ClUlIOO a Math ematical Journal 
Vol. 6, ~ 1, {11 6 -1 2 1). March 2004. 

On Brouwer's Fixed Point Theor em' 

N . Tarkhanov 
Un ivcrsitii.t Potsdam 

ln stitu t für Mathematik 
Postfach 60 15 53 

1441 5 Potsdam 
Ge rm any 

tark ha11 ov@math .uni -potsdam .de 

ABSTRACT 
Thc pa ¡>cr presc nts a u cx pli cit fo rmul a for t hc numb er oí fixcd points of a C"" 

nrn p oí a segmc11t [a , b] C R. Whi! c t hc formula can be derived frorn thc Lcíschct:1. 
flx«I point thoo rcm for gc ncrnl CW -comp lcxes , t hc nc w p roof is ins truct ivc and 
hi ghli ght.s th c co 11 t ribu tiom1 of dcgcncratc fi xcd points . 

1 Introduction 

111 19 10 Orouwe r (cf. IB ro l 2]} provc<l that a ny continuous map f: S --t S of a n 
n -dimensional slmplex has at least one fixed point p E S , i.e., J(p ) = p. This rcsult 
cxtc nds to cont.inuous maps of any compact convex bod y in a finite dimensional topo-
luf(kal v('C tor spacc and has numcrous app lications to proving exi stcnce t heorcms for 
divcrst~ equatio ns. Morcovc r, th c flxcd point t hcorem geucralises to infinite d imen-
~i01ml topologica1 vec to r spaccs. Whcn thi nking ovcr this res u\t, Lcfschct.z jLef'26] 
showcd a formul a whi ch cxprcsscs t hc numb cr of fixcd points of a map of a n oric nt ab lc 
lopological man ifold th rough t he traces o f thc cor res ponding pull-back operators 0 11 
t hc cohomology. Alt. ho ugh a fixcd point t hcorcm a la Lc fschetz has nowadays bccu 
k11own íor general CW -complcxcs , cf. [Dol72J , it gives no explicit. dcscri ption of t hc 
eonlrib ut1o n of n no n-interior fixcd point. In IBS9 1] a n explicit fo rmul a is obtaincd 
íor thr rontribu tio n of a simple boundary fixed point . A fi.xed point on the boundary 

1AM S SubJ«I Cla..~ification : pri nmry : 32850; seoondary: 58G 10. 
Kl.'y 1ll"O rd~ l.nd phrues: de ru1a111 co mpl cx , Lcfsch(l l i uumbcr. fixed poi nt.s . 

11 5 



116 N. Tarkbanov 

contri b utes t o t he Lefschetz number only in the case when it is attracting. Thc pur-
pose of t he present pa.per is to explicitly evaluate contributions for arbi t rary isolated 
tlxed points on the boundary. While the proof of [BS91J is an exposition of a rgumcnts 
of jAB67J in the context of Boutet de Monvel's algebra [BdM7l], t he present paper 
uses a constructive approach elaborated in !Tar95j. 

To make t he p roof complete!y transparent we restrict ourselves to the case 11 = l. 
Let M = [a , b] be a closed in ter val on the real axis, and f be a smooth map of [a, b] 
to itse lf. 

2 The Neumann problem 

In a rder to deri ve a n integral formula for the Lefschetz number oí f we need a 
parametrix of t he de Rbam complex on [a, bj 

O___, E[a,b] _!_., E1 [a,b[ ___,O (2. 1) 

where l" 1 [a , bj is t he space of d ifferential forms of degree l wi t.h C 00 coefficients on 
[a , bJ. 

Thc Neumann problem at extreme step 1 consists o f finding, for a givcn F E 
l' 1 [a, bJ, a d ifferential form u E l' 1 [a, bJ such t.hat 

ddº u = F in (a , b) , 
n(u) = O on 8(a , b) , (2.2) 

where el' is th e formal adjoint for d. Not.e t.hat t.he Neumann boundary condition 
n (u ) = O appears from the consi<lerat.ion of dº as t he Hilbert space adjoint., d. for 
ins tance § 4.2 in [Tar95]. 

