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Abstract Let $${\mathrm{Diff}}_{0}(N)$$ represent the subgroup of diffeomorphisms that are homotopic to the identity. We show that if $$N$$ is a closed hyperbolic 4-manifold, then $$\pi _{0}{\mathrm{Diff}}_{0}(N)$$ is not finitely generated with similar results holding topologically. This proves in dimension-4 results previously known for $$n$$-dimensional hyperbolic manifolds of dimension $$n\ge 11$$ by Farrell and Jones in 1989 and $$n\ge 10$$ by Farrell and Ontaneda in 2010. Our proof relies on the technical result that $$\pi _{0}{\mathrm{Homeo}}(S^{1}\times D^{3})$$ is not finitely generated, which extends to the topological category smooth results of the authors. We also show that $$\pi _{n-4} {\mathrm{Homeo}}(S^{1} \times D^{n-1})$$ is not finitely generated for $$n \geq 4$$ and in particular $$\pi _{0}{\mathrm{Homeo}}(S^{1}\times D^{3})$$ is not finitely generated. These results are new for $n=4, 5$ and $$7$$. We also introduce higher dimensional barbell maps and establish some of their basic properties.
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The Point-to-Set Principle and the dimensions of Hamel bases
We prove that, for every 0 ⩽ s ⩽ 1, there is a Hamel basis of the vector space of reals over the field of rationals that has Hausdorff dimension s. The logic of our proof is of particular interest. The statement of our theorem is classical; it does not involve the theory of computing. However, our proof makes essential use of algorithmic fractal dimension–a computability-theoretic construct–and the point-to-set principle of J. Lutz and N. Lutz (2018).
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- Award ID(s):
- 1900716
- PAR ID:
- 10515834
- Publisher / Repository:
- IOS Press
- Date Published:
- Journal Name:
- Computability
- Volume:
- 13
- Issue:
- 2
- ISSN:
- 2211-3568
- Page Range / eLocation ID:
- 105 to 112
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3233/COM-210383
