A METHOD OF COMBINING FIXED POINTS

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 51, Number 1, August 1975 A METHOD OF COMBINING FIXED POINTS ROGER WAGGONER ABSTRACT. It is now well known that in the category of finite polyhedra the fixed point property is not preserved by the operations of suspension, Cartesian product, adjunction along a segment, and join. Thus far none of the examples given have involved polyhedra of dimension 2. It is shown in this paper that two fixed points x and y of a self-map of a polyhedron K can be combined in a certain way if a certain criterion is satisfied by the /-image of a path from x to y. Several corollaries follow, one of which is that if K is a finite simply connected 2-polyhedron with no local separating points, H AK)?t 0, and K has a 2-simplex cr such that 17.(X Inter, z) is cyclic, then K fails to have the fixed point property. This eliminates many 2-dimensional polyhedra from consideration as examples. 1. Introduction. It is now well known that in the category of finite polyhedra the fixed point property is not preserved by the operations of suspension, Cartesian product, adjunction along a segment, and join. The reader may consult [l], [2], or [3] as general references. Thus far none of the examples given have involved spaces of dimension 2. It is shown in this paper that two fixed points x and y of a self-map / of a polyhedron K can be combined in a certain way if a certain criterion is satisfied by the /-image of some path from x to y. This is an improvement of a result of Shih [4]. A corollary is that if K is a finite simply connected 2-polyhedron with no local separating points, HAK) / 0, and K has a 2-simplex a such that 77.(K - Int cr, z) has a single generator, then K fails to have the fixed point property. This eliminates many 2-dimensional polyhedra from consideration as examples. A map /: X > Y is a continuous function from the space X to the space Y. If /: X»X is a map, then F(f) denotes the fixed point set of /, and Lif) is the Lefschetz number of /. The Euclidean metric of Em will be denoted by d. If x and y are points of Em, then (x, y), [x, y], and Received by the editors April 10, 1973 and, in revised form, May 12, AMS imos) subject classifications (1970). Primary 55C20, 55A20. Key words and phrases. Fixed point property, Lefschetz number, fixed point index. 191 Copyright 1975, American Mathematical Society

2 192 ROGER WAGGONER R\.x, y] denote, respectively, the open interval from x to y, the closed interval from x to y, and the closed ray from x through y. An 772-ball in Em is a set {x ri(x, xq) <r\ tot some fixed x r. An 72-disk in Em is a homeomorph of an 72-ball in E". The following two preliminary lemmas are concerned with n-balls in E". Lemma 1. Let D and > be n-balls in E" with D centered at the origin. Suppose f : D Dn z's a map such that f~ (O) = {x, y A and yq e Int D. Then there is a homotopy G: D x I D. for which Gix, t) = fix) if x Bd D or t = 0, and Gix, t) = 0 iff x = x or x = y = Zx. + (l - t)y.. Proof. Let h: D x I > D x I be a map satisfying: (i) hix, t) = (x, /) if t = 0 or x Bd D. (ii) My0, t) = (y<1 i). (iii) hiyt, 1) = xq. (iv) h\id x /) - {(y, 1)1 (D x /) - (xq, l) is a homeomorphism. Let r: D - {y A > Bd D be the radial projection, and let p: D x I > D be the projection onto the first factor. Then define G by: 0 if (x, t) = (y(, t) or if x = xq, /p/>-hx, i) if t <y2, "J /7775-Hx, f) if í > ^ and l/p/7-hx, Z) < \frph~\x, t)\, \(2 _ 2í) + Í2t-D\frph-Kx,t)\ \fph-\x,t)\, L fph Hx, i*) otherwise. Lemma 2. If D C D are n-balls in En with center z, and g: D» D z's a map with two fixed points yq Int D and x., then there is a homotopy H: D x I D, such that Hix, t) = gix) for x Bd D or t = 0, and Hix, t) = x iff x = xq or x = yt = tx + (l - t)y n. Proof. Let r be the radius of D Define /(x) = g(x) - x. Then for some Dn, /: D DQ as in Lemma 1. Define G as in Lemma 1, and let y(x, t) = max{ G(x, t) + x - z\/r, l\. Finally, define H by Hix, t) = igix, t) + x - z)/yix, t) + z.

