Catherine Lee

H Schirmer

M Arkowitz

M Shub

Nirattaya Khamsemanan

RF Brown

Robert F Brown

Sompong Dhompongsa

V Guillemin

Fixed Point Theory and Applications

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Khamsemanan et al. Fixed Point Theory and Applications 2014, 2014:97http://www.fixedpointtheoryandapplications.com/content/2014/1/97RESEARCH Open AccessA ﬁxed point theorem for smooth extensionmapsNirattaya Khamsemanan1*, Robert F Brown2, Catherine Lee3 and Sompong Dhompongsa4*Correspondence:nirattaya@siit.tu.ac.th1Sirindhorn International Institute ofTechnology (SIIT), ThammasatUniversity, Pathum Thani, ThailandFull list of author information isavailable at the end of the articleAbstractLet X be a compact smooth n-manifold, with or without boundary, and let A be an(n – 1)-dimensional smooth submanifold of the interior of X . Let φ : A → A be asmooth map and f : (X ,A) → (X ,A) be a smooth map whose restriction to A is φ . Ifp ∈ A is an isolated ﬁxed point of f that is a transversal ﬁxed point of φ , that is, thelinear transformation dφp – IA : TpA → TpA is nonsingular, then the ﬁxed point index off at p satisﬁes the inequality |i(X , f ,p)| ≤ 1. It follows that if φ has k ﬁxed points, alltransverse, and the Lefschetz number L(f ) > k, then there is at least one ﬁxed point of fin X \ A. Examples demonstrate that these results do not hold if the maps are notsmooth.MSC: 55M20; 54C20Keywords: smooth manifold; smooth map; extension of a map; ﬁxed point index;transversal ﬁxed point; inverse function theorem; Lefschetz number; Lefschetz-Hopftheorem1 IntroductionIt has been known at least since the work of Shub and Sullivan in  [] that the valuesof the ﬁxed point index of smooth maps are more restricted than they are for continuousfunctions in general. In [] it is proved that, given integers r and s, there is amap f : X → Xof a manifold with boundary ∂X that restricts to f |∂X = φ : ∂X → ∂X and an isolated ﬁxedpoint p of f such that the ﬁxed point indices are i(∂X,φ,p) = r and i(X, f ,p) = s. On theother hand, it is proved in that paper that if f : (X, ∂X)→ (X, ∂X) is smooth and p is a trans-verse ﬁxed point of φ, then either i(X, f ,p) =  or i(X, f ,p) = i(∂X,φ,p). A consequence ofthis result is that, under appropriate hypotheses on a smooth map f , it must have ﬁxedpoints on X \ ∂X, the interior of the manifold X. Those same hypotheses are shown to beinsuﬃcient to imply the existence of such interior ﬁxed points if the map f is not smooth.(See also [, ].)We will consider a somewhat diﬀerent setting, as follows. Let X be a compact smoothn-manifold, with or without boundary, and letA be a smooth (n–)-dimensional subman-ifold of the interior of X. As in [], we shall consider f : (X,A)→ (X,A) to be an extensionof its restriction f |A = φ : A → A. Suppose that p is a transverse ﬁxed point of φ, then asimple example will show that the relationship between i(X, f ,p) and i(A,φ,p) cannot beas close as it is when A = ∂X. However, we will prove that there is still a very strong re-striction on the value of i(X, f ,p), namely, that |i(f ,X,p)| ≤ . As a consequence, we obtaina condition on the Lefschetz number L(f ) of f that implies the existence of ﬁxed points of©2014Khamsemanan et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.Khamsemanan et al. Fixed Point Theory and Applications 2014, 2014:97 Page 2 of 6http://www.fixedpointtheoryandapplications.com/content/2014/1/97f in X \ A. We demonstrate by an example that the same Lefschetz number condition isnot suﬃcient to imply the existence of ﬁxed points in X \A for maps f : (X,A)→ (X,A) ingeneral.2 The index of ﬁxed points of smooth extensionmapsThe index theorem of [] is the following.Theorem  Let X be a compact smooth n-manifold with boundary ∂X. Given a smoothmap