NEW ZEALAND JOURNAL OF MATHEMATICS
Volume 22 (1993), 23-29
C H O R D T H E O R E M S , C IR C L E M A P S A N D
T H E B O R S U K -U L A M A N T IP O D A L T H E O R E M
M .M . D
odson
(Received April 1993)
Abstract. A chord theorem for lifts is used to show that for any map ip: S 1 —> S 1
and any z £ S 1, there exists a point £ G S 1 such that ip(z£) = z desv><p(£).
This
implies the two-dimensional Borsuk-Ulam theorem.
1. In tro d u ctio n
A map / : S n —> Sp is said to be antipode-preseruing if f { —x) — —f(x ) for all
x G S n; or in other words, if each pair of antipodal points x, —x is sent to the pair
of antipodal points f(x ), —f( x ) . The Borsuk-Ulam theorem says that a continuous
function / : S n —> S'71-1 cannot be antipode-preserving, i.e., there is always a point
x G S n for which f ( —x) / ~ f { x ) (see for example [5, Chapter 20]). Equivalently,
any continuous function g: S n —> R n sends at least one pair of antipodal points
x, —x to the same point g(x) = g (—x). In his monograph [2, p. 95], Boas points
out a connection between the universal chord theorem and the one-dimensional
Borsuk-Ulam theorem. In this note a new proof for the two-dimensional BorsukUlam theorem is given using a chord theorem for a class of real valued functions
of a real variable. This class includes lifts of circle maps and the result implies
that circle maps are homogeneous at some point (further details and definitions
are given below). This result is in turn used to prove the Borsuk-Ulam theorem in
the plane.
C h ord T h eorem s
Let C be a chord of length \C\ with endpoints in a path-connected subset S of
the plane. The chord is permitted to meet S in intermediate points as well, i.e.,
at points between the endpoints of the chord. The following remarkable result was
proved by Levy [6 ].
T h e U niversal C h ord T h eorem . Let C be a chord with endpoints in a pathconnected planar set S. For each positive integer n, there exists a chord parallel
to C with endpoints in S and having length \C\/n.
A proof is given in [8 , p. 16]. This theorem does not always hold if the integer
n is replaced by a real number (related results and counterexamples are given
in [2 , p. 93] and [4]). Levy’s example was the graph of the function f( x ) =
sin2(7ra:/a) — a;sin2(x /a ), where a ^ 1/n (see [2, p. 213]).
The path-connected set S will be restricted here to graphs of continuous real
functions. When the continuous function is periodic, it always has a horizontal
chord C say and hence by the chord theorem, horizontal chords of length \C\/n,
n = 1 ,2 ,___ But more is true; the graph of a continuous periodic function has
1991 A M S Mathematics Subject Classification: Primary 26A 24, Secondary 55M 25.
24
M.M. DODSON
horizontal chords of all lengths ([2 , pp. 91-2]). A simple generalisation of this result
is discussed below. For non-periodic functions the situation is quite different: there
are continuous functions / : [0,1] —> 1R with graphs having no horizontal chords at
all. However by the chord theorem, if such a graph has one horizontal chord C,
then it has infinitely many of length | C | / n n = l , 2 , . . . .
The horizontal chord result can be extended a little (in fact to the chord theorem
for the graph of a continous function). This is done in a nice paper by T.M. Flett
[4] who traced the result back to Ampere but the proof will be given here for
completeness. Let a < b and denote by C(a, b) the chord joining the point (a, /(a ))
to the point ( b ,f(b )) on the graph { ( x , / ( x ) ) : x E [a, 6]} of / . The slope of the
chord C(a, b) is (/(&) —f ( a ))/( b —a) and its length is ((/(& ) —/ ( a ) ) 2+ (6 —a)2) 1//2.
When /( a ) = /(&), the chord is horizontal with length b — a.
The Universal Chord Theorem for Continuous Functions. Let the function
f : [a, b] —■
>R be continuous. Then for each natural number n, there exists a chord
parallel to the chord C (a ,b ) joining f(a ) to f(b) and with length |C(a, b)\/n.
Proof. Let x(t) = tb + (1 —t)a and
0 (0 = / ( ( * + j^) b + ( J ~ t -
“ ) - /(<*> + ( ! - t)a) - i ( f ( b ) - /( o ) )
= / (X(t + n ) ) “
“ /(a)) '
Then
'
r= 0
\6
n /( £ )
=
~^
~ U^n~
= 0>
Thus either 9 (r /n ) = 0 for r = 0 , . . . , n — 1 or 6(t) changes sign in [0,1 — 1fn\. In
either case 0(t) vanishes for some t E [0,1 — 1/n] and so
/
+ “ ) ) “ / ( * ( 0 ) = “ ( f ( b) ~ /(«))•
Thus the chord joining (x(t), f ( x ( t ) ) ) to (x(t + 1/n), f ( x { t + 1/n))) has slope
n ( f ( x { t + l /n ) ) ) - f ( x ( t ) ) = f(b) - f(a )
(6 — a)
b —a
and so is parallel to the chord C (a , b) and has length \C(a, b)\/n.