3 COMBINING FIXED POINTS For the remainder of the paper, K will be a finite polyhedron with a given triangulation and f:k > K will be a map with finitely many fixed points, all contained in maximal simplices of K. There is no loss of generality in placing these conditions on /, since any g: K > K is homotopic to such an /. A homotopy will be constructed which "slides" a fixed point of / along a broken line from one maximal simplex into another. The carrier of a point x of K is oix). Let M = {(x, y) K x X cl a(x) O cl oiy) 4 0\. For A C K, a map f : K K is a proximity map on A if for all x e A, (x, fix)) M. It is shown in [4] that there is a map a: M x I >K satisfying: (i) cz(x, y, 0) = x and a(x, y, l) = y, (ii) a(x, x, i) = x for all t I, (iii) a(x, y, t) / x if x 4 y and / ^ 0, and a(x, y, t) 4 y if x y and Lemma 3. Let [«, è] be an interval in some maximal simplex a of K, a Int a and b Bd a, such that f is a proximity map on an e-neighborhood Ui[a, b], e) = U of [a, b] and Fif) n U = {a\. Then there is a homotopy H: K x I K satisfying: (i) Hix, t) = /(x) if t = 0 or x </u, (ii) //(x, i) = x for x U iff x = xt = (l - t)a + tb, 0 < t < 1. Proof. For convenience suppose that the -neighborhood of a, Ilia, e), is in Int a. Let y: I */ be a homeomorphism such that yio) = 0 and for any t 4 1, aifix), x, yit)) Int a. Let R: J [«, ] be a retraction such that Rix) = b iff x U - Int a. If Rix) = x, let t% = Í. For x U, write x = r /?(x) + (1 - r )5(x) where Six) Bd [/. Finally, define H l by!/(x) if x e/ij, a(/(x), x, rx yitx)) if x e U and r^ < ^s, aifix), x, r yi\js)) otherwise. Choose a disk D such that [a, b] C D C o n (J and [a, b] D Bd D = Bd a O Bd D = {/>}. For x e D let g(x) = aifix), x, yit)), and choose D small enough so that gid) C a- Construct a map H D x I > a as in Lemma 2. Let ß: K / such that /3(x) = 0 if x U, ßix) = y(/x) if x e D, and 0 < ßix) < 1 otherwise. Extend H2 to K x I by the definition H2(x, ) = a(/(x), x, /3(x)) if x e ii - D and H 2ix, t) = /(x) if x ///. Now define // by

4 194 ROGER WAGGONER H2ix, s) if tx < is + Vs)/2, Hix, s)= {HA[x, s) if tx> sjs, H2ix, 2s[iy/s - tx)/i\js - s)]) otherwise. Note that if x = x = s then Hix, s) = x because of the way in which // was defined. On the other hand, if s 4 0 and (s + \js)/2 < t <\Js then Hix, s) 4 x because 2si\Js - t )/(\/s - s) < s and t > s. Lemma 4. Let b be a fixed point of the map f : K» K. Then there is an interval [b, c], a neighborhood U of \_b, c], and a homotopy H: K x I > K satisfying: (i) H(x, t) = f(x) if x /(/; (ii) if x U then H(x, t) = x iff x = x = (l - t)b + tc. Proof. Let e > 0 such that if «"(x, y) < e then (x, y) e M. Choose 8 > 0 satisfying the following conditions: (i) If dib, y) <8 then diy, fiy)) < e. (ii) There is a map r/: K x I» K such that r/(x, t) = x if dix, b) > 2t8, 7?(x, t) = b if dix, ii) < t8, and -q ; K - V - K - {b\ is a homeomorphism, where V is the /S-neighborhood of b. (iii) No other fixed points of / lie within 2<5 of b. Let ßt: K [0, l] with ß'Ko) = K- V2t and ß~\l) = Vf. Define /f by!a(/(x), x, j8,(x)) if «"(x, /(x)) < S, fix) Define h : K K by t5;(x) = jj^" htiv) = 6. Now define H by otherwise. l(*). Then F(/b ) = Fif), hq = /, and Hix, t) Ífcfx) if x ^Vt, (r/(x, 6>- fc)/i5+(l dix, b)/ts)[il - t)b + te] otherwise, where \b, e] C K. Choose a point c on lb, e] such that dib, c) = i8 die, b))/is+ die, b)). Then for x Vf, Hix, t) = x iff x 6 [fe, e] and «"(//(x, i), 6) = dix, b), which means (l - dix, b)/ts) itdie, b)) = dix, b); hence dix, b) = (is«?(e, b))/i8 + die, b)). Thus Hix, t) = x iff x = (l - í) + License /c. or copyright restrictions may apply to redistribution; see