φ : ∂X → ∂X and a smooth map f : (X, ∂X) → (X, ∂X) extending φ, suppose that p ∈∂X is an isolated ﬁxed point of f and that dφp – I∂X : Tp(∂X) → Tp(∂X) is a nonsingularlinear transformation. Then either i(X, f ,p) =  or i(X, f ,p) = i(∂X,φ,p).We will modify the proof of Theorem  that was given in [] in order to obtain a result ofthis type in the setting of smooth extension maps on a pair consisting of an n-dimensionalcompact smooth manifold X and an (n – )-dimensional smooth submanifold A of theinterior of X. Although our index result is similar to Theorem , the following exampledemonstrates that we cannot expect that there will be as close a relationship betweenthe indices of φ and of f as there is in Theorem . Let f : (S,S) → (S,S) where S isviewed as the complex plane C compactiﬁed at inﬁnity, S is the unit circle, f (z) = z forz ∈ C and f (∞) = ∞. Let φ : S → S be the restriction of f . Then i(S,φ, ) = –, whereasi(S, f , ) = .Theorem  Let X be a smooth n-manifold, with or without boundary, and let A be an(n – )-dimensional smooth submanifold of the interior of X . Let φ : A → A be a smoothmap with smooth extension f : (X,A) → (X,A). Suppose that p ∈ A is an isolated ﬁxedpoint of f and that dφp – IA : TpA → TpA is a nonsingular linear transformation. Then|i(X, f ,p)| ≤ .Proof If the linear transformation dfp – I : TpX → TpX is nonsingular, then i(X, f ,p) =±, see []. Therefore, we assume that the linear transformation dfp – I : TpX → TpX issingular. Note that determining the index of a map f at a ﬁxed point p is a local problem,so we may choose a local coordinate system about p in which the smooth manifold Xis identiﬁed with Rn such that p is the origin in Rn and the smooth submanifold A isidentiﬁed with the subspace Rn–.Let us deﬁne G : Rn → Rn by G(x) = f (x) – x. To calculate the index of f at p, we needto determine the degree of –G restricted to a sphere around . More speciﬁcally, for  > suﬃciently small, we may consider the map E : Sn– →Rn – {} deﬁned byE(x) = –G(x).Since G is a C function with value  at the point p, we know from the deﬁnition of thederivative that∣∣G(x) – dGp(x)∣∣≤ o()uniformly over {x ∈ A| : |x| = }. Our goal is to deﬁne maps Pσ related to E such that Pσtakes each of the upper and lower half-plane into either the upper or the lower half-planeKhamsemanan et al. Fixed Point Theory and Applications 2014, 2014:97 Page 3 of 6http://www.fixedpointtheoryandapplications.com/content/2014/1/97and that will allow us to calculate the index i(X, f ,p). However, we need to make use ofa change of coordinate. Since dfp – I = dGp is singular but dGp|A is not, we know thatdGp(, . . . , , ) ∈ span{x, . . . ,xn–}. Now rotate the x, . . . ,xn– coordinate system so thatxn– points in the direction of dGp(, . . . , , ). Choose new coordinates y, . . . , yn so thatthe hyperplane yn =  is the same as xn =  and, for each j = , . . . ,n–, the transformationdGp takes the xj unit vector to the yi unit vector. Now we have{(x, . . . ,xn–, ) ∈Rn}={(y, . . . , yn–, ) ∈Rn},{(x, . . . ,xn–,xn) ∈Rn}={(y, . . . , yn–, yn) ∈Rn}and the positive yn axis is in the same half-space as the positive xn axis.With this change ofcoordinates, dGp, as a map from x coordinates into y coordinates, has the following form:dGp(α, . . . ,αn–,αn) = (α, . . . ,αn–, ) + αn(, . . . , ,B, )for some constant B. DeﬁneS+ ={(x, . . . ,xn) ∈Rn : xn ≥ }∩ Sn–, S– = {(x, . . . ,xn) ∈Rn : xn ≤ }∩ Sn–.It is easy to check the following:(i) The pointx+ =(, . . . , , – B( + B) , ( + B) )is the unique point of S+ with the property that dGp(x+) = . From the inversefunction theorem, there is a small neighborhood U+ of the point x+ such thatdGp|S+ is a nonsingular diﬀeomorphism of U+ onto a neighborhood of  in thehyperplane yn =  and (dGp|S+ )–(dGp(U+)) =U+.