Surfaces of infinite-type are non-Hopfian

Sumanta Das; Siddhartha Gadgil

doi:10.5802/crmath.504

Geometry and Topology

Surfaces of infinite-type are non-Hopfian

Sumanta Das ¹ ; Siddhartha Gadgil ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1349-1356

Abstract (Original language)
Résumé

We show that finite-type surfaces are characterized by a topological analogue of the Hopf property. Namely, an oriented surface $Σ$ is of finite-type if and only if every proper map $f : Σ \to Σ$ of degree one is homotopic to a homeomorphism.

Nous montrons que les surfaces de type fini sont caractérisées par un analogue topologique de la propriété de Hopf. A savoir, une surface orientée $Σ$ est de type fini si et seulement si toute application propre $f : Σ \to Σ$ de degré un est homotope à un homéomorphisme.

Received: 2023-01-03
Revised: 2023-03-17
Accepted: 2023-04-06
Published online: 2023-10-31

DOI: 10.5802/crmath.504

Classification: 57K20, 55S37

Author's affiliations:

Sumanta Das ¹; Siddhartha Gadgil ¹

¹ Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Sumanta Das and Siddhartha Gadgil},
     title = {Surfaces of infinite-type are {non-Hopfian}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1349--1356},
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Sumanta Das; Siddhartha Gadgil. Surfaces of infinite-type are non-Hopfian. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1349-1356. doi: 10.5802/crmath.504

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