5 COMBINING FIXED POINTS 195 Let a and c be fixed points for /. A regular combining homotopy from a to c is a map H: K x I K such that Hix, 0) = fix), and for / 4 0, Hix, t) = x iff /(x) = x and x 4 a, or x = p(y(i)) where >: / > K is a simple polygonal path from a to c and y: / > / is an onto map with yii.) < y(i,) whenever z" < i,. Recall that two fixed points a and c of f ate said to be equivalent if there is a path p: I > K from «to c such that fp and p ate fixed endpoint homotopic. Two fixed points «and c oí f ate said to be strongly equivalent if there is a simple polygonal path p from a to c and a homotopy G: Ix I >K such that Gis, 0) = fipis)) and Gis, t) = p(s) iff í = 1 or fpis) = p(s). The relation of strong equivalence is an equivalence relation on the fixed point set of /. Theorem 1. Let K be a finite polyhedron and let f: K > K be a map with finite fixed point set contained in the maximal simplices of K. The fixed points a and c of f are strongly equivalent iff there is a regular combining homotopy from a to c. Proof, Suppose that a and c ate strongly equivalent and let p and G be maps which satisfy the conditions of the definition. We may assume that the only fixed points of / which lie on pil) ate the points «and c. Let V be an open polygonal neighborhood (possibly infinite) of pil) - {«, c\ with {«, c\ C Bd V. Choose V in such a way that Cl V collapses to pil) with a map r: Cl V» pil), such that: (iii) if B(x) is the first point of the above ray on Bd V, then the function (i) if x Cl V then a(/(x), frix), t) is defined, (ii) the ray RÍrix), x] intersects Bd V, B is continuous. Let nix) = dix, 3Íx))/dirix), Ifix) B(x)). Define /: K -» K by if x e/v, aifix), frix), 2n(x)) if 0 < nix) < V2, Girix), (2-8)inix) - %)) otherwise. If 8 is chosen to be sufficiently small, / is a proximity map on pil). If V is then chosen to be sufficiently small, / has the same fixed point set as /. Now the endpoints of pil) lie in maximal simplices of K. We may assume that pil) intersects nonmaximal simplices of K in only finitely many points. License In that or copyright case, restrictions Lemmas may apply to 3 redistribution; and 4 see demonstrate a method of "moving" the fixed