(ii) The pointx– =(, . . . , , – B( + B) , – ( + B) )is the unique point of S– with the property that dGp(x–) = . Similarly, from theinverse function theorem, there is a small neighborhood U– of the point x– suchthat dGp|S– is a nonsingular diﬀeomorphism of U– onto a neighborhood of  in thehyperplane yn =  and (dGp|S– )–(dGp(U–)) =U–.Now consider the maps P deﬁned byP(x) = –E(x).The map P |Sn– converges in the C topology to –dGp|Sn– becaused(P(x))= d(–E(x))= d(–(–G(x)))Khamsemanan et al. Fixed Point Theory and Applications 2014, 2014:97 Page 4 of 6http://www.fixedpointtheoryandapplications.com/content/2014/1/97= ––d(G(x))= ––dGp(x)= –dGp(x).Consequently, we have the following analogues of (i)-(ii) above.() There is a unique point x+ ∈ S+ such that the y, . . . , yn– coordinates of P(x+ ) are all. Also, since p is an isolated ﬁxed point of f ,P(x+ ) = . In particular, the yncoordinate of P(x+ ) is nonzero. Hence we haveP(x+ ) = (, . . . , , y+n ),where y+n = .() There is a unique point x– ∈ S– such that the y, . . . , yn– coordinates of P(x– ) are all. Also, since p is an isolated ﬁxed point of f ,P(x– ) = . In particular, the yncoordinate of P(x– ) is nonzero. Hence we haveP(x– ) = (, . . . , , y–n ),where y–n = .If P(x) = (y, y, . . . , yn), deﬁne a map Pσ : Sn– →Rn – {} byPσ ={(y, . . . , yn–,σ +|yn|) if x ∈ S+,(y, . . . , yn–,σ –|yn|) if x ∈ S–\Rn–,where σ + = y+n /|y+n | =± and σ – = y–n /|y–n | =±.Notice that the values of Pσ |S+ lie entirely in the half-space where P(x+ ) is located withrespect to the y, . . . , yn coordinates and the values of Pσ |S– lie entirely in the half-spacewhere P(x– ) is located with respect to the y, . . . , yn coordinates. Furthermore, P and Pσare homotopic as maps into Rn – {} by the following homotopy:H(x, t) = tP(x) + ( – t)Pσ (x).For x ∈ S+, either (y, . . . , yn–) =  or x = x+ . If (y, . . . , yn–) = , then H(x, t) =  by deﬁni-tion. If x = x+ , then H(x, t) = P(x) which we know is never . A similar argument showsthat H(x, t) =  for x ∈ S–\Rn–.Since the map Pσ takes S+ into either the upper or lower half-planes with respect to they coordinates and takes S– into either the upper or lower half-planes with respect to they, . . . , yn coordinates, the map Pσ is homotopic either to a constant map, the suspensionof Pσ |Sn– = P |Sn– or the suspension of Pσ |Sn– = P |Sn– followed by a reﬂection about thehyperplane yn = . This homotopy tells us that either P has degree  or deg(P |Sn– ) or–deg(P |Sn– ).Although P is a map from x to y coordinates, we know that the x and y coordinates arerelated by a linear map, call it L, satisfying L(xi) = yi, for i = , . . . ,n. Thus, the map L– ◦PKhamsemanan et al. Fixed Point Theory and Applications 2014, 2014:97 Page 5 of 6http://www.fixedpointtheoryandapplications.com/content/2014/1/97takes x, . . . ,xn coordinates to x, . . . ,xn coordinates anddeg(L– ◦ P)= deg(L–) · deg(P) = deg(P).Thus the index i(X, f ,p) = deg(P) which is either  or ±. 3 Fixed point theorem for smooth extensionmapsWe can now use Theorem  to establish the existence of ﬁxed points in X \A.Theorem  Suppose that A is an (n – )-dimensional smooth submanifold of the interiorof a compact smooth n-manifold X . Given a smooth map φ : A → A and a smooth mapf : (X,A) → (X,A) extending φ, suppose that the ﬁxed points of φ are {x,x, . . . ,xk}, allof which are transversal, that is, the linear map dφxj – IA is nonsingular for each xj. If theLefschetz number L(f ) > k, then there must be at least one ﬁxed point in X \A.Proof Suppose that f only has ﬁxed points in A. This means the ﬁxed point set of f is{x, . . . ,xk}, the set of ﬁxed points of φ. Then, by the Lefschetz-Hopf theorem [], the Lef-schetz number of f isL(f ) =k∑j=i(X, f ,xj)≤ kbecause i(X, f ,xj)≤  by Theorem . This is contrary to the assumption