Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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C. Loeh - M. Moraschini - G. Raptis

On the simplicial volume and the Euler characteristic of (aspherical) manifolds

created by moraschini on 17 Sep 2021
modified on 15 Jul 2022

[BibTeX]

Published Paper

Inserted: 17 sep 2021
Last Updated: 15 jul 2022

Journal: Res. Math. Sci.
Volume: 9
Number: 44
Year: 2022

ArXiv: 2109.08115 PDF

Abstract:

A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov's question and collect strategies towards affirmative answers and strategies towards negative answers to this problem. Moreover, we put Gromov's question into context with other open problems in low- and high-dimensional topology. A special emphasis is put on a comparative analysis of the additivity properties of the simplicial volume and the Euler characteristic for manifolds with boundary. We explain that the simplicial volume defines a symmetric monoidal functor (invertible TQFT) on the amenable cobordism category, but not on the whole cobordism category. In addition, using known computations of simplicial volumes, we conclude that the fundamental group of the 4-dimensional amenable cobordism category is not finitely generated. We also consider new variations of Gromov's question. Specifically, we show that counterexamples exist among aspherical spaces that are only homology equivalent to oriented closed connected manifolds.

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