothoriented in this section. Two manifolds M1, M2 are said to be h-cobordant ifM1 − M2 = ∂W and M1, M2 are both deformation retracts of W . It defines anequivalence relation on manifolds.

The connected sum M1 =|| M2 is well defined up to orientation preserving dif-feomorphism. It is commutative, associative and compatible with the relation ofh-cobordism. All closed n-manifolds form a commutative monoid under =|| , withidentity the standard sphere Sn. We are interested in those closed manifolds whichhave the same homotopy type as a sphere, called the homotopy spheres. we have

(5.1) The set of h-cobordant classes of homotopy n-spheres form an abelian groupunder connected sum, denoted by Θn.

(5.2) Θn is finite.

By Smale’s h-cobordism theorem, which says “h-cobordant ⇒ diffeomorphic”,and the truth of generalized Poincare conjecture of dimension ≥ 5, we have

(5.3) Θn is the group of all smooth structures on Sn.

We will not actually use (5.2), (5.3) in the sequal, what we really concern isa smaller subgroup bPn+1 ⊂ Θn, to define it, we need the concept of parallelizablemanifolds. M is called parallelizable if TM is trivial, and called s-parallelizable ifTM ⊕ ε is trivial, where ε is the trivial line bundle over M .

We need the following basic facts:

(5.4) Lemma. Let ξ be a k plane bundle over Mn, k ≥ n. If ξ ⊕ εr is trivial, then

ξ is trivial.

Proof. Only have to consider the case r = 1. The isomorphism ξ ⊕ ε ∼= εk+1 givesrise to a bundle map

ξ −→ γk

y yM

f−→ Sk

where γk is the universal oriented k plane bundle over the oriented grassmannianG(k, k + 1) = Sk. Since k ≥ n, f is null homotopic, so ξ is trivial.

(5.5) Corollary. Let Mn be a submanifold of Sn+k, k ≥ n, then M is s-parallelizable

iff the normal bundle is trivial.

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Proof. The bundle T ⊕N ⊕ ε is always trivial, where ε is the (trivial) normal bundleof Sn+k in Rn+k+1. If the normal bundle N is trivial, apply (5.4) to (T ⊕ ε)⊕N , weget T ⊕ ε is trivial. Conversely, if M is s-parallelizable, apply (5.4) to N ⊕ (T ⊕ ε),we get that N is trivial.

(5.6) Corollary. A connected manifold with nonempty boundary is s-parallelizable

iff it is parallelizable.

Proof. We need Morse Theory to conclude that a smooth manifold admits a CWcomplex structure, and if the boundary is not empty, the dimension of this CWcomplex can be choosen to be < n = dim(M). In the proof of Lemma (5.4), we needonly k ≥ the CW complex dimension, so the result follows.

(5.7) Corollary. Any oriented submanifold M of Rn with ∂M 6= ∅ is parallelizable.

Proof. Such manifold has trivial normal bundle. If we take n large, then M becomess-parallelizable by (5.5). So it is parallelizable by (5.6).

Now we define the set bPn+1 ⊂ Θn: it consists of those homotopy n-spheres whichbound a parallelizable manifold. This condition depends only on the h-cobordismclass (This is clear if we use h-cobordism theorem). The main property we shouldknow is that bPn+1 is a finite cyclic group and its members can be classfied by simpletopological invariant. For simplesty we only consider bP4m, (m ≥ 2), the collectionof allparallelizable 4m manifolds with ∂M = (4m − 1)-sphere. The correspondingsignatures σ(M) form a non trivial subgroup of Z, denote it by σmZ where σm ≥ 0.Then the following structure theorems are known:

(5.8) Let Σ1, Σ2 be two 4m−1 homotopy spheres, ∂Mi = Σi, with Mi parallelizable.Then Σ1 is h-cobordant to Σ2 if and only if σ(M1) ≡ σ(M2)(mod σm). In anotherwords, the signature (mod σm) classifies the smooth structures on S4m−1.

So bP4m is a subgroup of Z/σmZ, later we will see that all such parallelizablemanifolds have signatures ≡ 0 (mod 8), so the order of bP4m divides σm/8. In factit equals, and its value is also determined by Bernoulli numbers:

(5.9) The determination of bPn is:

(1) bP2k+1 = 0

(2) bP4m−2 = Z/2Z if m 6= 1, 2, 4

(3) bP4m is cyclic of order σm/8, it equals

εm22m−2(22m−1 − 1) numerator(

4Bm

m

).

where εm = 1 if m is odd, = 2 if m is even.

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