As Flett points out in [4], more can be said when n > 3. In this case, either
6{r/ri) vanishes at each point r /n or the value of one of the 9 (r /n ) is positive, say
9(r' /n ) > 0 , then another, say 9 (r "/n ) must be negative and 9(t) must vanish at
some point t , strictly between r '/n and r " /n , so that t E (0,1 — 1/n). In either
case, it follows that 9(t) = 0 for some t E (0,1 — 1/n) corresponding to a point
x(t) E (a, 6), i.e., the chord C (x (t), x ( t + l /n ) ) is of length |C(a, 6) |/n and is parallel
to C(a, b).
CHORD THEOREMS, CIRCLE MAPS AND THE BORSUK-ULAM ANTIPODAL THEOREM
25
When / : [a, b] —>R is differentiable on (a, &), the theorem implies the first mean
value theorem. For by repeated application of the chord theorem with n = 3, a
sequence of nested subintervals [aj,bj\, j = 1, 2 , . . . of [a, b] can be chosen so that
bj — a,j = (b — a)3-J and
f(b j) - / ( o j ) = f(b) - / ( a )
bj — aj
b —a
By Cantor’s theorem, lim [bj — a,j] = {c } say, exists and c G [dj,bj] C (a, b) for
j —>oo
each ji = 1, 2,. .. . But
/(c)=
lim / M - / W .
c
v —u
/W -/(«).
b- a
u <c<v
The result that the graph of a continuous real periodic function has horizontal
chords of all lengths is now extended slightly to continuous functions h: R —> R
with the property that for each integer k,
h(t + k) = h(t ) + (h(l) - h(0))k.
( 1)
The graphs of such functions have chords parallel to the chord (7(0,1) of all lengths.
P rop ositio n 1. Let the continuous function h:TR —*■ R satisfy (1) and let a be
any real number. Then there exists a chord parallel to the chord C (0 ,1), of length
«|<7(0,1)|.
P ro o f. Given any real a,
f h(t
Jo
(
h(t dt Jof h(t a) dt—
Jo( h(t
Ja h(t dt Jo h(t) a
/ h(t dt J h(t)dt
—Jo h(t dt Jo h{t dt
r1
+ a) —
))
r1
=
) dt
+
/• 1 + a
=
/
/*1
)
— /
)
—
1 + a
ra.
p a
/
p a
+ 1)
— /
)
pOL
Jo
=a{h(l)-h(0)).
= /
Hence
(h(t) + h( 1) — h( 0 )) dt —
POL
Jo
h(t ) dt
«i
J
Jo
'o
(h(t + a) — h(t ) — a(h( 1) — h(0 ))) dt = 0 ,
whence the integrand h(t + a) — h(t) — qi(/i(1) — (0)) vanishes for some t
i.e., there exists a t G [0,1] such that
G
[0,1],
h(t + a) — h{t ) = a(/ i (l) — /i( 0 ))
and there is a chord parallel to the chord (7(0,1) joining (0, fo(O)) to (l, h( 1)) and
of length o;|(7(0,1)|.
Since t G [0,1], the chord begins in [0,1], so that if 0 < a < 1, the chord would
end at a point in [0 , 2).
26
M.M. DODSON
C ircle M aps
A circle map is a continuous function / : S 1 —> S'1. The simplest example is the
kth power map Pk: S 1 —> S'1, where k e Z , given by
Pk(z) = zk.
Each circle map / is associated with the loop (or closed path) i\ [0,1] —►S 1 given
by t = f o e with ^/(0) = ^/ (l) = /(1 ). Each continuous function g : [0,1] —> S 1 has
a lift g\ [0 , 1] —►R , i.e., a continuous function g which satisfies
e 27vig(s) = e ( ^ ( s ) ) =
where e2™* = e(£) (see [5, Chapter 16], [12, Chapter 6]). The lift is closely related
to the logarithm. The lift g is unique up to translation by an integer; there is just
one lift g which satisfies
9(0) = a0
where <?(0) = e2nia° .
Let
[0,1] —> R be the lift of ip o e: [0,1] —> S'1, so that for each s G [0,1],
3>(s) = ip~cT°{t) G R . Then e o $ = ipoe and the degree of ip: S 1 —* S 1 is defined by
deg<p = $ ( 1) — $ ( 0 )
(=deg</?oe).
