Comptes Rendus
no. 9
Topology/Differential topology
On maps which are the identity on the boundary

Presented by: Claire Voisin

Albert Fathi1
1 Georgia Institute of Technology and ENS de Lyon (Emeritus), School of Mathematics, 686 Cherry St NW, Atlanta, GA 30313, USA
Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 1022-1025.

The 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.

Le fait suivant ne semble pas être connu :

Soit F un sous-ensemble fermé de la variété connexe M (de dimension finie). Si f:F→M est 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.08.001

Albert Fathi 1

1 Georgia Institute of Technology and ENS de Lyon (Emeritus), School of Mathematics, 686 Cherry St NW, Atlanta, GA 30313, USA 
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Albert Fathi. On maps which are the identity on the boundary. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 1022-1025. doi : 10.1016/j.crma.2017.08.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.08.001/

[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, Volume 8 (1969), pp. 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, 2009 (ISBN: 978-0-8218-4915-6)

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