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 (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.
Euler Characteristic Tools For Topological Data Analysis
Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Mathematical Institute（数学研究所）；University of Oxford（牛津大学）
