A N N A L ES S O C IE T A T IS M A T H E M A T IC A E P O L O N A E Series I : C O M M E N TA TIO N E S M A T H E M A T IC A E X X I I I (1983) R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A T EM A T Y C Z N EG O
Sé ria I : P R A C E M A T EM A T Y C Z N E X X I I I (1983)
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a r o lS
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An inductive proof of the non-retractibility oî a cell onto its boundary
The non-existence of a retraction of an ^-dimensional ball to its boundary is equivalent, by a simple argument, to the Brouwer fixed point theorem. The classical proof of the theorem, due to Knaster, Kuratowski, and Mazurkiewicz [2 ] is based on the Sperner Lemma. A modern and elegant proof by Hirsch [1] depends on the assumption that the existence of a retraction implies the existence of a simplicial retraction, the proof of this complement being more troublesome than the proof of the theorem.
In this note we give a simple, inductive and self-contained proof of the theorem in a slightly more general formulation than that proved by Hirsch.
If a simplex s is a face of a simplex t, then we write s < t. In order to simplify the notation we assume that if sn denotes a simplex, then dims11
= n. Let К be an w-dimensional complex, where n > 0. The subcomplex dK ~ {s e K : there is tn~l e К such that s < tn~l and the number of all un e К satisfying tn~l < un is odd} is called the boundary of K . This is an evident generalization of the boundary of a pseudomanifold. I t can be proved (though we do not use this property) that the polyhedron of dK depends only on the polyhedron of К but not on the choice of a particular triangulation К . If the boundary dK is empty, then we say that К is without boundary.
Let K ' denote the barycentric subdivision of a complex K .
L
e m m a1 . 8{K') = (dK)'.
The lemma follows directly from the definitions of* the barycentric subdivision and of the boundary of a complex.
L
e m m a2. I f К is an n-dimensional complex, where n ^ l , then the bound
ary dK is either empty or is an (n — 1 )-dimensional complex without boundary.
P roo f. Let sn~2 e К and denote by s f" 1, s”-1, . .., s”-1 e К all (n — 1 ) dimensional simplexes in К satisfying sn~2 < sf-1 for i — 1, 2, . . . , p
9 — R o c z n ik i P T M — P r a c e M a te m a ty c z n e t. X X I I I
130 K. S ie k lu ck i
Let s”-1 be a face of k{ n-dimensional simplexes in K , where i — 1, 2, ..., p.
Suppose that the numbers k x, k 2, ..., kr are odd and that the remaining numbers kr+11 kr+2, . .. , kp are even.
If we assume that sn~2 e 8(8K), then the number r is odd. Consequently,
T p p
the sum k{ is odd and, since the sum kc is even, the sum k{ is
i = l i = r + 1 i= 1
odd. On the other hand, if sn~2 < sn, then there exist exactly two simplexes s^r1 and s^T1 satisfying sn~2 < s^r1 < sn and sn~2 <
s”
t\ < sn. Thus the sum
p
ki is even and this contradiction completes the proof of the lemma.
г= 1
Let \K\ denote the polyhedron of a simplicial complex K . We shall prove the following
T
h e o r e m. I f К is a simplicial complex, then \8K\ is not a retract o f \K\.
P ro o f. We proceed by induction with respect to n = dim W. If n — 0, then \8K\ — 0 and the theorem is evidently true. Let us suppose that the theorem is valid for n — 1, where n > 0, and let К be an ^-dimensional simplicial complex with a boundary. Suppose that there is a retraction r: \K\->\8K\.
By the Baire theorem there is a point p e \8K\ such that for any bary- centric subdivision JB№ of K , where q — 0 ,1 , . .., there is tn~l e 8K(e>
with p e inttn~l. In particular, let p e ints”-1, where sn~l e 8K. Choose a ô > 0 such that if æ, y e |K\,
q(
oc, y) < ô, and r(æ) = p , then r(y) e s”-1.
Denote by K iq) a barycentric subdivision of К such that mesh K^q) < <L Let p e i n t r -1, where tn~~l e £2£(9); evidently, tn~l <= sn~l.
Let F — г~г{р) and L — {v e Ж(з): there is un e K (q) such that v < un and unn F Ф 0 }. If vn~l е ! * 3’ satisfies vn~l c\F Ф 0 , then vn~l e 8L if and only if vn~l e 8K^q\ Indeed, if vn~l n F ф 0 , then for any u11
gK Ul) satisfying vn~l < un we have also unn F Ф 0 , i.e. un
gL ; thus the number of those un e L which satisfy vn~l < un is equal to the number of those un
eK (q) which satisfy vn~1 < un.
t I t follows from the property stated above that tn~l e 8L and that
\8L\nF — \8K^\nF. Since \8K(q)\nF = {p}, we have therefore \8L\c\F
= {pj. B y Lemma 2 the boundary 8L is an (n — l)-dimensional complex without boundary. Thus K Q = 8L\{tn~1} is an (^ — 1)-dimensional complex whose boundary 8K0 is the natural triangulation, of the geometric bound
ary in~l of f 1-1 ; moreover, |/
il0| n F = 0 .
From the definitions of à and of q it follows that r(\L\) с 5й-1; con
sequently, r(\K0\) cz sn_1. Since |И0|пР = 0 , we have r(|IT0|) c sn_1\{p}- Let us denote by r' the natural retraction of sn~l\ {p} to in~l. Then the map
ping r0 = rVj|2T0| is a retraction of j./f0| to in~l — \8K{)\, contrary to the
inductive assumption.
S on-retractïbïlity of a cell onto Us boundary 131
References
[ 1 ] W. H irsch , A proof of the nonretractibility of a cell onto its boundary, Proc. Amer.
Math. Soc. 14 (1963), 364-365.
[2] B. K n a s te r , K. K u ra to w s k i, S. M azU rk iew icz, E in Beweis des Eixpunktsatses fü r n-dimensionale Simplexe, Fund. Math. 14 (1929), 132-137.
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