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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.02560 |
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Table of Contents:
- We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to 4 when aspherical. We also show that closed $3$-manifold groups have asymptotic dimension at most 3. Our proof method yields that the asymptotic dimension of closed $3$-dimensional Alexandrov spaces is at most 3. We thus obtain that the Novikov conjecture holds for closed $4$-manifolds with such a geometric decomposition and closed $3$-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain $0$-surgered geometric $4$-manifolds and the existence of zero in the spectrum of aspherical smooth $4$-manifolds with a geometric decomposition.