Albert Fathi

Fathi, Albert

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1022–1025Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comTopology/Differential topologyOn maps which are the identity on the boundarySur les applications qui sont l’identité sur la frontièreAlbert Fathi 1Georgia Institute of Technology and ENS de Lyon (Emeritus), School of Mathematics, 686 Cherry St NW, Atlanta, GA 30313, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 2 December 2016Accepted after revision 29 August 2017Available online 6 September 2017Presented by Claire VoisinThe following fact seems to have been unnoticed until now:Let F be a closed subset of the (finite-dimensional) connected manifold M . If f : F → M is a proper continuous map which is the identity on the boundary ∂ F of F in M , then either f (F ) ⊃ F or f (F ) ⊃ M \ F .The proof is elementary and simple using degree theory.The statement has many deep consequences.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éLe fait suivant ne semble pas être connu :Soit F un sous-ensemble fermé de la variété connexe M (de dimension finie). Si f : F → Mest une application continue et propre qui est l’identité sur la frontière ∂ F de F dans M , alors, on a, soit f (F ) ⊃ F , soit f (F ) ⊃ M \ F .La preuve, qui utilise la théorie du degré, est élémentaire et simple.Ce fait a des conséquences profondes.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeLe fait suivant ne semble pas être connu :Théorème 0.1. Soit F un sous-ensemble fermé de la variété connexe (de dimension finie) M. Si f : F → M est une application continue qui est l’identité sur la frontière ∂ F de F dans M, alors, soit f (F ) ⊃ F , soit f (F ) ⊃ M \ F .De plus, si f est proprement homotope à l’inclusion F ⊂ M relativement à ∂ F , on a nécessairement f (F ) ⊃ F .E-mail address: albert.fathi@math.gatech.edu.1 Work supported by ANR-12-BS01-0020 WKBHJ.http://dx.doi.org/10.1016/j.crma.2017.08.0011631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.A. Fathi / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1022–1025 1023Rappelons que f est proprement homotope à l’inclusion F ⊂ M relativement à ∂ F , s’il existe une application H : F ×[0, 1] → M continue et propre telle que H(x, 0) = x, H(x, 1) = f (x) pour tout x ∈ F et H(x, t) = x pour tout (x, t) ∈ ∂ F ×[0, 1].Le résultat ci-dessus a des conséquences profondes. Nous en mentionnons quelques-unes ici.Corollaire 0.2. Soit F un sous-ensemble fermé, d’intérieur F̊ non vide, de la variété connexe M. Il n’existe pas de rétraction continue et propre de F sur sa frontière ∂ F .Corollaire 0.3. Soit F un sous-ensemble fermé de la variété connexe M. Si H : F × [0, 1] → M est une homotopie continue telle que H(x, 0) = x, pour tout x ∈ F , et H | F × {1} est constante, alors on a obligatoirement une des deux possibilités suivantes :(i) pour toute composante connexe relativement compacte C de M \ F , on a H(∂C × [0, 1]) ⊃ C ;(ii) il existe une composante connexe relativement compacte C de M \ F avec H(∂C × [0, 1]) ⊃ M \ C.En particulier, on a :(a) si M est compacte, alors H(F × [0, 1]) contient toutes les composantes connexes de M \ F , sauf peut-être au plus une ;(b) si M n’est pas compacte, alors H(F × [0, 1]) contient toutes les composantes connexes relativement compactes de M \ F .L’homotopie H du corollaire ci-dessus ne peut pas être propre, sauf si F est compact. Une version propre de ce corollaire est donnée par le résultat ci-dessous.Théorème 0.4. Soit F un sous-ensemble fermé de la variété connexe M. Si H : F × [0, 1[→ M est une homotopie continue et propre telle que H(x, 0) = x, pour tout x ∈ F , alors H(F × [0, 1[) contient toutes les composantes connexes de M \ F , sauf peut-être au plus une.Théorème 0.5. Soit h : M → M un homéomorphisme de la variété connexe M (de dimension finie). Si on note par Fix(h) l’ensemble des points fixes de h, on a :(i) soit h(C) = C pour toute composante connexe de M \ Fix(h) ;(ii) soit M \ Fix(h) a exactement deux composantes connexes qui sont permutées par h.De plus, si M est orientable et h préserve l’orientation, alors h(C) = C pour toute composante connexe de M \ Fix(h).La plupart des énoncés ci-dessus peuvent être améliorés, parfois substantiellement. Une version détaillée de ce travail contiendra ces améliorations.1. StatementsThe following fact seems to have been