that L(f ) > k, so fhas ﬁxed points in X \A. A consequence of this theorem is the following.Corollary  Let S be the complex plane C compactiﬁed at inﬁnity and S be the unitcircle. Suppose that φ : S → S is a smooth map deﬁned by φ(ζ ) = ζ k for some k ≥  andthat f : (S,S) → (S,S) is a smooth extension of φ. If f is homotopic to the suspension ofφ, then there is at least one ﬁxed point in S\S.Proof Since f is homotopic to the suspension of φ, then f is of degree k and thus L(f ) = + k. However, φ has only k –  ﬁxed points. Note here that the k –  ﬁxed points of φ onS are all transversal so that the hypotheses of Theorem  hold. The following example illustrates the fact that the corollary, and therefore Theorem ,require the hypothesis that the map f is smooth, by exhibiting a non-smooth mapf : (S,S)→ (S,S) homotopic to the suspension of φ that has no ﬁxed points on S\S.Example  Let S = C ∪ {∞} be the complex plane C compactiﬁed at inﬁnity and S bethe unit circle. Let φ : S → S be deﬁned by φ(ζ ) = ζ . Let B+ = {z ∈ C : |z| ≤ } and B– ={z ∈C : |z| ≥ } ∪ {∞}, so S = B+ ∪B– and B+ ∩B– = S. For z ∈ B– \ {}, there exist ζ ∈ Sand t ∈ [, ) such that z = t + ( – t)ζ . As in Lemma . of [], deﬁne f : B+ → B+ bysetting f (z) = t + ( – t)φ(ζ ) for z =  and f () = , then f has no ﬁxed points in B+ \ S. Inorder to extend f to a selfmap of B– so that there are no ﬁxed points on B– \ S, we deﬁneρ : S → S by ρ(z) = ρ(reiθ ) = r eiθ , ρ() =∞ and ρ(∞) = , so ρ(B–) = B+ and ρ(B+) = B–.Khamsemanan et al. Fixed Point Theory and Applications 2014, 2014:97 Page 6 of 6http://www.fixedpointtheoryandapplications.com/content/2014/1/97Then, for z ∈ B–, let f (z) = ρf ρ(z). The only ﬁxed point of f is z =  and thus there are noﬁxed points in S \ S.Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.Author details1Sirindhorn International Institute of Technology (SIIT), Thammasat University, Pathum Thani, Thailand. 2Department ofMathematics, University of California, Los Angeles, USA. 3School of Public Health, Harvard University, Cambridge, USA.4Department of Mathematics, Chiang Mai University, Chiang Mai, Thailand.AcknowledgementsThis work has been supported by the Thailand Research Fund MRG5580232 under the mentorship of Professor Robert F.Brown of the Department of Mathematics, UCLA and Professor Sompong Dhompongsa of the Department ofMathematics, Chiang Mai University. The author would also like to thank the referee for all the useful suggestions.Received: 26 December 2013 Accepted: 31 March 2014 Published: 16 Apr 2014References1. Shub, M, Sullivan, D: A remark on the Lefschetz ﬁxed point formula for diﬀerentiable maps. Topology 13, 189-191(1974)2. Brown, RF, Greene, RE, Schirmer, H: Fixed points of map extensions. In: Topological Fixed Point Theory and Applications(Tianjin, 1988). Lecture Notes in Math., vol. 1411, pp. 24-45. Springer, Berlin (1989)3. Brown, RF, Greene, RE: An interior ﬁxed point property of the disc. Am. Math. Mon. 101(1), 39-47 (1994)4. Schirmer, H: Nielsen theory of transversal ﬁxed point sets. Fundam. Math. 141(1), 31-59 (1992). With an appendix byRobert E Greene5. Guillemin, V, Pollack, A: Diﬀerential Topology. Am. Math. Soc., Providence (1974)6. Arkowitz, M, Brown, RF: The Lefschetz-Hopf theorem and axioms for the Lefschetz number. Fixed Point Theory Appl.2004, Article ID 465090 (2004)7. Schirmer, H: A relative Nielsen number. Pac. J. Math. 122, 459-473 (1986)10.1186/1687-1812-2014-97Cite this article as: Khamsemanan et al.: A ﬁxed point theorem for smooth extension maps. Fixed Point Theory andApplications 2014, 2014:97

A fixed point theorem for smooth extension maps

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A fixed point theorem for smooth extension maps

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