Vector fields on noncompact manifolds

Kato, Tsuyoshi; Kishimoto, Daisuke; Tsutaya, Mitsunobu

doi:10.2140/agt.2024.24.3985

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Vector fields on noncompact manifolds

Tsuyoshi Kato, Daisuke Kishimoto and Mitsunobu Tsutaya

Algebraic & Geometric Topology 24 (2024) 3985–3996

DOI: 10.2140/agt.2024.24.3985

Abstract

Let $M$ be a noncompact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$ . We establish a Poincaré–Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an application, we show that such a vector field must have infinitely many zeros whenever $G$ is amenable and the Euler characteristic of $M ∕ G$ is nonzero.

Keywords

vector field, noncompact manifold, bounded cohomology, Poincaré–Hopf theorem

Mathematical Subject Classification

Primary: 57R25, 58K45

References

Publication

Received: 21 December 2022

Revised: 6 September 2023

Accepted: 23 September 2023

Published: 9 December 2024

Authors

Tsuyoshi Kato
Department of Mathematics Kyoto University Kyoto Japan

Daisuke Kishimoto
Faculty of Mathematics Kyushu University Fukuoka Japan

Mitsunobu Tsutaya
Faculty of Mathematics Kyushu University Fukuoka Japan

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Volume 24, issue 7 (2024)