Wri te 
F = F1 (x)dx , 
u = u 1 (x) dx, 

Lhcn t he problem (2.2) becomes just. t.he Dirichlet. problem for t.he íunr.t ion u ¡(x), 
namely 

- u1{ = F1 in (a , b), 
u 1 (a) = u1 (b) = O. 

(2.3) 

T he general solution of - u.1i = F 1 is 

u 1 (x) = (Ax + B) - J.'( x - y ) F,(y)dy, 

and the substitution of th is cxpreMsion to the boundary conditio ns gives 

D - a A. 

A = J."~ F¡(y ) d y . 
" b - n 



On Brouwcr's Fixed Point Thcorem 117 

Tl1c !!Olulion 

r• b r 
u 1 (x) = (x- a)¡

0 
b =~ F1 (y) dy - J. (x-y) F 1 (y)dy 

is 1) 1la , bj ·Orthogona l t.o t.hc space OÍ solut io ns OÍ t he homogenoous p roblcm corre· 
s ponding i.o (2.3), for thi11 lat t cr is zcro. lt fo\lows t ha t t.he Ncumann o perator a t 1:1t ep 
l is 

NF(x) = [ F(y) ((x - a)!=~ - (x -y) 0 (x-y))dy, (2.4) 
t.lw intrgral b ing ovcr y E [r1, bJ. 

3 A parametrix of the de Rham complex 

Civcnany F e C1¡a, bJ,sct 

PF(x ) d' NF (x) 

J.' (e(x - y)-:=~)F(y), (3.1) 
i.c., P :: d"N . 

Le n:ima 3.1 A ., ílefinc<I in (9. 1}, thc opemtor P satisfie., 

Pdu u - Sou f orall u El'[a,bJ, 
dPF = F - S1 F foral/ F EC 1\a, bJ, 

(3. 2) 

wh ere 

i r' Sºu ;:: u-~ Jo u (y)dy, 
S1 F = O. 

Proof. lt sufficcs to provc o nly thc first cquaJit.y oí (3.2) , for t he second oue is 
obviou.s. To th is end, writc 

P du(x) { ' ( b - Y) J. 0(x - y) - ¡;-::--;; u' (y)dy 

( ) b - y ¡· J.' b - y u (x) - u (a) - b _ a u (y) 
0 
+ 

0 
u(y) d ¡;-::--;; 

i r• 
u(x) - ¡;-::--;;J. u (y)dy, 

a,.¡ des1red. 

o 



11 8 N . 1arkhanov 

4 A formula for the Lefschetz nurnber 

By t he Leíschetz num ber of the map f : ¡a , b] -+ [a , bJ is meant 

L(J) = t o· (H /)o - te (H f ), 

wh ere 
( H /) o H º (o·¡a, bj)--> H º(o '[a,bj) , 
( H f ), H 1 (o' [a, bj)--> H ' (o' [a,bj) 

are cndomorphis ms of t hc cohomology of th c de Rh a m co mp lex o n lc1, b], indu1,;cd 
by t hc pull-back o pera t or ¡n on differe nti a l fo rm s. Sincc Lh e cohomology is fi 11 it(' 
d im (' nsio nal al C\'e ry stcp , t hc t races of th ese endo mo rphi sms ar e well dcfi ncd . 

o. 
Note th at in fact. L ( f) = 1 in our specia l case , fo r Hº( l [a, bJ) ~ C a nd H 1 (l [a , b]) ~ 

Le mma 4 . 1 Sup11os e all jixcd¡minl s of ll1 e ma¡J f on [a , bJ are iso fot cci . Tlw11 

l b b " ) L(f ) = P·' · d( 0(J (y) - y) - b = . 
" a 

(4 . 1) 

Proo f. Appl ying t hc pu \1 -back opcrato r r t o both sid cs o f equalitics (3.2 ) WC g 1' I 
(f'P ) d f ' - f' Sº. 

d(f1P ) = f ' - f'S' . 

for ¡: and d rninmuLe. 
Si ncc ¡ : P maps E 1 [a , b] to ljri , b] wc ded uce t ha t / 1 a nd /' S are liomo to pi c· cndo-

mo r phi :-; ms of t h€' de Rl mm com pl cx . Hencc t hcy induce th e samc endomo rp hisms of 
t hc co homology, i.c ., fl f = fl (JDS). !t. fol lows t hat L(/ ) = L( P S). 