The degree of ip is the winding number of ip{Sl ) about the origin, so that
A
deg<p
= —1
V(z ) dz
,
/f -----2m Js i 2
[1, p. 115].
The lift $ = (pote: [0,1] —> R can be extended continuously from [0,1] to R by
defining
&(t + k) = $ (t) + fc($( 1) - $ ( 0 )) = $(£) + k deg ip
for each k G 2 Z . Thus the extended lift $>:R —> R satisfies (1) and an extended
lift of <p o e is periodic if and only if deg ip = 0 .
Evidently the extended lift satisfies the hypotheses of Proposition 1 and so the
graph of any extended lift 4>: R —> R has a chord C (t, t - f a ) of slope $(1) —$(0) =
deg ip of any length
^(<£(t + a) — $ ( i ) ) 2 -f a 2j
= a ( ( $ ( 1) - $ ( 0 ))2 + 1^
= a ((d e g <^)2 + l ) 1//2.
C orolla ry. For each circle map <p\ S 1 —►S 1 and any point z
G
S 1,
</>(<) = 2degM C )
( 2)
for some ( G 5 1.
P ro o f. Let 2 = e2nta and let $ be the extended lift of ip o e. By the definition
of degree,
satisfies (1). The lift of the left hand side of (2) is $ ( t -f a) where
£ — e27r^ whence by the above Proposition,
3>(a + t) = $ (t) + a(<3>(l) — $ ( 0 )) = $(£) + a deg ip
CHORD THEOREMS, CIRCLE MAPS AND THE BORSUK-ULAM ANTIPODAL THEOREM
27
for some t € [0,1]. Hence
e ( $ ( a +.£)) = y?(e(a + t )) = ip(z() = e($(£) + adegy?)
_
e 2 n i $ ( t ) e 2niadeg<p _
deg <p
Thus circle maps of degree k retain a vestige of the homogeneity of the k-th
power map Pk(z ) = z k.
The Borsuk-Ulam Theorem
These ideas can be used to give a simple proof of the Borsuk-Ulam theorem in
the plane. In its general form, the Borsuk-Ulam theorem states that no continuous
function / : S n —>•5 n_1 can be antipode-preserving, i.e., there exists a point x e S n
such that f ( x ) ^ —f ( —x ) ([3, p. 347], [5, Chapter 20], [7, Chapter 5, §9],' [9,
p. 104, p. 266]). The equivalent form of the theorem that for each continuous map
g : S n —•» H n, there exists a point x G S n such that g(x) = g (—x) (i.e., such that a
pair of antipodal points is mapped to the same point) is proved using the associated
map G: S n —> 5 n_1 given by
gfe) ~ 9
( , ) _ 1®(*)-S(-*)||
r (« \ -
(3)
(|| . || is the usual Euclidean norm). This map is antipode-preserving providing
g(x) ^ g (—x) for all x € S n.
The Borsuk-Ulam theorem has a number of amusing consequences, such as the
ham sandwich theorem (see [5, p. 159], [7, Chapter 7]). An account of the BorsukUlam theorem and related topics together with an extensive list of references is
given in Steinlein’s comprehensive survey article [10]. The one dimensional BorsukUlam theorem, where f : S 1 —> S °, is immediate since S ° = { —1, 1}. Boas points
out in [2 , p. 95] that the equivalent form for g: S 1 —> ]R can be proved using the
intermediate value theorem.
However the proof for the general case is much harder and involves n-dimensional
homotopy groups, homology or cohomology. One approach uses the fact that S n
is a double cover for real projective space R P n. Since an antipode-preserving
map is equivariant with respect to the 7Z 2 action, an antipode-preserving map
/ : S n —►5 n_1 induces in a natural way a map from R P n to R P n_1. The result
can be deduced from the homotopy or homology groups of ]RP n (see for example [9,
p. 266)]. In two dimensions, a knowledge of the fundamental groups of 5 1 (= IRP1)
and IRP2, the real projective plane, can be used ([5, pp. 157-9], [7, pp. 170-2]).
