Loeh, Clara and Moraschini, Marco and Raptis, George (2022) On the simplicial volume and the Euler characteristic of (aspherical) manifolds. RESEARCH IN THE MATHEMATICAL SCIENCES, 9 (3): 44. ISSN 2522-0144, 2197-9847
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A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed 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 four-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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | LOCALLY SYMMETRIC-SPACES; BOUNDED COHOMOLOGY; CURVED MANIFOLDS; GROMOV INVARIANT; MINIMAL ENTROPY; HOMOLOGY; HYPERBOLIZATION; THEOREM; |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Clara Löh |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 14 Nov 2023 13:29 |
| Last Modified: | 14 Nov 2023 13:29 |
| URI: | https://pred.uni-regensburg.de/id/eprint/56741 |
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