\\'(' no w ob:;c rvc t.hat J DS is a trac e dass end omorp hism of t. he de JUmm co mplcx . 
Dy thc nltcrn at.i ng su 111 fo rmul a (cf. for ins t.a.ncc Theo rcm 19 .1.1 5 of [HOr85 ]) wc 
obtai n 

L(f) Tr J' Sº - Tr J' S' 

[ ll.'(U x l )'Kso - (! x l )' K s • ) 
wh(' rr ~ stands for t. hc d iagonal m ap [a , bJ -+ [a . bl x lo ,bJ . an cl K s is t hr Schw<1rlz 
ke rn el of 

By as.-. ump1io n, th c se t Fix(f, [a, b]) is d isc rc tc. Since thc intcg ran cl is of cla."'~ 
l 1 j11. bJ. W(' gf'l 

L (f) = li"' 1. ll.'(11 x l )'K so - (/ x I )'Ks •) 
' 'º ¡ .. .1,¡\ U, 

wlu •rt• ( ' I!' l lw M't o f al l ¡wi1 1t s y E [a , bl wh osc d ist a ncc to Fix( / . [o, b]) is l t~ss t hai1 



On Drouwer's FLYed Poillt Theorem 119 

Wc no~ make use of cqualities (3.2) to evaluate Lhe integrand in the latter integnll. 
Nnmcly, tlhey imply that 

d~Kp = -Kso, 

d~Kp = -Ks• 

mvny from Lhc diagonal oí [a, b] x [a, b]. IL follows that 

holds on la, bJ \ Uc, whcnct: 

L(f) 

T hii; provt.'S thc formula. 

t-•(-d~(f x 1)'Kp+d.,(fx 1)' I<p) 

d(t>'U X 1)'Kp) 

lim { d(t:::..~ (f x l)~Kp) 
€-~u Í¡n,bJ\U, 

pv[ d(8(f(y)-y)-:=~) 

5 Fixed point theorem 

o 

'1'111• sccm1d term in t hc in Lngral (4. l) is eas ily evaluated , he11ce t.his for1nula t r m1sf111·n1s 

'" 
L(f) = l + p.v. [ d8(f(y) - y) . 

T heorem 5.1 Let f be 11 C 00 mav of tite sr.9m e11t !n , b] with isofolerl jix1~1i ¡min/ .. '1. 
'l'lie11 

L(f) = 1 + 8(/(71) - 11) r-+ L µ(p) 
n+ pEFix(/,(n,b) ) 

wlum~ ¡1(1' ) i.s thc loco/ dcgree of 1 - f at p. 

Proof. Writ.r r1 < p 1 < ... < PN < ¡, for 1.111' fixccl points oí f Lha1. lit• in l.lu~ 01w11 
i11t1•rv11I (u ,b}. Siurr 1.ht! f1111ct.\011 0(f(y) - y) is c:onstant away from tlw set. o f fixt~d 
poin!s uf f w1· ~1·t 

r1 - • N 1•~+1 -• b -t 

L(f ) = 1 + I d8(f(71) -y) + L: I d8(f(y)-y ) + I d8(f(y)- y) 
o+c k = l l'~ +c p ,, +r 

fur 1111 ( > O Rnmll r.nouRh. lfoncc it follows t hat 

n +r N 1•1+ t 
L(f ) = 1 - (-l(/(71) - y) I,_, - L: 8(/(y) - y ) I,.. _,. 

