Comptes Rendus
no. 9
Topology/Differential topology
On maps which are the identity on the boundary
Claire Voisin
Albert Fathi1
1Georgia 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 
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
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[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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