unnoticed until now.Theorem 1.1. Let F be a closed subset of the (finite-dimensional) connected manifold M. If f : F → M is a proper continuous map which is the identity on the boundary ∂ F of F in M, then either f (F ) ⊃ F or f (F ) ⊃ M \ F .Moreover, if f is properly homotopic to the inclusion F ⊂ M modulo ∂ F , we must have f (F ) ⊃ F .We recall that f is properly homotopic to the inclusion F ⊂ M modulo ∂ F if there exists a proper continuous map H : F × [0, 1] → M such that H(x, 0) = x, H(x, 1) = f (x), for all x ∈ F , and H(x, t) = x, for all (x, t) ∈ ∂ F × [0, 1].Theorem 1.1 has several deep consequences; we mention here some of them.Corollary 1.2. Let F be a closed subset with non-empty interior F̊ of the connected manifold M. There does not exist a continuous proper retraction of F on its boundary ∂ F .The corollary above is well known for the unit ball of an Euclidean space: it is one of the equivalent forms of Brouwer’s fixed point theorem. It is also known when the boundary of F is a smooth submanifold of M [3, Proposition 7.1, page 133]. Moreover, there is a simple proof in [1, 3.6, page 12] when F is a general compact subset of the Euclidean space M .This shows of course that the proof of the theorem must use some deep topological fact.Corollary 1.3. Let F be a closed subset of the connected manifold M. If H : F × [0, 1] → M is a continuous homotopy such that H(x, 0) = x, for all x ∈ F , and H | F × {1} is constant, then one of the following two possibilities holds:1024 A. Fathi / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1022–1025(i) for every relatively compact connected component C of M \ F , we have H(∂C × [0, 1]) ⊃ C;(ii) there exists a relatively compact connected component C of M \ F with H(∂C × [0, 1]) ⊃ M \ C.In particular, we have:(a) if M is compact, then H(F × [0, 1]) contains all the connected components of M \ F except at most one;(b) if M is not compact, then H(F × [0, 1]) contains all the relatively compact connected components of M \ F .Note that the homotopy H in Corollary 1.3 above cannot be proper unless F is compact. A proper version of this Corol-lary 1.3 is given by the following result.Theorem 1.4. Let F be a closed subset of the connected manifold M. If H : F × [0, 1[→ M is a continuous proper homotopy such that H(x, 0) = x, for all x ∈ F , then H(F × [0, 1[) contains all the connected components of M \ F , except at most one.Theorem 1.5. Let h : M → M be a homeomorphism of the connected (finite-dimensional) manifold M. If Fix(h), denotes the set of fixed points of h, we have:(i) either h(C) = C for every connected component of M \ Fix(h);(ii) or M \ Fix(h) has exactly two connected components, which are permuted by h.Moreover, if M is orientable and h preserves the orientation, then h(C) = C for every connected component of M \ Fix(h).Most of the statements can be improved. There are even more consequences of Theorem 1.1. We postpone these facts to a more detailed version of this work.2. ProofsProof of Theorem 1.1. Since f is proper, f (F ) is a closed subset. Assume that f (F ) does not contain M \ F . Then the open set U = M \ (F ∪ f (F )) is not empty. We extend f to a map f̂ : M → M by the identity on M \ F . This extension f̂ is continuous and proper, and has therefore a well-defined degree mod 2, [3, 6.1, page 124]. We now compute this degree using the points in U = M \ (F ∪ f (F )). In fact, since U is an open non-empty subset of M with f̂ −1(U ) = U , and f̂ is the identity on U , the degree must be 1. This in turn implies that f̂ is surjective, hence f (F ) ⊃ F .If H is a proper continuous homotopy from the inclusion F ⊂ M to f modulo ∂ F , we can extend it to a proper continuous homotopy Ĥ : M × [0, 1] → M by Ĥ(t, x) = x, for x /∈ F , t ∈ [0, 1]. This gives a proper homotopy from the identity of M to f̂ . Therefore, the degree of f̂ must be 1, the degree of the identity map of M , [3, 6.1, page 124]. This implies again that f̂ is surjective, and f (F ) ⊃ F . Proof of Corollary 1.2. Assume r : F → ∂ F is such a proper continuous retraction. Since F̊ is not empty, the image r(F ) = ∂ Fis also non-empty. This implies that the interior of M \ F is also non-empty. By Theorem 1.1, the image r(F ) contains either the interior of F or the interior of M \ F , which are both non-empty and disjoint from ∂ F = r(F ). This is a