Dugundji gives a simple proof of the n-dimensional Borsuk-Ulam theorem in
[3, Chapter 16, Corollary 6.2] using the degree of maps between n-spheres. The
degree deg / of an antipode-preserving map / : S n —> S n is odd (the proof of this
is not simple, see [3, Chapter 16, Theorem 6.1]). By considering 5 n_1 as the
equator of S n, an antipode-preserving map h: S n —►S'72-1 leads to an antipodepreserving map / = jo h : S n —> S n, where j : 5 n_1 —►S n is the inclusion which takes
(x i , . .. , x n) € S n~ 1 to ( x i , ... , x n,0) G S n. Thus deg/ is odd. But / : S n —» S n
28
M.M. DODSON
is not surjective and so deg / = 0, a contradiction. However the construction of
the degree is rather complicated and the argument appears to be incomplete (see
[11, pp. 101-2]). In [11], Verheul introduces the more general notion of a ‘sign
map’ to deduce in a simple way a number of classical results including Brouwer’s
fixed point theorem and the fundamental theorem of algebra. The Borsuk-Ulam
theorem, which is not included, would follow if it could be shown that an antipodepreserving map was a non-zero sign map but it is not clear how to do this.
The two dimensional case of the Borsuk-Ulam theorem can also be proved with­
out the use of covering or projective spaces by using the case n = 2 of the
chord theorem for graphs of continuous functions. (This special case can also
be proved directly by considering the sign of the function 9 given by 9(t) =
h (t + |) —h(t) — | (h( 1) —MO)) and using a simple intermediate value argument.)
With this result, only elementary degree and homotopy results are needed and in­
stead of the fact that the fundamental group Hi (H P 2) = 2Z 2, one only needs the
fact that n i ( S 2) = 0, i.e., that S 2 is simply connected.
Putting z — —I in the above corollary gives the
Lemma. For each circle map <p : S 1 —> S 1, there exists a point ( € 5 1 such that
f( 0 =
Corollary. Suppose degy? is even (resp. odd). Then for some £ in S 1,
{resp. — </?(—£)).
V > (0 = < P (-t)
Thus circle maps (p: S 1 —> S l of even or odd degree have an even or odd point
respectively.
The Two-dimensional Borsuk-Ulam Theorem. Suppose f : S 2 —* S 1 is con­
tinuous. Then for some point x E S'2,
/(* ) / - /( - * ) •
Proof. Suppose that on the contrary for each x = (x, y , z) € S 2, f(x ) — —f ( —x).
For each (x, y) G S 1, define <p : S 1 —* S 1 by
= f ( x , y , 0 ),
so that <p{x,y) — —<p(—x , —y). Hence by the contrapositive of the corollary to
Lemma 2, degt/? must be an odd integer. But the diagram
S1
S1
j \
/ S
52
where j : S 1 —> S 2: (x ,y ) 1—> (x ,y , 0), commutes. Since Hi (S'1) = 2Z and n i ( S 2) =
0 , the commutative diagram of fundamental groups and induced homomorphisms
becomes
7Z
^
7Z
j* x\
f*
0
CHORD THEOREMS, CIRCLE MAPS AND THE BORSUK-ULAM ANTIPODAL THEOREM
29
where ip*(k) = (deg ip)k. Since <p* = /* o j* — 0, deg<p = 0, which contradicts
deg (p being odd, and the theorem follows.
As has been noted, the equivalent version of the theorem for g : S 2 —» K 2 follows
by considering the function G : S 2 —> S 1 given by (3).
A ck n ow led gem en t. It is a pleasure to acknowledge some very helpful conversa­
tions with Chris Wood.
R eferen ces
1. L.V. Ahlfors, Complex Analysis, Second edition, McGraw-Hill, New York,
1966.
2. R.P. Boas, A Primer of Real Functions, Second edition, Carus Math Mono­
graphs, John Wiley, New York, 1981.
3. J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
4. T.M. Flett, Ampere and the horizontal chord theorem, Bull. Inst. Math. Appl.
11 (1975), 34.
5. C. Kosniowski, A First Course in Algebraic Topology, C.U.P, Cambridge, Eng­
land, 1980.
6 . P. Levy, Sur une generalisation d ’une theoreme de Rolle, Comp. Rend. Acad.
Sc., Paris 198 (1934), 424-425.
7. W.S. Massey, Algebraic Topology: An Introduction, ‘Harcourt, Brace and World,
1967.
8 . D. Rolfsen, Knots and Links, Publish or Perish Inc, 1976.
9. E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
10. H. Steinlein, Borsuk’s antipodal theorem and its generalisations: a survey, in
Topological methods in nonlinear analysis, Sem. Math. Sup., Presses Univ.
Montreal, Montreal, Quebec, 1985, pp. 166-235.
11. E.R. Verheul, Elementary proofs concerning results about functions on the
n-sphere, Topology and its applications 40 (1991), 101-116.
12. C.T.C. Wall, A Geometric Introduction to Algebraic Topology, Addison-Wesley,
New York, 1972.
M .M . Dodson
University of York
York Y O l 5DD
ENGLAND