k = l 



120 N. Th.rkhanov 

A passage to the limit when € ~ O+ g ives the desired formula, for the local dcgrcc of 
1 - f at. pis opposite to that of f - l . 

o 
lf a is a simple fixed poini of f then t he sign of (1 - / )(a+) already unic¡ucly 

determines the local degree of a ny smooth extens ion of l - f t.o a ncighbo urhoocl 
o f a . amely t he local degree of 1 - f at. a just. amounts to s ign (1 - /)(a+), or 
s ign {I - /'(a )} . The s ame rcasoning applies to the case where bis a si mple fixcd 
point off. Howcver these arguments no longer work if a or b is not s imple, for f can 
be extended to a smooth function in a ne ighbo urhood of a respecti vely b in di vcrse 
manners. Fa r this reason we need another s pecification of fixed points of f 0 11 thl.' 
boundary. Supposc / (a) = a. T hcn a is said to be an at.tracti ng fixed point off if 
( 1 - /)(a+) > O, a ncl repul!iing if ( J - f)(a+) < O. lf f(b ) == b thcn t he fixccl point 
bis called a t tracti ng if ( 1 - J)(b- ) < O, and repu\sing if ( 1 - j)(b- ) > O. F'or thc 
atLracting fi.xcd points on the boundary we define t he local degree o r 1 - f to be 1, 
and for thc repuls ing fixed points wc defi ne the loca l degree o f l - f to be - 1. Thcn 
Thcorcm 5. 1 can be reformulatcd in t hc following way. 

Corollary 5.2 Let f be a C 00 map of the segment [a, b] with isolated fixed 11oi11h. 
Th en 

L(f) = µ (p), 
11EFix( / ,(n,b))U Fix!• I( / ,O(o .b)) 

F'ix1ª 1(/,8(o, b)) being the set aj attracting fixed points o/f on tli e bo1mdary. 

For thc s mooth mapg or [a, b] the fi xed point theorcm of Bro uwer [Bro 12J is a n 
obvious consequcnce of Corollary 5.2 because L(/) == 1. 

References 

IAB67J M . F'. ATIYAH a nd R. BoTT, A Lefschetz fixed point formula for ellipt.ic 
complezes. 1, Ann. Mat h. 86 (1967), no. 2, 374-407. 

IBd~HIJ L . BouTET DE M ONVE L, Boundary probfem.t for pseudo-different.ial 011Cr· 
aton, Act a Math. 126 (197 1) , no. 1- 2, 11 - 5 1. 

ID 91J A . V . BllENNEn a nd M. A. S tt un1N, T l1 e Atiyah-Bott-Le/schetzform1Jlajor 
ell1pltc complexes 011 manifolds with botmdary, C ur rent. Problcms of Math-
e mat.ics. F\mdamental Directions. Vol. 38, VIN IT I, Moscow, 1991 , pp. 11 9 
183. 

jBro 12J L. 8 . J . B nouwEn, Über Abbildungen uon Mannig/alt1gke1 tcn, Math. Ann. 
7 1 ( 1912) , 305- 3 14 . 

IDo172) J\ . D OLO , l ccturcs 011 Algebrair Topology, Spring cr- Verlag, Bt•rlin c:L al.. 
1972. 



On Brouwer's Fixed Point Theorem 121 

[H0r85] L . H6RMANDER, 11~e Analysis o/ Linear Partia/ Diflerentfol Operators. 
Vol. 3: Pseudo-dif!erential operators, Springer-Verlag, Berlin et al., 1985. 

[Hop29J H. HOPF, Úber die algebraische Arizalll von Fi.xpu11kten, Math. z, .. 29 ( 1929~, 
493- 524. 

[Lcf26] S. LEFSCHETZ, Jntersectio11s and transformation.s o/ complexes and mani-
foldJ, Tra.ns. Amcr. Matlh. Soc. 28 (1926), 1-49. 

[Tar95] N. N. TAR.KHANOV, Comvlexes o/ Dif!erential Operotors, Kluwer Academic 
Publis hcrs, Dordrccht., NL, 1995.