contradiction. Proof of Corollary 1.3. We consider a relatively compact component C of the open set M \ F . Arguing by contradiction, suppose that H(∂C × [0, 1]) does not contain either C or M \ C . Fix x̂ ∈ C \ H(∂C × [0, 1]) and ŷ ∈ M \ (C ∪ H(∂C × [0, 1])). Note that x̂ = x0 and ŷ = x0, where {x0} = H(F × {1}). Since ∂C ⊂ F , the homotopy H is defined on ∂C × [0, 1], and we can extend it to a continuous map Ĥ : ∂C ×[0, 1] ∪ C̄ ×{1} → M by Ĥ(x, 1) = x0, for x ∈ C̄ . Since M is a manifold, we can extend Ĥ to a map Ĥ : W → M , where W is a neighborhood of the closed subset ∂C × [0, 1] ∪ ∪C̄ × {1} in C̄ × [0, 1]. Since neither x̂ nor ŷ are in H(∂C × [0, 1]), we can find a open neighborhood V of ∂C in C̄ such that V × [0, 1] ⊂ W , and neither x̂ nor ŷ are in Ĥ(V × [0, 1]). We then choose a continuous function ρ : C̄ → [0, 1], with ρ = 0 on ∂C and ρ = 1 outside V . We can define the continuous map f : C̄ → M by f (x) = Ĥ(x, ρ(x)), for x ∈ C̄ . The map f is the identity on ∂ C̄ = ∂C , and f (C̄)contains neither x̂ ∈ C nor ŷ ∈ M \ C̄ . Since C is compact, this contradicts Theorem 1.1. The proof of Theorem 1.4 is a more involved version of the proof given above. We postpone it to a more detailed version of this work.Proof of Theorem 1.5. Since h is a homeomorphism, it permutes the connected components of M \ Fix(h). Therefore, if C is a connected component of M \ Fix(h) such that C = h(C) then h(C) ∩ C = ∅. We can apply Theorem 1.1 to h | C̄ to conclude that M \ C̄ ⊂ h(C). Since h(C) ∩ C̄ = ∅, we get h(C) = M \ C̄ . It is not difficult to conclude that Fix(h) = ∂C and M \ Fix(h)has exactly two connected components C and h(C) which are permuted by C .A. Fathi / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1022–1025 1025To prove the last part, we argue by contradiction. If C is a component of M \ Fix(h) such that C ∩h(C) = ∅, we can define ĥ : M → M by ĥ | C̄ = h, and ĥ is the identity outside of C . Clearly h̄ is proper and ĥ(M) = M \ C . Since ĥ is not surjective, its degree is 0. Note that this degree is now with value in Z, since M is orientable, see [3, Chapter III, §1 & 3, page 133]. On the other hand, if x is in the open set h(C) then ĥ−1(x) = {x1, x2}, with x1 ∈ C and x2 ∈ h(C). We can compute the degree of ĥ using the point x. Since ĥ is a local homeomorphism preserving the orientation near x1 and x2, we obtain that the degree of ĥ is 2. This is a contradiction. It is possible to give another proof of the first part of Theorem 1.5 using Newman’s theorems on transformation groups, namely using [2, Theorem 1, page 204]. Note however that the proofs of Newman’s theorems requires degree theory. We sketch the proof. With the notations of the proof of Theorem 1.5, if C ∩ h(C) = ∅, we can define a homeomorphism h̃ :M → M of order 2, byh̃(x) =⎧⎪⎨⎪⎩h(x), if x ∈ C ,h−1(x), if x ∈ h(C),x, if x /∈ C ∪ h(C).Since h̃ is not the identity, and h̃ is the identity outside C ∪h(C), Theorem 1 in [2] implies that the closed set M \ (C ∪h(C))has empty interior, and therefore C and h(C) are the only connected component of M \ F .AcknowledgementsThe first result the author obtained, in Summer 2014, was Theorem 1.4, in the case where F was a compact smooth codimension-1 submanifold. This was used to study singularities of viscosity solutions to the Hamilton–Jacobi equation. After a colloquium the author gave in Gainesville in Fall 2014, Sasha Dranishnikov suggested that Theorem 1.4 was true for a general closed set.Theorem 1.1 was obtained by the author during Summer 2016 while he enjoyed the support of DPMMS, Cambridge University, and the hospitality of Downing College.References[1] K. Borsuk, Theory of Retracts, Monografie Matematyczne, vol. 44, PWN – Państwowe Wydawnictwo Naukowe, Warsaw, 1967.[2] A. Dress, Newman’s theorems on transformation groups, Topology 8 (1969) 203–207.[3] E. Outerelo, J.M. Ruiz, Mapping Degree Theory, Graduate Studies in Mathematics, vol. 108, American Mathematical Society, Real Sociedad Matemática Española, Providence, RI, Madrid, ISBN 978-0-8218-4915-6, 2009.

On maps which are the identity on the boundary

Comptes Rendus Mathématique

http://www.numdam.org/item/10.1016/j.crma.2017.08.001.pdf

On maps which are the identity on the boundary

Abstract

Similar works

Full text

Available Versions

Numérisation de Documents Anciens Mathématiques

Comptes Rendus Mathématique