6 196 ROGER WAGGONER point a along pil). By composing a suitable homotopy between / and / with those obtained from repeated applications of the lemmas, a regular combining homotopy from a to c can be constructed. The converse is not necessary for what follows, and the proof is omitted. 3. Applications. In Theorem 2 below, a condition on K is given which insures that any two equivalent fixed points of a map /: K > K be strongly equivalent. In that case, / may be altered with a homotopy to obtain a map /' which has exactly one fixed point for each essential fixed point class. The proof of the preliminary lemma, although quite technical, is elementary and is omitted. Lemma. Assume that K has no local separating points and let a and b be two fixed points of a map f: K > K. Let p be a path from a to b such that pit) 4 pit) for t 4 0, 1. Let [ij, 12] C [0, l]. Then there is a map H: I x I > K satisfying: (i) Hit, 0) = fp(t), for all t, (ii) Hit, s) = pit) iff t = 0,1, (iii) aihit, l), pit), s) is defined for all t lty *2I Throughout this section it will be assumed that K has no local separating points. Theorem 2. Let o be a maximal simplex of K, and let C be the cyclic subgroup of 77.(K - a) generated by a simple loop in Bd cr. If the sequence 0 _ C -. 77j(K -a) IL na\k) 0 z's exact, where i is induced by inclusion, then any two equivalent fixed points of a map f: K > K are strongly equivalent. Proof. Let «and b be equivalent fixed points of /: K > K. Let p be a simple path from a to b with p ^ p. Let r be a maximal simplex of some subdivision of K with a, b f r and r C Int a. We may assume that pil) meets Int T in a simple arc. Let [t, t ] C [0, l] such that pit, t ] = pil) D r. Use the preceding lemma to alter fp with a homotopy to obtain a proximity map on pil) - pit.) t ]. Let «be a path made up of a segment from fpit ) to y Bd r, a path y in Bd t from y. to y2, and a segment from y 2 to fpit A. Then /[/,, ta * «is a loop in K Int r. Now fip) * p~ ^ fit, t2] * q. Since fip) * p~ is inessential in K, fit., ta * q is equivalent in K - License Int or copyright r to restrictions a", where may apply to a redistribution; a loop see in Bd r which generates C. We may use a

7 COMBINING FIXED POINTS 197 homotopy between fit v 12] * q and a" in K - Int r, keeping y fixed throughout, to obtain a proximity map on pil). Notice that throughout these homotopies the image of pit) is equal to pit) iff / = 0, 1. It follows that a and b ate strongly equivalent. Corollary 1. //, with the hypotheses of Theorem 2, dim K > 3, then any two equivalent fixed points of f are strongly equivalent Proof. If dim a > 3, then C = 0. Theorem 1 and Corollary 1 give a result of Shih [4]. Corollary 2. // dim K = 2, tta\k) = 0, and for some a, tta[k - Int a) S C, /eerc any two equivalent fixed points of f are strongly equivalent. If, in addition, H AK) 4 0, then K admits a map of Lefschetz number zero [5]. In this case the fixed points of f may be eliminated completely. Consider the following example. Let a be a simple polygonal loop in the real projective plane RP2 which generates ît^rp2). Attach a 2-cell a to RP by mapping Bd a onto a with a map of degree 3. Call the resulting adjunction space K. It follows from Corollary 2 that K fails to have the fixed point property. Corollary 3 also follows immediately from Corollary 2. Corollary 3. // dim(k) = 2, tt^k) = 0, H2iK) 4 0, and a is a 2-simplex such that K Ci a is a l-simplex, then K U a fails to have the fixed point property. REFERENCES 1. R. H. Bing, The elusive fixed point property, Amer. Math. Monthly 76 (1969), MR 38 # R. F. Brown, The Lefschetz fixed point theorem, Scott, Foresman, Glenview, 111., MR 44 # E. R. Fadell, Recent results in the fixed point theory of continuous maps, Bull. Amer. Math. Soc. 76 (1970), MR 42 # Ken-hua Shih, Oil least number of fixed points and Nielsen numbers, Acta Math. Sinica 16 (1966), = Chinese Math.-Acta 8 (1966), MR 35 # Roger Waggoner, A fixed point theorem for in 2)-connected n-polyhedra, Proc. Amer. Math. Soc. 33 (1972), MR 45 #2699. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF SOUTHWESTERN LOUISIANA, LAFAYETTE, LOUISIANA 70501